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interactive activities, section summaries and projects, and chapter openers and reviews. For anyone who wants to see and understand how mathematics are used in everyday life.
Problem Solving: Probability and Statistics. Sets and Set Operations. Principles of Counting. Introduction to Probability. Computing Probability using the Addition Rule. Computing Probability using the Multiplication Rule. Bayes' Theorem and Its Applications | 677.169 | 1 |
Free Worksheet on Algebraic Proofs
Did you enjoy yesterday's video on algebraic proofs? Do you want to see if you remember everything from the lectures? Check out today's free Geometry worksheet covering the Algebraic Proofs video. It's two pages of word problems and equations in which you'll need to justify each step you take on your way to the solution. Don't forget the different properties of equality and congruence, as you'll be asked to identify them in a few examples.
In addition to videos and worksheets, students in our courses get interactive activities, notes, transcripts of all the lectures, quizzes, and tests. No stone is left unturned when it comes to ensuring you grasp the fundamentals of the course you take. So click on the worksheet below (PDF) and see what you think! | 677.169 | 1 |
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Mathematics
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CRC Press exhibits every year at more than 100 professional and academic conferences held across the world. At these events, you will have the chance to meet the authors and to get to know the CRC Press staff.
You can also take advantage of special discounts for convention attendees.
Visit us at the following conventions throughout the year. | 677.169 | 1 |
Make sleek and functional home furnishings from inexpensive plywood and other off-the-shelf materials using only basic hand and power tools. This unique building guide offers 73 innovative ideas for using plywood to make everything from desks and workstations to children?s playhouses. Projects for every need and skill level are presented with clear... more...
500 Ways to Achieve Your Best Grades We want you to succeed on your college algebra and trigonometry midterm and final exams. That's why we've selected these 500 questions to help you study more effectively, use your preparation time wisely, and get your best grades. These questions and answers are similar to the ones you'll find on a typical college... more...
An ideal course text or supplement for the many underprepared students enrolled in the required freshman college math course, this revision of the highly successful outline (more than 348,000 copies sold to date) has been updated to reflect the many recent changes in the curriculum. Based on Schaum's critically acclaimed pedagogy of concise theory... more...
This third edition of the perennial bestseller defines the recent changes in how the discipline is taught and introduces a new perspective on the discipline. New material in this third edition includes: A modernized section on trigonometry An introduction to mathematical modeling Instruction in use of the graphing calculator 2,000 solved problems... more...
Confusing Textbooks? Missed Lectures? Tough Test Questions... more...
Tough Test Questions? Missed Lectures? Not Enough Time? Fortunately for you, there's Schaum's,...Tensors and methods of differential geometry are very useful mathematical tools in many fields of modern physics and computational engineering including relativity physics, electrodynamics, computational fluid dynamics (CFD), continuum mechanics, aero and vibroacoustics and cybernetics. This book comprehensively presents topics, such as bra-ket notation,... more...
The Handbook illustrates a new interdisciplinary conceptualization of the category ?house? as evoked by current discussions in the social sciences and humanities. In addition, it offers readers a panorama of European research related to the historical dimensions of house, household, and domesticity. more... | 677.169 | 1 |
This course, which is intended for the mathematics major who is enrolled
in the secondary education program, provides a bridge and establishes
connections between the college level mathematics required of the
mathematics major and the mathematics of the secondary school curriculum.
It is often said that mathematics is a language. In this class you
will begin to learn to speak this language. Just like in an
introductory language course, we will start with the most fundamental
concepts and grammar rules. After we have some familiarity with the
language of formal mathematics, we will practice this language in the
setting of counting problems of different types. More like an
advanced language class, it will not suffice just memorizing the
vocabulary (in fact, I hope we can keep vocabulary to a minimum), but
rather you will be required to understand and speak clearly in this
language. The material learned here will help you understand the
mathematics you read and clarify the mathematics you write. Because
we are learning how to write mathematics, exposition will also be a
component in your evaluation.
The first main goal of this course is to connect the mathematics you have
learned (and some you haven't) with the history you have learned (and some
you haven't). The second main goal is to connect the mathematics you
have learned together.
Professional Activities
I am an active member of the Mathematical Association of America. In
particular, I am the liaison coordinator and the co-chair of the Program
Committee for the Seaway Section. I am also a member of the American
Mathematical Society.
Areas of research
Low-dimensional Topology
Knots, Links, and 3-manifolds
Current projects
I am currently pursuing several research projects. The newest of the
projects is an exploration of the role of Euclid's Fourth Postulate:
"All right angles are equal." The older of these projects
consists of investigating how the Casson-Walker-Lescop 3-manifold
invariant changes when modifying the presenting link for a 3-manifold.
This project has evolved into studying questions of the Ohtsuki invariants
of rational homology spheres, and questions of the space of finite type
invariants for links of three or more components. Another long-term
project is to study symmetries of links. In particular I am
examining a refinement of unlinking number accounting for which components
are involved in each of the crossing changes, a so-called coloured
unlinking number. Finally, I am examining comparisons and connections
between mathematician Evariste Galois and composer Hector Berlioz.
AMS sectional meeting in Las Vegas, April 21 - 22, 2001.
Co-organized special session on the topology of links.
MAA national MathFest in Madison, August 2 - 4, 2001.
"Infiltrating Preservice Elementary School Mathematics with History",
contributed paper session on the use of history in the teaching of
mathematics.
MAA national MathFest in Burlington, VT, July 31 - August 4, 2002
"Modern Geometry", contributed paper session on the use of recent history
of mathematics in teaching.
"Welcome to Mathematics: A Cornerstone Experience", contributed
paper session on the role of proof in teaching mathematics.
MAA national MathFest in Boulder, CO, July 30 - August 2, 2003
"Days are Numbers: The Mathematics of the Calendar", general
contributed paper session.
"Honesty is the Best Philosophy", contributed paper session on innovations
in quantitative literacy.
MAA national MathFest in Providence, RI, August 11 - 15, 2004
Co-organised session on "Extracurricular Mathematics"
MAA national MathFest in Albuquerque, NM, August 3 - 6, 2005
"Why Are We Math Majors?", contributed paper session on current issues in
mathematics education courses. "Greatest Hits of Mathematics", general
contributed paper session.
"Where are we from? - An entire class
project", contributed paper session on getting students to discuss
and to write about mathematics.
"Four dimensional tic-tac-toe on a torus - the game of SET", general
contributed paper session
"Euclid's Neglected Postulate",
contributed paper session on history of mathematics uses in the
classroom.
"Four different experiences", contributed paper session on first
year seminar / experience mathematics courses. | 677.169 | 1 |
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Precise numerical analysis may be defined as the study of computer methods for solving mathematical problems either exactly or to prescribed accuracy. This book explains how precise numerical analysis is constructed. The book also provides exercises which illustrate points from the text and references for the methods presented.
All disc-based content for this title is now available on the Web.
? Clearer, simpler descriptions and explanations of the various numerical methods ? Two new types of numerical problems; accurately solving partial differential equations with the included software and computing line integrals in the complex plane. | 677.169 | 1 |
12/27/02 REVIEW FOR ELEMENTARY ALGEBRAFINALEXAM In order to be prepared for the finalexam, students should be able to do all of the following problems and related ...
Algebra 2: Themes for the Big FinalExamFinal will cover the whole year ... for honors, rational for honors ... Algebra 2: Year in Review Solving: The first step to ... 2/Algebra 2 Semester 2 FinalExam Review Packet.pdf
Students should be able to solve all of the following problems. Algebra II Honors Midterm Exam Review The following are some suggested practice problem. ii honors midterm review.pdf | 677.169 | 1 |
DTW Algebra Lite
Warning! you don't need internet to use this app, preferece used only for check conection button.
Why I need this app?
This app can be useful to you, if you are an Algebra teacher, student, pupil or simply using algebra in your daily life, so DTW means Don't trouble with, this app help you do not have 'em
DTW Algebra Lite's review
DTW Algebra Lite is a powerful algebra tool.
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"Resolving complex equations is just a few taps away"
Is there a moment in our life that we need to resolve things like x³ + x ² + x = 0. Particularly if you use algebra in your daily life or are an algebra teacher. This app goes one step beyond and resolver quadratic equations, logarithms and matrices. The UI is easy and intuitive, but it might have been better a little more attractive. Touch response is somewhat weird and sometimes it force closes when you tap on an option that is only available in the paid version. You can look theorems up, which may be useful God knows when.
This is the free version, which has just a few options. The paid version adds lots of features such as differential, qubic and linear system equations. As there aren't too many apps to help people in their everyday algebra problems, maybe you should check the paid version | 677.169 | 1 |
Math is used by many different types of scientists to model phenomenon and evaluate data from an experiment. By building mathematical models scientists can understand how different physical, chemical, and biological processes are affected by different variables. The most important tools are: making a graph to give a visual representation of the relationships between your variables and making an equation to give a way of computing the relationships between your variables. Find a source of data,Want to send coded messages to your friends? Can you write a simple letter-substitution encryption program in JavaScript? How easy is it to break the simple code? Can you write a second program that "cracks" the letter-substitution code? Investigate other encryption schemes. What types of encryption are least vulnerable to attack?
Read more
Sudoku puzzles have become extremely popular over the past couple of years. You can find books of puzzles for beginners to experts, and many newspapers print Sudoku puzzles daily. This project challenges you to write a computer program to check if your Sudoku solution is correct.
Read more
CompSci_p023
+ More Details
- Less Details
Time Required
Average (6-10 days)
Prerequisites
An understanding of the material covered in [# ProjectIdea Name="CompSci_p002" Value="HtmlAnchor" #]
Students who are mathematically inclined can use the student version of a program like MatLab or Mathematica to convert a digital image into numbers, then perform operations such as sharpening or special effects. This is a great way to learn about image processing algorithms.
Read more
Research the famous collapse of the Tacoma Narrows suspension bridge.
What lessons were learned about the potentially damaging effects of wind on bridges? What structures stabilize a bridge against wind forces? Build models and use a wind tunnel to test your hypothesis.Take shots at a set distance from the basket, but systematically vary the angle to the backboard. For a basic project: How do you think your success rate will vary with angle? Draw a conclusion from your experimental results. A bar graph showing success rate at different angles can help to illustrate your conclusion. For a more advanced project: Use your knowledge of geometry and basketball to come up with a mathematical expression to predict your success rate as a function of angle…
Read more
Block off one-third of a soccer net with a cone, 5-gallon bucket or some other suitable object. Shoot into the smaller side from a set distance, but systematically varying the angle to the goal line. Take enough shots at each angle to get a reliable sample. How does success vary with angle? For a basic project: How do you think your success rate will vary with angle? Draw a conclusion from your experimental results. A bar graph showing success rate at different angles can help to…
Read more
Here's a puzzle you may have heard before which you can build as a simple electric circuit. First, the puzzle: a farmer is traveling to market with his cat, a chicken and some corn. He has to cross a river, and the only way to cross is in a small boat which can hold the farmer and just one of the three items he has with him. The problem is, he has to be very careful about what he chooses to leave behind at any time. If the cat and chicken are left alone, the cat will eat the chicken. If | 677.169 | 1 |
Find a Cheyney CalculusA continuation of Algebra 1 (see course description). Use of irrational numbers, imaginary numbers, quadratic equations, graphing, systems of linear equations, absolute values, and various other topics. May be combined with some basic geometry. Emphasis on the ideas that lie behind dates, facts and documents. | 677.169 | 1 |
Related software: 04/09/2015 12:59:35 Euclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the ancient Greek mathematician Euclid in Alexandria c. 300 BC. It is a ...Euclid's Elements: all thirteen books complete in one volume. The Thomas L. Heath Translation. Dana Densmore, Editor. Santa Fe, New Mexico: Green Lion Press, 2002.This dynamically illustrated edition of Euclid's Elements includes 13 books on plane geometry, geometric and abstract algebra, number theory, incommensurables, and ...Guide About the Definitions The Elements begins with a list of definitions. Some of these indicate little more than certain concepts will be discussed, such as Def.I ...Euclid, sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referred to as the "Father of Geometry". He ...EUCLID'S ELEMENTS OF GEOMETRY The Greek text of J.L. Heiberg (1883–1885) from Euclidis Elementa, edidit et Latine interpretatus est I.L. Heiberg, in aedibusGreen Lion Press has prepared a new one-volume edition of T.L. Heath's translation of the thirteen books of Euclid's Elements. In keeping with Green Lion's design ...Euclid' Elements Book I, 23 Definitions, a one-page visual illustration of the 23 definitions. College Geometry, SAT Prep. ElearningEuclid's Elements is the oldest mathematical and geometric treatise consisting of 13 books written by Euclid in Alexandria c. 300 BC. It is a collection of ...Euclid's Elements Reference Page, Book I. To draw a straight line from any point to any point euclids elements book 13 | 677.169 | 1 |
Ray's Differential and Integral Calculus. 442 pages. Begins with definitions. Careful attention has been given to the teaching of the doctrine of limits, which has been made the basis of both the... More > Differential and Integral Calculus. Problems are supplied in the bookThis is a text on elementary multivariable calculus, designed for students who have completed courses in single-variable calculus. The traditional topics are covered: basic vector algebra; lines,... More > planes and surfaces; vector-valued functions; functions of 2 or 3 variables; partial derivatives; optimization; multiple integrals; line and surface integrals.
The book also includes discussion of numerical methods: Newton's method for optimization, and the Monte Carlo method for evaluating multiple integrals. There is a section dealing with applications to probability. Appendices include a proof of the right-hand rule for the cross product, and a short tutorial on using Gnuplot for graphing functions of 2 variables.
There are 420 exercises in the book. Answers to selected exercises | 677.169 | 1 |
While open to anyone interested in refreshing math skills, this course is designed primarily for new or current college students, particularly those who have not completed their college math requirement. It will provide math refresher materials covering a wide range of mathematical conceptsThis course will cover the mathematical theory and analysis of simple games without chance moves. This course explores the mathematical theory of two player games without chance moves. We will cover simplifying games, determining when games are equivalent to numbers, and impartial games. | 677.169 | 1 |
"Provides step-by-step instructions for working through all the major types of word problems usually found in algebra texts, providing sample problems, detailed answer explanations, and a practice drill."
""With this easy-to-use pocket guide, solving word problems in algebra become almost fun. This anxiety-quelling guide helps you get ready for those daunting word problems, one step at a time. With fully explained examples, it shows you how easy it can be to translate word problems into solvable algebraic formulas{u00AD}{u00AD}and get the answers right! You get complete directions for solving problems that involve time, money distance, work, and more."--Provided by publisher."
Al Algebra Elementary solvingmathematWord problems (Mathematics | 677.169 | 1 |
Find a MedfordGraduate level would be dependent on the specific topic. As far as the college general education form of "finite math", I teach the course often at my college. We cover set theory, logic, numerical systems, group theory, and elementary number theory | 677.169 | 1 |
"The Teaching of Mathematics: What Changes Are on the Horizon?"
Abstract:
Over the past two decades, a spotlight has been on calculus, precalculus, and college algebra courses. In this talk, we will look at why this happened, what the challenges were, and how changes in curriculum, pedagogy, and technology have changed the course. What do we expect in the future? What techniques are there to encourage students to think deeply about what they are learning? How do we ensure our courses are responsive to the needs of faculty and students? | 677.169 | 1 |
Soquel TrigonometryJames MLouise A.
...Algebra simply uses an "x" instead of a number. Algebra 2 is the time in the development of our curriculum that takes the basic skills and puts them into context. More types of functions are introduced and applied during this year | 677.169 | 1 |
Find a BeniciaThe key concepts from this are used in EVERY other math class that you can take. So it is essential that this subject is understood completely. When I took this class in middle school, I was one of the few people in my class that actually understood it | 677.169 | 1 |
Top Ten Math Books on Number Theory
Number theory is a part of math that's devoted primarily to the study of the integers, or in terms understood by all, it's arithmetic. Distinguished authors have written and published books about number theory; one is as interesting as the next.
If you're a math enthusiast or someone excited to learn more about mathematics, here's a list of top ten math books on number theory. Each one is an excellent read and leaves the reader with plenty to think about and gives math students some interesting problems to solve.
George E. Andrews, noted mathematics scholar covers topics including fundamental theorem of arithmetic, combinatorial and computational number theory and a great deal more. Of note, the author has included exercises for constructing numerical tables on your own or by using a computer. Appropriate as a basic introduction to number theory books for beginners, the author also holds the interest of students of more advance levels.
Authored by Underwood Dudley, a popular and respected writer explains many of the basics in his elementary number theory books. This second edition book includes nearly 1,000 exercises and problems to challenge the reader. Elementary Number Theory: Second Edition is a solid book for beginners and good refresher for students ready to advance to the next level.
The author takes a compelling look at numerical topics and concepts and their roles. The book is well – written and speaks to a wide audience. Color illustrations, drawings and diagrams are used to show math problems and theories that involve the reader and help to make learning easier. James Pommersheim's book on Number Theory is true to its secondary title: A Lively Introduction with Proofs, Applications and Stories".
If you already like studying math, you might find yourself loving it after reading one of the best number theory books written. It's fun, educational and easy to comprehend. Diagrams and color pictures help learners visualize math concepts to understand how it all works. The Book of Numbers was written by John H. Conway and Richard Guy.
Fundamentals of Number Theory was written by William J. LeVeque about the basics of number theory as related to algebraic concepts. Well – written and comprehensive, LeVeque talks about early number theory and continues on through problems to solve, tables, graphs, symbols and more. This is among the good books on number theory that contain content that is appropriate for math majors as well as beginning grad students.
Written by Titu Andreescu and Dorin Andrica, Number Theory: Structures, Examples, and Problems gets a top rating from readers and students for its problem solving approach, depth of content and inclusion of sophisticated open problems and puzzles. Stimulating and challenging for beginner math students. Appropriate for high school and undergrad college students and any interested in learning more about number theory.
Written by William J. LeVeque, Elementary Theory of Numbers is a short book that should appeal to math students and anyone new to exploring number theory. Recommended as a self-tutorial but handy as a guide or refresher for students. Although it's only about 120 pages long it's considered one of the top ten math books on number theory.
Excursions in Numbers contains a variety of challenging math problems that include prime numbers, number patterns and many more. Great for anyone who loves challenging math problems and puzzles and has a high school math education. Fun and interesting read by Stanley Ogilvy and John T. Anderson.
Written by John J. Watkins, an award – winning teacher, Number Theory covers math topics from soup to nuts while telling stories about many of the great mathematicians responsible for theoretical discoveries and developments. Number Theory: A Historical Approach is full of compelling biographical facts and events. It's an enjoyable read for every math lover and interesting even to the non-mathematician.
Unsolved Problems in Number Theory was written by Richard Guy. Along with complex problems for solving by different levels of math students and scholars. The book also contains valuable references to Neil Sloane's Online Encyclopedia of Integer Sequences. Third edition promises to be as exciting and well received as the first two editions. | 677.169 | 1 |
Plan
1.
Game Mathematics
(10 Week Lesson Plan, Lesson 6 through 10)
2.
Lesson 6: Analytic Geometry I
Textbook: Chapter Six (pgs. 125 – 149)
Goals:
Beyond triangles, students will also need to understand other important constructs. In this
lesson, we will introduce analytic geometry as the means for using functions and polynomials
to mathematically represent points, lines, planes and ellipses. All of these concepts are vital in
game development since they are used in rendering and optimization, collision detection and
response, game physics, and other critical areas. We will start with points in space and move on
to simple 2D lines and their various forms (including the allimportant parametric
representation). We will look at intersection formulas and distance formulas with respect to
lines, points, and planes and also briefly talk about ellipsoidal intersections.
Key Topics:
Points and Lines
§ TwoDimensional Lines
§ Parametric Representation
§ Parallel and Perpendicular Lines
§ Intersection of Two Lines
§ Distance from a Point to a Line
§ Angles between Lines
§ ThreeDimensional Lines
Ellipses and Ellipsoids
§ Intersecting Lines with Ellipses
§ Intersecting Lines with Spheres
Planes
§ Intersecting Lines with Planes
Projects: Varied Exercises
Exams/Quizzes: None
Recommended Study Time (hours): 5 – 7
Lesson 7: Vector Mathematics
3.
Textbook: Chapter Seven (pgs. 151 – 174)
Goals:
In this lesson, students are introduced to vector mathematics – the core of the 3D graphics
engine. After an introduction to the concept of vectors, we will look at how to perform various
important mathematical operations on them. This will include addition and subtraction, scalar
multiplication, and the allimportant dot and cross products. After laying this computational
foundation, we will look at the use of vectors in games and talk about their relationship with
planes and the plane representation, revisit distance calculations using vectors and see how to
rotate and scale geometry using vector representations of mesh vertices.
Key Topics:
Elementary Vector Math
§ Linear Combinations
§ Vector Representations
§ Addition/Subtraction
§ Scalar Multiplication/Division
§ Vector Magnitude
§ The Dot Product
§ Vector Projection
§ The Cross Product
Applications of Vectors
§ Directed Lines
§ Vectors and Planes
o Backface culling
o Vectorbased Plane Representation
§ Distance Calculations (Points, Planes, Lines)
§ Point Rotation, Scaling, Skewing
Projects: Varied Exercises
Exams/Quizzes: None
Recommended Study Time (hours): 8 10
Lesson 8: Matrix Mathematics I
Textbook: Chapter Eight (pgs. 177 – 188)
Textbook: Chapter Nine (pgs. 191 – 210)
4.
Goals:
In this lesson, students are introduced to the concept of a matrix. Like vectors, matrices are
one of the core components of every 3D game engine and as such are required learning. In this
first of two lessons, we will look at matrices from a purely mathematical perspective. We will
talk about what matrices are and what problems they are intended to solve and then we will
look at various operations that can be performed using them. This will include topics like
matrix addition and subtraction and multiplication by scalars or by other matrices. We will
conclude the lesson with an overview of the concept of using matrices to solve systems of linear
equations. We will do this by lightly touching on the notion of Gaussian elimination.
We continue our discussion of matrix mathematics and introduce the student to the problem
that matrices are generally used to solve in 3D games: transformations. After introducing the
idea of linear transformations, we will take a brief detour to examine how an important non
linear operation like translation (used to reposition points in 3D game worlds) can be made
compliant with our matrix operations by introducing 4D homogenous coordinates. Once done,
we will examine a number of common matrices used to effect transformations in 3D games.
This will include projection, translation, scaling and skewing, as well as rotations around all
three coordinate axes. We will wrap up with the actual vector/matrix transformation operation
(multiplication) which represents the foundation of the 3D graphics rendering pipeline.
Key Topics:
Matrices
§ Matrix Relations
§ Matrix Operations
o Addition/Subtraction
o Scalar Multiplication
o Matrix Multiplication
o Transpose
o Determinant
o Inverse
§ Systems of Linear Equations
o Gaussian Elimination
Linear Transformations
§ Computing Linear Transformation Matrices
§ Translation and Homogeneous Coordinates
§ Transformation Matrices
o The Scaling Matrix
o The Skewing Matrix
o The Translation Matrix
o The Rotation Matrices
o The Projection Matrix
§ Linear Transformations in 3D Games
6.
In this lesson, students are introduced to quaternion mathematics. To set the stage for
quaternions, which are hypercomplex numbers, we will first examine the concept of imaginary
numbers and look at the various arithmetical operations that can be performed on them. We
will look at the similarities and differences with respect to the real numbers. Once done, we will
introduce complex numbers and again look at the algebra involved. Finally we will examine the
quaternion and its associated algebra. With the formalities out of the way we will look at
applications of the quaternion in game development. Primarily the focus will be on how to
accomplish rotations about arbitrary axes and how to solve the gimbal lock problem
encountered with Euler angles. We put this concept to use to create an updated world to view
space transformation matrix that is derived from a quaternion after rotation has taken place.
Key Topics:
Imaginary Numbers
§ Powers
§ Multiplication/Division
§ Addition/Subtraction
Complex Numbers
§ Addition/Subtraction
§ Multiplication/Division
§ Powers
§ Complex Conjugates
§ Magnitude
Quaternions
§ Addition/Subtraction
§ Multiplication
§ Complex Conjugates
§ Magnitude
§ Inverse
§ Rotations
§ WorldtoView Transformation
Projects: Varied Exercises
Exams/Quizzes: None
Recommended Study Time (hours): 10 12
Lesson 10: Analytic Geometry II
Textbook: Chapter Eleven (pgs. 229 – 251)
7.
Goals:
In this lesson, we will focus on some of the practical applications of mathematics. In this
particular case we will look at how analytic geometry plays an important role in a number of
different areas of game development. We will start by looking at how to design a simple
collision/response system in 2D using lines and planes as a means for modeling a simple
billiards simulation. We will continue our intersection discussion by looking at a way to detect
collision between two convex polygons of arbitrary shape. From there we will see how to use
vectors and planes to create reflections such as might be seen in a mirror. Then we will talk
about the use of a convex volume to create shadows in the game world. Finally we will wrap
things up with a look at the Lambertian diffuse lighting model to see how vector dot products
can be used to determine the lighting and shading of points across a surface.
Key Topics:
§ 2D Collisions
§ Reflections
§ Polygon/Polygon Intersection
§ Shadow Casting
§ Lighting
Projects: Varied Exercises
Exams/Quizzes: None
Recommended Study Time (hours): 8 10
Final Examination
8.
The final examination in this course will consist of 35 multiplechoice and true/false questions
pulled from the final six textbook chapters. Students are encouraged to use the lecture
presentation slides as a means for reviewing the key material prior to the examination. The
exam should take no more than 1.5 hours to complete. It is worth 100% of student final grade. | 677.169 | 1 |
mysteries lesson plans to your Bookmark Collection or Course ePortfolio
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A lot of events in our life seem random or impossible to predict. However, with probability theory we can learn more about...
see more
A lot of events in our life seem random or impossible to predict. However, with probability theory we can learn more about these things to solve interesting problems that range from the lottery to diagnosing medical diseases. By teaching you basic principles and more advanced topics about theorems and models, this class will give you the tools to see the world in a different way that may not be intuitive but is proved by the math behindability: Random Isn't So Random | Highlights for High School to your Bookmark Collection or Course ePortfolio
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Do you love math but get bored in math class? Then this is the course for you! Combinatorics is a fascinating branch of...
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Do you love math but get bored in math class? Then this is the course for you! Combinatorics is a fascinating branch of mathematics that applies to problems ranging from card games to quantum physics to the Internet. The only pre-requisite is basic algebra; however we will be covering a lot of material. A mathematically agile mind will be helpful. Introductory Video View an introduction from the instructor outlining the aims of the course. More introductory videos are available in Meet the Instructor Videos Combinatorics: The Fine Art of Counting | Highlights for High School to your Bookmark Collection or Course ePortfolio
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This course offers an introduction to the polynomial method as applied to solving problems in combinatorics in the last...
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This course offers an introduction to the polynomial method as applied to solving problems in combinatorics in the last decade. The course also explores the connections between the polynomial method as used in these problems to the polynomial method in other fields, including computer science, number theory, and The Polynomial Method | Mathematics to your Bookmark Collection or Course ePortfolio
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This course offers an introduction to the finite sample analysis of high- dimensional statistical methods. The goal is to...
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This course offers an introduction to the finite sample analysis of high- dimensional statistical methods. The goal is to present various proof techniques for state-of-the-art methods in regression, matrix estimation and principal component analysis (PCA) as well as optimality guarantees. The course ends with research questions that are currently open. You can read more about Prof. Rigollet's work and courses on his website other High-Dimensional Statistics | Mathematics to your Bookmark Collection or Course ePortfolio
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The goal of this class is to prove that category theory is a powerful language for understanding and formalizing common...
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The goal of this class is to prove that category theory is a powerful language for understanding and formalizing common scientific models. The power of the language will be tested by its ability to penetrate into taken-for-granted ideas, either by exposing existing weaknesses or flaws in our understanding, or by highlighting hidden commonalities across scientific996 Category Theory for Scientists | Mathematics to your Bookmark Collection or Course ePortfolio
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The subject of enumerative combinatorics deals with counting the number of elements of a finite set. For instance, the number...
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The subject of enumerative combinatorics deals with counting the number of elements of a finite set. For instance, the number of ways to write a positive integer n as a sum of positive integers, taking order into account, is 2n-1. We will be concerned primarily with bijective proofs, i.e., showing that two sets have the same number of elements by exhibiting a bijection (one-to-one correspondence) between them. This is a subject which requires little mathematical background to reach the frontiers of current research. Students will therefore have the opportunity to do original research. It might be necessary to limit enrollment66 The Art of Counting | Mathematics to your Bookmark Collection or Course ePortfolio
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This course, which is geared toward Freshmen, is an undergraduate seminar on mathematical problem solving. It is intended for...
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This course, which is geared toward Freshmen, is an undergraduate seminar on mathematical problem solving. It is intended for students who enjoy solving challenging mathematical problems and who are interested in learning various techniques and background information useful for problem solving. Students in this course are expected to compete in a nationwide mathematics contest for undergraduates34 Problem Solving Seminar | Mathematics to your Bookmark Collection or Course ePortfolio
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Saint
Louis University
Department
of Mathematics and Mathematical Computer
Science
Maple
worksheets for abstract algebra
Overview
These
worksheets are part of an ongoing project of mine to produce
courseware for using Maple to help students learn concepts in
abstract algebra. I want my students to have the power of Maple for
exploring difficult concepts and for gaining intuition. On the other
hand, I don't want my class to become a class in Maple. The model I
am using is that Maple is a programming language that the instructor
uses to produce specific user friendly applications that the student
can use to explore a particular concept.
The
worksheets are being developed with exercises to encourage the
students not only to follow the Maple computations, but to experiment
with variations from the worked examples. Nevertheless it is
important that the worksheets be available to those who don't
understand computers at all. Thus, the first time through the
worksheet, the student only has to hit enter to see the results of
the computations. The second time through, the student can use the
worksheet as a template, changing numbers to explore similar
problems. If the student finds the material interesting, he or she
can use the code as a model for using Maple for more work. The image
I have is not to create black boxes, but to create clear fiberglass
boxes. The user can ignore the inner workings, but they are visible
for inspection if the user wants to tinker.
This
project started on Maple V release 3 on a Macintosh. They have been
revised to run on Maple 9.5. As I converted old
worksheets and wrote new ones, I added exercises. I find that
this
encourages the students to explore, and gain better insight. I have
also added a number of references to the textbook (Abstract Algebra,
by Dummit and Foote) we have been using.
The first three worksheets are concerned with concepts and
examples that come up in the study of rings.
The first worksheet, FactoringExamples.mw,
is a preliminary worksheet to familiarize the student with the
peculiarities of the factor command in Maple.
The second worksheet, GaussInt.mw,
looks at standard ring concepts over the Gaussian integers. The
worksheet looks at the standard Euclidean norm, prime factorization,
the Euclidean algorithm, and the Chinese Remainder theorem. The
worksheet utilizes the GaussInt package.
The third worksheet,
QuadraticEuclidean.mw, looks at the this same norm over a number of
other quadratic extensions of the integers. It considers extensions
where the norm is Euclidean as well as extensions where the norm is not
Euclidean. Then the norm is Euclidean, the students use it in the
Euclidean algorithm.
The next two worksheets look at ideas that come up in the
study of simple extensions of fields.
The worksheet MinPloy.mw
explains a way to use Maple to find the minimum polynomial of an
element that is algebraic over the rationals. The algorithm uses the
Grobner package.
The worksheet ComputingInFields.mw
shows how to compute the inverse of an element in a simple extension of
the rational. (With inverses, one can also compute quotients.) The
process uses the extended gcd command , and hence is an application of
the Euclidean algorithm over the polynomial ring.
The next three worksheets look at automorphisms of field
extensions, the heart of Galois theory.
The worksheet
DefiningAutomorphisms.mw takes the student through the process of
defining automorphisms of finite extensions of the rationals by
defining actions on the generators of the extension. The question of
when such a definition defines an automorphism is addressed.
The worksheet
AutomorphismGroups.mw looks at finding the group structure of a set
of automorphisms of a finite extension of the rationals. The worksheet
explores finding the order of an automorphism and composing
automorphisms.
The worksheet
AutosOverFiniteFields.mw looks at the group of automorphisms of a
finite extension of a finite prime field. Since the fields involved are
finite, these examples allow exhaustive testing.
The next pair of worksheets look at the problem of finding
the Galois group of an irreducible polynomial over the rationals.
The worksheet
GaloisGroupOfPoly.mw breaks apart Maple's galois command and looks
at the algorithm Maple uses to find the Galois group of an irreducible
polynomial over the rationals of degree at most 7. The students then
use the algorithm to find the Galois group of a polynomial of degree 8.
The worksheet
GaloisPolys2.mw walks the students through a probabilistic argument
that can be used to find the Galois group of a polynomial of arbitrary
degree. | 677.169 | 1 |
Schaum's Outline of Beginning Calculus
Summary
Confusing Textbooks? Missed Lectures? Tough Test Questions? ..Fortunately ..This Schaum's Outline gives you:. . Practice problems with full explanations that reinforce knowledge . Coverage of the most up-to-date developments in your course field . In-depth review of practices and applications ..Fully compatible with your classroom text, Schaum's highlights all the important facts you need to know. Use Schaum's to shorten your study time-and get your best test scores!..Schaum's Outlines-Problem Solved.. | 677.169 | 1 |
Numerical Methods for Ordinary Differential Equations is a self-contained introduction to a fundamental field of numerical analysis and scientific computation. Written for undergraduate students with a mathematical background, this book focuses on the analysis of numerical methods without losing sight of the practical nature of the subject. It covers the topics traditionally treated in a first course, but also highlights new and emerging themes. Chapters are broken down into 'lecture' size pieces, motivated and illustrted by numerous theoretical and computational examples. Over 200 exercises are provided and these are starred according to their degree of difficulty. Solutions to all exercises are available to authorized instructors | 677.169 | 1 |
Introduction to Numerical methods with strong emphasis to applications in engineering, physical and natural sciences. The course has a strong programming component done in MATLAB. Case studies with applications relevant to each engineering or mathematics stream will reinforce the theory seen in class.
Course Information:
An application-oriented course in scientific computation. Numerical methods and approximations underlie much of modern science, engineering and technology, such as modelling structures, aircraft, geophysical situations, the spread of viruses, design of integrated circuits, and for image processing problems such as creating special effects for movies. The blend of theory, numerical methods, modelling and applications forms the basis for scientific computation.
The emphasis will be to survey a number of different numerical techniques rather than discuss any single topic in great detail. It will involve a mix of techniques from calculus and linear algebra, together with algorithmic and programming considerations. Programming exercises will be conducted using MATLAB. The interplay between mathematics, algorithmic concepts, the coding and numerical experiments is what makes scientific computation such a fascinating subject. | 677.169 | 1 |
What is Keyah Math?
Kéyah Math, is a project that developed a series of versatile, place-based, culturally-responsive, and technology-intensive modules in mathematical geoscience to enhance undergraduate geoscience courses, particularly for Native American students. The name, Kéyah, the Diné (Navajo) term for their home lands and the environment, emphasizes this connection. | 677.169 | 1 |
In this section, we consider the way calculus principles can be applied to objects that are larger than scalars. The objects of interest to us are Vectors and Tensors. More formally, we will try to give interpretations to the derivatives and integrals of tensor functions of orders 0,1,2,3 and 4. The arguments can also be tensors of all relevant orders.
Furthermore, we will look at tensor fields. These are tensors that are functions of the Euclidean Point Space that we will fully define. We will be free to refer the point space to Cartesian as well as general coordinates.
It appears a number of students don't want the class tomorrow and many are travelling. It will be cancelled unless there is unanimity in holding it. Let me have a feedback from the class on this please. | 677.169 | 1 |
Course description:
This course covers the fundamental techniques in classical analytic number theory. The objects of study are the natural numbers; the theorems sought are statistical statements about the distribution of primes, the number of divisors of integers, and similar multiplicative questions; the techniques involve both "by hand" real analytic estimation and contour integration of meromorphic functions. The successful student will be well-equipped to understand much of the current research literature in this area.
Prerequisites:
Students should have had a previous course in number theory (preferably MATH 537 here at UBC). It will be assumed that the student has had the usual undergraduate training in analysis (for example, MATH 320) and a strong course in complex analysis (preferably MATH 508). In particular, in complex analysis students should have a working knowledge of the residue theorem, logarithmic derivatives, and the argument principle. Students will also need to have a working knowledge of LaTeX, although this can be acquired along the way if necessary.
Course textbook: This course will require the book by Montgomery and Vaughan, Multiplicative Number Theory I: Classical Theory (Cambridge University Press, 2006). Please let me know if you encounter problems buying the textbook from the UBC bookstore. Here is a link to Hugh Montgomery's home page, at which you can access a list of errata for the book. If you find an error not on the list, you should email him!
Topics to be covered in this course:
Dirichlet series and the Mellin transform
Arithmetical functions and their summation and estimation
Prime counting functions and Chebyshev's and Mertens's estimates
The Riemann zeta function and its zeros
The prime number theorem and applications
Dirichlet characters and L-functions
The prime number theorem in arithmetic progressions
Use of the internet: We will be using Piazza for all class-related announcements and discussion. Piazza is a question-and-answer platform specifically designed to expedite answers to your questions, using the collective knowledge of your classmates and instructor. It has several features that facilitate discussion of mathematics, most notably LaTeX support. You are encouraged to answer your classmates' questions, or to brainstorm towards answers, every bit as much as you are encouraged to ask questions.
No handouts will be distributed in class. All homework assignments and any other course materials will be posted on Piazza in PDF format. I encourage you to ask questions on Piazza any time you're working towards understanding a concept; you can even do so anonymously if necessary. Part of the reason I don't have regularly scheduled office hours is that many questions can be answered through Piazza; in fact, I prefer using Piazza instead of email for questions related to the course (Piazza allows you to ask privately if necessary).
Evaluation: The course mark will be based on eight homework assignments, due approximately every five class days (a little more often than once every two weeks). Your homework will be marked on correctness, completeness, rigor, and elegance. A correct answer will not earn full marks unless it is completely justified, in a rigorous manner, and written in a logical sequence that is easy to follow and confirm. Survival tip: don't start these assignments the night before they're due! Anecdotal evidence suggests that each assignment could take as much as 15 hours or more to complete.
Homework solutions must be prepared in LaTeX and submitted in PDF format via email. I will supply LaTeX templates with each assignment. All homeworks are due before the beginning of class (9:59 AM) on the indicated days.
Homework #0: due Friday, January 13
Homework #1: due Wednesday, January 25
Homework #2: due Monday, February 6
Homework #3: due Friday, February 17
Homework #4: due Friday, March 9
Homework #5: due Monday, March 19
Homework #6: due Friday, March 30
Homework #7: due Monday, April 16
Students are allowed to consult one another concerning the homework problems, but your submitted solutions must be written by you in your own words. Students can be found guilty of plagiarism if they submit virtually identical answers to a question, or if they do not understand what they have submitted.
Because there are no exams, the lectures will continue into the beginning of the final exams period, ending on (probably) Wednesday, April 18.
I. Niven, H. S. Zuckerman, and H. L. Montgomery, An Introduction to the Theory of Numbers
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers
Page maintained by
and last modified on Mar. 05, 2012.
This page's URL is:
This page, and all files on this server linked from it, are ęGreg Martin and are not to be copied, used, or revised without explicit written permission from the copyright owner. (Some manuscripts have had their copyright assigned to the journals in which they are published). | 677.169 | 1 |
The Math Page
Welcome to the Math Page. Here you will discover several technology-based math lessons. Enjoy!
Mathematica - Secant and Tangent Lines.
From the curriculum materials page at the Institute for Technology
in Mathematics, this grade 12 Calculus lesson uses Mathematica to
recognize and work with graphs of secant lines and the two formulas
of slope.
Interactive Mathematics Miscellany And Puzzles
Teachers of K-12 math will find this collection of online games and
puzzles that take advantage of the latest interactive technology...
a fun approach to having students apply deductive reasoning skills
and math concepts in various math subjects.
Mathematics: Web-Linked Activities and Lesson Plans For Grades
1-6
Although designed to complement Mcgraw-Hill School Division
educational materials, you will find lesson plans in a variety of
subjects which can be easily modified to meet your needs. Solutions of Radical Equations.
From the curriculum materials page at the Institute for Technology
in Mathematics, this high school algebra plan has students graph and work
with radical functions using the TI-81/82 graphing calculator. Extreme Values.
From the curriculum materials page at the Institute for Technology
in Mathematics, this high school Calculus lesson plan works with such
concepts as maxima and minima and extreme values, using the TI-
81/82 graphing calculator. Parallel Lines
From the curriculum materials page at the Institute for Technology
in Mathematics, this high school Algebra lesson plan explores parallel
lines and properties of slope using the TI-81/82 graphing
calculator. Three Dimensional Sketching.
From the curriculum materials page at the Institute for Technology
in Mathematics, this high school Geometry plan compares perspective and
three dimensional sketching, and introduces the concept of
vanishing point using the Geometer's Sketchpad. An Animated Triangle Inequality.
From the curriculum materials page at the Institute for Technology
in Mathematics, this Plane Geometry high school lesson plan uses the
Geometer's Sketchpad to observe what happens when you manipulate a
triangle. Fixed Area of a Triangle.
From the curriculum materials page at the Institute for Technology
in Mathematics, this high school Geometry lesson plan uses the Geometer's
Sketchpad to help students discover what happens to the area of a
triangle when the base remains the same, but the third vertex is
moved. Visualizing Arithmetic and Geometric Means.
From the curriculum materials page at the Institute for Technology
in Mathematics, this high school Algebra lesson plan helps students apply
their knowledge of arithmetic and geometric means visually using
the Geometer's Sketchpad. Pythagorean Theorem Investigation.
From the curriculum materials page at the Institute for Technology
in Mathematics, this high school Mathematics lesson plan incorporates the
use of Geometer's Sketchpad to the learning of the Pythagorean
Theorem and related concepts. Math Lessons
Mathematics teachers from k-12 are invited to search these pages
containing individual lesson plans that integrate teaching
different math concepts with the use of various technologies: from
the web, to computer applications, to math software. Brain Teasers
From "Link 2 Learn," this primary level lesson has kids work on
problem solving and deductive reasoning, applying this to fun
puzzles on the web. Brain Aerobics
From "Link 2 Learn," this intermediate level lesson has students
apply reasoning and other critical thinking skills to solving
online puzzles. Math Mutt!
From "Link 2 Learn," this primary level lesson has kids practice
addition, then go online to play addition games with an internet
puppy. It's All In Proportion.
From "Link 2 Learn," this middle school lesson is about fractions,
percentages and decimals. Get Into Shape!
From "Link 2 Learn," this middle school lesson reviews basic
geometric concepts, with problem solving activities online. Math Aid
From "Link 2 Learn," this high school lesson introduces students to online
algebra and trigonometry tutorials. Seeing Double! Triple!
From "Link 2 Learn," this h.s. lesson has students go online to
learn about and experience stereograms, linking this to geometry. Flashcards for Kids
This site allows primary students to practice basic math skills. The functions they can practice include addition, subtraction, multiplication, and division and range in number size as well as complexity. | 677.169 | 1 |
Real World Mathematics
10 Jan 2003
Have you ever sat in your math class asking yourself, "Why do I need to
learn trigonometry and calculus? Who uses trigonometry and calculus in the real
world?"
Civil engineering professors and students from the University of Hawaii
College of
Engineering visited McKinley High School and Kaimuki High School
to teach
upper division math students how civil engineers use trigonometry and calculus
to design and construct buildings, bridges, and other structures.
Dr. Robertson demonstrating the shake table.
At each high school, students were shown three exciting, hands-on civil
engineering demonstrations, where students got up close and personal with various
civil engineering equipment and testing tools.
In one of the demonstrations, civil engineering associate professor
Dr. Ian Robertson and civil engineering graduate student Alison Agapay demonstrated
how shake tables and determining a structure's natural frequency are used in the
building of a structure.
When designing and constructing a building, engineers have to consider
many factors. They have to figure out how a structure will respond to things like
earthquakes, wind, blasts, and extreme impacts. These events cause structures to
vibrate. If these vibrations happen to be the same as the natural vibration
frequency of the structure, the vibrations
can become very large and cause the structure to collapse. Using
mathematics, engineers can determine the frequency of a structure.
To demonstrate this Dr. Robertson had rods of various stiffness, height,
and mass on a computer controlled shake table. When a certain frequency is inputted
into the computer, only the rod that is attuned to that frequency will vibrate.
Change the frequency and another rod will vibrate.
A Kaimuki student using the surveying equipment.
Another demonstration dealt with surveying. You've probably seen
surveyors at
construction sites measuring various things, where one person is holding a
pole up and
another is looking through a piece of equipment from far away. The job of
surveyors is
to determine the height of things, distances, and slopes with their
equipment.
Graduate student Matt Fujioka gave presentations at McKinley High
School. Doctoral
candidate Lin Zhang, graduate student Kainoa Aki, and juniors Jason Chee
and Eric
Tashima gave surveying presentations at Kaimuki High School. Surveying
equipment
was brought in for the high school students to use. The tools of the
surveyor include a
theodolite, which is used to measure horizontal and vertical angles, and a
level, which measures elevations of the land relative to sea level. Trigonometry is a
big part of
surveying, since it deals with angles and altitudes.
At Kaimuki High School, Dr. Horst Brandes and at McKinley High School,
Randy
Akiona helped explain the forces of compression and tension using a
three-truss bridge.
Compression is the force that forms when two things are pushed together
and tension is
the force that is formed when something is pulled apart. Imagine pushing
a spring
together, it will try to get back to its original state, that's
compression. Now imagine
pulling that same spring apart, the force that wants to bring the spring
back to its original
state is called tension.
Randy Akiona working with McKinley students.
All structures have to take into consideration these forces of compression
and tension.
Students volunteered to construct the bridge and after they were done,
some of them
where able to cross it. They also learned when engineers design bridges
they also have to
think about factors like weight, load, and the wind. "It was interesting
because it was a
hands-on activity. We were able to see the structure being built and the
elements of
compression and tension playing their parts," McKinley student Diana Wan
said.
These demonstrations opened the eyes of many of these students to the
exciting world of
engineering. "After the presentations, I got a better understanding of
what engineers
really do," McKinley student Jessica Mau said. "I realize that engineers
thrive on
challenges and the love of their job makes it fun, yet serious. The
feeling of satisfaction
after the end product must really be rewarding." | 677.169 | 1 |
Free Coursera Calculus course with hand-drawn animated materials
From the Boing Boing Shop
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Robert Ghrist from University of Pennsylvania wrote in to tell us about his new, free Coursera course in single-variable Calculus, which starts on Jan 7. Calculus is one of those amazing, chewy, challenging branches of math, and Ghrist's hand-drawn teaching materials look really engaging.
Calculus is one of the grandest achievements of human thought, explaining everything from planetary orbits to the optimal size of a city to the periodicity of a heartbeat. This brisk course covers the core ideas of single-variable Calculus with emphases on conceptual understanding and applications. The course is ideal for students beginning in the engineering, physical, and social sciences. Distinguishing features of the course include:
the introduction and use of Taylor series and approximations from the beginning; signed up for this course, rather than another introductory calculus course offered through Coursera, because Ghrist's approach seems radically different than the industry standard. His Funny Little Calculus Textbook starts off with functions, but immediately jumps to Taylor series, assuming the reader knows how to take the derivatives of simple polynomials. In other words, it seems more like a course about understanding calculus than doing calculus.
Even as a programmer I've always felt my basic calc was a bit rusty, and while I could probably just take a two-session refresher course and jump straight to Calc 2, this course looked fun. Now… hopefully I can stick with the schedule better than I could with "The History of the World Since 1300." Who would have guessed that 700 years of history would require lots of reading and lectures? (The course was very good, and I made it through four weeks on-schedule, but in the end I didn't have nearly enough time.)
Holy moley! It's still as densely unapproachable to me as I remember it being when I flunked it in high school! Even with cartoons, which always grab my attention! I was lost by 1:03 seconds in! Thank goodness there are other people that can do this stuff and put it to practical use. I'll just stay in the kitchen, if anyone needs a sandwich.
I did a computer science course on Coursera in the spring that I thought was very good. I then tried another and didn't like the way it was done and gave up on it. I definitely need a calculus refresher and this looks good so I'm sold on this one, I hope it turns out well. | 677.169 | 1 |
Find a TaminaThomas and received an A in the course. Linear Algebra is the study of matrices and their properties. The applications for linear algebra are far reaching whether you want to continue studying advanced algebra or computer scienceIf a teacher or tutor creates an environment where the student is "allowed" to learn, the results quickly change. This is not soft, lese faire approach. It does require that the work be done and learning happen, however, my techniques involve bringing out the learning rather than inculcating it on the child. | 677.169 | 1 |
Elementary Differential Equations with Boundary Value Problems integrates the underlying theory, the solution procedures, and the numerical/computational aspects of differential equations in a seamless way. For example, whenever a new type of problem is introduced (such as first-order equations, higher-order equations, systems of differential equations, etc.) the text begins with the basic existence-uniqueness theory. This provides the student the necessary framework to understand and solve differential equations. Theory is presented as simply as possible with an emphasis on how to use it. The Table of Contents is comprehensive and allows flexibility for instructors.
Table of Contents
Preface
xi
Introduction to Differential Equations
1
(14)
Introduction
1
(1)
Examples of Differential Equations
1
(7)
Direction Fields
8
(7)
First Order Differential Equations
15
(92)
Introduction
15
(4)
First Order Linear Differential Equations
19
(12)
Introduction to Mathematical Models
31
(10)
Population Dynamics and Radioactive Decay
41
(7)
First Order Nonlinear Differential Equations
48
(6)
Separable First Order Equations
54
(9)
Exact Differential Equations
63
(7)
The Logistic Population Model
70
(7)
Applications to Mechanics
77
(12)
Euler's Method
89
(18)
Review Exercises
100
(1)
Projects
101
(6)
Second and Higher Order Linear Differential Equations
107
(106)
Introduction
108
(7)
The General Solution of Homogeneous Equations
115
(6)
Constant Coefficient Homogeneous Equations
121
(6)
Real Repeated Roots; Reduction of Order
127
(5)
Complex Roots
132
(10)
Unforced Mechanical Vibrations
142
(12)
The General Solution of a Linear Nonhomogeneous Equation
154
(4)
The Method of Undetermined Coefficients
158
(10)
The Method of Variation of Parameters
168
(6)
Forced Mechanical Vibrations, Electrical Networks, and Resonance
174
(14)
Higher Order Linear Homogeneous Differential Equations
188
(7)
Higher Order Homogeneous Constant Coefficient Differential Equations
195
(6)
Higher Order Linear Nonhomogeneous Differential Equations
201
(12)
Review Exercises
206
(1)
Projects
206
(7)
First Order Linear Systems
213
(104)
Introduction
213
(10)
Existence and Uniqueness
223
(5)
Homogeneous Linear Systems
228
(10)
Constant Coefficient Homogeneous Systems; the Eigenvalue Problem
238
(9)
Real Eigenvalues and the Phase Plane
247
(9)
Complex Eigenvalues
256
(10)
Repeated Eigenvalues
266
(11)
Nonhomogeneous Linear Systems
277
(11)
Numerical Methods for Systems of Linear Differential Equations
288
(12)
The Exponential Matrix and Diagonalization
300
(17)
Review Exercises
310
(1)
Projects
311
(6)
Laplace Transforms
317
(74)
Introduction
317
(12)
Laplace Transform Pairs
329
(15)
The Method of Partial Fractions
344
(6)
Laplace Transforms of Periodic Functions and System Transfer Functions
350
(9)
Solving Systems of Differential Equations
359
(9)
Convolution
368
(9)
The Delta Function and Impulse Response
377
(14)
Projects
385
(6)
Nonlinear Systems
391
(80)
Introduction
391
(9)
Equilibrium Solutions and Direction Fields
400
(13)
Conservative Systems
413
(11)
Stability
424
(9)
Linearization and the Local Picture
433
(15)
Two-Dimensional Linear Systems
448
(10)
Predator-Prey Population Models
458
(13)
Projects
466
(5)
Numerical Methods
471
(44)
Introduction
471
(2)
Euler's Method, Heun's Method, and the Modified Euler's Method
473
(6)
Taylor Series Methods
479
(14)
Runge-Kutta Methods
493
(22)
Appendix 1: Convergence of One-Step Methods
506
(1)
Appendix 2: Stability of One-Step Methods
507
(3)
Projects
510
(5)
Series Solutions of Linear Differential Equations
515
(50)
Introduction
515
(12)
Series Solutions Near an Ordinary Point
527
(9)
The Euler Equation
536
(6)
Solutions Near a Regular Singular Point and the Method of Frobenius
542
(8)
The Method of Frobenius Continued: Special Cases and a Summary
550
(15)
Projects
561
(4)
Second Order Partial Differential Equations and Fourier Series
565
(94)
Introduction
565
(5)
Heat Flow in a Thin Bar; Separation of Variables
570
(10)
Series Solutions
580
(9)
Calculating the Solution
589
(11)
Fourier Series
600
(16)
The Wave Equation
616
(12)
Laplace's Equation
628
(13)
Higher-Dimensional Problems; Nonhomogeneous Equations
641
(18)
Project
655
(4)
First Order Partial Differential Equations and the Method of Characteristics | 677.169 | 1 |
Browse related Subjects ...
Read More activities are found in her accompanying Activities Manual , which comes with every new copy of the text. As a result, students engage, explore, discuss, and ultimately reach a true understanding of mathematics. The new Active Teachers, Active Learners DVD helps instructors enrich their classroom by expanding their knowledge of teaching using an inquiry-based approach. The DVD shows Beckmann and her students discovering various concepts, along with voiceover commentary from Beckmann. This DVD is the ideal resource for instructors who are teaching with an inquiry-based approach for the first time, and for instructors who seek new ideas to integrate into their course. The table of contents is organized by operation rather than number type to foster a more unified understanding of the math concepts. Throughout the text, students learn why the math works, rather than just the mechanics of how it works. In this new edition the contents have been updated and rearranged for a more natural organization good. [Activity Manual included] [Authentic USA edition] [I will ship immediately] Book in standard used condition: may have minimal or no markings, slightly worn covers and edges, and good binding.
Good. This is a hard cover book. The dust jacket shows normal wear and tear. The cover has visible markings and wear. Book cover has some corner dings. The pages have normal wear | 677.169 | 1 |
> It has some of the same elements as algebra, but lacks the transformative elements that promote reuse and generalization.
What I mean by that is that algebra focuses on the art of solving itself, in general. It is more concerned with solution forms rather than a particular instance of a solution. Arithmetic and pre-algebra lack that focus. Don't get me wrong, you form strategy in arithmetic, but it isn't organized and studied like it is in algebra. | 677.169 | 1 |
Abstract Algebra
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Read More exercises to aid the reader's understanding. In this way, readers gain an appreciation for how mathematical structures and their interplay lead to powerful results and insights in a number of different settings. The emphasis throughout has been to motivate the introduction and development of important algebraic concepts using as many examples as possible | 677.169 | 1 |
Description
This text offers a comprehensive presentation of the mathematics required to tackle problems in economic analyses. To give a better understanding of the mathematical concepts, the text follows the logic of the development of mathematics rather than that of an economics course. The only prerequisite is high school algebra, but the book goes on to cover all the mathematics needed for undergraduate economics. It is also a useful reference for graduate students. After a review of the fundamentals of sets, numbers, and functions, the book covers limits and continuity, the calculus of functions of one variable, linear algebra, multivariate calculus, and dynamics. To develop the student's problem-solving skills, the book works through a large number of examples and economic applications. This streamlined third edition offers an array of new and updated examples. Additionally, lengthier proofs and examples are provided on the book's website. The book and the Web material are cross-referenced in the text. A student solutions manual is available externally, and instructors can access online instructor's material that includes solutions and PowerPoint slides. Visit for complete details.
Review quote
`Mathematics is the language of economics, and this book is an excellent introduction to that language.` George J. Mailath , Walter H. Annenberg Professor in the Social Sciences and Professor of Economics, University of Pennsylvania `While there are many mathematics texts for economics available, this one is by far the best. It covers a comprehensive range of techniques with interesting applications, and the numerous worked examples and problems are a real bonus for the instructor. Teaching a course with this book is enjoyable and easy.` Kevin Denny , University College Dublin
About Thanasis Stengos
Michael Hoy is a faculty member in the Economics Department at the University of Guelph. John Livernois is a faculty member in the Economics Department at the University of Guelph, Ontario. Chris McKenna is a faculty member in the Economics Department at the University of Guelph, Ontario. Ray Rees is a faculty member at the Ludwig Maximilians University, Munich. Thanasis Stengos is a faculty member in the Economics Department at the University of Guelph, Ontario. | 677.169 | 1 |
Fundamentals of PrecalculusFundamentals of Precalculusis designed to review the fundamental topics that are necessary for success in calculus. Containing only five chapters, this text contains the rigor essential for building a strong foundation of mathematical skills and concepts, and at the same time supports studentsrs" mathematical needs with a number of tools newly developed for this revision. A student who is well acquainted with the material in this text will have the necessary skills, understanding, and insights required to succeed in calculus.
Mark Dugopolski was born in Menominee, Michigan. After receiving a B.S. from Michigan State University, he taught high school for four years, and then went on to receive an M.S. in mathematics from Northern Illinois University. He also received a Ph.D. in the area of topology from the University of Illinois at Champaign-Urbana. Mark has been teaching at Southeastern Louisiana in Hammond, LA, ever since. Mark has been writing textbooks for about fifteen years. He is married and has two daughters, and enjoys playing tennis, jogging, and riding his bicycle in his spare time. | 677.169 | 1 |
MatBasic Desciption:
The MatBasic is the language of mathematical calculations. Strong mathematical base: full complex arithmetic's, linear algebra and operations, nonlinear methods and graphical visualization.
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MatBasic is a calculating, programming and debugging environment using special high-level programming language designed for solving mathematical problems. MatBasic programming language allows execution of difficult mathematical calculations, involving an exhaustive set of tools for the purpose of creation of algorithmic programs. It also allows a user to abstract his mind from the type of working data which can be either real-valued, or complex numbers, or matrices, or strings, or structures, etc. The MatBasic supports both the text and the graphical data visualization.
MatBasic is fast language interpreter and its environment application field is wide: from solving the school problem to executing different engineering and mathematical computations. The MatBasic programming language combines; simplicity of BASIC language, flexibility of high-level languages such as C or Pascal and at the same time turns up to be a powerful calculation tool. By means of a special operating mode, Matbasic it is possible to use as the powerful calculator. Also the MatBasic can be used for educational purpose as a matter of studying the bases of programming and raising algorithmization skillsDifferential Equations is a handy application designed to help you solve equations with minimum effort. The program enables you to specify the coefficients by using the keyboard on the main window.It is designed to calculate the solution to homogeneous...
Linear Algebra Decoded is a program designed to assist students in the subject of Linear Algebra, although it has features for professors, including the ability to generate tests where problems are customized and solutions are in the field of integers....
This script defines the Matrix class, an implementation of a linear algebra matrix. Arithmetic operations, trace, determinant, and minors are defined for it. This is a lightweight alternative to a numerical Python package for people who need to doThe Bluebit .NET Matrix Library provides classes for object-oriented linear algebra in the .NET platform. It can be used to solve systems of simultaneous linear equations, least-squares solutions of linear systems of equations, eigenvalues and...
Diofantos is a library for the solution of equations that arise in physics. It deals with ordinary differential equations (ODE), partial differential equations (PDE), including grid generation, and integral equations.
centralApp Controller was built as a small and useful app that uses Ordinary Differential Equations to find a solution to a body under a central force.centralApp Controller was developed with the help of the Java programming language and can run | 677.169 | 1 |
This
chapter presents first an overview of the topics pertinent to this study.The first section describes the role that
multiple representations have played in the teaching and learning of
mathematics.The second section includes
the importance that the study of functions has in mathematics curricula.The last section discusses how technology
has been used in order to enhance and promote a better understanding of
mathematics.The statement of the
problem, the purpose of the study and the research questions complete this
chapter.
Multiple
Representations
One of the most important
issues that arises in mathematics education scenarios is the fact that ways
need to be found to promote understanding in mathematics (Hiebert and
Carpenter, 1992).In order to fulfill
this goal, teachers, administrators, curriculum designers and researchers have
suggested and implemented different ideas, based on mathematical learning
theories.As cited in Porzio (1994) and
based on research done by Hiebert and Carpenter, Kaput (1989a) and Skemp
(1987), "an emerging theoretical view on mathematical learning that has been
growing in significance is that multiple representations of concepts can be
utilized to help students develop deeper, more flexible understanding" (p. 3).
The role and use of multiple
representations have been constituted as an emerging research and extensive
discussion area during the last years in the mathematics education
community.Most recently, the National
Council of Teachers of Mathematics (NCTM, 2000), facing a new millennium, has
included the uses of representations as one of the new standards in mathematics
teaching and learning. The representation standard states:
Instructional
programs from prekindergarten through grade 12 should enable all students to
create and use representations to organize, record, and communicate
mathematical ideas; select, apply, and translate among mathematical
representations to solve problems; and to use representations to model and
interpret physical, social, and mathematical phenomena. (p. 67)
This educational guide, illustrated by this standard
in its three aspects, confirms the important and transcendental role and the
urgent need of using representations in teaching mathematics at all levels,
grades K through 16.
It
has been extensively discussed that mathematics, by its own nature, is one of
the academic subjects where multiple representations are currently used (De
Jong, et al., 1998).Mathematics as a
"collection of languages" (Kaput, 1989a, p. 167) and characterized, the
majority of the time by the presence of symbols and abstractions, is one of the
fields where representations could be used widely due to their capabilities to
enhance "understanding and for communicating information" (Greeno and Hall,
1997, p. 362).Due to this extensive
use of symbols, abstractions, rules, definitions, it is also known that
students in mathematics are confronting real troubles trying to understand,
internalize, apply, and communicate important concepts in their mathematics
school levels.Because of this, it is
right and necessary to think about the ways that mathematical ideas are being
currently represented, due to the understanding of these concepts and the use
of the ideas depend on how these representations are being used (NCTM, 2000).
Dufour-Janvier,
Bednarz, and Belanger (1987) have classified the term representation in two
major categories: internal representations and external representations. Each
of them possesses a considerable amount of sub-themes exposed to more and
deeper research linked with other fields.According to them, the first category deals with "more particularly
mental images corresponding to internal formulations we construct of reality".The second area deals with "all external
symbolic organizations" (p. 109), illustrated frequently in the forms of
symbols, schema, and diagrams.Özgün-Koca (1998) states "multiple representations are defined as
external mathematical embodiments of ideas and concepts to provide the same
information in more than one form" (p. 1).On the other hand, NCTM (2000) affirms that the "term representation
refers both to process and to product –in other words, to the act of capturing
a mathematical concept or relationship in some form and to the form itself" (p.
67).This research project, in order to
fulfill its objectives, proposes to limit the term representation to its
external category.
The capabilities
of using these representations in mathematics teaching and learning have also
been discussed and illustrated by the literature.Özgün-Koca (1998) suggested that the use of multiple
representations in mathematics could provoke an appropriate and healthy
environment for students to abstract and understand major mathematical
concepts.Moreover, Dufour-Janvier and
colleagues (1987) expressed their motives for using external representations in
mathematics.They argued that first,
representations are an inherent part of mathematics; second, representations are
multiple concretizations of a concept; third, representations are used locally
to mitigate certain difficulties; and last, the representations are intended to
make mathematics more attractive and interesting (p. 110-111).Porzio (1994) calls "obvious" all of the
benefits that the use of multiple representations can give to mathematical
teaching and learning (p. 47).In
addition, as cited in the same study, Kaput (1992) says that the use of more
than one representation or notation system help to illustrate a better picture
of a mathematical concept or idea."Complex ideas are seldom adequately represented using a single notation
system.The ability to link different
representations helps reveal the different facets of a complex idea explicitly
and dynamically" (p. 542).In summary,
mathematics at all levels needs the use of representations in order to
communicate appropriately ideas, and more importantly, to transmit, meaning,
sense and understanding.
Other studies have
supported the use of representations in mathematics in order to enhance concept
understanding.Hiebert and Carpenter
(1992) state that the process of the learning of mathematics with understanding
"extends beyond the boundaries of mathematics education" (p. 65).They define understanding as the way certain
information can be represented and structured.Moreover, they affirm that "mathematics is understood if its mental
representation is part of a network of representations" (p. 67).Kaput (1989a), as well as, Keller and Hirsch
(1998) found that the use of multiple representations provide diverse
concretizations of a concept, carefully emphasize and suppress aspects of
complex concepts, and promote the cognitive linking of representations.Furthermore, Moschkovich, Schoenfeld and Arcavi
(1993) explored in their research the fact that there are multiple ways to
solve a given problem and that solving a problem calls for making connections
across representations and for employing both the process and object
perspectives (p. 94).In this way, NCTM
(2000) states,"representations should
be treated as essential elements in supporting students' understanding of
mathematical concepts and relationships; in communicating mathematical
approaches, arguments, and understandings to one's self and to others; in
recognizing connections among related mathematical concepts; and in applying
mathematics to realistic problem situations through modeling" (p. 67).In summary, it has been showed that the use
of multiple representations is a useful tool to promote better understanding of
key concepts in the mathematics curricula.
Functions have a
key place in the mathematics curriculum, at all levels of schooling;
particularly in secondary and college levels where they get their maximum
expressions and representations.The
concept of function has been usually introduced early in algebra courses,
starting in the majority of the cases with the linear form.As a result, NCTM (2000) has placed the
concept of function as one of the cornerstones of mathematics curricula:
algebra.The algebra standard states
that students from prekindergarten through twelfth grade should understand
patterns, relations, and functions (p. 37).Thorpe (1989) proposed the use of functions "as the centerpiece of
algebra instruction" (Gningue, 2000, p. 28).The literature in mathematics education possesses a vast amount of
research concerning functions and their teaching and learning.Dubinsky and Harel (1992), and Cooney and
Wilson (1993) have agreed to say that functions should be located at the center
of the mathematics curricula.Lastly,
Selden and Selden (1992) point out that functions play a central and unifying
role in mathematics (Poppe, 1993, p. 2).
By their nature,
functions are one of the best examples in which to use multiple representations
in the teaching and learning process.Researchers have agreed that functions can be represented in the
following forms: algebraic or formulas, tables, and graphs (Brenner, et al.,
1997; Greeno & Hall, 1997; Iannone, 1975; Janvier, et al., 1993; Mevarech
& Kramarsky, 1997; and others)."These forms of representation – such as diagrams, graphical displays,
and symbolic expressions – have long been part of school mathematics" (NCTM,
2000, p. 67).In the same document,
NCTM continues saying that one of the major goals of algebra is that students
should "understand the relationships among tables, graphs, and symbols and to
judge the advantages and disadvantages of each way of representing
relationships for particular purposes" (p. 38).Furthermore, Leinhardt and colleagues (1990) and Moschkovich, et
al. (1993) affirm that using multiple representations to teach functions, that
is, numeric, graphic, and symbolic, will enhance a broad understanding of
functions.In summary, the use of
representations in mathematics consists of a rich and varied group of
alternatives that students can use, whenever they want, in order to promote a
better achievement of a particular topic.
Technology in all
of its manifestations plays an important and primary role in introducing and
supporting multiple representations in mathematics.It has served to engage students in a harmonious process of
teaching and learning mathematics.Through the use of technology, multiple representations can be
introduced more powerfully as well as, in an interactive and attractive way
(Confrey, et al., 1991).Fey (1989)
proposed the use of calculators and computers to introduce algebraic concepts
like functions.Porzio (1994) assures
that "instructional practices that involve the use of multiple representations
are not employed simply because technology now makes multiple representations
more readily accessible, but because of the potential benefits associated with
their use" (p. 4).Fey (1989),
Goldenberg (1987), and Kaput (1992) have agreed that due to the advancements
and advantages of technology, the chance to provide students better access to
the use of representations have considerably increased.In summary, the appropriate use of
technology, represented in this case by graphing calculators, computers,
software packages, like spreadsheets, without doubts, brings an invaluable
direction to the acquisition and understanding of mathematical concepts, such
functions, at the same time, emphasizing the varied representations that
functions have (Schwarz, Dreyfus, and Bruckheimer, 1990; Browning, 1991; and
Hart, 1991).
Following calls
for reform according to Keller & Hirsch (1998), current precalculus and
calculus reform projects are attempting to incorporate numeric, graphic, and
symbolic representations into the curriculum.The Calculus Consortium at Harvard (2001), a group of recognized
scholars established in the late 1980's, started a revolution in the teaching
and learning of mathematics, particularly in calculus courses at the college
level.One of the guiding principles of
this consortium is based on the 'Rule of Four' where mathematics topics are
introduced geometrically, numerically, analytically, and verbally (Hart, 1991;
Hughes-Hallet, 1991; Megginson, 1995 & Porzio, 1994).
During the past
decade, with the purpose to "consider the needs of all undergraduates attending
all types of United States two- and four-year colleges and universities", the
National Science Foundation (NSF) issued the report Shaping the Future
on new expectations for undergraduate education in science, mathematics,
engineering, and technology (George, et al. 1996, p. ii).The goal of this report was that:
All students have access to supportive, excellent
undergraduate education in science, mathematics, engineering, and technology,
and all students learns these subjects by direct experience with the methods
and processes of inquiry. (p. ii)
As part of
this report, the NSF emphasized the importance of the effective use of technology
to enhance learning (p. iv) recommending to institutions of higher education
its incorporation into the curriculum of science, mathematics, engineering, and
technology.
The proposition
that mathematics teaching and learning, at all levels of education, is divorced
from major curricular trends is still alive.In many mathematics education scenarios, both processes are going in
opposite directions, disregarding the calls and movements for reform.It is also true that antique methods and
strategies that are strictly traditional instruction. In many instances they
are based on the idea that teachers are the authority and transmitters of
knowledge.And those students are but
passive recipients predominates in our classrooms.Therefore, the mathematics curriculum continues to be strictly
limited, in the majority of the cases, to the prescribed textbook, when
available.The problem solving process
is limited to the use of paper and pencil, without the initiatives to experiment
with innovative changes such the use of technology like calculators and
computers.Moreover, the textbooks
currently used in some mathematics classrooms are not offering to students the
use of multiple representations of transcendental concepts, like functions
(Rodríguez-Ahumada, et al., 1997; Angel, 2000).In these traditional settings, teachers and students are
experiencing functions without an appropriate emphasis on multiple
representations, and moreover, the linking process that should exist between
them is missing (Kaput, 1989a).
Greeno and Hall
(1997) state that "under the pressure to cover the prescribed curriculum,
teachers often feel that there is not enough time to teach students what
representations are for and why the forms are useful and effective" (p.
362).Hart (1991) affirms that students
who use multiple representations along with technology can acquire richer
concept images than those who do not have the same experience (p. 45).In addition, Hart has shown that students
exposed to the use of technology and representations "had better conceptual
understanding than those students not having this exposure" (p. 46).
In summary, the
literature on representations in mathematical teaching and learning has shown
that the appropriate use of multiple representations, supported by technology,
seems to be helpful in promoting understanding and the acquisition of a broader
achievement of important mathematics concepts, like functions.
Functions
are very important in the mathematics curriculum.The use of multiple representations of functions, strongly
supported by technology, has not reached all corners of mathematics
education.In many courses the uses of
calculators and computers have been nonexistent. In other cases, where some
kind of technology is implicitly allowed, it has been classified as optional.
The main purpose
of this study is to develop computer-based algebra lessons using spreadsheets
about linear functions and their related topics where multiple representations
can be emphasized in order to determine if these learning activities can help
college students achieve a broad understanding of linear functions.
In order to
fulfill this purpose, an experiment was carried out in which a portion of
subject matter dealing with linear functions was developed using multiple
representations as basis for instruction.A control group was also used, wherein the same subject matter was
taught.Figure 1 below shows how
spreadsheets supporting multiple representations were handled in this study.
Figure 1.
Multiple representations of a linear function using spreadsheets.
Research Questions
This
study investigates the following research questions:
1.How did the students in the two groups, experimental and
control, compare in the prior achievement and attitudes, and their experiences
with technology?
2.What relationships appear to exist between attitudes and
achievement in the learning of linear functions activities?
3.At which level and in what ways, can the use of multiple
representations be supported by spreadsheets learning activities to better
promote understanding of linear functions in students at college level algebra?
4.How well does the medium of a powerful spreadsheet like Excel,
lend itself to promoting instruction through multiple representations?
The next chapter
consists of a review of the research literature pertinent to this study.It will include a review about the uses of
multiple representations in mathematics, learning theories dealing with
multiple representations, technology and multiple representations, and
functions and their representations. | 677.169 | 1 |
Precalculus: A Graphing Approach
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Read More and innovative resources, make teaching easier and help students succeed in mathematics. This edition, intended for precalculus courses that require the use of a graphing calculator, includes a moderate review of algebra to help students entering the course with weak algebra skills.
Read Less
Fair. 0618394664 | 677.169 | 1 |
HSF-IF.C.7
7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
Next to magic tricks and juggling flaming torches, being able to sketch a graph just by looking at the equation is pretty much the coolest party trick around. We don't know what kinds of parties your students go to, but that's the gospel truth as far as we're concerned.
With all the different types of functions out there, knowing how to sketch them all is more than just a skill—it's a brag-worthy talent. | 677.169 | 1 |
Key Biscayne Algebra Pre-algebra, as a subject in the university curriculum, exists to prepare students for college algebra. | 677.169 | 1 |
Details about Mathographics:
"A wealth of intriguing and lovely ideas." — Information Technology & Learning. While the beauty of mathematics is often discussed, the aesthetic appeal of the discipline is seldom demonstrated as clearly as in this intriguing journey into the realms where art and mathematics merge. Aimed at a wide range of ages and abilities, this engrossing book explores the possibilities of mathematical drawing through compass constructions and computer graphics. Compass construction is an extremely ancient art, requiring no special skills other than the care it takes to place a compass point accurately. For the computer graphics part of the present work, however, readers will need some familiarity with basic high school mathematics-mainly algebra and trigonometry. Still, much of the book can be enjoyed even by "mathophobes," for it is about lines and circles and how to put them together to make various patterns, both abstract and natural. One hundred and six full-page drawings, ranging from totally abstract to somewhat pictorial, demonstrate the possibilities of mathematical drawing and serve as inspiration to readers to carry out their own creative investigations. Among the illustrations are such intriguing configurationsas a five-point egg, golden ratio, 17-gon, plughole vortex, blancmange curve, Durer's pentagons, pentasnow, turtle geometry, and many more. In guiding students toward the comprehension and creation of such figures, the author explains helpful basic principles (of number, length and angle) as well as reviewing relevant fundamentals of trigonometry. In addition, he has provided numerous useful exercises (with answers} at the ends of the chapters, together with recommended further reading, detailed in the bibliography. 211 black-and-white illustrations. Bibliography. Index.
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Rent Mathographics 1st edition today, or search our site for other textbooks by Robert Dixon | 677.169 | 1 |
GeometryMeasure angles, prove geometric theorems, and discover how to calculate areas and volumes in this interactive course! More than 2000 years ago, long before rockets were launched into orbit or explorers sailed around the globe, a Greek mathematician measured the size of the Earth using nothing more than a few facts about lines, angles, and circles.
How can robots perceive the world and their own movements so that they accomplish navigation and manipulation tasks? In this module, we will study how images and videos acquired by cameras mounted on robots are transformed into representations like features and optical flow. Such 2D representations allow us then to extract 3D information about where the camera is and in which direction the robot moves. You will come to understand how grasping objects is facilitated by the computation of 3D posing of objects and navigation can be accomplished by visual odometry and landmark-based localization.
The PACE Mathematical Foundations MOOC is an online program designed to enhance your mathematics skills in the areas of Number Theory, Algebra, Geometry, Probability and Statistics. This MOOC is designed to help develop the skills needed to be successful in college-level mathematics. | 677.169 | 1 |
1878975itative Toolkit for Economics and Finance
This primer covers mathematical methods used in both economics and finance courses at the undergraduate and MBA levels. It describes the purpose and techniques, both algebraic and geometric, associated with basic economic model building and interpretation. For students with a weaker background in mathematics, this book provides a intuitive explanation of the basic mathematical tools utilized in economics. For students with a stronger background, this book simplifies the application of those tools through numerous economic examples. These attributes make this book useful as either a supplementary text or as a reference source for many finance and economics courses | 677.169 | 1 |
Learn Math
CURRENT AND PROSPECTIVE STUDENTS: I use free software and ebooks for my online classes. If you want to create an account before class begins to try out the software and get a feel for what the system is like, go Then enter any of the courses and try them out. If you want to learn/study/review any topics from Arithmetic, Algebra or Trigonometry by watching my videos on YouTube, see Option 2 below.
If you want to learn some math at your own pace, you may do that on your own, without being in a class at MiraCosta College. There are options below including free or inexpensive classes where you may learn the material online. Once you feel confident you have mastered the material, you may take a placement test to pass into a higher level math class.
There is a Bridge to Success in Math Program for students at MiraCosta to help prepare students to take the placement exam. Email studentsuccess@miracosta.edu for info, or contact the Testing Office.
OPTION 1: There is a free resource online for you to learn courses on your own by reading a free textbook online, watching videos, and trying online problems which are immediately graded. Click on to learn more. Go
I've created this website showing which of my videos correspond to the chapters in the Beg/Int Algebra book by Tyler Wallace in MyOpenMath: | 677.169 | 1 |
Details about Introduction to Differential Topology:
This book is intended as an elementary introduction to differential manifolds. The authors concentrate on the intuitive geometric aspects and explain not only the basic properties but also teach how to do the basic geometrical constructions. An integral part of the work are the many diagrams which illustrate the proofs. The text is liberally supplied with exercises and will be welcomed by students with some basic knowledge of analysis and topology.
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Rent Introduction to Differential Topology 1st edition today, or search our site for other textbooks by T. Brocker. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Cambridge University Press.
Need help ASAP? We have you covered with 24/7 instant online tutoring. Connect with one of our tutors now. | 677.169 | 1 |
Description:
From polynomials to rational expressions, the world of algebra can be a befuddling place. Of course, trigonometry presents certain challenges as well, what with its sine graphs and inverse functions. Students and teachers of these subjects need fear no longer, as this site offers a guide for smooth sailing through all of these thorny matters. These mathematics resources were designed to complement a textbook authored by Robert Blitzer, and they include a number of interactive quizzes and tests. The materials on the site cover topics like exponents, matrices, and conic sections. Some of the materials are meant to be used as stand-alone educational materials, although visitors should note that some require the actual textbook. | 677.169 | 1 |
am familiar with these tests as well as the new common core standards. Topics in Algebra 2 include continuing to work with expressions, equations and inequalities,factoring, quadratic formula, and graphing. New topics include: complex numbers, logarithms, linear quadratic and exponential functions, solving rational equations, analyzing functions,and trigonometry. | 677.169 | 1 |
This page is designed to provide a guide to a planned implementation of The Math You Need, When You Need It.
It will change as the implementation proceeds at this institution.
Please check back regularly for updates and more information.
The Math You Need in Geology 1401- Earthquakes and Natural Disasters at Baylor University
Challenges to using math in introductory geoscience
Baylor University is a private 4 year institution in central Texas with a total enrollment of 14,900 with 12,575 undergraduates, 3,033 of which are freshmen. Our student population is diverse and accomplished, originating from every state and 71 countries around the world. Baylor's median SAT score range is 1120-1300 (math and reading) and 74% of our freshmen are in the top 25% of their graduating class. Despite their achievements, they often still struggle with quantitative skills in our freshmen introductory courses. Most students are required to take 8 hours of lab science courses (2 classes) as part of their general education requirements, and choose geology over physics or chemistry based on their perception of how easy it is. The majority of them will not take another math or science class beyond the 1000 level.
More about your geoscience course
Geo 1401 is a service course, taken mostly by freshman non-science majors to fulfill a lab science requirement, although our geology majors may chose it also. There is a required lab component, which is where I anticipate using TMYN most frequently. The lab is taught by TA's, who currently spend a great deal of lab time reminding and in some cases teaching students the math skills utilized in the labs. The course is taught by two different professors that alternate semesters; there is no online component.
Inclusion of quantitative content pre-TMYN
Each lab addresses quantitative content. The labs are designed so that the students perform a series of experiments, observe and calculate results, and employ critical thinking to draw conclusions. My goal in using TMYN is to remind students of the math skills they have forgotten before the lab is performed, reducing the use of lab time necessary to address confusion and frustration. We currently spend as much as 1/4 to 1/3 of the lab time addressing quantitative skills, and students often are so frustrated by the math, they lose sight of the content we are trying to communicate.
Quantitative content in the course includes rearranging and solving algebraic equations, plotting and interpreting a variety of graphs, and taking a variety of measurements including velocity, period, and displacement.
Selected modules for fall semester include rearranging equations, graphing, order of operations, and unit conversions.
Strategies for successfully implementing The Math You Need
Planned strategies for fall 2012 include students taking an assessment test during the first week of class so that their level of preparation and understanding of quantitative skills may be gauged. This pre assessment will be mandatory, and graded on a pass/fail basis, and not included in the class grade. A total of three modules will be used in the lab class, and assigned as follows:
Each of the three selected modules will be worth 2% of their lab grade. After lab 7, students will take a post assessment test identical to the pre assessment that will be worth 4% of their lab grade. The total of post module assessments and the final assessment will be worth a total of 10% of their lab grade. As our students work in groups during lab, this also provides an additional measure of individual effort and comprehension. | 677.169 | 1 |
Editors' Review: 04/14/2015 23:50:22 Agree in principle. I have taught Middle school & HS math & science. In Algebra II, students are confused, and can't connect concepts week to week.Mathematician Chris McKinlay hacked OKCupid to find the girl of his dreams. Emily Shur. Chris McKinlay was folded into a cramped fifth-floor cubicle in UCLA's math ...Wonder7/23/2014 · The Common Core should finally improve math education. The problem is that no one has taught the teachers how to teach it.Free math lessons and math homework help from basic math to algebra, geometry and beyond. Students, teachers, parents, and everyone can find solutions to their math ...Math homework help. Hotmath explains math textbook homework problems with step-by-step math answers for algebra, geometry, and calculus. Online tutoring available for ...< Prev; Random; Next > >| Permanent link to this comic: Image URL (for hotlinking/embedding): 3.7. Don't Take Notes on a Computer By the way, I do not recommend taking notes on a laptop computer during class. Certainly you should not do this unless you ...Welcome to my online math tutorials and notes. The intent of this site is to provide a complete set of free online (and downloadable) notes and/or tutorials for ... how to do algebra | 677.169 | 1 |
Details about Analysis:
By introducing logic and by emphasizing the structure and nature of the arguments used, this book helps readers transition from computationally oriented mathematics to abstract mathematics with its emphasis on proofs. Uses clear expositions and examples, helpful practice problems, numerous drawings, and selected hints/answers. Offers a new boxed review of key terms after each section. Rewrites many exercises. Features more than 250 true/false questions. Includes more than 100 practice problems. Provides exceptionally high-quality drawings to illustrate key ideas. Provides numerous examples and more than 1,000 exercises. A thorough reference for readers who need to increase or brush up on their advanced mathematics skills.
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Rent Analysis 4th edition today, or search our site for other textbooks by Steven R. Lay. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Pearson. | 677.169 | 1 |
Details about Calculus and Its Applications:
Calculus and Its Applications is aimed at university students with varied levels of mathematical knowledge. Although it requires a functional understanding of algebra, past readers have rated the text as easy to comprehend, and many say its inclusion of an algebra review section is extremely helpful for those who haven't dealt with the topic in a while.
This book incorporates a number of visual examples designed to help students understand the usefulness of the calculus exercises it covers as well as the reasoning behind the methodologies it teaches. In addition, it employs an organized chapter layout that features "Objectives" sidebars and footnotes with important definitions for easy reference. "Quick Check" sidebars provide basic in-chapter exercises that readers can use to confirm they're keeping up with the material, and many of the examples are prefaced by brief narrative sections that present key concepts in alternative contexts.
Other potentially useful features include basic training sections on the use of graphing calculators and similar tools. Readers can look for "Technology Connection" sidebars throughout the text to learn how to apply their graphing calculator knowledge to the different types of problems they'll encounter. Because the applications in the text draw from many different fields and multiple appendices cover critical rules for derivation, integration and similar topics, students of life sciences, economics and other disciplines may all find this book a useful fallback.
While this book is also notable for featuring a large number of practice questions in each section, these range in difficulty. Some recommend purchasing the accompanying solutions guide "Student Solutions Manual for Calculus and Its Applications," as the answers section at the end of the main text doesn't present step-by-step solutions.
The authors of Calculus and Its Applications, Marvin Bittinger, David Ellenbogen and Scott Surgent, all taught various university-level mathematics in the United States. The publisher, Hall H Pearson Education, also offers a MyMathLab interactive learning module that calculus students can use to gauge their progress.
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Rent Calculus and Its Applications 10th edition today, or search our site for other textbooks by Marvin L. Bittinger. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Pearson. | 677.169 | 1 |
game explores functions in a different way: a and b = f(a) are drawn in a unique numerical line. When the user changes a, b = f(a) changes following a rule. The objective of the game is to discov... More: lessons, discussions, ratings, reviews,...
Enter a set of data points and a function or multiple functions, then manipulate those functions to fit those points. Manipulate the function on a coordinate plane using slider bars. Learn how each co | 677.169 | 1 |
97801302275gebra: A Combined Approach
The engaging Martin-Gay workbook series presents a reader-friendly approach to the concepts of basic math and algebra, giving readers ample opportunity to practice skills and see how those skills relate to both their lives and the real world. The goals of the workbooks are to build confidence, increase motivation, and encourage mastery of basic skills and concepts. Martin-Gay enhances users' perception of math by exposing them to real-life situations through graphs and applications; and ensures that readers have an organized, integrated learning system at their fingertips. The integrated learning resources program features book-specific supplements including Martin-Gay's acclaimed tutorial videotapes, CD videos, and MathPro 5. This book includes key topics in algebra such as linear equations and inequalities with one and two variables, systems of equations, polynomial functions and equations, quadratic functions and equations, exponential functions and equations, logarithmic functions an equations, rational and radical expressions, and conic sections. For professionals who wish to brush up on their algebra skills | 677.169 | 1 |
Hey guys,I was wondering if someone could explain answers for book c of middle school math with pizzazz ? I have a major assignment to complete in a couple of months and for that I need a thorough understanding of problem solving in topics such as evaluating formulas, angle-angle similarity and factoring. I can't start my project until I have a clear understanding of answers for book c of middle school math with pizzazz since most of the calculations involved will be directly related to it in one way or the other. I have a question set, which if somebody could help me solve, would help me a lot.
Algebrator is a good software to solve answers for book c of middle school math with pizzazz questions. It gives you step by step solutions along with explanations. I however would warn you not to just paste the answers from the software. It will not aid you in understanding the subject. Use it as a reference and solve the problems yourself as well.
Interesting! I don't have that much time to hire someone to tutor me so I think this would be just fine. Is this something bought from a mall? Do they have a website so that I can see more details about the program?
proportions, adding exponents and binomials were a nightmare for me until I found Algebrator , which is really the best math program that I have come across. I have used it through several algebra classes – Intermediate algebra, Pre Algebra and College Algebra. Simply typing in the algebra problem and clicking on Solve, Algebrator generates step-by-step solution to the problem, and my algebra homework would be ready. I really recommend the program. | 677.169 | 1 |
Top positive review
5.0 out of 5 starsLots of illustrations and exercises-with answers at the back
ByCraig Mattesonon January 10, 2004Top critical review filtering reviews right now. Please try again later.There was a problem loading comments right now. Please try again later. loading comments right now. Please try again later.
This is Coxeter best book. Introduction to Geometry covers a wide range of topics and is the first book that I will look at for any geometry topic. It is now a little dated but only in the topics that it does not cover. Like all of Coxeter works each topic is clear and to the point. If you only buy one book on geometry this is it.
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This is the best book I've seen covering geometry at this level. Coxeter was known as an apostle of visualization in geometry; many other books that cover this material just give you page after page of symbols with no diagrams. He motivates all the topics well, and lays out the big picture for the reader rather than just presenting a compendium of facts. This is a survey of a huge field, but he does a great job of focusing on the most important results. As other reviewers have noted, this book is not "introductory" in the sense of high school geometry; it's introductory in the sense of being the kind of book a college math major would use in his/her first upper-division geometry course. It doesn't presuppose a great deal of mathematical knowledge, but it probably isn't a book that one could appreciate without having already developed quite a high level of mathematical maturity.
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This is one of those books that's listed in the bibliography of almost every other geometry text I've read -- and rightly so. Reading through it, you'll find some absolute gems of geometric insight. So why am I giving it only three stars? Primarily because it misrepresents itself as an "Introduction," which it isn't. It's much more like one of those fast-paced "ten countries in five days" package tours offerred by various travel agents. In a mere 412 pages, Coxeter zips through a vast number of topics -- each of them actually a specialty area in the larger field of geometry. It simply isn't possible for a book of this length to give the reader any kind of serious grounding in this material.
In addition, some of the topics are ones at which Coxeter himself admitted he wasn't very skilled. During his career, his main areas of interest were symmetry, n-dimensional Euclidean geometry, projective geometry, and higher-dimensional polygons. Things like topology and differential geometry were outside his territory, so the treatment of these topics in "Introduction" is not as engaging as his discussion of various isometries.
This book originally grew out of a set of lectures that Coxeter gave to college-level math majors and math teachers. By all accounts, Coxeter was a very lively and engaging teacher; I imagine it must have been wonderful to listen to those lectures, and then have Coxeter's own lecture notes (i.e., this book) as a reminder of everything that he said. Unfortunately, I don't think the book stands as well on its own as a teacher; it needs Coxeter himself to fill in the gaps between the words and bring it to life.
So if you already know this material, and you just want to discover some wonderful Coxeterian pearls of wisdom about the subject, then go for it; pay the hundred dollars. As your knowledge deepens, you'll always be able to return to this book and find some new insight that you missed on a previous reading. There's also a wonderful "visual" quality to the way Coxeter thinks about geometry -- something that's missing from many other texts.
On the other hand, if your goal is to learn the material covered in this book, you'll need other books to do it. This is definitely not a good choice for a first exposure to the subject.
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Euclid said "there is no royal road to learning geometry". However geometry for the boomer generation has been easier to learn because of the classic H.S.M. Coxeter "Introduction to Geometry". This 1980 final edition simply perfects the 1961 first edition that helped me start my career in computer graphics. The minor typographical errors have been fixed. One such defect in the first edition for equation 18.21 had stopped my progress in tensor notation for weeks before I could confidently mark it as a printing error. Almost all the exercise answers have been improved. And the four color map proof is mentioned. Returning to Coxeter now in 2014 I see it as a practical review for the serious physics enthusiast to get from the geometry of Euclid to that of the Einstein Field Equations. An alternate more modern and more difficult book covering this mathematics would be Penrose's "Road to Reality" first published in 2004. Everyone must find their own road through geometry, this book might help you.
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The only reason I give it a 4/5 is because the diagrams need to be labeled. It's pretty hard to keep up with the conversation when the author refers to a poorly labeled complex diagram. I think Coxeter's other book does a great job and you can download it for free or buy it for under $10. It is a great introduction to college level geometry; and introduction because it doesn't really go into too much depth, but is not shallow either. I would recommend it, but buy it used because it's just not one of those books you'll constantly look back at for help in the future...it's pretty much a one read and that's it.
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H.S.M. Coxeter is a legend in Geometry, primarily for his work on higher dimensional objects...polytopes, and the Physics applicable, study of Symmetry. The book covers the gamut of Geometry, touching on the peaks and covering the entire range. No book could cover the entire subject, in one volume. This is an "Introduction" for someone who has already made peace with Mathematics....its not an elementary text. The proofs are rigorous and many details are not presented as the volume was written for university students. I reccomend it for true Math Philes. Dr.Pratt.
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And not really to modern geometry, but rather to Coxeter-style geometry. I studied with Coxeter as an undergraduate (he was a very good teacher), and am a professional geometer, but I have never liked this book. Unfortunately, there is no good "Introduction to Geometry", and hence, 2500 years after Archimedes, still no royal road to it. I would very much recommend Thurston's notes (or his book, which is a little easier going, but has a lot less content), or (on a more basic level) Geometric Transformations by Yaglom.
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As usual, Coxeter shows that he is the master of the revival of geometry !
Look at the table of contents to see how "rich" this book is... while every subject is treated in a wonderfully comprehensive way.
This book is part of Coxeter's geometry SUM : Introduction to Geometry, The real Projective Plane, Projective Geometry, Geometry Revisited, Non-Euclidean Geometry... to be included in the collection of anyone interested in mathematics. | 677.169 | 1 |
fully worked-out solutions to all of the odd-numbered exercises in the text, giving students a way to check their answers and ensure that they took the correct steps to arrive at an answer. (Solutions for sections 12.2-12.3 and Chapter 13 are available via the web.) | 677.169 | 1 |
Woburn Englishawn RThorough understanding of the theoretical underpinnings of this powerful tool can be left to the math majors. Those who ask for help in a calculus course are most often taking it as a requirement for a technical field. Here, the practical application of derivatives and integrals are what is important. | 677.169 | 1 |
Basic Algebra Introduction
Basic Algebra is where we finally put the algebra in pre-algebra. The concepts taught here will be used in every math class you will take from here on. We'll introduce you to some exciting stuff like drawing graphs and solving some complicated equations.
Don't let the word "algebra" intimidate you. You actually have been using this type of math for years. In fact, many people find algebra to be one of the easier types of math to learn, since it is full of common sense rules and ideas.
"As long as algebra is taught in school, there will be prayer in school. " – Cokie Roberts | 677.169 | 1 |
Education:
Education means the change in behaviour and
attitude.
2. Mathematics:
It is the branch of science which deals with
calculations known as mother of science.
Its branches are Mathematic, Algebra and
Geometry.
1.
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4.
Terminology:
A specific word that is used for a specific thing
is known as terminology.
3.
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5.
1.
Specific Objectives:
To make the student familiar with the
definitions of mathematical terms.
To inculcate the concept of these
terms.
To make the students understand
these terms completely and make
them able to use these terms in
mathematical applications.
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6.
2.
General Objectives:
To enhance the mathematical
understanding of the students.
To enhance the mathematical skills of
the students.
To provide the students mathematical
practice regarding these terms.
Sajjad Ahmad Awan PhD Research Scholar
TE DTSC Khushab
3/4/2014
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7.
Symbol:
Generally a symbol refers to some image or
objects which suggests to some thing else.
For example:
Basic Symbols used in Mathematics:
= These two parallel lines are named as
symbol equal to.
≠ This named as not equal to.
- This symbol is named as Subtract.
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8.
X It is a multiplication sign.
< This sign named as greater than.
>This sign named as less than.
Є This is named as the member of.
U This symbol shows the union of two
sets.
∩ This symbol shows the intersection of
two sets.
( ) This sign is known parenthesis.
@ It is taken "At the Rate of".
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9.
Set:
A set is collection of well
defined objects
Sub Set:
If A and B are two sets and
every member of set A is the member
of set B, then set A is as sub set of set
B.
Union of two Sets: The union of two
sets A and B is set consisting of all the
elements which are in set A or in set B.
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10.
Intersection of Two Set:
The
intersection of two sets A and B is a set
consisting of all the common elements of
sets A and B.
Power Set: If A is any set then the set
consisting of all the subsets of the set A
is called power of A.
Null Set:
If there is no element
present in a set it is called a null set.
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11.
Ratio:
A relation between two quantities
in same units i.e.
a:b
Proportion:A relation between two
ratios is called proportion. i.e. a:b:: c:d
Direct Proportion:
If a proportion
between two ratios is such that with the
increase one ratio the second ratio may
rise is called direct proportion.
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12.
Inverse Proportion:
If a proportion
between two ratio is such that with
the rise of one ratio the second
decreased that is called inverse
proportion.
Percentage:
Percentage is
such common fraction, in which
denominator consists 100 i.e. 12%
means 12/100.
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14.
Algebra:
It is the branch of
mathematics which deals generalized
arithmetic to solve the complicated
problems.
Constant:
The value which remains fixed
and does not change is called constant. e.g:
2,3,5,8.
Variable:
The value which is not fixed
and varies is called variable. e.g: X,Y,Z.
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15.
Co-efficient:
The number which is multiply
by variable is called co-efficient. e.g:
3x,5y,etc.
Equation:
An open sentence which
shows "=" among two expressions is called
equation.
e.g: 3x=12, x+2=9.
In-equation: Such sentences in which
symbols < or > used are called in-equation.
e.g: 3x>12, x<7.
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16.
Geometry:
Geometry means the
measurement of earth.
Square:
The shape having four
equal sign with four right angles is called
square.□
Rectangle: The shape which consists of
four sides having opposite sides equal is
called rectangle.
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18.
The students will understands all
these terminologies very well.
The students will be able to improve
their knowledge.
The well qualified students will take
active part in the progress of Pakistan.
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19.
The post graduate teachers should be
provided in primary school of rural areas
of Punjab.
AV aids should be provided in school of
remote areas for the better quality of
education.
Teachers training courses should be
conducted after 3 years for newly
recruited and senior teachers.
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None of the lesson in this series have been reviewed.
Below are the descriptions for each of the lessons included in the
series:
Local Max and Min; Increasing/Decreasing Functions
Increasing/Decreasing , Local Maximums/Minimums - The basic idea!
In this video, I just give a graph and discuss intervals where the function is increasing and decreasing. I also discuss local maximums and minimums.
This video is available to be viewed online for free on PatrickJMT's YouTube channel or his main site. Here, you can purchase access to download a version of it for offline viewing.
The Mean Value Theorem
This video is available to be viewed online for free on PatrickJMT's YouTube channel or his main site. Here, you can purchase access to download a version of it for offline viewing.
Finding Critical Numbers – Ex 1
Finding Critical Numbers - Example 1. In this video I show how to find the critical numbers of a rational function.
This video is available to be viewed online for free on PatrickJMT's YouTube channel or his main site. Here, you can purchase access to download a version of it for offline viewing. | 677.169 | 1 |
Forcing the Red Badge of Courage and the Great Gatsby on students is a waste also. Have you ever need to know anything about those books?
When people talk about algebra, they usually mean polynomials or something like "find the dimension of a rectangle whose length is 9 less than twice its width if the perimeter is 120 cm."
Math is about identifying and understanding patterns. That you can use the same variable for length and width is important. That you can take information that initially looks unrelated and solve a problem is important.
Can you really teach statistics without an understanding of algebra?
Probably, yes. I'd argue that that's more of a sociology or political science class than a math class, if you're focusing on interpreting numbers or defining terms.
NO. Statistics is about drawing conclusions based on the distribution of points. Recognizing the function that governs the distribution and thus defines what properties we can expect from the points. mean X = (1/n) Sum(X) is algebra standard dev X= (1/n-1)Sum(X-meanX) is algebra Don't argue with me that this is high level statistics as mean and standard deviation is very very basic stuff, and putting one formula into another is introduced in college algebraWhat world do you live in that everything is related linearly? You have space and velocity but no acceleration?
I think there might be some value to introducing them earlier, mostly because the students will be introduced to calculators and computers at an early age regardless of how the school approaches it. There's no harm in showing students how use tools properly.
I tutored a home schooled girl who's mother let her use a calculator. She could not progress past 5th grade level because she could not recognize the patterns numbers make. She could not identify 36 as a square or tell me the roots of 12 because she'd always used a calculatorWhen I was asked "when am I ever going to use this?" by a student, I'd answer "I don't know, Tell me exactly what your future holds and I'll only teach the math that you need. Math, Logic and Pattern Recognition are powerful tools, Since you don't know what your future holds, don't you think you should get as many tools as possible?"
no you dont. there are no scientists that cant write. writing is a huge component of being a scientist. historians should be analyzing history, not data anyway. people that like learning will do so no matter what. It is not a university's job to "round me" it is their job to provide specialized high tech training with resources I cant find elsewhere. I can buy lit books and biographies on my own thanks.
You are thinking of trade school/apprenticeships. Universities ARE supposed to round you. While I'll concede that there are scientists who can't write, there are few if any SUCCESSFUL scientist that can't write. If the reader can't figure out what you are saying, your papers will not be published and your proposals will not be funded. It doesn't matter how brilliant you are if it can not be communicated, and it doesn't matter who well you can diagram a sentence if you have nothing to say. The rounding done at universities allows people to communicate with others not directly in their field. Cross disciplinary work leads to new insights in both fields. I can give you references (from the field of information management) if you like.
So, I've scanned the thread and come up with two additional reasons that this is a horrible idea that people have not touched on yet:
1. First, even if you don't use a discipline regularly, and even if you forget a lot of it, that fact that you were once familiar with it gives you a huge advantage if you ever find yourself in a situation where you need to use it to solve a problem. Ask any engineer: Most days he won't go around applying Green's theorem to a closed curve or evaluating optical transmission matrices with in materials with anisotropic complex-valued permittivity, but the fact that he studied how to do it at some point means that if he ever finds himself confronted with a similar problem, he at least has a starting point for how to approach the problem. He may not remember exactly what to do, but at the very least, he can remember that he has a textbook somewhere with a chapter dealing with this very thing.
Having studied algebra at some point just might mean the difference between thinking "Hey, I could solve this if I looked up the quadratic formula" and "Huh, I have no idea what to do here, it must not be that important."
2. The more important reason that eliminating the algebra requirement is dumb, though, is that high school is supposed to provide a broad education, in part because most high-schoolers haven't specialized yet. They haven't been exposed to enough different fields to really decide what they even want to do yet. An important part of high school is to introduce kids to enough of a variety of subjects that they can intelligently pick which ones they'd want to focus on - sometimes forcing a kid to take a year of algebra or a year of world history or a year of english lit can expose him to ideas that he might wind up liking. I understand that not everyone is college-track, and that's fine, but I am horrified at the trend of allowing kids to deprive themselves of future choices earlier and earlier, and with less and less knowledge about what they're even choosing not to do.
Everything that exists in the universe is, basically, a giant math problem.
But why learn even the basics of the language of all creation when you could just pound out a degree in political science and get paid to expand stupid questions in the New York Times into a thousand word screed against basic competency, right?
Almost every well-paying long-term career requires both the ability to write and the ability to deal with numbers.
If one can't write, one can't document, can't deal with contracts, can't defend one's work if someone accuses that the work is sub-par. If one can't even read well, then performing basic tasks will be difficult.
If one can't do numbers, one can't calculate costs, can't try out different models for paying workers (hourly/billable-hourly/salaried/contract), can't estimate supplies, or follow technical documents. This even applies to the trades, like plumbing, electrical, and certainly to electronics and low voltage. It OBVIOUSLY applies to engineering and manufacturing.
Those who do badly at math or at writing will find themselves working for someone else, or will find someone else getting the better-paying job who can understand the job. That is literally it.
My job doesn't require a college degree, but I have to deal with numbers and with instructions daily. There are others who I know who would like to be in this field, but they really never will make it unless they're just doing the grunt portion of the job, which pays less.
EvilEgg:ForThe fact that this is even a debate is proof that we're completely farked as a country. How the hell can the US compete with non-derpy countries if we can't understand the most basic of abstract mathematics?
Seriously, people. We must teach it because it's hard....even though it really isn't.
Also, while I understand the pervasiveness of calculators and computers today (my TI83 got me through Trig and Calc), calculators need to stay out of the classroom until at least high school. I'm not a math whiz, but I can make change in my head. When the power went out in the WalMart I worked at in College, half of the cashiers had to use their phones to calculate change amounts because they couldn't do it manually.
I hope that mathematics departments can also create courses in the history and philosophy of their discipline, as well as its applications in early cultures. Why not mathematics in art and music - even poetry
no there is no need to teach ANY of these useless topics. history? worthless. science? useless. philosophy is for morons. literature, writing, reading? only librarians would need these useless skillz.
school should require only recess, lunch, sex, internet stuff and how to get cheat codes for video games.
/LOLOLOLOL I love when morons think that the topics that they hate are worthless
downstairs:
1yingtong:Dude, seriously.. it's been 80 years since Kurt Godel proved beyond any shadow of a doubt that math is not the language of the universe. Either you get a system strong enough to verify every possible truth (but it also verifies certain untruths), or you get a system that only verifies things that are actually true (but can't verify certain truths). Math as we know it belongs to the second category (which is A Good Thing).
Godel's Incompleteness Theorem states that in any sufficiently advanced axiomatic system, there will be truths that are unprovable and falsehoods that are provable. That doesn't mean that "math isn't the language of the Universe". That's like saying because it's impossible to determine if a given C program will ever halt, C isn't a programming language. It is entirely possible that, while mathematics is an incomplete system, the Universe itself is describable using math from the consistent subset. What's more likely, in my opinion, is that if we do end up with a theory of everything (which I believe we will), we won't be able to be absolutely sure that it is a theory of everything.
wingedkat:downstairs: As a completely random example, but something that irks me personally... so many people cannot uderstand crime statistics. Not even to the point 1I totally disagree with this. I think high school teachers should have at minimum a bachelor's degree in Math, not some baloney education degree. The PRAXIS test for mathematics content should be a breeze. I thought it was incredibly easy, but I got a BS in math. The fact that people had to study for and struggled to pass that test, and go on to become math teachers, frightens the hell out of me.
I think a big obstacle to math is that it isn't taught in a way that kids can readily see useful applications, so a lot of kids think it's useless and put in the effort they think it deserves. When kids see adults, like the author of TFA, saying math isn't very useful it only reinforces that view.
CSB:
My dad used to be a carpenter. When I was a kid, around 7 or so, I remember I complained about having to do math homework on the basis there was no point to it in the real world. My dad was like "O RLY?" then set about going through a series of exercises with me along the lines of:
"I'm building a roof, the room is 20 feet long and the joists are 16" apart, how many will I need" then "OK, so the roof is on a 2:1 slope, how long do the joists have to be" then "OK, a beam X feet long costs Y, how much will the roof joists cost" and so on. I spent more than a few afternoons getting run through a series of questions like that, where there was a clear and real goal.
I guess it worked, I ended up acing math from elementary school all the way through to the end of an engineering degree.
FTFA: It's not hard to understand why Caltech and M.I.T. want everyone to be proficient in mathematics. But it's not easy to see why potential poets and philosophers face a lofty mathematics bar.
Sure..... failing to teach algebra and geometry might not hobble a future poet but it will sidetrack a future engineer or physicist. Since we don't wear our future occupations like badges on our foreheads we are forced to educate everyone in the basics of our civilization. Sure, everyone may not "need" such an education (by a very narrow definition of need) but we must damn well make sure that the ones who do get it.
"What of the claim that mathematics sharpens our minds and makes us more intellectually adept as individuals and a citizen body? It's true that mathematics requires mental exertion. But there's no evidence that being able to prove (x² + y²)² = (x² - y²)² + (2xy)² leads to more credible political opinions or social analysis."
Author clearly does not know what actual mathematics is. The author was taught mathematics improperly and thinks math consists of just "solving for x" in various ways according to the formulas/algorithms on their cheat sheet. If you're completely ignorant about a topic then your opinion on whether or not it should be taught is pretty much worthless.
Mathematics has allowed me to learn physics/comp sci incredibly efficiently and quickly because I was already used to thinking abstractly and logically. Math is a little more than multiplying polynomials, if you're not a complete idiot. It's about thinking critically and analyzing situations in creative and complex ways. This may actually lead to "more credible political opinions or social analysis."
And kids are not good deciders of what they're going to do years down the line (hell, many kids think shooting to be a sports star or to make it in music with no backup plan is a good career path, rather than the one-in-a-hundred-thousand shot it is), so you cannot just leave it up to them to decide whether they want to study hard at a particular subject that could be critical to them.
In a world where more and more, the good jobs rely on applying your brains, it would be criminal to abdicate responsibility to teach something as fundamental as algebra.
We're not even talking about calculus! It's farking algebra! Without learning that kind of basic skill of mental abstraction, good luck at ever being a scientist, engineer, or do any sort of serious programming! And this isn't the sort of stuff that they'll have time to pick up later (nor even the ability to learn as well, later).
Will all of them need it? NO. But if you don't make it mandatory, a lot of kids who don't even know yet that they'll need it to reach they're full potential won't learn it.
Of course, the ability to use math correctly is necessary in any scientific field. However, when scientists get stuck in the box, they just keep repeating mistakes without seeing the possibilities in front of them. The ability to think outside the box is where the major innovations come from. We have to be able to see outside of what is expected, to see what is actually happening.
The Problem with American schools is that we basically have the same program of study for 12 years.
I'm in a PhD program, and the Americans are 10% of the cohort. The Other 90% have at least 8-12 more semesters of math than we do. They get it Earlier and they get it deeper. At Age 10 or so, most of them are asked to pick from three or four "focus streams" that direct them towards jobs in Technical skills, arts, literature or Theory. They then cut the items that are 'least useful' to their focus pool and double-up on the items that are more important. Doing this while students still have enormous mental plasticity Allows a level of achievement in those realms that is genuinely surprising.
It's not that they have taken more math - they are acculturated to mathematical culture.
While I think the goal of having well-rounded students is important, I think you do that by letting them take what they want, not taking subjects that they don't like and will probably stink at. I hated taking lots of stupid pointless classes in College and High School. I wanted to take another language, or art, or shop or Math. Screw Literature - I'd been reading at a college level since 3rd grade.and Screw the Hell out of Gym. More music classes? I'd love thatThis.
As someone in a STEM field, have I used "algebra" much in my career? No. Have I used deductive logic that was introduced to me at a young age through the vehicle of Algebra? Yes, hourly.
But on average, they don't. Regardless, the point is that if you don't want a well-rounded education and just want to learn a trade, there are trade schools available. Colleges are not and should not be in the business of cranking out tradesmen.
However, if you want to be one of the big-timers - a Fortune 500 CEO, a neurosurgeon, a high-powered lawyer - you need a strong, well-rounded background and a flexible mind and that's what colleges are for. Colleges give you those things by exposing you to a wide variety of topics that require a wide variety of mental skills to understand and absorb. Part of that involves maths that you may not, ultimately, have any practical use for. The point isn't the maths, that point is the exposure to that sort of thinking so that you have that general flexibility.
Higher level math should be taught in public school to expose kids to their options. It should be taught across all college programs to some extent to expose students to the type of thinking required. To argue that it should be pulled back because some people aren't good at it is absurd. The purpose of high school and college isn't good grades.
buckler:Icaramba421:The solution should be to start shaming people that are innumerate. People that don't understand maths should be paraded through the streets with "RETARD" painted on their backs. Since most women respond negatively to reduced social status, the innumerate will no longer be able to get laid. The problem will solve itself after a couple of generations.
I would say that we teach advanced mathematics (beyond "counting out change") because by the time one student out of fifty decides he wants to study something actually challenging, it's too late to start teaching him real math. If everyone gets algebra crammed into their skulls in middle school, the ones who discover they need calculus and statistics in high school will be ready to take them.
Kimothy:TheirWell said. And the author is an idiot for thinking you can teach stats without any math background... unless one is a social scientist who likes playing with numbers without understanding how methodologies work. I know plenty of social scientists who love playing with quantitative models, but when you ask basic questions about their logic and causality, everything falls apartMaybe, but everyone should be forced to take Probability and Stats in college.
If critical thinking is a goal of algebra education then we'd probably do better by replacing it with formal and informal logic. It might cut down on the series of fallacies I read on Fark, or what are commonly called arguments by morans.
pciszek:Can you really teach statistics without an understanding of algebra?
You can't teach stats without a basic understanding of algebra. But you can teach basic stats to someone without a complete mastery of algebra. You can also use statistics to drive understanding of algebra.
Babwa Wawa:I suppose my suggestion would be having statistics drive the education around algebra.
That could work. Maybe something more interesting. I'd advocate for a variety of applied math classes, like "interior design" "carpentry" "sports statistics" etc, followed by a mandatory "life statistics and applied probabilities" class of some sort, which would cover population, media, and political statistics. Let people do and learn things that are interesting to them while also learning algebra.
Thoguh:Kimothy: Most people aren't using algebra in their everyday lives.
Hamburger meat is $2.99 a pound, how much is three pounds?
Oh shiat! You just used algebra!
This is an algebraic word problem:I personally think the requirement is a bit weird - sure you want people heading off to university to have no less than trig, and you'll probably want calc once you get there.
But if someone just wants to go to nursing school or whatever, a mastery of basic stats is far more useful than a mastery of abstract algebra
Grand_Moff_Joseph:Proving theorems is not just there to help you understand calculus. Mathematical reasoning is useful for other things. Physics, or computer science. Not to mention, of course, other fields of math besides algebra or calc. To say that very few people use mathematical reasoning after graduating would be wrong. To say that "No one (and I mean NO ONE) cares" why is wrong. So, so wrong. Almost dirty.
buckler:namatad: The radio, public speaking and other related work. Likewise, several others have had great monetary success with art and music.
--- Ballet is what made me get into physics and kinesiology. Body mechanics is math heavy. Even to get a good personal trainer certification, you have to get through all of the metabolic calculations (nothing but algebra). While you laugh at my dance major, keep in mind that right out of college I was making $50 an hour teaching ballet. I use my ballet classes as a medium to teach kids physics and to generate interest in science. I have made girls and boys who thought they were "dumb" turn into scientists or at the very least show interest in these subjects in school. I am currently back in school working towards getting into a doctor of physical therapy program. The psychotically strict work ethic and discipline that dance taught me has been useful in every aspect of my life.
The stagehand skills I learned as a dance major landed me a job in audio broadcast engineering. Now, I work in radio and get to learn how to fix transmitters. I also mix concerts and get to work with famous musicians.
On the side, I get stagehand gigs. While the pay is not astronomically high, getting $17-$24 an hour as a part time job sure as hell beats min wage at McDonald's. Loading in equipment for a rock legend beats flipping burgers or folding sweaters at the Gap any day.
I would not have had these opportunities if I had not majored in dance.
Algebra needs to stay in schools. This article made me feel like a math genius. Are people really failing out of school because they can't do algebra (says the fine arts/dance major)?
umad:YouBWrong: Requiring high school students to learn algebra while cutting funding for arts, and foreign language studies implies that algebra is somehow universally useful. It is not.
But art is? LOL. Math is waaaaay more useful.
Both are universally useful, and no child--none--should grow to adulthood without at least basic exposure to both.
There was once a time when people of means took pride in being learned and proficient in as many different disciplines as possible. For one thing, exploring multiple disciplines helps one see the ways that they overlap. It helps them conceive novel solutions to difficult problems that a complete specialist would never think of developing. Now that far more people have this opportunity than before, we seek to neuter it.
Trigonometry? Well... actually, I can see where this should be a specialty class. I mean, it's not so much that it's HARD, it just doesn't have much application unless you're going into drafting or astronomy.
Trigonometry is a necessary prerequisite for calculus. How can you integrate 1/sqrt(1-x^2) if you don't know any trigonometry?
Trig is also hugely relevant to complex arithmetic, and signal processing. JPEG images are based on the discrete cosine transform, which like the Fourier transform decomposes a signal into a set of trigonometric functions.
However, it is reasonable to require high school graduates to have a little bit of all of these subjects, and to handle the core subjects with at least a "high school" level of competence. And "high school" algebra is actually pretty basic: many kids cover it in 8th grade, and spend high school taking trig, calc, etc.
The argument that only a tiny handful of people need higher math is silly: only a tiny handful of people need to know Shakespeare for their jobs, and only a tiny handful of people need to know the details of the Louisiana Purchase. It's easy to argue against anything taught in school using this reasoning, and hence one should question the reasoningJohn Allen Paulos, who wrote the very insightful Innumeracy (and the funny and also insightful Humor and Mathematics) is a math teacher by vocation. In the first book, he explained how he felt that kids found math too abstract, because it's taught that way, so they don't understand or appreciate how it can be useful and interesting. To solve this, he routinely took his students on math 'field trips' around the school, where he'd have them directly apply math to solve real-life questions, such as, "How many bricks are on this face of the wall?" He wouldn't tell them the answers or how to solve the problems, but knew that someone with some algebra knowledge would instinctively set to an algebraic solution rather than try to just count all the damn bricks. He might follow up with, "How much money did the school department spend on just those bricks, if a brick weighs x and costs y per pound?" Over time, these exercises build up in kids a solid grasp of what math is in the adult world, how it's used everyday, and why it's useful to know.
doglover:Because People in power are Stupid: I was speaking about a phobia in the same context that people are math phobic.
So am I. You can't just reduce your argument's assumption to "I was assuming cases where I am correct."
Math is a symbolic language. We don't grow up speaking it. We don't need it to learn the basic skills it uses, as they are cross disciplinary. So someone who is "math phobic" is more like someone learning English as a second language than you realize.
Again, my assumption was that you were smart enough to remember what is germane to the discussion. The subject is whether or not America should continue teaching mathematics and not English as a second language and people from foreign countries that are "scared" to learn English.
My assertion is that people who are "phobic" about learning math don't belong in regular classes. They should go to "special" classes and learn with the developmentally disabled or whoever else rightfully belongs there. "Math phobia" is a refuge for people who are too lazy to make an effort and are looking for excuses for their own behavior.
It is not an excuse to remove Algebra altogether from American curriculum as the article is attesting.
The Voice of Doom: I spent two days preparing for the exam in one "major" (chemistry).. .. while my preparation for the other (math) consisted of hanging out with a girlfriend whose exams were already over (different "majors"=different days) and 20 minutes of checking that I still had the most important theorems memorized correctly. With the constant repetition and solving one problem of final exam caliber each week for a year, there was only shiat that you knew you could do and shiat you simply couldn't prepare for (15% of the exam was to solve a kind of problem completely new to you, e.g. having to find a proof for a theorem that you've never heard before)
That's the way exams should be. Difficult enough that there's no way you can cram and with something on it that you haven't seen before that is based on the same principles as what you've been doing.
You can't do Multivariable Calculus if you don't know Calculus. You can't possibly do Calculus if you don't know Algebra. You can't do Algebra if you don't know basic math and order of operations. Passing the test in a math class is pointless if you don't actually understand the material. You'll just fail when you attempt to apply the next level of learning on top of it.
I think that's why the US is so poor at math. People (including educators) don't understand that you can't just study for the test. You actually have to learn it as you go along.
Russky: I'm not ignoring it at all, the point being there is a higher demand right now for scientific degrees but people aren't taking those.
Perhaps if there were, I dunno, JOBS at the other end of all the bullsh*ttery that one needs to go through to get the degree perhaps more students might take the courses. If there were as much "demand" as some people are whining about (which is just more bullsh*ttery so they can H1-B and outsource this country to death) then unemployment rates in STEM-related careers would be virtually ZERO, and we know that's not happening.
I just completed my degree, and in those years I took pre-algebra (hadn't seen a classroom for over 15 years, needed the update), algebra, macroeconomics, statistics and logic... all passed with A's... and I was a Graphic Design major. Took the logic course because I liked it. If I'm going to bust my ass to be good at something, I'm going to do it for one of three reasons: 1. I like doing it, 2. I'm getting paid phat cheddar for doing it, or 3. a combination of 1 and 2. I'm not going to do it because someone "demands" I do it, I'm going to do it for my own selfish reasons and no others.
We need to make sure there are jobs for the people we are encouraging to take these courses, and they should pay enough to be worth the effort.
Babwa Wawa:If you really think that a person needs calculus before being merely introduced to the concepts and some of the math associated with probability and statistics, then you are part of the problem.
How do you teach about the normal distribution without algebra? Standard deviation? Even the ground-level stuff like probability distributions require the concept of area under a curve, which you really need calc to deal with properly.
Babwa Wawa:Oh, and by the way, you don't "learn" anything in a 100 level course. 100-level courses are introductions - surveys at best.
All_Farked_Up:Problem with american schools? Lack of 2 parents and or lack of involvement.
If parents have to spend hours every night teaching their children they may as well home school. If teachers expect parents to do their jobs for them they should not be paid. I never had a job where someone did my work for me.
Jim_Callahan:FloydA: Maybe you could encourage the under-performing students by throwing acid at those who fail. After all, if it's good enough for women in politics, it's good enough for students, right?
Nothing to say on this topic, eh? Did it occur to you that maybe the best plan in that case might be not to post anything? I know that's not algebra as such but there's some logic involved, at least.
Has it occurred to you that maybe when you said that we should "hurl some acid" at people,and that it was "just a colloquialism," someone might have taken offense at that, for some reason?
Has it ever occurred to you that saying that your your political opponents deserve incredibly horrible torture and permanent disfiguration might cause offense?
Did it ever occur to you, even once, that saying "Let's hurl some acid at those female democratic Senators who won't abide the mandates they want to impose on the private sector." ~Jay Townsend
might not be acceptable political speech?
Or are you going to stick with the "he didn't really mean it" bullshiat and continue to defend acid attacks?
This is really a test of your character here. If you continue to say that it's acceptable for people to advocate "hurling acid at those female democratic senators," then you are beneath contempt.
You made it very clear that you think advocating throwing acid at your political opponents is just fine because, in your opinion, it's just "idiomatic phraseology not being meant to be taken literally."
I've made my opinion clear, that this type of threat is beyond the pale of acceptable political speech.
Until you say "Oh, yeah, that is over the line," I have to assume that you meant what you said.
So, are you willing to condemn acid attacks on your political opponents, or do you still think it's just an "idiomatic phraseology not being meant to be taken literally"?
I'm not going to let this go. In your opinion, is it, or is it not, acceptable to advocate throwing acid at people?
Not true in experimental physics either. Without some creative thinking you'd never figure out why your experiment wasn't working. The problem is that these non-scientists think it's all follow the recipe experiments like in high school science.
I'd argue that "outside the box" thinking is critical for the hard sciences for THIS reason and many others.
To those who argue about whether Math majors should teach math, my experience is that my worst math teachers were those for whom math came easily. The best ones had to work at it and had some understanding of different learning styles.
My brother is one of those for whom math is easy, and always has been. It was so obvious to him; that made it hard for him to understand how students could struggle with problems/proofs that he found so simple.
Isildur:This bears repeating. The author's plan would put many more talented high-school freshmen on a math track that would take them away from being to go into math, natural science, engineering, or economics later on, and I think that's too early for that.
Come to think of it, if they would teach basic computer programming skills starting from a very young age (ie as they're learning how to use computers), i have a feeling people would be MUCH better at algebra. I seem to recall that most students were frustrated by the fact that algebra seems abstract... its hard to apply it to "real life" from a childs mind. Programming would bridge that gap nicely, as the students would understand manipulating variables etc in a very logic driven way.
/Disclaimer: ive always been a computer nerd, from the first time i touched one in elementary school
John P. Smith III, an educational psychologist at Michigan State University who has studied math education, has found that "mathematical reasoning in workplaces differs markedly from the algorithms taught in school."
Perhaps then we should cancel English class: the Shakespeare we read in school differs markedly from the stuff we have to read in our cubicles at work.
And pretty much everything else in this guy's article could be applied to Shakespeare as well.
Nilatir:Because People in power are Stupid: Nilatir: To a certain extent this is true. Think back to college and you'll likely notice that the better a person is at very abstract concepts the worse they are in explaining those concepts to others.
That's the problem. You expect your teacher to "explain" something that is fundamentally visually based.
These expectations are really YOUR issue and not an issue with your teachers.
Blame the teacher if you can't persuade them to give you the grade that you want.
My degree requires courses up through Difficult Equations (with Stats and Combinatorial courses out to the side) so I understand what you mean by visual. But still "to teach" is to pass on information and skills and if the people teaching lack the skills to do that then I can see why, unless you come into a class already understanding the material (which many Engineering and Math students do), it could be frustrating for an Arts and Crafts major to overcome.
"Teaching" is a separate skillset than "knowing math". Put the two together and you find that the teacher is particularly challenged by lazy students making ridiculous assumptions about how the teacher is supposed to impart the knowledge in a book.
Generally one doesn't just "know math" which is what lazy American students seem to believe. It comes from drilling and doing work.
The problem is also Pavlovian. Everytime the students see the teacher -the teacher gives them work to do. Psychologically, the students then associate the teacher with this unpleasantness and subsequently blame the teacher when they fail to make an effort to get their homework done.
But that is not the hard part about teaching math. There are hardly those that are "English Phobic" or "Political Sciences Phobic". However, it is openly acceptable to be "math phobic". It's one of the many cop outs that lazy people come up with to blame the teacher, blame the subject and blame everyone except for themselves for not doing the work required to be good at math.
dericwater:ThNot algebra. That's simple arithmetic.
It depends. Are you actually multiplying .09 by three, then multiplying .9 by three and adding it in, then multiplying 2 by three and adding it in again? Because that's the arithmetic approach and it's kind of a pain in the ass.
If you're adding a cent so that it's 3$, multiplying that by 3, then subtracting 1 cent *3 on the understanding that the addition and subtraction operations cancel each other out, then you're doing algebra.
As with most examples of the practical difference between Arithmetic and Algebra, it's a matter of understanding how the problem works in a broader sense versus just grinding blindly through something you've memorized, with the end result being saving a bunch of time and effort to reach the same answer. This is what we're talking about when we say it's a basic practical skill more on par with being able to sound out words or recognize that a period ends a sentence than some vague academic form of literary analysis.
Mimic_Octopus:Babwa Wawa: slno you dont don't. there are no scientists that who cant can't write. writing is a huge component of being a scientist. historians should be analyzing history, not data anyway. people that who like learning will do so no matter what. It is not a university's job to "round me∨." it is their job to provide specialized high∨-tech training with resources I cant find elsewhere. I can buy lit books and biographies on my own∨, thanks.
[these are in addition to capitalization errors]
It's generally accepted that many people in science and tech fields are not especially great written communicators. Just ask my dad, who taught basic mechanical engineering at RPI for a couple years and said that some of the kids couldn't write a one-page paper. People don't say, "I are an engineer" for no reason. (On the reverse side of that coin, some liberal arts majors can't do simple math without the aid of a calculator).
At most universities, you can place out of certain requirements (like basic writing and basic math) with a minimum score on certain standardized or university-administered tests. If you can't, you take the classes. The universities, for selfish and obvious reasons, don't want to put out retards in the world who can't properly punctuate their sentences or calculate the tip on a bill.
/you want specialized training? go to a trade school. you want an education? go to college.
buckler:Agent Smiths Laugh: But I digress, I'm not trying to bust your balls but to show you that you're probably better at it then you've convinced yourself you are, because you already have and use the skills needed for it.
Thanks for that clarification. Sorry I jumped the gun. I was made to feel defensive.
It's alright. The big trick I've found in teaching someone a subject like math is first getting them over the fear and self-doubt stage. Some people quickly convince themselves that they suck at math the first time they run into a tricky problem, or get one wrong, when they really don't. They have the skills, they just haven't had the rules explained to them in a way they digested. They've often been sabotaged by fear and self-doubt from previous mistakes, which can create a destructive feedback loop as continued mistakes feed that fear, until they just give up.
I think it's pretty much the same with any intellectual discipline.
That's why I find analogy and example to be so useful in teaching. It provides people with alternate ways to look at something that often prove more "digestible" to them.
It also helps bridge the fear gap once they start realizing that it isn't really as hard as they assumed it was, once they have a way of looking at it that they can get a foothold on.
I once tutored someone in electrical theory by teaching him to think of electron flow as a river.
lockers: You are confused about when schools teach algebra. In both my and my daughter's school it starts in 8th grade for the ADVANCED classes. No, algebra is a highschool class.
He may be confused by the fact that we kind of run out of names for classes that lie within the vocabulary of students of the appropriate grade level, so we tend to get lazy and just name them after the class before or after and add "pre" to the beginning or "II" to the end.
For instance, eighth-grade math, a class that would probably be called something like "long-form operations (basic)" were an adult in a technical field to name it, is usually called "pre-algebra" in Texas, despite having nothing to do with Algebra. The high-school trigonometry course is called "pre-calculus" despite having nothing to do with calculus, and the mathematical rhetoric/proofs class is called "geometry" despite being only baaaaaarely about geometry in any recognizable fashion.
So someone that hasn't been an actual student or teacher in a while (like, say, a parent, or someone looking it up on the internet) can pretty easily look over a list of course names, see the word "algebra", and go "holy shiat, they're teaching kids algebra in junior high now? That just seems unnecessary." Not so much the reader's fault as our* fault for naming things in a lazy/retarded fashion. Poor documentation usually ends like that in basically every field.
buckler:lockers: slayer199: umad &math skills in the real world because if you can't communicate, how will anyone know about your wonderful engineering skills.
You use algebra all the time. You just don't call it that because you don't write things out formally. Algebra as taught just takes all the stepwise arithemetic you do in your head and gives it expressive written form. Everytime you do something as simple as figuring out your portion of a meal out with friends, you are doing algebra. You just don't organize it systematically as they tried to teach you. That isn't just it's uselessness, norjustyour teachers failure. It's also a failure to recognize the tool or it's usefullness.
I see that as akin to the idea that you're doing complex physics calculations in your mind every time you catch a baseball, though it may not be in the time-consuming, written notation you use in the classroom.
Of course, writting things out formally is often a waste of time. But you don't learn stepwise problem solving until you hit algebra. But you still need concepts from algebra to be successful in any kind of life where you don't depend on others. Modern financial life demands that of you, and depending on others for that requires a never ending string of luck to stop you from disaster. Yes, I am a software architect, so I do get paid professionally to do algebra. But that also makes me appreciate how often people do that informally.
You NEED the written notation, because it is the language the subject is taught. English doesn't have the formalism needed, which is why you do word problems. It teaches youhow tobridge the gap between english and algebra. Without that its like saying we should teach film theory in spanish. Or political science in german. Or if you want to really be pedantic, teaching philosphy in symbolic logic. The problem with that human language is ambigious. It's a poor tool for the job in the same way a hammer is a poor tool for a screwdriver.
Well, it is in a way. You're coming into a policy argument that we've been having literally since the founding of the nation (it pops up in the federalist papers) of equal ability versus equal opportunity and the relationship between the two.
Public education is an equalizer in the "opportunity" sense. It doesn't automatically put you on par with the rich dude who can be privately tutored in everything, but it gives you the opportunity to make up the remaining gap through work, luck, and sheer awesomeness by making resources available to you.
And in the more general sense, it diversifies our upper economic brackets and helps keep us from slipping into a class system-- if there are people that worked their way up in there along with the folks that were born rich, and some people that got rich by getting lucky with property deals, we're not in any particular danger of being dominated thoroughly enough by any one group to end up with hereditary lordships, robber barons, or a technocracy. All of which would suck for the losers more than the current system.
I could not agree more. I teach remedial math at a university. It is the stuff that they should have gotten in High School but didn't. Most of my students hate math because someone made it a miserable experience for them. I majored in Zoology and I hated math when I was an undergrad. I joke with my students that people go into Biology because they don't like math. I only started to understand math when I started helping other people with it. Now I teach a class that students have fun in. They learn the math and I don't make them feel stupid. I usually say the correct math term and what it means every time. For example I will say "The denominator, the number on the bottom." I try to do this every time. They don't feel stupid for not remembering what a denominator is and it eventually sticks in their head.
I never ever blame my students for their lack of understanding in math. I blame their teachers. Almost every one of them can give me the name of the person who made them hate math. I tell my students that my class is not about math, but about problem solving, and it is. I don't want them to memorize formulas. I have had students who can rattle of a formula perfectly, but have no idea how to use it. I tell my students to look at the problems and figure out what they can do with it. If they can't multiply 7 and 8, I don't care. I tell them to just add stuff up to get the answer. I can't multiply 7 and 8. I tell them to look for patterns and develop tricks that always work. The sad thing is, students are so scared of doing things the wrong way that they are afraid to even try. I tell them that as long as they get the answer there is no wrong way (except cheating, that is wrong). I give them unlimited time to do their work so they don't freak out so much and shut down mentally. I have sat with a student for 5 hours doing a test. I don't accept blank answers and I will give partial credit for pretty much anything written in the answer space (aside from IDK). I gave them 5 points extra credit on the final if they could tell me who Henry Rollins is because we talked about him one day in class.
I am not an awesome teacher. I am a pretty crappy teacher if you go by the standards. I just understand the fear these students have and try to make things a little less stressful so they can focus on what I want them to learn instead of having to guess what it is I want.
buckler:umad: buckler: umad: Because People in power are Stupid: buckler: IWhich is exactly why this article was written.
My friend, I think you hit the nail on the head.
Not really. For me, it isn't about "feeling good" about an answer, it's about the difficulties in comparing one way of structuring things vs. another. Please don't drag that "everyone's a winner" crap into this. It doesn't apply.
Where the hell did you get "everyone's a winner" from either post? You can't talk your way out of a wrong answer in math. That pisses people off when they get away with it everywhere else.
From your agreement with the post that you were responding to. Look, I know that math is an objective field. I understand that. I was expressing my amusement with the fact that, when approached from another perspective, the answers to those problems are all correct. Like I said, it's a clash of disciplines that gives sometimes surprising results. When you use words to express a problem, it puts it into the purview of language, which may come up with interesting responses to what would otherwise be a purely mathematical problem. If you wrote an equation on a board, putting X's in certain spaces, and asking for students to solve for X, English would have no way to touch it. By using words, it falls squarely into the domain of English as much as it does math, so I find the creative answers to be amusing. That's all.
They are amusing. Amusing and wrong. That is the beauty with math. A problem can be approached from many perspectives, but there will still only be one correct answer.
I (optimistically) believe that anyone can understand anything if whatever's being taught can be done so in a way that relates to something that the student already understands.
Now that I am studying mathematics for its own sake, I can offer one thing that would've helped me immensely as a youngster when it came to learning math.
All those stupid, pointless, boring "exercises?" I could not understand why I'd ever use quadratic equations in life as a youngster, and that's what I thought the exercises were for. What would've helped me out is if someone had told me that doing those exercises is a lot like practicing a musical instrument, or practicing using a shop tool. Doing mathematical exercises is all about getting used to the "feel" of a certain tool. Imagine using a chiseling tool for whittling on some wood. Each time you do that, your goal isn't to carve out David. It's to get used to how the tool feels in your hand, how the wood responds to different pressures and angles, etc. Over time, you can sort of mindlessly do it, much like driving a stick shift. You do it without even thinking about it. The point isn't getting good at calculus or trig and applying it later on to a specific thing, per se, it's all about becoming comfortable with using the various tools. Like a craftsman.
Mathematical exercises are the EXACT same thing in my mind: to get used to handing and wielding the tool effectively, not grinding mindlessly on some super-abstract idea that has zero appreciable impact on your life. It's not so much the ends as it is understanding the means and getting good with manipulating the tool itself. And of all the tools available to us on this planet, none is more pervasive or useful as mathematics.
I gently urge anyone out there who believes themselves (as I once was) to be "not a math person," to give the subject another chance. The hardest part about math is finding a learning resource that resonates with how you naturally learn things.
slayer199:umad > math skills in the real world because if you can't communicate, how will anyone know about your wonderful engineering skills.
We can tell you use no math is life by your posts in the politics threads regarding the economy.
It's great we have a population with strong opinions and a belief that they understand macroeconomics and that these people also admit that middle school math has them scratching their heads.
pushpinder:Christ, did a cow crap in here? Figures the article would come from a liberal arts major. Know what, take David Copperfield and shove it up your bung hole! If you can't learn a concept that is a few hundred years old, you're an idiot. Math, at its core, is about problem solving whether it is useful for you in life or not, it builds cognitive skills in looking at a problem, breaking it down and finding a solution. It trains the brain to solve problems. Painting happy trees every day will not help you tackle problems you might encounter in the workplace (though they will make your cubicle friendlier).
The author teaches political science. That's a school of belief that you can do anything you want and be successful, without regard to history or science, so long as you keep trying and throw enough money at the problem. Logical thinking would only get in the way.
That's what math can be reduced to. Your ability to follow instructions. Being able to visually recognize certain symbols and knowing what instructions to follow when you see them.
That's why math is not taught properly.
People with an innate grasp of math teach it. That's why everyone thinks it's hard. If you start fencing against Zorro, Dartanian, and Ingio Montoya and they don't take it easy on you, you're gonna think OMFG FENCING IS IMPOSSIBLE. That's what math class does. It's a bunch of people who automatically get it because of a natural propensity for the skill with years of experience yelling at you for not being born into an artificial way of thinking.
Math class cuts all the important and real life parts of math out and presents it in the least useful, most boring, and an entirely haughty way. And they we act surprised when the only people who can do math really well are boring and haughty and not very practical. You get what you teach.
Like I said in the redlit thread: we shouldn't stop teaching math, we should stop teaching math like we do. Instead of hard rules for making integers have sex for an hour a day, teach real world examples and introduce practical applications from day one. Don't just say "You can use the area of a square to measure your floor." make all the problems "You must carpet this house. Here is the price per square foot per carpet. You have $2000. Which carpet can you afford?"
Witthout that real world anchor right away, most people will never get it.
Certification programs for veterinary technicians require algebra, although none of the graduates I've met have ever used it in diagnosing or treating their patients.
Let m be the mass of the dachshund you're treating. Let reff be the volume of medicine per kilogram of dachshund required to effectively treat its ailment. Let rmdk be the volume of medicine per kilogram of dachshund that would cause the patient's brain to explode.
Guess what basic, every-moron-should-know-it skill the vet can use to solve this problem? Hint: It isn't "oh, this much looks about right".
-History (so you remember enough not to vote for things that didn't work the first time) -Stats (so you can interpret data for yourself.) -Comprehensive reading (so you can understand if a study or article has a logical flaw.)
Yes, you can get away with knowing the bare minimum and still living your day to day life....but politicians, businesses and other such folks who know more will be able to run your environment into the ground without you realizing it.
rumpelstiltskin:
^^^ This.
If you're able to reason logically, then there's no reason to resort to algorithms to do agebra. Algebra is logical, and it makes perfect sense to anyone who can think logically.
It isn't that we are teaching too much math. It is that we are failing to teach it properly. Teaching to the test (so that kids pass the state tests) is not beneficial to anyone. People educated under this system lack the ability to use critical thinking, logic, analysis or evaluation techniques.
The problem is that we treat education like several things it's not, and no one wants to have an honest conversation about them.
1. Education is not the great equalizer. We waste inordinate amounts of money trying to get everyone the same education, as if the only reason that kids can't all be the next Einstein or the next Mark Twain is that we're just not trying hard enough. Bullshiat. Some people lack the brainpower to aspire to the intellectual or professional class. Some kids don't care to learn, and some are so farked up by poverty, drugs and abuse that they are lost. 50 years ago we would have shaken our head, washed our hands of it, and hoped that the ones that couldn't make it to college would find a useful trade or at least not end up a burden on the system. Now EVERYONE has to go to college. And so we try to teach everyone algebra. Well, guess what? Some of those kids are going to end up as assistant manager at the Kroger down the street, and they don't need trigonometry or calculus to tally out the cash registers at the end of the night. And it's only made worse by (sorry to say this) Affirmative Action. We can't have too many poor black kids failing, so we rig the grading system and teach the test and automatically pass kids to grades and to subjects for which they're not prepared.
2. Education is not a day-care. I understand that times are different and that most mothers have to work at least part time. That's just reality, I guess. But back in the day when most middle-class families had a stay-at-home mom, you had someone to tutor the kids and to teach them basic life lessons. Take the stay-at-home mom out of the equation, and is it any wonder that our schools are full of struggling students and troubled kids? We've tried to shift the responsibility to raise children to teachers, who are doing the best they can just to teach the kids enough so that they pass the No Child Left Behind tests.
3. Education is not a trade school. If you want to run a trade school, run a trade school. That's what Germany does after age 12 or so and they're getting along just fine. College prep and college for the kids who want/are prepared for it, trade school and apprenticeships for the kids that are more inclined to work with their hands. High schools and colleges shouldn't have to have official academic programs and majors for medical billing specialists or communications hacks or marketing. If you've got a good, well rounded education, you can figure those things out on the job.
red5ish:Oznog: red5ish: Russ psychology are tied at 10.9%.
Do they also publish the % of people graduating in these disciplines"LIBRARY SCIENCE"?? How is that even a thing? Libraries are already obsolete. We don't organize information this way anymore. I'm not saying that in any way we don't NEED books, but we don't get physical books out of a lending library. Even publishers don't care for that anymore. I don't see any use for a "Library Scientist" unless the term is grossly misleading and describes something else altogether.
The term is grossly misleading and describes something else altogether in many ways. Don't kid yourself though, there are still huge libraries that require librarians, and a lot of library science is learning how to do research which is quite useful.
As a librarian myself, I'll take this:
A, libraries are hardly obsolete. Maybe they are for how you used them - big box of encyclopedias and paper journals for school papers - but there are other options. But lots of people use libraries for lots of reasons. Libraries are community meeting places. They're a place many people go - job seekers, parents, children, seniors, people with limited budgets, people who need specialized information, all of them go to libraries. A good public library is the heart and mind of a neighborhood. It teaches its children, informs its public servants, entertains its citizens and enriches everyone.
B, Even how you used libraries still exists. Now-a-days, rather then being gatekeepers for the Encyclopedia, now librarians work to help filter the masses of information. In a world where a google search finds you 100 thousand hits, and maybe a few are relevant and accurate, a lot of people need help, especially when the target is academic or business-related, where accuracy is more important then on Fark.
C, And that's completely ignoring the dozens of non-public/school libraries there are - corporate libraries, for instance. Many companies have internal libraries to look up information related to their business and that of their industry. Archives, Records Management, keeping a history of a place, a company, an organization, that's all under the purview of the librarian. (yes, I know there are RM people who hate being lumped with librarians - tough shat, it's my post).
D, You are right in that "Library Science" is not the most preferred term - some people really hate being called a library scientist or a librarian. These people prefer the term "Information Scientist" - indeed, the school I went to was called the "School of Library and Information Sciences" (SILS for short). Librarian is a more... human word to me, so I prefer it. But I can dig where they come from.
And finally , E, you think we handle just books? Ha. Maybe 30% of my day is paper on a busy day. The rest is digital. Subscription databases, e-magazines, journal databases, e-Lending libraries, digital archives, I handle more tech then some IT guys. Remember: we brought this. We've lived in this world for 20 years now where most people just got it 6 or 8. We've adapted, and will adapt however it goes forward.
So please, don't step. We've been here, we ain't going no where. As long as there is data, as long as it has to be sorted, portioned and doled out, as long as there's students who need to know facts, and business reports to be written, while there's paper that needs to be preserved and digital files that need to be kept, we are going no where. We're Librarians.
State regents and legislators - and much of the public - take it as self-evident that every young person should be made to master polynomial functions and parametric equations.
For the record, this argument applied to English would be:
"State regents and legislators - and much of the public - take it as self-evident that every young person should be made to master two-syllable words and telling a verb from a noun."
This is pretty basic stuff that's vital to basic functioning in society here. It's the technical version of functional literacy. Things like calculating your gas mileage and creating a personal budget so you don't go into debt require algebra, which makes the "this isn't personal finance" comment rather puzzling as well.
//A 700 on an SAT subject test isn't quite the unreachable high bar the idiot writer seems to think it is, either. It's decent, yes, but the SAT is a literacy test, not a competence test, and to get into programs that actually specialize in some form of math you're not getting anywhere without an 800. 700 for a general program is a bit high for a general knowledge requirement, but only a bit high. I didn't get out of the English proficiency requirement for general knowledge for going into a chemistry program, the logic of requiring some roundedness of students isn't limited to the liberal arts.
Gyrfalcon:buckler: IThe ability to think "outside the box" doesn't count in hard science.
Math isn't hard science. The "proofs" side of algebra, i.e. proving that 2 + 2 = 4, is all about being creative.
Babwa Wawa:slThis.
If you think that the failure rate of college algebra is negatively affecting your retention, your priorities are totally backwards. The point isn't to get more people out the door with degrees, it's to get people out the door with worthwhile degrees. For instance, most social science majors need to understand stats, and any stats class requires mastering algebra. But go ahead and keep graduating economics majors without any math skills.
I may not use calculus directly in everyday life, but the understanding is there and I feel I understand the world a bit better because of it. Algebra even more so. Just because most adults don't solve for X on a daily basis, doesn't mean there is not a benefit from a fundamental understanding of it that influences their thoughts and actions. It is utterly irresponsible to deny kids that same learning. "Because it's hard" is simply not a valid excuse.
Mathematical literacy is even more important than ever in day to day life. Companies routinely obscure costs with tricks (cellphone and cable companies...I'm look at you, you assholes!). Your employers no longer give a damn about your retirement via pensions...here's a 401k program...good luck to you!
/I can't believe someone is trying to make a case for getting rid of any math education.
red5ish:Russ psychology are tied at 10.9%.
Do they also publish the % of people graduating in these disciplinesIt's a mistake to view this exclusively in the light of % of people getting JOBS exclusively in that discipline. You don't score the value of people knowing history by the % of jobs created in the "History" field.
One thing I heard Sandra Day O'Connor lament on the Daily Show was that NCLB had placed value exclusively on math and reading, to the detriment of civics. Consequently it appears fewer and fewer people understand the basic structure of US govt, that the POTUS does not direct the Supreme Court, nor does he "make laws". And that "activist judges" is truly an absurd term indicative of a basic misunderstanding of the Judicial Branch. "Activist Judges" determined the very principle of segregation was inconsistent with the US Constitution, despite a quagmire of laws created by Legislative and signed by Executive, all with popular support. To say that they should not overrule Legislative/Executive decisions is to nullify their basic check-and-balance power and basically say that "a law cannot be wrong", because legislature is infallible. Like the Pope.
Voiceofreason01:If college algebra and trig are to complicated for you, you have no business having a degree. If high school algebra and trig are too complicated for you, you have no business being a high school graduate. Saying you don't need basic numeracy is like saying you don't need literacy. But since this fail-the-children ideology permeates modern America, the percentage of people needing remedial english and math at universities is on the rise.
i think the best way to compete with the chinese is to produce more polysci majors. oh wait, i just realized that there is not a single thing that the modern consumer wants, that a polysci major can produce.
red5ish:How much does your IP attorney use calculus, or does s/he just charge you $800/hour and call it good?
Part of being a successful attorney is having a well-rounded education. Part of having a well-rounded education includes taking classes that don't necessarily have anything to do with your career.
You don't take most of your college courses to learn the facts in the courses, you take them to exercise your mind and make you aware of the larger world outside your own life. If you don't want to do that, go to a trade school and spend the rest of your life as a welder making "good money" at twenty bucks an hour.
buckler:IMath is a fundamental aspect of life. I would argue that it's absolutely required for critical thinking and long term success.
The problem is how it's taught. The teachers are either complete morons that don't even really want to do math themselves or they're so focused on the subject that they're no good at teaching it to normal people. The issue runs very deep, the education system itself failed to educate the educators properly.
By the time I reached high school, Carl Sagan had convinced me that what I really wanted to be was a cosmologist. However, I found that I just didn't have the chops for math, so instead of doing science, I ended up interpreting science for others. I found I was good at that, and enjoyed it immensely.
weiserfireman:This is how I learn math down to a tee. Understand the ideas and methodology, and then practice the heck out of it until it clicks. After a little while I tend to get a a ha moment and it is easily understood from then on.
Most people at college didn't want to sit down and practice. They wanted a life and chase girls. I was married, so that wasn't a problem.
FTA: "And if there is a shortage of STEM graduates, an equally crucial issue is how many available positions there are for men and women with these skills. A January 2012 analysis from the Georgetown center found 7.5 percent unemployment for engineering graduates and 8.2 percent among computer scientists. "
Let's see... bust your balls taking the hardest courses, the most units (and some of the highest debt because you don't have time to work) and still get paid like sh*t while watching your back (if you can even find a job) because your idiot potential employers would rather H1-B or outsource your ass as soon as they can... or skate though taking business courses, have a life and work on Wall Street for moar money than gawd...
"The problem with American schools is that they don't teach too much math anything to anyone"
FIFY
I went overseas to an international school for my sophomore year in high school. It was waaaaay harder than my HS here in the states (I guess those damn Europeans wanted their kids to be educated or something). When I returned for my junior year, I discovered that my Sophomore English text book over there was the Senior AP English text here.
slayer199:EvilEgg: ForI had to take college bound English before I went to college to major in Engineering. You can take a little bit of math, cupcake. It won't kill you.
At what point will brake application result in insufficient reduction of momentum to avoid collision requiring you to calculate the proper trajectory and starting velocity in which to disembark the train with statistically the least likely result being farked up beyond all recognition?
University education, especially liberal arts education, is more important than you might think. Requiring engineering students to learn a foreign language and understand philosophy makes them better engineers because it allows them to think about things in different ways (at a potentially fundamental level), as well as just making them better people in general. Besides, engineers need to be able to write well and read well to perform research.
Also, many 18 year olds don't even know what they want to do yet. Some aren't even aware that there are options out there. I know plenty of people who started out as CS majors and turned into pure math majors, or EEs or philosphy majors (and not just to get an "easy degree"). I'm lucky in that I've known what I wanted to do for as long as I can remember, but that's not true for everyone.
If you only want to take classes that are relevant to your career, fine, go to ITT Tech. The problem is that you'll miss out on the whole universe of knowledge and information that would have (a) made your life richer and (b) made you a better network engineer in incalcuable (but real) ways.
Kimothy:I'll bet you use algebra more than you think. Any time you see a package of 10 somethings for y dollars you might think about how each one of those things costs y/10. That's algebra, Bud.
Graffito:Kimothy: TheirLearning math is really learning problem solving, the numbers are almost irrelevant. When I taught prep for the GMATand GRE I found the students who couldn't do math were the students who could never solve any of the verbal problems they didn't immediately know. They seemed to lack the ability to break down a question and figure out how to solve it.
Mmm, no. Here's the thing: an alarmingly large proportion of the middle school kids I've encountered in the past while have often come in with huge gaps in their math abilities, often operating several years behind where they should be.
It should be noted that a BSc-Math degree doesn't qualify one to start an elementary certification program under typical state NCLB standards. You would need an extra year or so of general arts credits beyond your degree to qualify. Basically, you'd have to hybrid into the equivalent of a BA (Math). Math majors generally certify at middle or high school.
The converse is not true. One or two math credits are sufficient for an Arts or History major.
Worse, most teacher college professors appear to have been drawn from the huge pool of English/History majors. You're very lucky if you have a math or science background professor who can teach that aspect of education to the elementary school teacher candidates.
I hated math, became an English major, got out into the real world, and landed my first job in banking. That evolved into analytics, performance tracking, and statistical analysis & modeling. I use algebra every day. I'm damn glad I received the broad, liberal education that included algebra, stats, logic and computer science.
I'm one of the few in my part of the corporation who is a solid writer. Probably the only one who both understands the complex issues discussed and is capable of communicating effectively. Job security rocks...
/the math is there to teach you how to effectively approach abstract and uncomfortable challenges...pretty useful, in general...Ah, so this is an argument about degrees of algebra.
man, I hate useless word problems like this one. There has *got* to be a better way to get this same point across in a useful manner. I mean, are there any situations when I wouldn't know I had $7.62 last week, but I would know the relative difference in price and the amount I bought? That is sooo... backwards.
I guess, you could make it like a detective story: "A detective is investigating a robbery and the suspect was seen leaving the supermarket and throwing away the receipt, which would have his finger prints. There are 4 receipts, but they only indicate the price spent/item. The clerk doesn't remember the price of the meat, but does remember that the suspect bought 3 pounds of beef, currently $3.99, which cost 15% less the day on the crime. Which receipt has the suspect's fingerprints?"
That's probably too long and complicated, but at least more interesting.
Babwa Wawa:I went into that article thinking you could get rid of algebra if you replaced it with something more relevant like statistics.
The nation would be much better off if everyone had a basic understanding of stats.
That's funny... I was just having a discussion maybe 2 days ago about the reasoning behind why stats isn't a required part of a high school education. Not necessarily a whole semester of stats, but all the basics. I even discussed a single semester of algebra and stats combined. Advanced material from either one of them is all but useless to most students, but the basics learned from both carry on to a number of things in the job market that are not science related.
EngineerBoy TheseIf you can't explain it simply, you don't understand it well enough. If you can't explain it at all, you probably teach high school.
namatad:The radio, public speaking and other related work. Likewise, several others have had great monetary success with art and music.
Babwa Wawa:Thoguh: BabWhen people talk about algebra, they usually mean polynomials or something like "find the dimension of a rectangle whose length is 9 less than twice its width if the perimeter is 120 cm."
I don't normally nitpick on terminology, but in this case it's important. Of course the author was not saying that HS grads shouldn't need to be able to solve for x in 2.99 * 3 = xI'm not arguing that algebra shouldn't be taught - I said the problem was with the way it's currently taught, with an emphasis on testing, not application+1 The important skills to learn in algebra are how to manipulate numbers, not just how to solve for x. I work in accounting and finance, and while I seldom use actual algebra, I constantly manipulate numbers in ways that I learned while being taught algebra. While algebra itself might be replaceable in schools, some form of intermediate mathematics needs to take its place. I got through Calc2 in college and I have only once used that knowledge for any practical purpose. I would note that I described that application during the interview for the great job/career I have now - so perhaps I should give higher math a bit more credit. Having mathematical skills does make an employee more valuable.
I am terrible at math. I tried and tried in school, but I just couldn't wrap my head around it. My brain just isn't wired that way. However, I excel when it comes to language and interpretive arts, and I did very well in visual arts. Aside from the occasional grammar-Nazi snark here, I don't put down those who don't do well in English or related fields, because I know my own limits when it comes to math. I had a roommate who admitted he never learned to read, and I helped tutor him until he had at least the basic skills.
The important thing for me is that I was necessarily exposed to both fields. I found I did well in one, and not so much the other; I would expect to find that there are those who excel in math, but maybe not so much in language skills. I don't value them less that anyone else. Indeed, these people are vital in the STEM fields, which our country needs people in now more than ever. This guy's thesis is bunkumIMHO, algebra is too often incorrectly taught as a series of steps rather than a concept.
Thoguh:Babwingedkat:Wait. How exactly do you propose to teach statistics without algebra?
downstairs pretty much captured it. Yes, college-level stats course would and should need at least some foundation in algebra. But not all stats, and certainly not all interpretation of statistics needs an algebraic foundation. And interpretation of stats is something that everyone in every walk of life can benefit from.
I suppose my suggestion would be having statistics drive the education around algebra.
wingedkat:Babwa Wawa: I went into that article thinking you could get rid of algebra if you replaced it with something more relevant like statistics.
The nation would be much better off if everyone had a basic understanding of stats.
Wait. How exactly do you propose to teach statistics without algebra?
I get your point, but my point would be (from my experience in high school in the 1990s) is that most everything is taught much more rote than practical real-world situations. Yeah, you need some rote learning (2x - 4 = -3... solve for x)... but it would be better to move on to some sample real world situations.
All in all I just remember never having real-world situations taught to me in high school.
It's not that we teach too much math, it just seems like a third of every year of math is spent reviewing the previous year. If we'd just go with a "get it or don't" mentality, we might be able to teach something useful someday. | 677.169 | 1 |
introduction to a concept, practice of skills and understandings, applications of a concept or technique
Other Comments:
Great for California standards, 4th grade and up. Will hold student interest in seeing relationships between x and y, constants and coefficients, positive and negative slope.
What math does one need to know to use the resource?
Function tables, coordinate graphing, equation solving
What hardware expertise does one need to learn to use the resource?
none
What extra things must be done to make it work?
nothing
How hard was it for you to learn?
Very Easy
Explanation:
It would be very easy for students to work by trial and error (which would be sufficient at 4th and maybe 5th), but much more valuable for the teacher to support students by helping them formalize their generalizations | 677.169 | 1 |
FLY through Algebra for FLY Pentop Computer
Wow! If only I'd had a dynamite handheld pentop computer loaded with this program when Mr. Darnell was torturing me with all that algebra gooblygook back in the day!
Imagine being an algebra beginner stuck on an equation like 3x² x 22 x +117 = 35x. Say what? But thanks to the FLY Pentop Computer and the education-minded folks at LeapFrog, algebra no longer has to be the problem it once was. Just snap the Fly Through Algebra Flyware cartridge onto the back end of the FLY Pentop Computer and you've got an automated algebra coach checking each step of your homework.
Keep in mind that this is not a game. And it's not a full-blown pedantic lesson in how to do algebra. Instead, this computerized pen simply assists students who use it in solving algebra problems they've been assigned in school. Remember the talking car in the TV series "Knight Rider"? Well, this is sort of the talking-pen equivalent for tackling troublesome algebra equations.
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A former feature writer and consumer columnist at The Washington Post for 22 years, Don Oldenburg is the Director of Publications and Editor of the National Italian American Foundation, in Washington, D.C. He regularly reviews books for USA Today and is the coauthor of "The Washington DC-Baltimore Dog Lovers Companion" (Avalon Travel). The proud father of three sons, he lives with his journalist-author wife, Ann Oldenburg, in McLean, VA. | 677.169 | 1 |
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Almost every student has to study some sort of mathematical proofs, whether it be in geometry, trigonometry, or with higher-level topics. In addition, mathematical theorems have become an interesting course for many students outside of the mathematical arena, purely for the reasoning and logic that is needed to complete them. Therefore, it is not uncommon to have philosophy and law students grappling with proofs.This book is the perfect resource for demystifying the techniques and principles that govern the mathematical proof area, and is done with the standard "Demystified" level, questions and answers, and accessibility.
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Rent Math Proofs Demystified 1st edition today, or search our site for other textbooks by Stan Gibilisco | 677.169 | 1 |
C.I NCI N IN AT.i :
LIBRARY
OF THE
UNIVERSITY OF CALIFORI^
Received </J f ~~
Accession No. 6 7 J <
1
i
ECLECTIC EDUCATIONAL SERIES.
JKAY'S ARITHMETIC, SECOND BOOK.
INTELLECTUAL
ARITHMETIC
BY
INDUCTION AND ANALYSIS.
BY JOSEPH RAY, M. D.,
LATB PROFESSOR OP MATHEMATICS IN WOODWARD COLLEGE.
ONE. THOUSANDTH EDITION-IMPROVED,
CINCINNATI :
WILSON, HINKLE & OOo
PHIL'A: CLAXTON, REMSEN & HAFFELFINGEB.
NEW YORK: CLARK & MAYNARD.
y_
RAY'S MATHEMATICAL WORKS.
TYPE ENLARGED NEW ELECTROTYPE PLATES.
Each BOOK of Ray s Arithmetical Course, also of the Algebraic*
is a complete work in itself, and, is sold separately.
FIRST BOOK.
PRIMARY LESSONS AND TABLES; simple and progressive
Mental Lessons, for little learners.
SECOND BOOK.
INTELLECTUAL ARITHMETIC, by Induction and Analysis;
a thorough course on Intellectual Arithmetic.
THIRD BOO K.
PRACTICAL ARITHMETIC, by Induction and Analysis ; a sim-
ple, thorough work for schools and private students.
~J3Y TO RAY'S ARITHMETIC, THIRD BOOK.
RAY'S HIGHER ARITHMETIC. Principles of Arithmetic, ana-
lyzed and applied. For advanced students and business men.
KEY TO RAY'S HIGHER ARITHMETIC.
ELEMENTARY ALGEBRA,
RAY'S ALGEBRA, FIRST BOOK, for Common Schools and
Academies; a simple, progressive, elementary treatise.
HIGHER ALGEBRA.
RAY'S ALGEBRA, SECOND BOOK, for Academies, and for Col-
leges ; a progressive, lucid, and comprehensive work.
KEY TO RAY'S ALGEBRA, FIRST AND SECOND BOOKS.
Entered according to Act of Congress, in the year Eighteen Hundred and Fifty-
Seven, by WINTHROP B. SMITH, in the Clerk's Office, of the District Court of the
United States, for the Southern District of Ohio.
Entered according to Act of Congress, in the year Eighteen Hundred and Sixty,
by W. B SMITH, in the Clerk's Office of the District Court of the United States,
for the Southern District of Ohio.
PREFACE.
Few works on Intellectual Arithmetic have received more
nnqualified approbation, and a more extensive patronage, than
this, which, for several years, has been published under the
title Ray's Arithmetic, Part Second.
The numerous editions demanded, have again rendered
necessary a renewal of the plates, which has afforded an
opportunity for REMODELING the work.
Many important improvements have been introduced, with
a design to impart completeness, and give a concise and pro-
gressive course of arithmetical analysis.
The volume in its present form, embraces,
1st. Exercises on the primary principles and their applica-
tions, interspersed with appropriate models of analysis and
frequent reviews.
2d. A progressive and comprehensive presentation of Frac<
tions, intended to render the subject intelligible and attractive
to the pupil.
3d. A General Review, designed to test the pupil's knowledge
of principles, preparatory to the applications of mental analysis
which follow.
4th. Percentage, Gain and Loss, Interest, and their appli-
cations.
The value of Intellectual Arithmetic is so highly appreciated
by instructors, that little need be said in its commendation.
When 'properly taught, it is one of the most useful and
interesting studies in which pupils can engage, and should be
omitted by no one.
By its study, learners are taught to reason, to analyze, ie
think for themselves; while it imparts co'nfidence in their ows
reasoning powers, and strengthens the mental faculties.
HINTS TO TEACHERS.
LET the pupils be classified with reference to their attainments
and abilities. The recitation should be short and spirited, every
pupil being required to give undivided attention to the question
before the class.
Generally, while reciting, the pupils should be permitted to have
their books open before them the test of having properly studied
the lesson, being the readiness and accuracy with which the several
questions are analyzed and answered.
The explanations and operations termed ANALYSIS, are intended
as Model Solutions, pointing out to the learner the manner in which
the questions in the lessons are to be solved and explained.
The pupil should be required to furnish a similar explanation
to each of the jucceeding questions, and to give, not only a cor-
rect answer, but also, the reason for the method by which he
obtained it.
A method of solving questions in Mental Arithmetic, now mucb
used, is the following, called the "Four Step Method."
ILLUSTRATIONS. First step, James gave 7 cents for apples and
8 cents for peaches; how many cents did he spend? Second
step, as many as the sum of 7 cents and 8 cents. Third step, 1
cents and 8 cent 1 ? are 15 cents. Fourth step, hence, if James gave
7 cents for apples, and 8 cents for peaches, he spent 15 cents.
Again: First step, 4-fifths of 25 are how many times 6? Second
ttep, as many times 6 as 6 is contained times in 4-fifths of 25.
Third step, 1-fifth of 25 is 5, 4-fifths are 4 times 5, which are 20;
6 in 20 is contained 3 and 2-sixths times. Fourth step, therefore,
4-fifths of 25 are 3 and 2-sixths times 6?
As a means of keeping the attention of the class directed to each
question, it. will be proper for the instructor occasionally to read
an example aloud, and, having allowed sufficient time for the
answer, to call upon some one for the result, and then for the
analysis. By this means, each one is obliged to solve the example^,
not knowing but that he may be required to answer it.
INTELLECTUAL
ARITHMETIC.
SECTION I. NUMERATION.
Pupils who have studied Ray's Arithmetic, First Book, maj
commence with Lesson IV, on page 14.
NUMERATION is naming Numbers.
Learn the name and form of these figures:
1 2 3 4 5 G 7 .8 9 10.
One, two, three, four, five, six, seven, eight, nine, ten.
An Illustration of the Increase of Numbers.
l
2
3
4
5
6
7
8
9
10
10 KAY'S INTELLECTUAL ARITHMETIC.
SECTION II. ADDITION.
LESSON I.
1. James had one apple, and his brother gave him
one more : how many had he ? Why ?
Ans. Because 1 and 1 are 2.
2. Henry had two cents, and his sister gave him one
more : how many had he ? Why ?
Ans. Because 2 and 1 are 3.
3. A hoy had one marble, and found three more :
how many did he then have ? Why ?
Ans. Because 1 and 3 are 4.
4. Thomas had 4 cents, and his mother gave him
one more : how many cents had Thomas ? Why ?
5. Samuel had two cakes, and his father gave him two
more : how many did he then have ? Why ?
6. Three oranges and two oranges are how many
oranges ? Why ?
7. James had three apples, and his brother gave him
three more : how many apples had James then ? Why ?
8. John had four plums, and his sister gave him two
more : how many did he then have ? Why ?
9. Daniel had three cents ; his brother gave him two,
and his sister one : how many did he then have? Why?
10. Mary had four pears, and her brother gave her
three more : how many did she then have ? Why ?
11. George recites daily four lessons perfectly, one
imperfectly, and is absent from one recitation : how many
daily lessons has he ? Why ?
12. How many fingers have you on one hand ? How
many on both? Why?
13. Ida had four cents ; her mother gave her three more
at one time, and one at another : how many did she then
have? Why?
ADDITION. 1 1
14. Three cakes, and one cake, and two cakes, are how
many, cakes ? Why ?
15. Four cents, and three cents, and two cents, are how
many cents ? Why ?
16. Five oranges, and one orange, are how many
oranges ? Why ?
17. Henry had five cents, and his mother gave him two
more: how many did he then have? Why?
18. Five boys, and one boy, and two boys, are hoi
many boys ? Why ?
19. Oliver has five dollars, Henry three, and Samuel
one : how many dollars have all together ? Why ?
20. Three peaches, and six peaches, are how many
peaches ? Why ?
21. A lady paid one dollar for gloves, three dollars fo>*
a shawl, and five dollars for a dress : how much did si
spend? Why?
22. Four cents, and three cents, and two cents, and
one cent, are how many cents ? Why ?
23. If a man buy six pounds of sugar at one time,
and four at another, how much does he buy ? Why ?
24. Seven oranges, and one orange, and two oranges,
are how many ? Why ?
25. George has two cents, his sister three cents, and
his brother five cents : if they were all given to George
how many would he have ? Why ?
26. How many are 4 and 3 and 3?
27. James has 4 cents, and Joseph 2 more than James:
how many has Joseph ? How many have both ? Why ?
LESSON II.
1. One and 1 are how many? 1 and 2? 3 and 1?
4 and 1? 1 and 3? 1 and 5? 1 and 6? 6 and 1?
1 and 7? 8 and 1? 9 and 1 ? 1 and 8?
12 RAY'S INTELLECTUAL ARITHMETIC.
2. Two and 2 are how many? 2 and 3? 2 and 2
and 1? 1 and 2 and 3? 4 and 1 and 2?
3. Four and 2 are how many ? 4 and 3 ? 2 and 4
andl? 4 and land 2? 3 and 4? 5 and 4?
4. Six and 2 are how many? 6 and 1 and 2? 6
and 3? 6 and 4? 6 and 1 and 3? 1 and 2 and 6?
1 and 3 and 6?
5. Eight and 2 are how many? 8 and 3? 8 and
2 and 1? 2 and 8 and 1 ? 1 and 4 and 3? 8 and
4? 7 and 4? 6 and 7? 7 and 1 and 3* 1 and 6
and 2 ? 1 and 8 and 2 and 2 ?
6. Nine and 2 are how many? 9 and 3? 9 and
1 and 2? 8 and 1 and 3 ? 5 and 6 ? 1 and 4 and
5? 2 and 4 and 6? 3 and 4 and 5? 4 and 6 and
2 ? 5 and 4 and 2 and 1 ?
7. How many are 1 and 9? 7 and 3? 4 and 6?
9 and 1? 5 and 5? 6 and 4? 2 and 8 ? 3 and 7?
and 10? 8 and 2? 10 and 0? What two numbers
added together make 10?
8. Which is the greater, 7 and 2, or 6 and 4? Why?
one and 3 and 6, or 8 and and 2 ? Why ?
9. Begin at 4, and add 2 each time, up to 10.
10. Begin at 1, and add 3 each time, up to 13.
11. What two numbers added together will make 12?
What three numbers?
12. Seven and 5 and 2 are how many?
13. One and 7 and 3 and 4 are how many?
14. If 3 be added to 3, and that sum to 4, what will be
the result?
15. Tf you add 3, to 3 and 1 more, and then add 7,
what will be the amount?
16. I have in one basket 8 dozen eggs, in another 4
dozen, in another 3 dozen: how many in all?
ADDITION. 13
17. A lady bought 2 yards of tape for 3 cents, some
pins for 1 cent, and received 2 cents change : how many
cents had she at first?
18. Two and 1 more, and 3 and 4 more, are together
how many ?
10. One and 3 and 4 and 5 are how many? 5 and
one and 3 and 4 ?
20. Two and 1 and 3, taken from a certain number,
leave 2 : what is that number ?
21. A boy bought 3 cents worth of marbles, and 2
cents worth of candy, and received 5 cents change : how
muck money had he ?
22. I bought 3 hams for 8 dollars, and 10 bushels of
apples for 3 dollars : how much did I spend ?
23. If 6 yards of cloth will make 2 coats, and 4 yards
will make 2 pairs of pants, how many yards must I buy
for 2 coats and 2 pairs of pants?
24. Oliver has 4 cents in one hand, 3 in the other,
and 4 in his pocket : how many cents has he ?
LESSON III ADDITION TABLE.
2 and are 2
O and U are O
4 and are 4
2 and 1 are O
3 and 1 are 4
4 and 1 are 5
2 and 2 ,re 4
3 and 2 are 5
4 and 2 are 6
2 and 3 are 5
3 and 3 are 6
4 and 3 are 7
2 and 4 are 6
8 and 4 are 7
4 and 4 are 8
2 and 5 are 7
8 and 5 are 8
4 and 5 are 9
2 and 6 are 8
3 and 6 are 9
4 and 6 are 10
2 and 7" are 9
3 and 7 are 10
4 and *7 are H
2 nd 8 are 10
3 and 8 are 11
4 and 8 are 12
2 and 9 are H
3 and 9 are 12
4 and 9 are 13
2 and 10 are 12
3 and 10 are 13
4 and 10 are 14
9 and 11 a,e 13
3 and 11 are 14
4 and 11 are 15
2 and 12 are 14
3 and 12 are 15
4 and 12 are 16
14
RAY'S INTELLECTUAL ARITHMETIC.
5 and are 5
6 and are 6
7 and are 7
5 and 1 are 6
6 and 1 are 7
7 and 1 are 8
5 and 2 are 7
6 and 2 are 8
7 and 2 are 9
5 and 3 are 8
b and 3 are 9
7 and 3 are 10
5 and 4 are 9
6 and 4 are 10
7 and 4 are 11
5 and 5 are 10
6 and 5 are H
7 and 5 are 12
5 and 6 are H
6 and 6 are 12
7 and 6 are 13
5 and 7 are 12
6 and 7 are 13
7 and 7 are 14
5 and 8 are 13
6 and 8 are 14
7 and 8 are 15
5 and 9 are 14
6 and 9 are 15
7 and 9 are 16
5 and 10 are 15
6 and 10 are 16
7 and 10 are 17
5 and 11 are 16
6 and 11 are 17
7 and 11 are 18
5 and 12 are 17
6 and 12 are 18
7 and 12 are 19
8 and are O
9 and are 9
10 and are 10
8 and 1 are 9
9 and 1 are 10
10 and 1 are 11
8 and 2 are 10
9 and 2 are 11
10 and 2 are 12
8 and 3 are 11
9 and 3 are 12
.10 and 3 are 13
8 and 4 are 12
9 and 4 are 13
10 and 4 are 14
8 and 5 are 13
9 and 5 are 14
10 and 5 are 15
8 and 6 are 14
9 and 6 are 15
10 and 6 are 16
8 and 7 are 15
9 and 7 are 16
10 and 7 are 17
8 and 8 are 16
9 and 8 are 17
10 and 8 are 18
8 and 9 are 17
9 and 9 are 18
10 and 9 are 19
8 and 10 are 18
9 and 10 are 19
10 and 10 are 20
8 and 11 are 19
9 and 11 are 20
10 and 11 are 21
8 and 12 are 20
9 and 12 are 21
10 and 12 are 22
LESSON IV.
NOTE. In the following exercises let the numbers be added
both ways: thus, 2 and 8 are 5, 3 and 2 are 5; 4 and 3 are
7, 3 and 4 are 7 ; etc. In adding such numbers as 16 and 12 ;
say 16 and 10 are 26, and 2 are 28.
1. Three and 8 are how many? 3 and 10?
2. Four and 7 are how many? 4 and 9? 4 and
eleven? 4 and 10? 4 and 12?
ADDITION. 15
3. Five and 7 are how many? 5 and 6? 5 and 9?
five and 12? 5 and 10? 5 and 8 ? 5 and 11?
4. Seven and 6 are how many? 7 and 10? 7 and
eight? 7 and 12? 7 and 9 ? 7 and 11?
5. Nine and 11 are how many ? 9 and 9 ? 9 and 12 ?
nine and 7 ? 9 and 10 ? 9 and 8 ? 9 and 11 ?
6. Ten and 6 are how many? 10 and 8? 10 and
ten? 10 and 12? 10 and 11? 10 and 5? 10 and 7?
ten and 9 ?
7. Eleven and 2 are how many ? 11 and 4? 11 and
six? 11 and 5? 11 and 7? 11 and 9? 11 and 11?
eleven and 10?
8. Twelve and 5 are how many? 12 and 4? 12
and 6? 12 and 8? 12 and 10? 12 and 7? 12
and 9? 12 and 11? 12 and 12?
9. Thirteen and 4 are how many ? 13 and 6 ?
13 and 5? 13 and 7 ? 13 and 9? 13 and 10 ?
13 and 8? 13 and 11? 13 and 12?
10. Fourteen and four are how many? 14 and 6?
14 and 8? 14 and 5 ? 14 and 7? 14 and 10?
14 and 9? 14 and 11 ? 14 and 12 ?
11. Fifteen and 5 are how many? 15 and 7? 15
and 9? 15 and 4? 15 and 8? 15 and 10? 15 and
12? 15 and 11?
12. Sixteen "and 4 are how many? 16 and 6? 16
*nd 8? 16 and 5? 16 and 7? 16 and 9? 16 and 11?
16 and 10 ? 16 and 12 ?
13. Seventeen and 6 are how many? 17 and 4? 17
a^d 7 ? 17 and 5 ? 17 and 9 ? 17 and 8 ? 17 and
10 ? 17 and 12 ? 17 and 11 ?
14. Eighteen and 10 are how many? 18 and 4? 18
and 7? 18 and 5? 18 and 8? 18 and 6? 18 and 9?
18 and 11 ? 18 and 12 ?
15. Nineteen and 5 are how many? 19 and 3? 19
and 2? 19 and 7? 19 and 9? 19 and 8? 19 and 10?
19 and 6? 19 and 12 ? 19 and 11?
16 RAY'S INTELLECTUAL ARITHMETIC.
16. How many are 9 and 2 ? 19 and 2? 29 and 21
49 and 2 ? 6 ( J and 2 ? 39 and 2 ? 59 and 2 ? 79 and
2 ? 99 and 2 ?
17. How many are 9 and 3? 3 and 19? 29 and 3?
3 and 4 ( J ? 5U and 3? 3 and 39? 69 and 3? 3 and
79? 3 and 89? 99 and 3?
18. How many are 9 and 7 ? 29 and 7 ? 7 and 49?
39 and 7 ? 7 and 59 ? 79 and 7 ? 7 and 69 ? 89 and
7? 7 and 99?
19. How many are 9 and 8? 29 and 8? 49 and 8?
39 and 8? 8 and 69? . 59 and 8? 79 and 8?
20. How many are 9 and 9? 19 and 9? 9 and 29?
49 and 9? 69 and 9? 59 and 9? 79 and 9? 89 and
9? 9 and 99?
21. How mar- are 8 and 3 ? 28 and 3? 48 and 3?
68 and 3? 88 and 3? 98 and 3?
22. How many are 8 and 7 ? 28 and 7 ? 7 and 38 ?
48 and 7? 68 and 7 ? 58 and 7? 88 and 7?
23. How many are 7 and 7 ? 17 and 7? 27 and 7?
47 and 7 ? 57 and 7 ? 37 and 7 ? 67 and 7 ? 87 and
7 ? 77 and 7 ? 97 and 7 ?
24. How many are 7 and 10? 17 and 10? 27 and
10 ? 47 and 10 ? 37 and 10 ? 57 and 10 ?
25. How many are 6 and 5 ? 16 and 5? 15 and 6?
26 and 5? 25 and 6? 24 and 6? 26 and 4? 36 and
6? 48 and 6? 45 and 6? 57 and 6? 59 and 6?
66 and 6? 75 and 6? 86 and 6?
26. How many are 13 and 8? 17 and 3 ? 23 and 8?
24 and 8? 33 and 8 ? 3 and 37? 8 and 43? 47 and
3? 7 and 53? 58 and 3 ? 67 and 3? 3 and 87 ? 97
and 3? 88 and 3? 3 and 98 ?
27. How many are 14 and 9? 9 and 24? 25 and 9?
9 and 34 ? 36 and 9 ? 9 and 44 ? 9 and 47 ? 54 and
9? 9 and 56? 9 and 64? 74 and 9? 9 and 72? 8*
and 9 ? 86 and 9 ? 94 and 9 ?
ADDITION. 17
28. How many are 10 and 6? 6 and 21 ? 10 and
26? 46 and 10? 10 and 35? 10 and 55? 56 and
10? 10 and 66? 10 and 69? 76 and 10? 10 and
86? 96 and 10?
29. How many are 11 and 6? 11 and 16? 11 and
27? 25 and 11? 11 and 23? 31 and 11? 11 and
35? 37 and 11? 11 and 59? 46 and 11? 11 and
48? 52 and 11? 11 and 63?
30. Fifty-six and 1 are how many? 56 and 3? 56
and 5? 2 and 56? 4 and 56? 7 and 56 ? 56 and 6?
8 and 56 ? 56 and 9 ? 10 and 56 ?
31. Ninety-eight and 1 ? 3 and 98? 5 and 98? 98
and 2 ? 4 and 98 ? 98 and 6 ? 7 and 98 ? 98 and 8 ?
10 and 98? 98 and 9?
32. How many are 1 and 2 and 4 and 5 ? 5 and 1 and
four and 2 ? 2 and 1 and 5 and 4 ?
33. How many are 12 and 8 ? 9 and 2 and 9? 4 and
2 and 9 and 5 ? 4 and 2 and 5 and 5 and 4 ? 8 and 4
and 6 and 2 ? 8 and 9 and 6 ?
34. If you add 3 to 3, and 4 to 4, and 5 to 5, and add
those sums together, what number will you have ?
35. If to 10 you add 6, 7, and 9, how much will you
then have ?
LESSON V.
NOTE. The numbers in the following examples should be
added aloud; and if preferred, the "Four step method" can be
applied to all the questions.
Take for example the second question ; Four and 5 and 7 are
how many? As many as the sum of 4 and 5 and 7; 4 and 5
are 9, 9 and 7 are 1C ; therefore, 4 and 5 and 7 are 16.
1. Three and 6 and 4 are how many ?
2. Four and 5 and 7 are how many ?
3. Five and 6 and 2 are how many ?
4. Six and 4 and 5 are how many?
18 RAY'S INTELLECTUAL ARITHMETIC.
5. Seven and 3 and 5 and 2 are how many ?
6. Eight and 2 and 3 and 4 are how many ?
7. Nine and 2 and 4 and 3 are how many ?
8. Two and 9 and 5 and 4 are how many ?
9. Three and 9 and 5 and 4 are how many ?
10. Four and 8 and 3 and 5 and 2 and 6 and 3 and
one are how many ?
11. Five and 7 and 2 and 3 and 4 and 6 and 5 and 2 are
how many ?
12. Two and 4 and 3 and 5 and 6 and 2 and 7 and 4 are
how many ?
13. Three and 2 and 4 and 5 and 4 and 6 and 3 and
seven and 5 are how many ?
14. Four and 3 and 5 and 7 and 6 and 8 and 2 and 4 are
how many ?
15. Four and 9 and 3 and 5 and 6 and 7 and 8 and 9 are
how many ?
16. Five and 8 and 5 and 8 and 5 and 8 and 5 and 8 are
how many ?
17. Six and 8 and 7 and 3 and 5 and 4 and 7 and 1 and
nine are how many ?
18. Seven and 9 and 5 and 4 and 6 and 3 and 8 and
five and 9 are how many ?
19. Eight and 7 and 6 and 5 and 4 and 9 and 3 and
seven and 8 are how many ?
20. Nine and 6 and 7 and 4 and 5 and 3 and 8 and
two and 9 are how many ?
21. Seven and 6 and 5 and 8 and 7 and 9 and 8 and
four and 9 and 8 are how many ?
22. Nine and 8 and 7 and 5 and 8 and 9 and 5 and
four and 7 and 3 and 9 and 8 are how many ?
23. Twelve and 11 and 7 and 4 and 9 are how many ?
24. Thirteen and 10 and 8 and 6 and 4 and 10 are how
many?
ADDITION. 19
25. Fourteen and 16 and 7 and 5 and 9 and 8 and 9 and
Bix and 4 are how many ?
26. James gave 7 cents for apples, and 8 cents for
peaches : how many cents did he spend ?
27. Seven dollars, and 5 dollars, and 3 dollars, are how
many dollars ?
28. David had 11 books ; he bought 7 more, and his
brother gave him 5 ; how many had he then ?
29. A man gave 13 dollars for a cart, 6 for a plow, and
5 for a harrow : how many dollars did he spend ?
30. James has 8 marbles in one pocket, 5 in another^
6 in another, and 7 in another : how many in all ?
31. If a pound of butter cost 18 cents, of lard 7 cents,
and of cheese 9 cents, how much will all cost?
32. A man owes to one person 8 dollars, to another
5 dollars, to another 3 dollars, and to another 7 dollars:
how much does he owe ?
33. A boy gave 19 cents for a spelling-book, 8 cents
for a slate, and 6 cents for pencils : how many cents did
he spend?
34. A drover bought hogs as follows : of one man 17,
of another 9, of another 7, of another 8 : how many did
he buy ?
35. A little girl gave 10 cents for thread, 7 cents for
pins, 6 cents for needles, and 9 cents for tape : how
many cents did she spend ?
36. William has 7 cents, Thomas 10 cents, David 9
cents, and Moses 8 cents: if the other boys give their
cents to Moses, how many will he have?
37. The age of Thomas is 8 years ; Frank, 5 years ;
and William is as old as both together : what "is the sum
of all their ages?
38. Joseph has 4 marbles, William has 2, and David
has twice as many as Joseph : how many do they all
have?
20 RAY'S INTELLECTUAL ARITHMETIC.
39. Begin with 2, and count one hundred by adding
2 successively. Thus, 2 and 2 are 4, and 2 are 6, and 2
are 8, and 2 are 10, and so on.
40. Begin with 3, and count ninety-nine by adding 3
successively. Thus, 3 and 3 are 6, and 3 are 9, and 3
are 12, and so on.
41. Begin with 4, and count one hundred by adding
four successively.
42. Begin with 5, and count one hundred by adding
five successively.
43. Begin with 6, and count one hundred and two by
adding 6 successively.
44. Begin wit 1 7, and count ninety-eight by adding
seven successive^ .
45. Begin with 8, and count one hundred and four by
adding 4 successively.
46. Begin with 9, and count ninety-nine by adding
nine successively.
47. Begin with 1, and count one hundred by adding
three successively.
48. Begin with 3, and count one hundred and three
by adding 4 successively.
49. Begin with 2, and count one hundred and two by
adding 5 successively.
50. Begin with 5, and count one hundred and seven
by adding 6 successively.
51. Begin with 6, and count one hundred and four by
adding 7 successively.
52. Begin with 7, and count one hundred and three by
adding 8 successively.
53. Begin with 8, and count one hundred and seven by
adding 9 successively.
54. If an apple cost 3 cents, and an orange cost 2 cents
more than an apple, what will be the cost of 2 apples
and one orange?
ADDITION. 21
55. James has 5 marbles ; Henry, 2 more than James ;
and Samuel, as many as both James and Henry : how
many have all ?
56. Mary had a certain number of peaches ; she gave
5 to her sister, 7 to her brother, and then had 10 re-
maining : how many had she ?
57. A boy bought a sled for 20 cents, and paid 5 cents
for repairing it: what was it then worth?
53. If a hat cost 4 dollars, and a coat as much as
three hats, what do they both cost?
59. How many times must 10 be added to fifteen, to
make fifty-five?
60. A boy's father gave him 5 cents ; his mother, one
cent more than his father ; and his brother, two cents
more than his father: how many did he then have ?
61. What is the exercise of putting numbers together
called? Ans. Addition.
62. What is simple Addition?
Ans. Simple Addition is finding the SUM of two or
more numbers of the same denomination.
63. What is meant by same denomination f
Ans. Same denomination means all of the same name;
that iSj all pounds, or all dollars, &c.
SECTION III. SUBTRACTION.
LESSON I.
1. James had 2 apples, and gave 1 to his brother:
how many had he left? Ans. 1. Why? Because 1 and
I are 2.
2. Then 1 from 2 leaves how many ?
3. Jeseph had 3 apples and lost 1 : how many had lia
Jeft ? Ans. 2. Why ? Because 1 and 2 are 3.
2d Bk. 2
22 KAY'S INTELLECTUAL ARITHMETIC,
4. Then 1 from 3 leaves how many?
5. Thomas had 4 cents, and gave 1 to Frank: hoi*
many had he left? Why?
6. Then 1 from 4 leaves how many ?
7. One from 5 leaves how many? From 6? 7? 8?
9?*10?
8. John had 4 cents and gave his sister 2 : how many
had he left? Why?
9. Then 2 from 4 leaves how many ?
10. James had 5 apples, and gave his brother 2 : how
many had he left ? Why ?
11. Then 2 from 5 leaves how many?
12. Two from 6 leaves how many? From 7? 8? 9?
10? 11?
13. Thomas had 5 cents and lost 3 : how many had he
left? Ans.2. Why?
14. Then 3 from 5 leaves how many ?
15. Three from 6 leaves how many? From 7? 8?
9? 10? 11? 12?
16. Joseph had 9 marbles- and lost 4: how many had
he left ? Why ?
17. Then 4 from 9 leaves how many ?
18. Four from 10 leaves how many? From 11 ? 12?
13? 14? 15?
19. William had 10 apples and gave Joseph 5: how
many had he left ? Why ?
20. Then 5 from 10 leaves how many ?
21. Five from 11 leaves how many ? From 12? 13?
14? 15? 16?
22. James had 11 marbles and lost 6 : how many had
he left? Why?
23. Then 6 from 11 leaves how many ?
SUBTRACTION. 23
24. Six from 12 leaves how many? From 13? 14?
15? 16? 17? 18?
25. William had 12 cents and lost 7 : how many had
he left? Why?
26. Then 7 from 12 leaves how many?
27. Seven from 13 leaves how many? From 14? 15?
16? 17? 18? 19?
28. James had 13 apples and gave his sister 8 : how
many had he left? Why?
29. Then 8 from 13 leaves how many?
30. Eight from 14 leaves how many ? From 15 ? 16?
17? 18? 19? 20?
31. Thomas had 13 apples and gave his sister 9: how
many had he left ? Why?
32. Then 9 from 13 leaves how many?
33. Nine from 14 leaves how many? From 15? 16?
i7? 18? 19? 20?
34. Henry had 14 cents and lost 5 : how many had he
remaining ?
35. Mary is 15 years old, and Anna is 8 : how much
older is Mary than Anna ?
36. Sold a load of corn for 17 dollars; received for it
a barrel of flour worth 6 dollars, and the rest in money :
how much money did I receive ?
37. A boy had 18 marbles and lost 10 : how many had
he then ?
38. Nineteen is 11 more than what number ?
39. Went shopping with 20 dollars ; spent 10 dollars in
one store, and 5 in another : how much had I left ?
In*
mer edition, will obviate all confusion.
24
RAY'S INTELLECTUAL ARITHMETIC.
LESSON II. SUBTRACTION TABLE.
1 from
1
leaves
2 from 2
eaves
3
from
3
leaves
1 from
2
leaves
1
2 from 3
eaves
1
3
from
4
leavea
1
1 from
3
leaves
2
2 from 4
eaves
2
3
from
5
leaves
2
1 from
4
leaves
3
2 from 5
eaves
3
3
from
6
leaves
3
1 from
5
leaves
4
2 from 6
eaves
4
3
from
7
leaves
4
1 from
6
leaves
5
2 from 7
eaves
5
3
from
8
leaves
5
1 from
7
leaves
6
2 from g
eaves
6
3
from
9
leaves
6
1 from
8
leaves
7
2 from 9
eaves
7
3
from
10
leaves
7
1 from
9
leaves
8
2 from 10
leaves
8
3
from
11
leaves
8
1 from
10
leaves
9
2 from 11
leaves
9
3
from
12
leaves
9
1 from
11
leaves
10
2 from 12
leaves
10
3
from
13
leaves
10
1 from
12
leaves
11
2 from 13
leaves
11
3
from
14
leaves
11
4 from
4
leaves
5 from 5
leaves
6
from
6
leaves
4 from
5
leaves
1
5 from 6
leaves
1
6
from
7
leaves
1
4 from
6
leaves
2
5 from 7
leaves
2
6
from
8
leaves
2
4 from
7
leaves
3
5 from 8
leaves
3
6
from
9
leaves
3
4 from
8
leaves
4
5 from 9
leaves
4
6
from
10
leaves
4
4 from
9
leaves
5
5 from 10
leaves
5
6
from
11
leaves
5
4 from
10
leaves
6
5 from 11
leaves
6
G
from
12
leaves
6
4 from
11
leaves
7
5 from 12
leaves
7
6
from
13
leaves
7
4 from
12
leaves
8
5 fr 01 * 13
leaves
8
6
from
14
leaves
8
4 from
13
leaves
9
5 from 14
leaves
9
6
from
15
leaves
9
4 from
14
leaves
10
5 from 15
leaves
10
6
from
16
leaves
10
4 from
15
leaves
11
5 from 16
leaves
11
6
from
17
leaves
11
7 from
7
leaves
8-
irom O
leaves
9
from
9
leaves
7 from
8
leaves
1
8 from 9
leaves
1 )
9
from
10
leaves
1
7 from
9
leaves
2
8 from 10
leaves
2
9
from
11
leaves
2
7 from
10
leaves
3
8 from 11
leaves
3
9
from
12
leaves
3
7 from
11
leaves
4
8 from 12
leaves
4
9
from
13
leaves
4
7 from
12
leaves
5
8 from 13
leaves
5
9
from
14
leaves
5
7 from
13
leaves
6
8 from 14
leaves
6
9
from
15
leaves
6
7 from
14
leaves
7
8 from 15
leaves
7
9
from
16
leaves
7
7 from
15
leaves
8
8 from 16
leaves
8
9
from
17
leaves
8
7 from
16
leaves
9
8 from 17
leaves
9
9
from
18
leaves
9
7 from
17
leaves
10
8 from 18
leaves
10
9
from
19
leaves
10
7 from
18
leaves
11
8 from 19
leaves
11
9
from
20
leaves
11
SUBTRACTION. 25
To TEACHERS. Instead of requiring pupils to recite the Sub-
traction Table regularly, some of the best instructors omit it, and
connect the exercises with addition, thus:
1 and 2 are 3 2 and 3 are 5
2 and 1 are 3 3 and 2 are 5
1 from 3 leaves 2 2 from 5 leaves 3
2 from 3 leaves 1 3 from 5 leaves 2
In case the Table should not be omitted, the above exercises
tfill be found very profitable.
1. Two from 7 leaves how many?
2 from 12?
3 from 8?
3 from 10?
4 from 9 ?
4 from 12 ?
4 from 13 ?
5 from 14?
5 from 17?
6 from 11 ?
6 from 15?
6 from 18?
7 from 11 ?
7 from 15 ?
7 from 17?
7 from 19?
8 from 13?
8 from 15 ?
8 from 17 ?
9 from 13?
9 from 15?
9 from 17?
2. What number must be added to 8 to make 10 ?
To 12 to make 15? To 9 to make 15?
To 17 to make 20? To 16 to make 20?
To 14 to make 19? To 19 to make 21?
To 13 to make 16? To 12 to make 20?
To 11 to make 20 ? To 13 to make 20 ?
LESSON III.
1. A boy gave 9 cents for a spelling-book, worth
only 7 cenis : how much did he pay for it more than it
was worth? Why?
2. A inan having 16 dollars lost 12 : how many had
he left? Why?
3. Bought a book for 12 cents, and a top for 7 cents:
how much did the book cost more than the top ?
26 RAY'S INTELLECTUAL AKITHMETIC.
4. Thomas had 18 cents given him by two boys ; one
gave 9 : how many did the other give ?
5. Bought a book for 14 cents, and gave the shop-
keeper 20 cents : how much change did he return me ?
6. William has 19 apples ; in one pocket he has 15;
how many are in the other ?
7. A man has 25 miles to travel : when he has gone
nineteen, how many will he still have to travel ?
8. A boy gave 24 cents for a book, and sold it for
sixteen cents : how much did he lose ?
9. James had 24 marbles; he gave 19 to his brother:
how many had he left ?
10. A man bought a horse for 19 dollars, and sold
him for 27 dollars : how much did he gain ?
11. A man owing 26 dollars, paid 18 : how many did
he then owe ?
12. Frank had 26 cents given him by William and
Thomas. William gave him 17 : how many did Thomas
give ? How many more did William give than Thomas ?
13. If you had 10 apples, and should give 2 to John,
and 6 to your sister, how many would you have left?
14. Abel had 36 cents, and his mother gave him
pnough to make 40 : how many did she give him ?
15. George had 40 marbles; he lost 20, and found
five : how many did he then have ?
16. A man had 100 barrels of flour; he sold 50, and
afterward bought 10 : how many did he then have?
17. A farmer had 35 bushels of grain ; a part having
been wasted, he found there were but 22 bushels remain-
ing : how much was wasted ?
18. John's father is 36 years old ; John is 12 : in how
paany years will he be as old as his father now is ?
19. I had 65 cents ; spent 20 cents for a book and
ten for a slate : how many had I left ?
SUBTRACTION. 27
. If you take 10 from the sum of two numbers there
be 8 left : what is their sum ?
21. If you take 16 from the difference of two numbers
there will remain 12 : what is their difference ?
22. The sum of two numbers is 20 : what number musl
be added to make their sum 30 ?
23. The sum of two numbers is 16 more -than thei/
difference ; if their difference is 4, what is their sum ?
24. The greater of two numbers is 12, and their differ-
ence 5 : what is the less?
25. The sum of two numbers is 21 ; the less number is
eight : what is the greater ?
26. If you take one number from another, what is the
operation called ? Ans. Subtraction.
27. If you add one number to another, what is the
operation called? Ans. Addition.
28. In what respect do Addition and Subtraction differ?
Ans. One is exactly the reverse or opposite of the other.
29. When you take one number from another, what
do you call that which is left?
Ans. Difference or Remainder.
30. The remainder and less number, added together
are always equal to what ? Ans. The greater number.
SECTION IV. REVIEW.
LESSON I.
1. James had 13 marbles; he gave 2 to Henry, and
three to Thomas : how many had he left ?
ANALYSIS. He bad as many left as the difference between
13 marbles, and the sum of 2 marbles and 3 marbles ; the sum
of 2 marbles and 3 marbles is 5 marbles; 5 from 13 leaves 8
marbles ; therefore, if James had 13 marbles, and gave 2 to
Henry and 3 to Thomas, he had 8 left.
28 KAY'S INTELLECTUAL ARITHMETIC.
2. A merchant had 40 barrels of flour; he sold to
one man 9, to another 21 : how many had 2le left?
3. On Christmas day, William had 36 cents given
him; ho spent 6 cents for apples, 9 cents for cakes,
and 10 cents for candy : how many had he left?
4. A man paid 30 dollars for a horse, the keeping
cost 9 dollars, and he sold him for 29 dollars : how many
dollars did he lose ?
5. A man having 34 dollars, bought a barrel of
molasses for 15 dollars, and a bag of coffee for 10 dol-
lars : how many dollars had he left ?
6. A grocer bought some oranges for 9 dollars, some
lemons for 7 dollars, some prunes for 5 dollars, and some
figs for 9 dollars, and then sold them for 41 dollars:
how much did he gain ?
7. A lady bought a comb for 25 cents, some pins for
10 cents, tape for 7 cents, thread for 6 cents, and toy
books for 5 cents ; she gave 60 cents to the shopkeeper :
how much change ougjht she to receive ?
a o
8. Two boys commenced playing marbles ; each had
18 when they began ; when they quit, one had 25 : how
many had the other ? *
9. Thomas has 7 marbles, David 5, and Moses 11;
how many have they altogether ? How many have Moses
and D?vid together more than Thomas?
10. Three boys commence playing marbles : Thomas
had 20, David 10, and Moses 4; when they quit, David
had 6 and Moses 8 : how many had Thomas ?
11. A farmer had 24 sheep: 9 of them were killed by
wolves, 5 of them were stolen, and 6 he sold : how many
had he left?
12. A grocer bought sugar for 12 dollars, flour for
six dollars, and coffee for 5 dollars ; he sold the whole
for 30 dollars : how much did he make ?
13. A lady had 50 cents ; she spent 25 cents for butter,
and 10 cents for eggs : how much had she left?
REVIEW. 29
14. A man is indebted to A, 5 dollars; to B, 6 dollars;
and to C, 10 dollars: he has cash to the amount of 20
dollars, and goods valued at 10 dollars : should he pay
his debts, how much would he be worth ?
LESSON II.
1. Four and 3, less 2, are how many?
ANALYSIS. As many as the difference between 2, and Hit
sum of 4 and 3 : 4 and 3 are 7 j 7 less 2 are 5 ; therefore, 4
and 3 less 2 are 5.
2. Five and 6 and 2, less 8, are how many?
3o Seven and 4 and 3, less 5, are how many ?
4. Eight and 5 and 4, less 3, are how many ?
5. Two and 3 and 5, less 7, are how many ?
6. Six and 4 and 3, less 6, are how many ?
7. Six and 3 and 5, less 7, are how many ?
8. Seven and 4 and 6, less 5, are how many ?
9. Eight and 5 and 4 and 6, less 5, are how many ?
10. Four and 7 and 6 and 5, less 8, are how many ?
11. Five and 8 and 4 and 9, less 7, are how many ?
12. Seven and 5 and 8 and 5, less 6, are how many ?
13. Eight and 4 and 3 and 9, less 5, are how many ?
14. Nine and 5 and 8 and 3 and 1, less 7, are how
many?
15. Eight and 6 and 5 and 2 and 4 and 3, less 4, are
how many?
16. Seven and 4 and 8 and 5 and 6 and 2 and 5, less
eight, are how many?
17. Nine and 7 and 5 and 3 and 6 and 8 and 7, less
six, are how many?
18. Eleven and 4 and 6 and 5 and 7 and 9, less 3, are
how many ?
80 RAY'S INTELLECTUAL ARITHMETIC.
19. Twelve and 5 and 7 and 6 and S and 3 and 4,
less 7, are how many ?
20. Twelve and 7 and 6 and 4 and 5 and 8 and 2,
less 7, are how many ?
21. Eleven and 6 and 5 and 3 and 6 and 8 and 6,
less 9, are how many?
22. Eleven and 7 and 5 and 4 and 5 and 9 and 6,
less 8, are how many ?
23. Eleven and 9 and 8 and 7 and 6 and 5 and 4,
less 3, are how many ?
24. Twelve and 9 and 7 and 6 and 5 and 3, less 7,
are how many ?
25. Thirteen and 4 and 5 and 7 and 8 and 6 and 9,
less 5, are how many?
26. Thirteen and 7 and 3 and 9 and 8 and 6 and 5,
less 8, are how many ?
27. Eighteen and 9 and 10 and 8 and 7, less 9, are
how many ?
28. Twenty-one and 5 and 6 and 7 and 8 and 9 and
10 and 9, less 8, are how many?
29. Seventy, less 10 and 9 and 8 and 6 and 5 and 4 and
3 and 2 and 4 and 5 and 6, are how many ?
LESSON III.
1. Henry had 24 cents, and spent all but 15 : how
many did he spend ?
2. A man bought a cask of wine containing 27 gal-
lons ; after selling 10 gallons he found there were but 9
gallons remaining, the rest having leaked out : how much
did he lose ?
3. If from 20 you take 12 less 3, how many will
remain ?
4. If from the sum of 19 and 10, you take the differ-
ence between 17 and 10, what will be left?
REVIEW. 3j
5. A man owed 60 dollars : lie paid at one time 20
dollars, and at another 30 dollars: he afterward borrowed
5 dollars : how much does he still owe ?
6. A maft paid 38 dollars for a horse, and 20 for a
colt: he afterward sold the colt for 10 dollars, and the
horse for 65 : how much did he make by the transaction ?
7. Twenty-four less 8, and 12 less 5, are together how
much less than 25 ?
8. Engaged to do a piece of work for 60 dollars : had
an assistant 25 days at a dollar a day, and paid 20 dollars
for materials : how much did I clear ?
9. If from the sum of 8 and 9 and 10 and 11, you
take the sum of 4 and 5 and 6 and 7, what will you have
remaining ?
10. A jeweler bought a watch for 40 dollars, a chain
for 15 dollars, and a key for 3 dollars : he sold them for
63 dollars : what did he gain ?
11. A drover bought sheep as follows : of one man 10 ;
of another, 12 ; of another, 5 ; of another, 3 : he sold at
one time 15 ; and at another, 5 : how many were left ?
12. A gentleman having 40 dollars, purchased a suit of
clothes : his pants cost 7 dollars ; vest, 5 dollars ; coat, 25
dollars : how much had he left?
13. What number must be added to 25, to make a
gum 14 less than 45 ?
14. What number must be taken from 62, to give a
result which shall be 12 more than 45 ?
15. If from the sum of 25 and 10 and 12, you take the
difference between 28 and 19, what will remain ?
16. A man bought a horse for 40 dollars : and after
paying 15 dollars for keeping him, sold him for 75 dol-
lars : how much did he make ?
17. A gentleman engaged in trade with 75 dollars :
after losing at one time 10 dollars, and at another 5, he
gained 20 dollars : how much did he then have ?
32 RAY'S INTELLECTUAL ARITHMETIC.
SECTION V. MULTIPLICATION.
LESSON I.
1. A boy gave 2 cents for one lemon, and 2 cents foi
another : how many cents did he give for both ?
2. How many, then, are 2 times 2? Why? Because
2 and 2 are 4.
3. A boy gave 3 cents for one peach and 3 cents for
another : how many cents did he give for both ?
4. How many, then, are 2 times 3? Why?
5. At 4 cents apiece, what will 2 pears cost?
6. How many are 2 times 4? 4 times 2? Why?
7. At 3 cents apiece, what will 3 peaches cost ?
8. How many are 3 times 3? Why?
9. At 3 cents apiece, what will 4 apples cost?
10. How many are 4 times 3 ? Why ?
11. At 3 cents apiece, what will 5 pears cost? Hov
many are 5 times 3? 3 times 5 ?
12. At 4 cents apiece, what will 4 lemons cost? How
many are 4 times 4 ?
13. At 5 dollars a yard, what will 4 yards of cloth
cost? Htfw many are 4 times 5 ? 5 times 4?
14. At 6 dollars a barrel, what will 4 barrels of flour
cost? How many are 4 times 6 ? 6 times 4?
15. At 5 cents apiece, what will 5 oranges cost? How
many are 5 times 5 ?
16. At 6 cents a yard, what will 5 yards of tape cost?
How many are 5 times 6 ? 6 times 5 ?
17. At 6 cents apiece, what will 6 oranges cost? How
many are 6 times 6 ?
18. At 7 cents a yard, what will 2 yards of tape cost?
How many are 2 times 7 ? 7 times 2 ?
MULTIPLICATION, 33
19. At 7 cents apiece, what will 3 lemons cost? How
many are 3 times 7 ? 7 times 3 ?
20. At 7 cents apiece, what will 4 oranges cost? How
many are 4 times 7 ? 7 times 4 ?
21. If 1 marble is worth 7 apples, how many apples are
worth 5 maizes ? How many are 5 times 7?
22. If 1 peach is worth 8 apples, how many apples are
worth 2 peaches ? How many are 2 times 8 ?
23. If 1 orange cost 8 cents, how many cents will
three oranges cost? How many are 3 times 8?
24. If 1 barrel of flour cost 8 dollars, how many dollars
will 4 barrels cost? How many are 4 times 8?
25. If 1 orange is worth 8 apples, how many apples
are 5 oranges worth ? How many are 5 times 8 ?
26. At 9 cents apiece, what will 2 oranges cost? How
many are 2 times 9 ? 9 times 2 ?
27. At 9 cents a yard, what. will 3 yards of ribbon cost?
flow many are 3 times 9 ? 9 times 3 ?
28. At 10 cents a quart, what will 3 quarts of chest-
nuts cost? How many are 3 times 10?
29. If 1 yard of cloth cost 1 1 dollars, what will 3 yards
cost? How many are 3 times 11?
30. At 10 cents a bunch, what will 4 bunches of grapes
cost? How many are 4 times 10?
31. How many are 5 times 10? 10 times 5 ?
SUGGESTION. The exercises given under the subtraction table
can be extended to multiplication. One great use of such exer-
cises is to aw-iken an interest in the recitation.
Take, for example, the numbers 6 and 6. Five and 6 are 11
6 and 5 are 11 ; 5 from 11 leaves 6 ; 6 from 11 leaves 5; 5 times
$ are 30; 6 times 5 are 30,
RAY'S INTELLECTUAL ARITHMETIC.
LESSON II. MULTIPLICATION TABLE.
1 time 1 is 1 ,
1 time 2 is 2
1 time 3 is 3
2 times 1 are 2
2 times 2 are 4
2 times 3 are 6
3 times 1 are 3
3 times 2 are 6
3 times 3 are 9
4 times 1 are 4
4 times 2 are 8
4 times 3 are 12
5 times 1 are 5
5 times 2 are 10
5 times 3 are 15
6 times 1 are 6
6 times 2 are 12
6 times 3 are 18
7 times 1 are 7
7 times 2 are 14
7 times 3 are 21
8 times 1 are 8
8 times 2 are 16
8 times 3 are 24
9 times 1 are 9
9 times 2 are 18
9 times 3 are 27
10 times 1 are 10
10 times 2 are 20
10 times 3 are 30
11 times 1 are H
11 times 2 are 22
11 times 3 are 33
12 times 1 are 12
12 times 2 are 24
12 times 3 are 36
1 time 4 is 4
1 time 5 is 5 j 1 time 6 is 6
2 times 4 are 8
2 times 5 are 10
2 times 6 are 12
3 times 4 are 12
3 times 5 are 15
3 times 6 are 18
4 times 4 are 16
4 times 5 are 20
4 times 6 ar& 24
5 times 4 are 20
5 times 5 are 25
5 times 6 are 30
6 times 4 are 24
6 times 5 are 30
6 times 6 are 36
7 times 4 are 28
7 times 5 are 35
7 times 6 are 42
8 times 4 are 32
8 times 5 are 40
8 times 6 are 48
9 timeg 4 are 36
9 times 5 are 45
9 times 6 are 54
10 times 4 are 40
10 times 5 are 50
10 times 6 are 60
11 times 4 are 44
11 times 5 are 55
11 times 6 are 66
12 times 4 are 48
12 times 5 are 60
12 times 6 are 72
1 time 7 is 7
1 time 8 is 8
1 time 9 is 9
2 times 7 are 14
2 times 8 are 16
2 times 9 are 18
3 times 7 are 21
3 times 8 are 24
3 times 9 are 27
4 times 7 are 28
4 times 8 are 32
4 times 9 are 36
5 times 7 are 35
5 times 8 are 40
5 times 9 are 45
6 times 7 are 42
6 times 8 are 48
6 times 9 are 54
7 times 7 are 49
7 times 8 aro 56
7 times 9 are 63
. 8 times 7 are 56
8 times 8 are 64
8 times 9 are 72
9 times 7 are 63
9 times 8 are 72
9 times 9 are 81
10 times 7 are 70
10 times 8 are 80
10 times 9 are 90
11 times 7 are 77
11 times 8 are 88
11 times 9 are 99
12 times 7 are 84
12 times 8 are 96
12 times 9 are 108
MULTIPLICATION.
35
1 time 10 is 10
1 time 11 is 11
1 time 12 is 12
2 times 10 are 20
2 times 11 are 22
2 times 12 are 24
3 times 10 are 30
3 times 11 are 33
3 times 12 are 36
4 times 10 are 40
4 times 11 are 44
4 times 12 are 48
5 times 10 are 50
5 times 11 are 55
5 times 12 are 60
6 times 10 are 60
6 times 11 are 66
6 times 12 are 72
7 times 10 are 70
7 times 1 1 are 77
7 times 12 are 84
8 times 1Q are 80
8 times 11 are 88
8 times 12 are 96
9 times 10 are 90
9 times 11 are 99
9 times 12 are 108
10 times 10 are 100
10 times 11 are HO
10 times 12 are 120
11 times 10 are HO
11 times 11 are 121
11 times 12 are 132
12 times 10 are 120
12 times 11 are 132
12 times 12 are 144
EXERCISES ON THE TABLE.
7 times 4? 9 times
7 times 5? 6 times 7?
How many are 4 times 7?
3? 6 times 5? 8 times 2?
? times 8?
Eight times 6 ? 6 times 9 ? 9 times 7 ? 8 times 5 ?
7 times 3? 9 times 8? 7 times 7?
Nine times 6 ? 8 times 7 ? 7 times 6 ? 9 times 9 ? 7
times 9? 10 times 5? 4 times 11? 12 times 3? 8
times 8 ? 6 times 11 ? 7 times 10 ? 8 times 9 ?
How many are 2 times 2 ? 3 times 2 ? 4 times
2 ? 3 times 3 ? 5 times 2 ? 4 times 3 ? 7 times 2 ? 5
times 3 ? 4 times 4 ? 9 times 2 ?
Five times 4? 3 times 7? 11 times 2? 8 times 3?
5 times 5? 9 times 3? 7 times 4? 10 times 3? 8
times 4? 11 times 3?
Seven times 5 ? 9 times 4? 4 times 10? 7 times 6?
11 times 4? 9 times 5 ? 8 times 6 ?
Seven times 7 ? 10 times 5 ? 9 times 6 ? 11 times 5 ?
8 times 7? 5 times 12? 9 times 7? 8 times 8? 11 times 6?
Ten times 7 ? 9 times 8? 11 times 7? 8 times 10?
9 times 9? 7 times 12? 8 times 11? 9 times 10? 8
times 12? 11 times 9? 10 times 10 ?
36 BAY'S INTELLECTUAL ARITHMETIC,
LESSON III.
1. At 2 cents each, what will 7 oranges cost?
ANALYSIS. Seven oranges will cost 7 times as much as i
orange. If 1 orange cost 2 cents, 7 oranges will cost 7 times
2 cents, which are 14 cents; therefore, 7 oranges at 2 cents
each will cost 14 cents.
2. At 7 cents each, what will 3 melons cost?
3. At 6 cents a dozen, what cost 5 dozen apples?
4. At 6 cents a pound, what cost 7 pounds of beef?
NOTE. The dollar sign, $, will now be used in place of the
word dollar : thus, $-5, $(3 ; read 5 dollars, t> dollars.
5. At $6 a pound, what cost 8 pounds of opium?
6. At $3 a barrel, what cost 9 barrels of cider?
7. At $4 a pair, what cost 7 pairs of boots ?
8. At 8 cents a dozen, what cost 10 dozen pens?
9. What cost 6 yards of cloth at $7 a yard ?
10. What cost 8 barrels of flour at $5 a barrel?
11. If a man travel 7 miles an hour, how far will lie
travel in 8 hours ?
12. On a chessboard are 8 rows of squares, and 8
squares in each row: how many squares on the board?
13. An orchard has 11 rows of trees, and 7 trees in
each row : how many trees in the orchard?
14. In 1 cent are 10 milis ; how many mills are there
in 3 cents? In 4? In 5? 6? 7? 8?
15. In 1 pint are 4 gills ; how many gills are there in
2 pints? In 3? In 4? 5? 6? 7? 8?
16. In 1 bushel are 4 pecks ; how many pecks are
there in 2 bushels? In 3? In 4? 5? 6? 7 ?
17. In 1 peck are 8 quarts ; how many quarts ara
&ere in 2 pecks ? In 3? In 4? 5? 6? 7?
MULTIPLICATION. 37
18. In 1 bushel how many quarts? Why ?
19. What will 9 yards of cloth cost at $6 a yard?
20. What will 9 oranges cost at 8 cents each?
21. Two men start from the same place and travel in
apposite directions : one travels 2 miles an hour, the other
4 miles : how far will they be apart at the end of 1 hour?
At the end of 2 hours ? 3 hours ?
22. If 2 men can do a job of work in 3 days, how
many days will it take 1 man to do it ?
ANALYSIS. It will require 1 man twice as long as 2 men.
If it take 2 men 3 days, it will take 1 man twice 3 days^
which are 6 days; therefore, if 2 men do a job of work in
3 days, 1 man will do it in 6 days.
23. If 3 men can do a piece of work in 4 days, in how
many days can 1 man do it ?
24. If 4 men can do a piece of work in 6 days, in how
many days can 1 man do it?
25. If a quantity of bread serve 8 men 4 days, how
many days will it serve 1 man ?
26. If a man can earn $6 in 1 week, how many dol-
lars can he earn in 8 weeks ?
27. A person has a job of work which 6 men can ao
in 9 days ; but it is necessary to do it in one day : how
many men must be employed ?
28. If 2 barrels of cider last 6 persons 4 weeks, how
many weeks will it last 1 person ?
29. If $9 worth of provisions last 8 persons 11 days,
how many person^ will it last 1 day ?
30. If the interest of $1 is 6 cents a year, what will be
the interest for 2 years? For 3 .years? For 4? 5?
For 6? 7? 8? 9? 10?
31. I bought 6 barrels of apples at $2 a barrel, and 4
barrels of sugar at $11 a barrel : how much did they
both cost?
2d Bk. 3
38 HAY'S INTELLECTUAL ARITHMETIC.
32. Four and 4 are 8, and 4 are 12, and 4 are 16, and
4 are 20 : here we find that 4 taken five times makes 20.
What is this operation called? Ans. Addition.
33. When we say 5 times 4 make 20, what do we call
the operation? Ans. Multiplication.
34. How, then, would you define Multiplication ? Ans.
Multiplication is a short method of performing several ad-
ditions of the same number.
35. The number to be multiplied, 4, is called the mul-
tiplicand ; the number you multiply by, 5, the multiplier;
and the answer, 20, the product.
LESSON IV.
1. Bought 2 apples at 2 cents each, 2 pears at 3 cents
each, and an orange for 5 cents : what did they cost ?
2c Two men start from the same place and travel in
the same direction ; one, 5 miles an hour ; the other, 7
miles : how far will they be apart in 10 hours ?
3. If, in the above question, the men travel in opposite
directions, how far will they be apart in 12 hours?
4. A lady went shopping with $15; she bought 4 yards
of cloth at 82 a yard ; 2 pairs of gloves at $1 a pair ;
and a shawl for $2 : what did they cost, and how much
had she left?
5. A man bought 4 peaches at 5 cents each, 3 pears
at 3 cents each, and 2 pints of chestnuts at 5 cents a
pint : how much did they cost ?
6. What will be the sum of 3 and 9 and 7 ? less the
sum of 8 and 6 and 1 ?
7. If a man earn 5 shillings a day, and a boy 3 shil-
lings, how much will both earn in 7 days ?
8. A drover gave $10 and 7 sheep, valued at $4 *
head, for a cow and calf: how much did they cost?
MULTIPLICATION. 39
9. A merchant sold cloth at $7 a yard : a tailo* bought
of this cloth, at one time, 5 yards, and at another, 3 yards:
what was the amount of his bill ?
10 .Two brothers, Henry and Rufus, each received for
their work 3 dimes a day : how much did they both re-
ceive for 6 days' work ?
11. If 12 horses can be sustained in a pasture 10
months, how many horses will it feed 1 month?
12. What is 3 times the difference between 15, and tho
sum of 5 and 2 ?
13. The sum of two numbers is 23; the smaller is 11 :
what is 5 times the larger?
14. The difference between two numbers is 7 : if the
larger be 12, what will 8 times the smaller be ?
15. If a boy buy apples at 1 cent each, and sell them
for 3 cents each, what would he make if he purchase 10
cents' worth of apples?
16. George bought a book for 50 cents and sold it
for $1 : what would he have made, had he bought 2 books,
and sold them at the same rate as the first ?
17 .Albert has 5 times two marbles less than 50, and
Edward has 5 times two more than 50 : how many has
each ?
18. If $1 gain $3 in a year, what will $12 gain in
double the time ?'
19. A man bought a cask of wine containing 20 gallons,
at $1 a gal. ; 5 gal. having leaked out, he sold the remain-
der at $2 a gal. : how much did he make ?
20. Two men start from the same place, at the same
time, and tiavel the same way ; if they travel at the same
rate, how far will they be apart at the end of 10 hours?
If one goes 10 miles an hour, and the other 7, how fai
will they be apart in 7 hours ?
If they go in opposite directions, each at the rate of 5
miles per hour, how far will they be apart in 9 hours ?
40 RAY'S INTELLECTUAL ARITHMETIC,
21. A stage starts from a certain town, and travels at
the rate of 8 miles per hour : at the same time, another
starts from the same place, and travels in the same direc-
tion, 4 miles per hour : how far will they be apart at the
end of 12 hours ?
22. A grocer bought 10 pounds ot tea at 7 shillings
a pound ; after using 3 pounds, he sold the remainder
at 10 shillings a pound : how much did the 3 pounds
which he used cost him, in the end ?
23. Bought 6 bushels of corn at 5 dimes a bushel ;
sold 4 bushels at 6 dimes a bushel, and 2 bushels at 4
dimes a bushel : how much did I make ?
24. If an orange cost 5 cents, and an apple 2 cents,
what will 2 oranges and 4 apples cost ?
25. If pork is 8 cents, and beef 10 cents a pound, what
cost 7 pounds of pork and 6 pounds of beef?
26. If an orange cost 5 times as much as an apple, how
much more will 6 oranges cost than 25 apples, if an apple
is worth 1 cent?
27. If a pound of sugar cost 5 cents, and a pound of
coffee 3 times as much, less 3 cents, what will be the cost
of 3 pounds of sugar and 2 of coffee ?
*28 .Bought, at one time, 5 yards of muslin at 10 cents
a yard ; at another, 10 yards at 5 cents a yard : how
much did it all cost?
29. When salt is 4 cents a quart, and molasses 3 times
as much lacking 2 cents, what would be the cost of 3
quarts of molasses and 2 of salt ?
30. If a man earn $15 per week, and spend $11 a
week, how much will lae save in 3 weeks? How much
can he save in two months of 4 weeks each ?
* In-
Bier edition, will obviate all confusion.
DIVISION, 41
SECTION VI. DIVISION
LESSON I.
1. At 1 cent each, how many cakes can you buy fot
4 cents ? 1 in 4 how many times ? Why ? Ans. Be-
cause 4 times 1 are 4.
2. At 2 cents each, how many apples can you buy
for 4 cents ?
ANALYSIS. You can buy as many apples as 2 cents are
contained times in 4 cents ; 2 cents are contained in 4 cents
2 times ; therefore, at 2 cents each, you can buy 2 apples for
4 cents.
Two in 4 how many times ? Why ? Because 2 times
2 cure 4.
3. Among how many boys can 6 apples be divided,
giving to each boy two apples ? 2 in 6 how many times ?
Why ? Because 3 times 2 are 6.
4. At 2 cents each, how many apples can you buy
for 8 cents ? 2 in 8 how many times ? Why ?
5. At 3 cents each, how many peaches can you buy
for 6 cents ? 3 in 6 how many times ? Why ?
6. At 3 cents each, how many pears can you buy
for 9 cents ? 3 in 9 how many times ? Why ?
7. At 2 cents each, how many cakes can you buy
for 10 cents ? 2 in 10 how many times ? Why ?
8. At 2 cents each, how many marbles can you buy
for 14 cents ? 2 in 14 how many times ? Why ?
9. At 5 cents each, how many lemons can you buy
for 15 cents ? 5 in 15 how many times ? Why?
10. A boy has 16 marbles, and wishes to divide them
into piles of 2 each : how many piles will there be 1
How many twos in 16 ? Why ?
42 RAY'S INTELLECTUAL ARITHMETIC.
11. At 3 cents each, how many peaches can you buy
for 18 cents ? 3 in 18 how many times ?
12. At 5 cents each, how many oranges can you buy
for 20 cents ? 5 in 20 how many times ?
13. At $3 a yard, bow many yards of cloth can yov
buy for $21 ? 3 in 21 how many times ?
NOTE. The dollar sign, $, is used ie. place of the word dollar ;
thus, $4, $7 ; read 4 dollars, 7 dollars.
14. A lady spent 22 cents for tape, at 2 cents a yard:
how many yards did she buy ?
15. At 6 cents each, how many oranges can you buy
for 24 cents ? How many at 8 cents each ? 6 in 24 how
many times ? 8 in 24 how many times ?
16. In an orchard of 25 apple trees there are 5 rows;
how many trees in each row ?
ANALYSIS. One tree in each row requires 5 treec ; hence,
there will be as many rows, as 5 trees are contained times in
25 trees ; 5 trees in 25 trees, 5 times. Ans. 5 rows.
17. If a man can travel 3 miles in an hour, how many
hours will it take him to travel 27 miles ? 3 in 27 how
many times ?
18. A man gave 28 for sheep, at $4 a head : how many
did he buy ? 4 in 28 how many times ?
19. If you had 30 cents, how many marbles could you
buy at 3 cents each ? 3 in 30 how many times ?
20. There are 32 cents on a table, in 4 piles : how many
in eacn pile? 4 in 32 how many times?
21. In an orchard containing 35 apple trees, there are
5 rows : how many trees are there in each row ? 5 in 35
how many times ?
22. Six men receive $36 for a job of work: what is
each man's share ? 6 in 36 how many times ?
DIVISION.
43
23. Four quarts make 1 gallon : how many gallons in
36 quarts ? 4 in 36 how many times ?
24. If a man travel 10 miles in 1 hour, in how many
hours will he travel 40 miles ?
25. Forty-two cents were diyided equally among 6 boys;
how many cents did each boy receive ?
26. Forty-two are how many times 7 ?
27. If you divide 45 apples equally among 9 boys, how
many apples will each boy receive ?
LESSON II -DIVISION TABLE.
2 in 2 1 timo
3 in 3 1 .time
4 in 4 1 time
2 in 4 2 times
3 in 6 2 times
4 in 8 2 times
2 in 6 3 times
3 in 9 3 times
4 in 12 3 times
2 in 8 4 times
3 in 12 4 times
4 in 16 4 times
2 in 10 5 times
3 in 15 5 times
4 in 20 5 times
2 in 12 6 times
3 in 18 6 times
4 in 24 6 times
2 in 14 7 times
3 in 21 7 times
4 in 28 7 times
2 in 16 8 times
3 in 24 8 times
4 in 32 8 times
2 in 18 9 times
3 in' 27 9 times
4 in 36 9 times
2 in 20 10 times
3 in 30 10 times
4 in 40 10 times
2 i n 22 11 times
3 in 33 11 times
4 in 44 '11 times
2 in 24 12 times
3 in 36 12 times
4 in 48 12 times
5 in 5 1 timo
6 in 6 1 time
7 in 7 1 time
5 in 10 2 times
6 in 12 2 times
7 in 14 2 times
5 in 15 3 times
6 in 18 3 times
7 in 21 3 times
5 in 20 4 times
6 in 24 4 times
7 in 28 4 times
5 in 25 5 times
6 in 30 5 times
7 in 35 5 times
5 in 30 6 times
6 in 36 6 times
7 in 42 6 times
5 in 35 7 times
6 in 42 7 times
7 in 49 7 times
5 in 40 8 times
6 in 48 8 times
7 in 56 8 times
5 in 45 9" times
6 in 54 9 times
7 in 63 9 times
5 in 50 10 times
6 in 60 10 times
7 in 70 10 times
5 in 55 11 times*
6 m 66 11 times
7 in 77 11 times
5 in 60 12 times
6 in 72 12 times
7 in 84 12 timeq
44 .
BAY'S INTELLECTUAL ARITHMETIC.
8 in 8 1 time
9 in 91 time
10 in 10 1 time
8 in 16 2 times
9 in 18 2 times
10 in 20 2 times
8 in 24 3 times
9 in 27 3 times
10 in 30 3 timea
8 in 32 4 times
9 in 36 4 times
10 in 40 4 times
8 ia 40 5 times
9 in 45 5 times
10 in 50 5 times
8 in 48 6 times
9 in 54 6 times
10 in 60 6 times
8 in 56 7 times
9 in 63 7 times
10 in 70 7 times
8 in 64 8 times
9 in 72 8 times
10 in 80 8 times
8 in 72 9 times
9 in 81 9 times
10 in 90 9 times
8 in 80 10 times
9 in 90 10 times
10 m 100 10 times
8 in 88 11 times
9 in 99 11 times
10 in 110 11 times
8 in 96 12 times
9 in 108 12 times
10 in 120 12 times
11 in 11 1 time
11 in 55 5 times 11 in 99 9 times
11 in 22 2 times
11 n 66 6 times
11 in 110 10 times
11 in 33 3 times
11 n 77 7 timea
11 in 121 11 timos
11 in 44 4 times
11 n 88 8 times
11 in 132 12 times
12 in 12 1 time
12 n 60 5 times
12 in 108 9 times
12 in 24 2 times
12 n 72 6 times
12 in 120 10 timea
12 in 36 3 times
12 n 84 7 times
12 in 132 11 times
12 in 48 4 times
12 in 96 8 times
12 in 144 12 tiroes
NOTE. The four simple operations of Arithmetic may now be
combined in a single example : thus,
10 and 5 are 15 10 times 5 are 50
5 and 10 are 15 5 times 10 are 50
10 from 15 leaves 5 10 in 50 5 times
5 from 15 leaves 10 5 in 50 10 times
LESSON III.
1. Two in 12 how many times? 2 in 16? 2 in 24?
Sin 9? 3 in 15? 3 in 21 ? 3 in 27 ? 4 in 8 ? 4 in
20 ? 4 in 28 ? 4 in 36 ? 4 in 48 ?
2. Five in 15 how many times ? 5 -in 30 ? 5 in 45 ?
5 in 60 ? 6 in 18 ? 6 in 24 ? 6 in 36 ? 6 in 42 ? 6
in 54 ? 6 in 66 ?
DIVISION. 45
3. Seven in 14 how many times ? 7 in 28 ? 7 in
42? 7 in 56? 7 in 63 ? 7 in 84? 8 in 24? 8 in
40 ? 8 in 56 ? 8 in 72 ? 8 in 96 ?
4. Nine in 18 how many times ? 9 in 27 ? 9 in
45? 9 in 54? 9 in 63 ? 9 in 81 ? 9 in 108 ? 10 in
20? 10 in 60? 10 in 90 ? 10 in 100 ?
5. Eleven in 55 how many times? 11 in 77 ? 11
in 99? 11 in 110? 11 in 121 ? 12 in 24 ? 12 in 48 ?
12 in 60? 12 in 72? 12 in 96? 12 in 108? 12 in
120 ? 12 in 144 ?
6. If 12 peaches be divided equally among 3 children,
how many will each have ?
7. Four boys gave their sister 24 apples, each an equal
number : how many did each give ?
8. A mother divided 20 cents equally between her 2
little girls : how many did each receive ?
9. Five books cost 35 cents: how much is that
apiece ?
10. A man has $40 : if he spend $5 a week, how long
will it last ?
11. There are 3 feet in 1 yard: how manv yards are
there in 21 feet? In 27 feet? In 36 feet?
12. Four quarts make 1 gallon : how many gallons in
28 quarts ? In 16 ? In 32 ? In 36 ? 44 ? 48 ?
13. If 5 apples are worth 1 pear, how many pears are
worth 25 apples ? 35 apples ? 45 apples ?
14. If 6 pears are worth an orange, how many oranges
can you get for 30 pears ? For 42 pears? For 54 pears?
For 66 pears?
15. If 1 man do a piece of work in 42 days, how many
days will it take 7 men ?
16. If 1 man eat a certain quantity of provisions in 56
days, how many days will it last 7 men ?
17. If 1 pipe empty a cistern in 63 hours, in how many
hours will 9 pipes of the same size empty it ?
46 BAY'S INTELLECTUAL ARITHMETIC.
18. Eight quarts make a peck : how many pecks in 24
quarts ? In 40 ? In 56 ? In 72 ?
19. If hay is worth S9 a ton, how many tons can be
bought for 27 ? For $45 ? For 54 ? For 63 ?
20. Ten men bought a horse for 60 : how much did
each one pay?
21. If 11 ounces cost 88 cents: what cost 1 ounce?
22. A man paid 108 for 12 Saxony sheep : how inucJ*
was that apiece ?
23. In an orchard there are 120 trees in 10 rows : how
many trees in each row ?
24. A man earns 144 in 12 weeks : how much is that
a week ? How much a day, allowing 6 working days to
the week ?
25. If 6 men earn 84 in 7 days, how much do they
all earn in 1 day?
26. If 9 men earn 108 in 3 days, how much does i
man earn ? How much does each man earn in a day ?
27. If 2 from 6 leaves 4, 2 from 4 leaves 2, and 2 from
2 leaves 0, how many times is 2 taken from 6 ? Ans. 3.
What do you call the operation ? Subtraction.
28. Two is contained in 6, 3 times ; what do you call
the operation? Ans. Division.
29. How would you define Division? Ans. Division
is subtracting the same number several times; or, finding
koto many times one number is contained in another.
30. The number to be divided, 6, is called the dividend;
the number you divide by, 2, the divisor ; and the answer,
8, the quotient.
LESSON IV.
1. Twelve are how many times 2? 3? 4? 6?
2. Twenty -four are how many times 3? 6? 8? 12?
3. Seventy-two are how many times 12? 8? 0? 9?
DIVISION. 47
4. How many oranges at 5 cents each, must be given
for 10 pears at 2 cents each ?
5. A wheel is 10 feet in circumference ; what distance
Will it move in making one revolution ? how many revolu-
tions will it make in going 120 feet?
6. An orchard contains 10 rows of trees, and 7 trees
in a row ; if there were but 5 rows, how many trees would
there be in a row ?
7. I have three times as many marbles as the sum
of 1, 2, and 3, is contained times in 60 : how many
have I ?
8. Bought 6 hats at $5 apiece, and 4 yards of cloth
at $3 a yard ; gave in exchange flour at $6 a barrel : how
many barrels did it take ?
9. If a man gain 6 miles in 3 hours, how long will
it take to gain 24 miles ?
10. Two times 6 are contained how many times in the
sum of 36 and 12 ?
11. If 60 be divided by some number, the result will
be 10 : what is that number ?
12. I have a number in my mind which, divided by 3,
gives 2 times 6 : what is the number ?
13 .If I purchase lemons at the rate of 2 for 6 cents,
and sell 7 for 28 cents, how much do I gain ?
14. A man has a job of work which 9 men can per-
form in 2 days ; he desires to complete it in 3 days : how
many men must he employ ?
15. Five times the sum of two numbers is equal to 60;
if 7 is one of them, what is the other?
16. Henry has 6 dimes ; Thomas twice as many less 2;
and Samuel 3 times as many as Henry : how many have
they together ?
17 .If to the number of times 4 is contained in 12, you
add 3 ? and subtract the result from 9, what will remain ?
48 RAY'S INTELLECTUAL ARITHMETIC.
18. Five oranges were sold for 25 cents, and 10 cents
were gained : what did each cost ?
19. What number subtracted from 17, will leave double
the remainder that 5 from 9 leaves ?
20. A boy said that 10 taken from the number of
apples he had, left twice as great a remainder as the dif-
ference between 1 dozen and 8 : how many had he ?
21. A certain number multiplied by 10, is 5 less
than 45 : what is that number ?
22. If you multiply any number, 10, by any other
number, 5, and divide the product by the same num-
ber, 5, what will be the result ?
23. If 2 oranges are worth 5 apples, how many apples
are worth 12 oranges ?
ANALYSIS. As many times 5 apples, as 2 oranges are con*
tamed times in 12 oranges.
24. One man goes 10 miles while another goes 7 ;
when the first has gone 90 miles, how far will the
second have gone ?
25. James earns 8 cents while John earns 12; when
John has earned 60, how many has James earned ?
26 .George learns 5 lessons while Charles learns 4:
how many lessons will both have learned when Charle?
has learned 20 ?
27 . A man can earn 9 while a boy earns 5 : how
many dollars will both have earned when the man has
earned $36 ?
28. How many times will the hammer of a common
clock strike, from noon till 6 o'clock in the evening?
29. Two numbers added together make 30 ; if the
greater number was 5 more, and the less 3 more, what
would their sum then be ?
30. William can count 11 while James counts 7: low
many will James count while William is counting 77 ?
PARTS OF NUMBERS. 49
31. When flour was 5 cents a pound and sugar 11
cents, 10 pounds of flour and 5 pounds of sugar were
given for a box of eggs at 5 cents a dozen : how many
dozen were in the box ?
32. The sum of two numbers is 24 ; if they were both
equal to the greater, their sum would be 28 : what are
the numbers?
SEC. VII. PARTS OF NUMBERS.
LESSON I.
If a unit or whole thing is divided into TWO equal parts,
one of the parts is called ONE-HALF.
1. When an apple is divided into two equal parts, what
is one part called? Ans. One-half of an apple.
- 2. How many halves in 1 apple?- Ans. Two halves.
3. How many halves in 2 apples ? In 3 ? In 4 ?
[n5? In 6? In 7? In 8? In 9? In 10?
ANALYSIS. In 2 apples there are two times as many halves
as there are in 1 apple; but in 1 apple there are 2 halves, and
in 2 apples there will be 2 times 2 halves, or 4 halves ; there-
fore, there are 4 halves in 2 apples.
4. How many halves in 2 apples and one-half of an
apple ? In 3 apples and one-half? In 4 and one-half?
5. What do you mean by one -half of anything?
If any thing or any number is divided into THREE
equal parts, one of the parts is called ONE-THIRD of the
thing or number ; two parts are called TWO-THIRDS and
three parts, THREE-THIRDS, or the whole.
6. When an apple, or any thing is divided into three
equal parts, what is one part called ? What two parts ?
7. How many thirds in 1 apple ? In 2 ? In 3 ? In
4? In 5? In 6? In 7? In 8? In 9?
50 RAY'S INTELLECTUAL ARITHMETIC.
8. How many thirds in 1 apple and one-third of **
apple ? In one apple and two-thirds of an apple ?
9. How many thirds in two apples and one-third?
In 2 apples and two-thirds ?
10. What do you understand by one -third of any
thing? By two-thirds? Ans. That a whole thing has
been divided into 3 equal parts and 2 of those parts taken.
If any thing is divided into 4 equal parts, 1 of the parts
is called ONE-FOURTH ; two parts are called TWO-FOURTHS,
and so on.
When any thing is divided into 5 equal parts, 1 of the
parts is called ONE-FIFTH; two parts, TWO-FIFTHS; three
.parts, THREE-FIFTHS, and so on.
11. When an apple is divided into four equal parts, what
is one part called ? What two parts ? Ans. Two-fourths.
What three parts ? Am. Three- fourths.
12. How many fourths in 1 apple ? In 2 apples ? In
3 apples? In 4? In 5 ? In 6 ? In 7?
13. How many fourths in one apple and one-fourili?
In 1 and two -fourths ? In 1 and three-fourths ?
14. In 2 apples and one-fourth how many fourths?
In 2 apples and two-fourths ? In 2 and three -four ths ?
15. What are fourths often called ? Ans. Quarter*.
16. What do you understand by one-fourth of any
thing ? By two -four ths ? By three-fourths ?
17. When an apple is divided into 5 equal parts, what
is one part called? What are 2 parts called? What 3
parts? What 4 parts? Ans. Four-fifths.
18. How many fifths in 1 apple? In 2 ? In 3 ?
In 4 ? In 5 ? In 6 ? In 7 ? In 8 ? In 9 ?
19. How many fifths in 1 apple and 1 -fifth of an
apple ? In 1 and 2-fifths? 1 and 3-fifths? 1 and 4-fifths?
20. How many fifths in 2 apples and 1 -fifth ? In 2 and
2-fifths ? 2 and 3-fifths ? 2 and 4-fifths ?
PARTS OF NUMBERS. 51
21. What do you understand by 1 -fifth of any thing?
By 2-fifths ? By 3-fifths ?
22. When an apple is divided into 6 equal parts, what
is 1 part called ? What are 2 parts called ? What 3 parts ?
W^hat 4 parts ? What 5 parts ?
23. How many sixths in 1 apple ? In 2 apples ? In
3 apples? In 4? In 5 ? In 6? In 7? In 8?
24. How many sevenths in 1 apple ? In 2 apples ?
Ln3? In 4? In 5? In 6? In 7? In 8 ?
25. How many eighths in 1 apple? In 2? In 3 ?
In 4? In 5? In 6? In 7? In 8 ? In 9?
26. How many ninths in 1 apple? In 2 ? In 3 ?
In 4? In 5? In 6? In 7 ? In 8? In 9?
LESSON II.
1. If a yard of tape is worth 2 cents, what is one-
half of it worth ? Ans. One cent.
2. What is one -half of 2 cents ? Ans. One cent.
Why ? Ans. Because if you divide 2 cents into two equal
parts, one of the parts is 1 cent.
3. One is what part of 2 ? Ans. One is the half of 2.
4. If you can buy an apple for 2 cents, how many
can you buy for 3 cents? Ans. One apple and l-half.
5. Three are how many times 2 ? Ans. Once 2 and
l-half of 2.
6. Four are how many times 2 ?
7. If 2 cents will buy 1 yard of tape, how many
yards will 5 cents buy ?
8. Five are how many times 2? -4ns. Two times 2
and l-half of 2.
9. 6 are how many times 2 ? How many 2's ?
10. 7 are how many times 2 ? How many 2's ?
11. 8 are how many times 2? How many 2's?
52 RAY'S INTELLECTUAL ARITHMETIC.
12. 9 are how many times 2 ? How many 2's ?
13. 10 are how many times 2 ? How many 2's ?
14. If an apple is worth 3 cents, what is 1 -third of it
worth ? What are 2-thirds of it worth ?
15. What is 1-third of 3? What are 2-thirds of 3?
16. If an orange is worth 3 cents, what part of it will
1 cent buy ? What part will 2 cents buy ?
17. One is what part of 3? Ans. One is I -third of 3.
18. Two is what part of 3 ? Ans. Two is 2-thirds of 3,
or 2 times I -third of 3.
19. If a yard of cloth cost S3, how much can you buy
for 84 ? How much for $5 ?
20. Four are how many times 3? Ans. Once 3 and
1-third of 3.
21. Five are how many times 3 ? Ans. Once 3 and
2-thirds of 3.
22. 6 are how many times 3 ? 7 are how many 3's ?
23. 8 are how many times 3 ? 9 are how many 3's ?
24. 10 are how many times 3 ? 11 are how many 3's ?
25. If a lemon is worth 4 cents, what is 1 -fourth of it
worth ?
26. What is 1-fourth of 4 ? What are 2-fourths of 4?
What are 3-fourths of 4 ?
27. If you buy a yard of cloth for $4, what part of it
can you buy for 91 ? For 82 ? For 83 ?
28. What part of 4 is 1 ? Ans. 1-fourth of 4.
29. What part of 4 is 2 ? Ans. 2-fourths of 4.
30. -What part of 4 is 3 ? Am. ^-fourths of 4.
31. If a yard of tape cost 4 cents, how much can yoa
buy for 5 cents ? For 6 cents ? For 7 cents ?
32. Five are how many times 4? Ans. Once 4 and
one-fourth of 4.
33. Six are how many times 4 ? Ans. Once 4 and
two-fourths of 4.
PARTS OF NUMBERS. 53
34. Seven are how many times 4? Ans. Once 4 and
three-fourths of 4.
35. 8 are how many times 4 ? 9 are how many 4's ?
36. 10 are how many times 4 ? 11 are how many 4's ?
37. 12 are how many times 4 ? 13 are how many 4's ?
38. 14 are how many times 4 ? 15 are how many 4's ?
39. 16 are how many times 4 ? 17 are how many 4's f
40. 18 are how many times 4 ? 19 are how many 4's ?
41. 20 are how many times 4 ? 21 are how many 4's ?
42. If a melon is worth 5 cents, what is one-fifth of
it worth? What are two-fifths worth? What are three-
fifths worth ? What are four-fifths worth ?
43. What is 1-fifth of 5 ? 2-fifths of 5 ? 3-fifths of
57 4-fifthsof5?
44. If a melon is worth 5 cents, what part of it can
you buy for 1 cent ? For 2 cents ? For 3 ? For 4 ?
45. 1 is what part of 5 ? Ans. I -fifth of 5.
46. 2 is what part of 5 ? Ans. 2-fifths of 5.
47. 3 is what part of 5 ? Ans. ^-fifths of 5.
48. 4 is what part of 5 ? Ans. ^-fifths of 5.
49. If flour is worth $5 a barrel, how many barrels
can you buy for $6 ? For $7 ? For $8 ? For $9 ? For
810? For
50. 6 are how many times 5 ? Once 5 and 1-fifth of 5.
51. 7 are how many tinies-5 ? Once 5 and 2-fifthsof5.
52. 8 are how many times 5 ? Once 5 and 3-fifths of 5.
53. 9 are how many times 5 ? 10 are how many times 5?
54. 11 are how many times 5? 12 are how many times 5?
55. If 1 barrel of apples cost $6, what is 1 -sixth worth?
What 2-sixths ? What 3-sixths ? 4-sixths ? 5-sixths ?
56. What is 1-sixth of 6? What are 2-sixths of 6?
S-sixths of 6? 4-sixths of 6? 5-sixths of 6?
2d Bk. 4
54 KAY'S INTELLECTUAL ARITHMETIC.
57. One is what part of 6 ? 2 is what part of 6 ? 3 is?
what part of 6 ? 4 is what part of 6 ? 5 ?
58. How many yards of cloth at $6 a yard, can you
buy for 7? For 8? For $9? For $10? For 811 ?
For $12 ? For 813 ? For $14 ?
59. Seven are how many times 6 ? Ans. One time 6
&nd I -sixth of 6.
60. 8 are how many times 6 ? 8 are how many 6's ?
61. 9 are how many times 6 ? 10 are how many 6's?
62. 11 are how many times 6? 12 are how many 6's?
63. 13 are how many times 6 ? 14 are how many 6 T s ?
64. 15 are how many times 6 ? 16 are how many 6's?
65. At 7 cents a yard for ribbon, what is 1 -seventh
worth ? What are 2-sevenths worth ? What are 3-sev-
enths? 4-sevenths? 5-sevenths? 6-sevenths?
66. What is 1 -seventh of 7 ? What are 2-sevenths of
7 ? What are 3-sevenths ? 4-sevenths ? 5-sevenths ?
6-sevenths ?
67. One is what part of 7 ? 2 is what part of 7? 3
is what part of 7 ? 4 is what part of 7 ? 5 ? 6 ?
68. How many barrels of flour at $7 a barrel, can you
buy for $8? For $9? For $10? For $11? For $12?
For $13? For $14? For $15 ? For $16?
69. 8 are how many times 7 ? 9 are how many 7's ?
70. 10 are how many times 7 ? 11 are how many 7's ?
71. 12 are how many times 7 ? 13 are how many 7's ?
72. 15 are how many times 7 ? 16 are how many 7's ?
73. 17 are how many times 7 ? 18 are how many 7's ?
74. If a melon cost 8 cents, what is 1 -eighth worth?
What are 2-eighths worth ? 3-eighths worth ? 4-eighths
worth? 5-eighths? 6-eighths? 7-eighths?
75. What is 1 -eighth of 8 ? What are 2-eighths of 8T
3-eighths of 8? 4-eighths? 5-eightlw? 6-eighths?
7-eighths ? 8-eighths ? 9-eighths ? 10-eigfcths ?
PARTS OF NUMBERS. 55
76. One is what part of 8 ? 2 is what part of 8 ? 3 is
what part of 8 ? 4 is what part of 8 ? 5 is what part
of 8 ? 6 is what part of 8 ? 7 ?
77. How many gallons of beer, at 8 cents a gallon, can
you buy for 9 cents ? For 10 cents ? For 11 cents ?
For 12 cents ? For 13 cents ? For 14 cents ? Foj 15
cents ? 16 cents ? 17 cents ? 18 cents ?
78. 9 are how many times 8 ? Once 8 and \-eiglitTi of 8.
79. 10 are how man^ times 8 ? 11 are how many 8's ?
80. 12 are how many times 8 ? 13 are how many 8's ?
81. 14 are how many times 8 ? 15 are how many 8's ?
82. 16 are how many times 8 ? 17 are how many 8's ?
83. Eighteen are how many times 8 ? Ans. Two time*
8 and 2-ciglitlis of 8.
84. If an orange cost 9 cents, what is 1 -ninth of it
worth ? What are 2-ninths worth ? 3-ninths ? 4-ninths ?
5-ninths ? 6-ninths ? 7-ninths ? 8-ninths ?
85. What is 1-ninth of 9 ? What are 2-ninths of 9 ?
What are 3-ninths of 9 ? 4-ninths of 9 ? 5-ninths of 9?
6-ninths of 9 ? 7-ninths of 9 ? 8-ninths of 9 ?
86. One is what part of 9 ? 2 is what part of 9 ? 3 is
what part of 9 ? 4 is what part of 9 ? 5 is what part
of 9 ? 6 ? 7 ? 8 ?
87. If cloth is $9 a yard, how much can you buy for
$10? For $11? For $12? For $13 ? For $14? For
$15? For $16? For $17? For $18? For $19?
For $20 ?
88. If apples cost 10 cents a bushel, what is 1 -tenth
of a bushel worth? 2-tenths? 3 -tenths ? 4-tenths ?
- 5-tenths ? 6-tenths ? 7-tenths ? 8-tenths ? 9-tenths ?
89. One is what part of 10? 2 is what part of 10?
3 is what part of 10 ? 4 is what part of 10 ? 5 is what
part of 10 ? 6 is what part of 10 ? 7 is what part of
10 ? 8 is what part of 10 ? 9 ? 11 ?
56 RAY'S INTELLECTUAL ARITHMETIC.
90. When cloth is 10 a yard, how much can you buy
for 11 ? For 12 ? For 13 ? For 14 ? For 15 ?
For 16 ? For 17 ? For 18 ? For 19 ?
91. What do you understand by 1 -fifth of any thing?
By 3-fifths? By 4-fifths?
92. What do you understand by 3-sevenths of any
thing? By 2-eighths?
93. If I cut an apple into 6 equal parts, and give you
4, what part of the whole apple would I give you ?
94. If I divide an orange into 8 equal parts, and give
you 5, what part of the orange would I give you ?
95. What is meant by 1 -ninth of any thing ? 2 -ninths ?
96. What is meant by 1-tenth of any thing ? 2-tenths ?
LESSON III.
1. If you had 7 cents, how many cakes could you
buy at 2 cents each? Ans. 3 cakes and 1-half.
2. Seven are how many times 2 ? Ans. 3 times 2 and
l-half of 2.
3. If you had 11 cents, how many pears could you
buy at 2 cents each ? At 3 cents ?
4. Eleven are how many times 2 ? How many times
3 ? Ans. Three times 3 and 2-thirds of 3.
5. If you had 15 cents, how many cakes could you
buy at 2 cents each ? At 3 cents ? At 4 cents ? At 5 ?
6. Fifteen are how many times 2? 3? 4? 5?
7. If you had 17 cents, how many oranges could you
buy at 2 cents each ? At 3 cents each ? At 4 cents each ?
At 5 cents ? At 6 cents ? At 7 cents ?
8. 24 are how many times 2 ? 3? 4? 5? 6? 7? 8?
9. 25 are how many times 2 ? 3? 4? 5? 6? 7? 8?
10. 26 are how many times 3 ? 4? 5? 6? 7? 8? 9?
11. . 27 are how many times 3 ? 4? 5? 6? 7? 8? 9?
12. 28 are how many times 3 ? 4 5? 6? 7? 8? 9?
PARTS OF NUMBERS. 5-7
13. 29 are how many times 3 ? 4? 5? 6? 7? 8? 9*
14. 30 are how many times 3? ,4? 5? 6? 7? 8? 9?
15. 31 are how many times 3? 4? 5 ? 6? 7? 8? 9?
16. 32 are how many times 3? 4? 5? 6? 7? 8? 9?
17. 33 are how many times 3 ? 4? 5? 6? 7? 8? 9?
18. 34 are how many times 3? 4? 5? 6? 7? 8? 9?
19. 35 are how many times 3 ? 4? 5? 6? 7? 8? 9?
20. 36 are how many times 3? 4? 5? 6? 7? 8? 9?
21. 37 are how many times 3 ? 4? 5? 6? 7? 8? 9?
22. 38 are how many times 3? 4? 5? 6? 7? 8? 9?
23. 39 are how many times 4? 5? 6? 7? 8? 9?
24. 40 are how many times 4? 5? 6? 7? 8? 9?
25. If you had 41 cents, how many oranges could you
buy at 4 cents each ? At 5 cents eac-h ? At 6 cents ?
At 7 cents ? At 8 cents ? At 9 cents ? At 10 cents ?
26. 42 are how many times 4? 5? 6? 7? 8? 9?
27. 43 are how many times 4 ? 5? 6? 7? 8? 9?
28. 44 are how many times 4 ? 5? 6? 7? 8? 9?
29. 45 are how many times 4? 5? 6? 7? 8? 9?
30. 46 are how many times 4? 5? 6? 7? 8? 9?
31. 47 are how many times 4? 5? 6? 7? 8? 9?
32. 48 are how many times 4? 5? 6? 7? 8? 9?
33. 49 are how many times 4? 5? 6? 7? 8? 9?
34. If you had 56 cents, how many peaches could you
buy at 5 cents each ? At 6 cents each ? At 7 cents ?
At 8 cents ? At 9 cents ? At 10 cents ?
35. 56 are how many times 5? 6? 7? 8? 9? 10?
36. 57 are how many times 5? 6? 7? 8? 9? 10?
37. 58 are how many times 5? 6? 7? 8? 9? 10?
38. 59 are how many times 5? 6? 7? 8? 9? 10?
39. 60 are how many times 5? 6? 7? 8? 9? 10?
58 KAY'S INTELLECTUAL ARITHMETIC.
40. 61 are how many times 6? 7 ? 8? 9? 10?
*? 9? 10?
41. 63 are how many times 6 ? 7 ?
42. 65 are how many times 6 ? 7? 8? 9? 10?
43. 66 are how many times 6? 7? 8? 9? 10?
44. 68 are how many times 6? 7? 8? 9? 10?
45. 69 are how many times 6? 7? 8? 9? 10?
46. If 70 hours be required to perform a piece of
work, in how many days can it be done by working 6
hours a day ? By working 7 hours a day ? 8 hours a
day ? 9 hours ? 10 hours ?
47. 71 are how many times 6? 7? 8? 9? 10?
48. 72 are how many times 6? 7? 8? 9? 10?
49. 74 are how many times 6?' 7? 8? 9? 10?
50. 76 are how many times 6? 7? 8? 9? 10?
51. 77 are how. many times 6 ? 7? 8? 9? 10?
52. 79 are how many times 7? 8? 9? 10? 11?
53. 80 are how many times 7 ? 8? 9? 10? 11?
54. 81 are how many times 7 ? 8 ? 9 ? 10 ? 11 ?
55. 82 are how many times 7 ? 8? 9? 10? 11?
56. 83 are how many times 7 ? 8 ? 9 ? 10 ? 11 ?
57. 85 are how many times 7 ? 8? 9?^ 10? 11?
58 . 86 are how many times 7? 8? 9? 10? 11?
59. 87 are how many times 7? 8? 9? 10? 11?
60. 88 are how many times 7 ? 8 ? 9 ? 10 ? 11 ?
61. 89 are how many times 7 ? 8? 9? 10? 11?
62. 90 are how many times 7? 8? 9? 10? 11?
63. 91 are how many times 8 ? 9 ? 10 ? 11 ? 12 ?
64. 92 are how many times 8 ? 9 ? 10 ? 11 ? 12 ?
65. 93 are how many times 8 ? 9? 10? 11? 12?
66. 94 are how many times 8 ? 9? 10? 11? 12?
67. 95 are how many times 8 ? 9 ? 10 ? 11 ? 12 ?
68. 96 are how many times 8 ? 9 ? 10 ? 11 ? 12 ?
PARTS OP NUMBERS. 59
69. 98 are how many times 8 ? 9 ? 10 ? 11 ? 12 ?
70. 99 are how many times 8 ? 9 ? 10 ? 11 ? 121
71. 100 are how many times 8 ? 9 ? 10 ? 11 ? 12 ?
SECTION VIIL
LESSON I.
1. At 2 cents each, what will 2 apples and 1-half of
an apple cost?
ANALYSIS. If 1 apple cost 2 cents, 2 apples will cost
twice 2 cents, which are 4 cents; and 1-half of an apple
will cost 1-half of 2 cents, which is 1 cent; and 4 cents and
1 cent are 5 cents ; therefore, if 1 apple cost 2 cents, 2 apples
and l-half will cost 5 cents.
2. Two times 2 and 1-half of 2 are how many?
3. At 2 cents each, what will 3 pears and 1-half of a
lear cost?
4. Three times 2 and 1-half of 2 are how many?
5. At $3 a yard, what will 3 yards and 1-third of a
yard of cloth cost?
6. Three times 3 and 1-third of 3 are how many?
7. At $3 a barrel, what will 3 barrels and 2-thirds of
a barrel of flour cost?
SUGGESTION. To find 2-thirds of any number, first find 1-third,
then multiply by 2.
8. Three times 3 and 2-thirds of 3 are how many?
9. At S3 a yard, what will 4 yards and 2-thirds of a
yard of cloth cost?
ANALYSIS. If 1 yard cost $3, 4 yards will cost 4 times
$3, which are $12; and 1-third of a yard will cost 1-third
of $3, winch is $1 ; and 2-thirds of a yard will cost twice
as much as 1-third, that is, twice $1, which are $2; and $12
and $2 are $14. Ans. $14.
bU KAY'S INTELLECTUAL ARITHMETIC.
10. If an orange cost 4 cents, what will 3 oranges
and 1 -fourth of an orange cost ?
11. 3 times 4 and 2 -fourths of 4 are how many?
12. 4 times 4 and 3-fourths of 4 are how many ?
, 13. At 85 a barrel, what will 4 barrels and 2-fifths of
a barrel of flour cost?
14. 4 times 5 and 2-fifths of 5 are how many ?
15. 5 times 5 and 3-fifths of 5 are how many ?
16. If a man spend $6 a week, how much will he spend
in 4 weeks and 1 -sixth of a week ?
17. 4 times 6 and 2-sixths of 6 are how many ?
18. 5 times 6 and 3-sixths of 6 are how many ?
19. 6 times 6 and 4-sixths of 6 are how many ?
20. 7 times 6 and 5-sixths of 6 are how many ?
21. At 7 cents a yard, how much will 3 yards and
2-sevenths of a yard of tape cost ?
22. 4 times 7 and 3-sevenths of 7 are how many ?
23. 5 times 7 and 4-sevenths of 7 are how many ?
24. 6 times 7 and 6-sevenths of 7 are how many ?
25. If oranges are worth 8 cents each, how much are
3 oranges and 2-eighths of an orange worth ?
26. 4 times 8 and 3-eighths of 8 are how many ?
27. 5 times 8 and 4-eighths of 8 are how many ?
28. 6 times 8 and 7-eighths of 8 are how many ?
29. If a yard of muslin cost 9 cents, what will 2 yards
and 2-ninths of a yard cost ?
30. 3 times 9 and 5 -ninths of 9 are how many ?
31. 5 times 9 and 6 -ninths of 9 are how many ?
32. 7 times 9 and 8-ninths of 9 are how many ?
33. If a pound of sugar cost 10 cents, what will 2
pounds and 2-tenths of a pound cost ?
PARTS OF NUMBERS. Q\
34. 2 times 10 and 3-tenths of 10 are how many ?
35. 5 times 10 and 6-tenths of 10 are how many ?
36. 6 times 10 and 8-tenths of 10 are how many ?
37. 9 times 10 and 7-tenths of 10 are how many ?
38. 10 times 9 and 8-ninths of 9 are how many ?
39. 12 times 11 and 9 -elevenths of 11 are how many ?
SECTION IX.
LESSON I.
1. A boy bought 3 apples at 4 cents each : how much
did they cost? He paid for them with oranges, at 6 cents
each : how many oranges did it take ?
ANALYSIS. First. 3 apples cost 3 times the price of 1 apple,
that is, 3 times 4 cents, or \2 cents. Second. It took as many
oranges as 6 cents are contained times in 12 cents; 6 cents
are contained in 12 cents 2 times, Ans. 2 oranges.
2. A man bought 3 yards of cloth at $4 a yard : how
many dollars did it cost? He paid for it with cider, at
$2 a barrel : how many barrels did it take ?
3. A boy bought 8 apples at 2 cents each : how much
did they cost? He paid for them with pears, at 4 cents
each : how many did it take ?
4. Bought 9 marbles at 2 cents each ; paid for them
with tops, at 6 cents each : how many did it take ?
5. Bought 10 yards of cloth at $2 a yard; paid for
it with flour, at $4 a barrel : how many bar, did it take?
6. Bought 8 pints of cherries at 3 cents a pint j paid
for them with apples, at 6 cents a dozen : how many dozen
did it take ?
7. How many barrels of flour, at $3 a barrel, must be
given for 2 yards of cloth, at $7 a yard ?
S2 RAY'S INTELLECTUAL ARITHMETIC.
8. 5 times 3 are how many times 4 ? 6? 7 ? 8?
9. 4 times 4 are how many times 3? 5? 6? 7 ?
10. 5 times 4 are how many times 3? 6? 7? 8?
11. 2 times 11 are how many times 3? 4? 5? 6?
12. 5 times 5 are how many times 3? 4? 6? 7?
13. 8 times 4 are how many times 3 ? 5? 6? 7?
14. 7 times 5 are how many times 4? 6? 8? 9?
15. 6 times 7 are how many times 4? 5? 8? 9?
16. 6 times 9 are how many times 5? 7? 8? 10?
17. 8 times 7 are how many times 5? 6? 9? 10?
18. 6 times 10 are how many times 5 ? 7 ? 8 ? 10 ?
19. 9 times 7 are how many times 6 ? 8 ? 10 ? 11 ?
20. 8 times 8 are how many times 7 ? 9 ? 10 ? 11 ?
LESSON II.
1. Bought 4 boxes and 3-fifths of a box of raisins, at
85 a box : how much did they cost ? Paid for them with
flour, at 6 a barrel : how many did it take ?
ANALYSIS. It will take as many barrels of flour as $6, the
price of 1 barrel, are contained times in the cost of 4 boxes
and %-fifths of a box of raisins, at $5 a box.
2. Bought 4 gallons and 4-sixths of a gallon of wine,
for 6 a gallon, and paid for it with raisins, at 5 a box :
how many boxes did it take ?
3. Bought 5 kegs and 4-sevenths of a keg of tobacco,
for 7 a keg, and paid for it with paper, at 86 a ream :
how many reams did it take ?
4. Five times 5 and 3-fifths of 5 are how many
times 3? 4? 6? 7? 8? 9? 10?
5. Seven times 4 and 3-fourths of 4 are how many
times 3? 5? 6? 8? 9? 10?
6. Five times 6 and 5-sixths of 6 are how many
times 3? 4? 7? 8? 9? 10?
PARTS SF NUMBERS. 63
7 Seven times 5 and 3-fifths of 5 are how many
times 4? 6? 8? 9? 10?
8. Five times 8 and 1 -eighth of 8 are how many
times 4? 6? 7? 9? 10?
9. Seven times 6 and 2 -sixths of 6 are how many
times 5? 8? 9? 10?
10. Nine times 5 and 2 -fifths of 5 are how many
times 6? 7? 8? 10?
11. Nine times 5 and 4-fifths of 5 are how
times 6? 7? 9? 10?
12. Six times 8 and 3 -eighths of 8 are how many
times 5? 7? 9? 10?
13. Seven times 7 and 4-sevenths of 7 are how many
times 5? 6? 8? 9? 10?
14. Nine times 6 and 3-sixths of 6 are how many
times 5? 7? 8? 10?
15. Eight times 7 and 3 -sevenths of 7 are how many
times 5? 6? 9? 10?
LESSON III.
1. Bought 4 apples at 3 cents each: paid for them
lemons at 6 cents each : how many did it take ?
2. Bought 7 yards of tape, at 2 cents a yard : how
many pears at 3 cents each, will it take to pay for it ?
3. If 2 apples cost 4 cents, what cost 3 apples ?
ANALYSIS. Three- apples will cost 3 times as much as 1
apple, and 1 apple will cost l-half as much as 2 apples, that
is, l-half of 4 cents, \-half of 4 cents is 2 cents, and 3 times
2 cents are 6 cents. Ans. 6 cents.
4. If 3 yards of cloth cost $9, what cost 4 yards ?
5. If 3 oranges cost 15 cents, what cost 5 oranges?
6. If 4 barrels of flour cost $24, what cost 7 barrels ?
7. If 2 kegs of lard cost $8, what cost 9 kegs ? 11 kegs ?
G4 RAY'S INTELLECTUAL ARITHMETIC.
8. If 5 dozen eggs cost 35 cents, what will 3 dozen
eost ? What will 8 dozen cost ?
9. If a man get 14 for 7 days' work, how much will
he get for 9 days ? For 3 days ?
10. Bought 4 yards and 2-thirds of a yard of cloth, at
$3 a yard, and paid for it with cheese, at 87 a hundred
weight : how many hundred weight did it take ?
11. Bought 4 pounds and 4-fifths of a pound of nails,
at 5 cents a pound, and paid for them with eggs, at 3
cents a dozen : how many dozen did it take ?
12. Bought 7 pounds and 5-sevenths of a pound of
sugar, at 7 cents a p<?>md, and paid for it with chickens^
at 9 cents each : how any did it take ?
13. Bought 9 pounds and 2-sevenths of a pound of
sugar, at 7 cents a poand, and paid for it with eggs, at 6
eents a dozen : how many dozen did it take ?
14. How many pounds, at 7 cents a pound, must you
give for 8 and 2-ninth yards, at 9 cents a yard ?
15. How many barrels, at $6 a barrel, must be given in
exchange for 4 and 5 -seventh yards, at 7 a yard ?
16. Bought 5 pounds and 3-sevenths of a pound of
butter, at 7 cents a pound, and paid for it with raisins, at
6 cents a pound : how many pounds did it take ?
17. How many apples, at 2 cents each, can you buy
for 6 cents? How many for 15 cents?
18. How many pears, at 3 for 7 cents, can you buy
for 21 cents ? For 35 cents ?
19. If 6 pears are worth 2 oranges, how many oranges
ean you buy for 21 pears ?
20. A man bought 15 yards of cloth, at the rate of 3
yards for $5 : how many dollars did it cost ?
21. If a man receive 5 for 4 days' work, how many
dollars will he get for 12 days' work ?
22. How many pears, at 3 for 5 cents, can you buy
for 25 cents ?
PARTS OF NUMBERS. 65
23. If 2 pears are worth 6 cents, how many pears must
be given for 4 oranges, at 6 cents each ?
24. What will be the cost of 3 barrels and 9-elevenths
of a barrel of sugar, at $11 a barrel?
25. What will be the cost of 3 boxes and 4-fifths of
a box of butter, at $5 a box ?
26. Find the cost of 4 and 5-sixth tons of hay, at $6
a ton.
27. How many dozen eggs, at 12 cents a dozen, will
pay for 10 and 10-eleventh pounds of sugar, at 11 cents
a pound ?
SECTION X.
LESSON I.
1. If 1-half of an orange cost 3 cents, what will the
whole orange cost?
ANALYSIS. 1 orange will cost twice as much as \-Jialf of
an orange, that is, twice 8 cents. But twice 3 cents are 6 cents ;
therefore, if 1-half of an orange cost 3 cents, the whole orange
l cost 6 cents.
2. Three is 1-half of what number?
ANALYSIS. 3 is I-half of twice 3; but twice 3 are 6;
Cher ef ore, 3 is l-half of 6.
3. If 1 -fourth of a barrel of cider cost $2, what will
a barrel cost?
4. Two is 1 -fourth of what number?
5. If a man can walk 2 miles in 1 -third of an hour,
how far can he walk in an hour ?
6. Two is 1 -third of what number?
7. The age of Charles is 1 -third that of Thomas;
Charles is 4 years old : how old is Thomas ?
6o AAY'S INTELLECTUAL ARITHMETIC.
8. Four cents is the 1-third of what number of cents!
9. David has 5 marbles, which is only 1 -fourth as
many as Henry : how many has Henry ?
ANALYSIS. 5 is I -fourth of four times 5; but 4 times 5
are 20: therefore, 5 is 1-fourth of 20.
10. Six is 1 -fourth of what number ? 8 is 1 -fourth of
?rhat number ? 9 is 1 -fourth of what?
11. Six is 1-third of what number? 7 is 1-third of
what number? 9 is 1-third of what?
12. Three is 1-fifth of what number? 5 is 1-fifth of
what number? 9 is 1-fifth of what?
13. Five is 1-sixth of what number? 7 is 1-sixth of
what number?. 9 is 1-sixth of what?
14. Two is 1 -seventh of what number? 5 is 1 -seventh
of what number ? 8 is 1 -seventh of what?
15. Five is 1-eighth of what number? 7 is 1-eighth
of what number? 9 is 1-eighth of what?
16. Seven is 1-ninth of what number? 8 is 1-ninth
of what number? 9 is 1-ninth of what?
LESSON II.
1. James had 4 apples, and gave his brother 1-half
of them : how many did he give him ?
2. If you divide 6 apples equally between 2 boys,
what part of them must each have? Ans. \-lialf.
3. If you divide 6 apples equally between 3 boys,
what part of them must each have? Ans. 1-third.
4. If 3 yards of cloth cost $9, what part of 9 will 1
yard cost ? What part of 89 will 2 yards cost ?
5. If 4 oranges cost 12 cents, what part of 12 cents
will 1 orange cost? What part of 12 cents will 2 cost?
What part will 3 cost?
6. What is 1 -fourth of 12? What are 2-fourths
c<f 12? 3-fourthsof 12?
PAETS OF LUMBERS. 67
7. If 5 barrels of flour cost $30, what part of $30
will 1 barrel cost? What part of $30 will 2 barrels cost?
What part will 3 bar. cost ? What part will 4 cost ?
8. W T hat is 1-fifth of 30? What are 2-fifths of 30?
3-fifths of 30? 4-fifths of 30 ?
9. If 1 apple cost 4 cents, what will 1-half of an
apple cost ? What will 3-halves cost ?
10. If 2 oranges cost 8 cents, what part of 8 cents
will 1 orange cost ? 3 oranges ? 5 ?
11. If 3 barrels of cider cost $12, what part of $12 will
1 barrel cost? What part of $12 will 2 barrels cost? 4
barrels ? 5 barrels ?
12. What is 1 -third of 12 ? What are 2-thirds of 12?
4-thirds of 12? 5-thirds of 12?
13. What is 1 -fourth of 24? 2-fourths? 3-fourths?
5-fourths ? 6-fourths ? 7-fourths ? 9-fourths ?
14. What is 1-fifth of 25? 2-fifths? 3-fifths?
4-fifths? 6-fifths? 7-fifths? 8-fifths? 9 - fifths ?
15. What is 1-sixth of 24? 2-sixths ? 3-sixths ?
4-sixths ? 5-sixths ? 6-sixths ? 7-sixths ?
16. What is 1-seventh of 56? 2-sevenths? 3-sev-
enths? 4-sevenths? 5 -sevenths? 7-sevenths?
17. What is 1 -eighth of 72 ? 2-eighths ? 5-eighths?
8-eighths ? 10-eighths ? 11-eighths ?
18. What is 1-ninth of 54? 3-ninths ? 5-ninths?
7-ninths ? 9-ninths ? 11-ninths ? 12-ninths ?
19. What is 1-tenth of 60? 3-tenths ? 7-tenths?
9-tenths? 11-tenths?
20. What are 2-thirds of 6 ?
ANALYSIS. Two-thirds of 6 are twice l-third. One-third
vf 6 is 2, and twice 2 are 4 ; therefore, 2-thirds of 6 are 4.
21. What are 2-thirds of 12? 3-fourths of 12?
22. What are 4-fifths of 20 ? 5-sixths of 30 ?
23. What is 1-ninth of 27 ? 1 -fourth of 36 ?
RAY'S INTELLECTUAL ARITHMETIC.
24.
What
are
3-sevenths
of
28?
2-fifths
of
20?
25.
What
are
4-sixths
of
24?
3-fourths
of
20?
26.
What
are
5-ninths
of
18?
6 -sevenths
of
21?
27.
28.
29.
What
What
What
are
are
are
6 -sevenths
5 -eighths
6 -sevenths
of
of
of
49?
40?
63?
3-eighths
4-ninths
7-eighths
of
of
of
24?
54?
56?
30.
What
are
3 -halves
of
18?
4-thirds
of
24?
31.
What
are
7-fourths
of
12?
8-fifths
of
30?
32.
What
are
8-sixths
of
42?
9-sevenths
of
63?
33.
34.
What
What
are
are
9-eighths
11 -tenths
of
of
56?
50?
10 -ninths
12-tenths
of
of
81?
40 >
LESSON III.
1. A man having 12 bushels of grain, divided 5-sixths
of it equally among 3 poor persons : how many bushels
did each receive?
2. Five-sixths of 12 are how many times 3?
3. A boy having 25 apples, kept 1 -fifth himself, and
divided *he other 4-fifths equally among 6 of his com-
panions how many did each receive ?
4.
4-fifths
of
25
are
how
many
times
6?
5.
3-fourths
Of
24
are
how
many
times
9?
6.
7-fourths
of
24
are
how
many
times
8?
7.
8-thirds
of
18
are
how
many
times
6?
8.
7-thirds
of
27
are
how
many
times
10?
9,
3-fifths
of
60
are
how
many
times
7?
10.
5-sixths
of
54
are
how
many
times
8?
11.
8-sixths
of
48
are
how
many
times
9?
12.
3-sevenths
of
56
are
how
many
times
9?
13.
9-sevenths
of
63
are
how
many
times
10?
14.
5 -eighths
of
64
are
how
many
times
6?
15.
9-eighths
of
40
are
how
many
times
7?
PARTS OF NUMBERS. 69
16. 11 -sevenths of 49 are how many times 8 ?
17. 3-ninths of 54 are how many times 7 ?
18. 10 -ninths of 63 are how many times 8 ?
19. 8-ninths of 54 are how many times 5 ?
20. 9-sevenths of 42 are how many times 8 ?
LESSON IV.
1. James gave his brother 2 apples, which were
Vthird of all he had: how many had he left?
2. Thomas gave his brother 5 cents, which were
1 -fourth of all he had : how many had he ?
3. If, in traveling, I walk 1 -fifth of my journey in 2
hours, at that rate, in what time can I complete the re-
maining 4-fifths ?
4. One pint is 1 -eighth of a gallon : if a pint of wine
cost 7 cents, what will a gallon cost ?
5. A boy found a purse containing $12, and received
1 -sixth of the money for returning it to the owner : how
much did he receive?
6. If $42 be equally divided among a number of
men, giving each man 1 -sixth of the money, how many
men would there be, and what would each receive ?
7. Thomas had 28 marbles: he gave 1 -fourth of them
to James, and twice as many to William as to James :
how many did each receive ?
8. William had 24 apples: he gave 1-half to Thomas,
and 1 -third to James: how many did he give to both?
How many had he left?
9. A boy had 12 cents: he spent 1 -third of them fcaf
apples, and 1 -fourth for cakes : how many had he left?
10. A little girl received 20 cents from her mother;
her brother gave her 1 -fifth as many as her mother, and
her sister 1-half as many as her brother : how many did
she then have ?
2d Bk. ' 5
70 RAT'S INTELLECTUAL ARITHMETIC.
11. A boy having 40 cents gave 3-fifths of them foi
2 arithmetics : what was the price of 1 arithmetic ?
ANALYSIS. 3-Jifths are 3 times \-fifih: \-fifih of 40 cents
is 8 cents, and 3 fifths are 3 times 8 cents, ivhich are 24 cents.
If 2 books cost 24 cents, 1 book will cost \-Jialf of 24 cents >
which is 12 cents. Ans. 12 cents.
12. James had 14 cents, and gave 4-sevenths of them
to his sister : how many cents had he left ?
13. John had 15 pears: he gave 1-third to Frank,
and 3-fifths to Harry : how many had he left ?
14. A man had 30 yards of cloth, and sold 2-fifths of
it for 48 : how much was that a yard?
15. John had 25 cents, and gave 3-fifths for peaches,
at 2 cents each : how many did he buy ?
16. A boy having 54 chestnuts, divided 5 -ninths of
them among 3 girls : how many did each receive ?
17. A man had 28 barrels of flour, and sold 2-sevenths
of them for 24 : what was that a barrel ?
18. Bought 7 yards of cloth for 42 : I gave 3 yards
for 9 barrels of cider : how much was it a barrel ?
19. A man had 40, and lost 3-fifths of them: he ex-
pended the remainder in flour, at 84 a barrel : how many
barrels did he buy ?
20. I had 10 oents, and lost 1 -fifth : spent the rest for
apples, at 2 cents each : how many did I buy ?
21. James had 48 cents : he gave 3-eighths to his
brother, and spent the rest in chestnuts, at 5 cents a
quart : how many quarts of chestnuts did he buy ?
22. Thomas had 28 cents: he gave 1-fourth to his
sister, and 3-sevenths to his brother, and with the remain
der he bought 3 books : what did each cost ?
PARTS OF NUMBERS. 71
SECTION XI.
LESSON I.
1. If 2-thirds of a melon cost 4 cents, what will
1 -third cost?
ANALYSIS. l-third is I -half of 2-thirds: if 2-thirds cost
4 cents, l-third will cost l-half of 4 cents, or 2 cents, Ans.
2. Four is 2 times what number? Ans. Four is 2
times l-half of 4, which is 2.
3. If 3-fourths of a yard of cloth cost $6, what will
1 -fourth cost?
4. Six is 3 times what number?
5. If 2-thirds of a barrel of flour cost $8, what will
l-third cost?
6. Eight is 2 times what number?
7. If 3-fifths of a pound of butter cost 9 cents, what
will 1 -fifth cost?
8. Nine is 3 times what number ?
9. If 4-fifths of a pound of coffee cost 16 cents, what
will 1 -fifth cost?
10. Sixteen is 4 times what number?
11. If 5 -sixths of a gallon of wine cost 35 cents, what-
will 1 -sixth cost?
12. If 6-tenths of a yard of '4oth cost 30 cents, what
will 1 -tenth cost?
13. If 4-sevenths of a yard of muslin cost 28 cents,
what will 1 -seventh cost?
14. If 2-thirds of an orange cost 4 cents, what cost
l-third? If l-third cost 2 cents, what cost the whole?
15. Four is 2-thirds of some number: what is l-third
of that number? 2 is l-third of what number? Then
4 is 2-thirds of what number ?
16. If 2-thirds of a yard of cloth cost $6, what cost
1 -third of a yard ? If l-third cost $3, what eo3*-a yard ?
72 KAY'S INTELLECTUAL ARITHMETIC.
17. Six is 2-thirds of some number: what is 1 -third
of that number?' 3 is 1 -third of what number? Then
6 is 2-thirds of what number ?
18. If 3-fourths of a barrel of flour cost $6, what will
1-fourth cost? If 1-fourth cost $2, what cost a barrel?
19. Six is 3-fourths of some number : what is 1-fourth
of the same number? 2 is 1-fourth of what number?
Then 6 is 3-fourths of what number ?
20. If 2-fifths of a melon cost 8 cents, what cost 1-fifth?
If 1-fifth cost 4 cents, what cost the whole ?
21. Eight is 2-fifths of some number : what is 1-fifth
of the same number? 4 is 1-fifth of what number?
Then 8 is 2-fifths of what number?
22. If 3-fifths of a pound cost 9 cents, what cost 1-fifth?
If 1 -fifth cost 3 cents, what cost a pound ?
23. Nine is 3-fifths of some number : what is 1-fifth
of that number? 3 is 1-fifth of what number? Then 9
is 3-fifths of what number ?
24. If 4-fifths of a pound cost 8 cents, what cost 1-fifth?
If 1-fifth cost 2 cents, what cost a pound?
25. Eight is 4-fifths of some number: what is 1-fifth
of that number? 2 is 1-fifth of what number? Then
8 is 4-fifths of what number ?
26. If 3-fourths of a dozen eggs cost 9 cents, what
fcost 1-fourth ? If 1-fourth cost 3 cents, what cost a dozen ?
27. Nine is 3-fourths of some number, what is 1-fourth
of that number? 3 is 1-fourth of what number? Then
9 is 3-fourths of what number ?
28. If 2-thirds of a yard of tape cost 20 cents, what
cost 1- third? If 1 -third cost 10 cents, what cost a yard?
29. Twenty is 2-thirds of some number : what is
1 -third of that number? 10 is 1 -third of what number?
Then 20 is 2-thirds of what number ?
30. Six is 3-fourths of what number?
PARTS OF NUMBERS. 73
ANALYSIS. 1-fourth is l-third of 3-fourths : if 3-fourths
are 6, 1-fourth is l-third of 6, which is 2, and 4-fourths 3 (or
the whole number,) are 4 times 2, which are 8, Am.
31. 9 is 3-sovenths of what number ?
32. 10 is 2-S3venths of what number ?
33. 14 is 7 -eighths of what number ?
34. 15 is 3-eighths of what number ?
35. 16 IB 4-ninths of what number ?
36. 18 is 6 -tenths of what number ?
37. 20 is 5-fourths of what number?
38. 15 is 5-fourths of what number?
39 . 33 is 11 -sixths of what number?
40. 22 is 2-elevenths of what number?
41. 81 is 9-thirds of what number?
LESSON II.
1. If 3-fourths of a pound of raisins cost 9 cents,
how much will a pound cost ? How many lemons, at 2
cents each, will pay for 1 pound of raisins ?
ANALYSIS. First. 1-fourth will cost l-third as much as
3-fourths, and 4-fourths, or a pound, will cost 4 times as
much as 1-fourth ; if 3-fourths cost 9 cents, \-fourth will cost
l-third of 9 cents, or 3 cents, and 4-fourths will cost 4 times
3 cents, or 12 cents.
Second. At 2 cents each, it will require as many lemons as
2 cents are contained times in 12 cents; 2 cents are contained
in 12 cents 6 times. Ans. 6 lemons.
2. If 2-thirds of a pound of sugar cost 16 cents,
much will a pound cost ? How many oranges, at 4 cents
each, will pay for 1 pound of sugar ?
3. If 7-eighths of a barrel of wine cost $42, how
many bar. of cider, at $6 a bar., will pay for 1 bar. of wine?
74 KAY'S INTELLECTUAL ARITHMETIC.
y
4. If 3-fifths of a hogshead of sugar cost $24, how
many barrels of flour, at $4 a barrel, will pay for 1 hogs-
head of sugar?
5. Sold a horse for 25, which was 5-eighths of his
cost : how much did he cost me ? I paid for him with
sloth, at $6 a yard : how many yards did I give ?
6. Thirty is 5-eighths of how many times 5 ?
Ans.^Of as many times 5, as 5 is contained times in the
number of which l-fifth of 30 is \-eiglitli.
ANALYSIS. If 30 is 5-eighths, \-fiftli of 30, which is 6, is
i-eighth, and the number is 8 times 6, which are 48; 48 is 9
times 5 and 3-fifths of 5.
7. 12 is 4-sevenths of how many times 5?
8. 18 is 3 -eighths of how many times 9 ?
9. 16 is 2 -sevenths of how many times 9 ?
10. 36 is 4-sevenths of how many times 8 ?
11. 45 is 5 -ninths of how many times 7?
12. 24 is 4-thirds of how many times 5 ?
13. 72 is 8-fifths of how many times 7 ?
14. 81 is 9-fourths of how many times 3 ?
15. 50 is 10-sevenths of how many times 4?
16. 63 is 7-sixths of how many times 5 ?
LESSON III.
1. James gave his brother 4 marbles, which were
2-thirds of all he had : how many marbles had he ?
2. Thomas sold a knife for 15 cents, which were
3 fifths of its cost : how much did it cost ?
3. William lost 6 marbles, which were 3-eighths of
all he had : how many had he ?
4. I sold a horse for S42, which were 6-fifths of hi?
cost: how many dollars did I gain?
PARTS OF NUMBERS. 75
5. A grocer sold a lot of flour for $40, which were
5 -fourths of the cost : what was the cost ?
6. Sold a horse for $56, which were 8-fifths of the
cost : paid with flour, at $4 a barrel : how many barrels
did I give?
7. A man sold a watch for $28, which were 4-thirds
of its cost : how much did it cost ?
8. A man purchased a horse : after paying 3-fifths of
the price, he owed $20 : what was. the cost?
9. Alexander sold a book for 25 cents, and lost
2-sevenths of the cost : what was the cost ?
10. If a boy can earn 32 cents in a whole day, how
\nuch can he earn in 5-eighths of a day?
11. If 7-ninths of a cask of wine cost $42, how much
flour, at $8 a barrel, will pay for 1 cask of wine ?
12. If there are 10 links in 5 -ninths of a chain, how
many links are there in the whole chain?
13. In an orchard there are 12 cherry-trees : the re-
maining 5-sevenths of the orchard are apple-trees : how
many apple-trees are there?
14. There is a pole, 4-fifths of which is under water,
and 6 feet out of water : how long is the pole ?
15. There is a pole, 3-fifths of which is in the earth,
and 12 feet in the air : how long is the pole ?
16. One -fifth of a pole is in the mud, 2 -fifths in the
water, and 14 feet in the air : how long is the pole ?
17. The age of Joseph is 25 years, which is 5-eighths
of the age of his father : the father's age is 10 times that
of his youngest son : what is the age of the father? whal
is the age of his youngest son?
18. A man sold a horse for $45, which were 5 -thirds
of the cost : how much did he gain ?
19. A man paid $24 for a watch, and sold it for
7-fourths of the cost : he was paid in cloth, at $5 a yard :
how many yards did he receive?
76 RAY'S INTELLECTUAL ARITHMETIC.
20. A watchmaker sold a watch for 18, and lost 2-fifths
of its value : how much did he lose ?
ANALYSIS. Since he lost 2-fifths, he sold it for ^-fifths of
its value; then ^-fifths are $18, and \-fifth is l-third of
$18, or $6; 2-fifths are 2 times $6, or $12, Ans.
21. A watchmaker sold a watch for 8-15, and gained
2-sevenths of the cost : what was the cost ?
22. A boy spent 3-sevenths of his money, and had
12 cents left : how much had he ?
SECTION XII.
LESSON I.
1. If you have 12 cents, and 3-fourths of your money
equal l-third of mine, how many cents have I ?
ANALYSIS. \-fourth of 12 is 3, and 3-fourths are 3 times
3, or 9 ; 9 is l-third of 3 times 9 : 3 times 9 are 27, Ans.
2. 2-thirds of 9 are 1 -fifth of what number?
3. 4-fifths of 10 are 1-half of what number?
4. 3-sevenths of 14 are 1 -sixth -of what number?
5. 5-sixths of 12 are 1-fourth of what number?
6. 3-eighths of 16 are 1 -fifth of what number?
7. 2-fifths of 20 are how many thirds of 15 ?
8. 4-sevenths of 21 are how many fifths of 25 ?
9. 5-sixths of 30 are how many fourths of 32 ? ,
10. 7 -ninths of 45 are how many sevenths of 21 ?
11. 3-fourths of 36 are how many thirds of 18 ?
12. 5-eigl^ths of 40 are how many ninths of 36 ?
13. Divide 1-fourth of 32 by 2-thirds of 9. Three-
fourths of 28 by 2-thirds of 12.
PARTS OF NUMBERS. 77
14. Divide 3-fifths of 40 by 3-sevenths of 14. Five*
sevenths of 42 by 3-eighths of 24.
15. Divide the 7-ninths of 54 by 5-sixths of 12.
16. Nine-elevenths of 88 are how many times 5-ninths
of 18?
LESSON II.
1. William says to Frank, " Your age is 15 years,
and 4-fifths of your age are 6 -sevenths of mine : what
is my age ?"
ANALYSIS. l-fifth of 15 is 3, and 4-fifths are 4 times 3,
which are 12; if 12 is Q-sevenths, l-sixth of 12, which is 2,
is l-seventh, and if 2 is l-seventh, the number is 7 times 2,
which are 14. Ans. 14 years.
2.
3-fourths of
8
are
2-thirds
of what number ?
3.
4-fifths of
20
are
8-ninths
of what number ?
4.
3-fifths of
20
are
4-fifths
of what number?
5.
5-sixths of
36
are
10-thirds
of what number ?
6.
6 -sevenths of
28
are
3-fifths
of what number?
7.
&- eighths of
32
are
4-sevenths
of what number ?
8.
8-ninths of
45
are
10-elevenths
of what number ?
9.
Five-sevenths
of
21
are 3-fourths
of 5 times what ?
10.
Four-ninths of 36 are 8-sevenths
of 7 times what ?
11.
Three-tenths
of 40
are 6-fifths of 10 times what?
12.
Seven-eighths
of
64
are 7-ninths of 12 times what?
13.
Three -fourths
of
16 are 2-thirds of how many
times 1 -fifth of 20?
ANALYSIS. \-Jourih of 16 is 4, and 3-fourths are 3 times
4, which are 12 ; if 12 is 2-thirds, I -half of 12, which is 6,
is 1-third, and 6 is l-third of 3 times 6, ivJiich are 18 ; l-Jifth
of 20 is 4, and 18 is 4 times 4 and 2-fourths of 4. Ans. 4
and 2-fourths times.
78 RAY'S INTELLECTUAL ARITHMETIC.
14. Four-fifths of 20 are 8-sevenths of how many
times 1 -third of 15 ?
15. Six-sevenths of 21 are 2-fifths of how many
times 1 -fifth of 30?
16. Four-ninths of 27 are 2-thirds of how many
times 1 -seventh of 35 ?
17. Five-sevenths of 42 are 5 -ninths of how many
times 1 -eighth of 56 ?
18. Three-fourths of 24 are 9-tenths of how many
times 2-fifths of 10 ?
19. Five-sixths of 12 are 2-sevenths of how many
times 3-fourths of 8 ?
20. Five-eighths of 32 are 4-ninths of how many
times that number of which 2 is 1 -third?
21. Six-sevenths of 28 are 8-elevenths of how many
times that number of which 4 is 2-fifths ?
22. Eight-ninths of 45 are 10-thirds of how many
times that number of which 14 is 7-fourths?
23. Seven-tenths of 50 are 5-ninths of how many
times that number of which 8 is 4-fifths ?
24. Nine -sevenths of 56 are 12 -fifths of how many
times the number of which 6 is 6 -sevenths ?
25. One-third of a certain number is 2 more than
1-half of 12 : what is the number ?
26. One-fourth of a certain number is 3 less than
1 -fifth of 30: what is the number?
27. Two-fifths of 20 is 6 less than how many thirds
of 21 ?
28. Three-fourths of 24 is 6 more than 2-thirds of
what number ?
29. Five-sixths of 30, increased by 4, is 1 less than
3-fourths of some number : what is that number ?
30. Three -fifths of 40 is just 3 less than 9 -sevenths of
how many times 7 ?
MISCELLANEOUS REVIEW. 79
SECTION XIII. REVIEW.
LESSON I.
1 William had 23 cents : Thomas gave him 8 cents
more, treorge 6, James 5, and David 7: he gave 15 cents
for a book: how many cents had he left?
2. A grocer paid $12 for sugar, $9 for coffee, $5 for
tea, $7 for flour, and had 10 left : how many dollars had
he ai first?
3. A boy had 11 cents : his father gives him 9 cents,
his mother 6, and his sister enough to make 34 : how
many cents did his sister give him ?
4. Five men bought a horse for $42 : the first gave
$13, the second 7, the third 5, and the fourth 9: how
many dollars did the fifth give ?
5. A man purchased 8 sheep at $4 a head, 5 barrels
of flour at $3 a barrel, 4 yards of cloth at $3 a yard,
and 5 ounces of opium at $1 an ounce : how much did
he spend?
6. A boy lost 25 cents : after finding 15 cents, he had
25 cents : how many cents had he at first ?
7. A man owed a debt of $28, and paid all but $9 :
how much did he pay?
8. Borrowed $56 : at one time paid $23 ; at another,
all but $7 : how much did I pay the last time ?
9. James borrowed 37 cents : at one time he paid 5,
at another 8, and the third time, all but 15.: how many
cents did he pay the third time ?
10. A farmer sold 2 cows at $9 each, and 5 hogs at
$3 each, and received in payment 3 sheep at $3 each, and
the rest in money : how much money did he receive ?
11. A farmer sold 12 barrels of cider at $3 a barrel :
he then purchased 5 barrels of salt at $3 a barrel, and
some sugar, for $8 : how many $'s had he left?
80 BAY'S INTELLECTUAL ARITHMETIC.
12. A merchant purchased 13 hats at 4 each, 5 pairs
of shoes at 2 a pair, and an umbrella, for 7 : what
must he sell the whole for, to gain $9 ?
13. If 1 barrel of flour cost S3, what cost 7 barrels?
14. If 2 pounds cost 16 cents, what cost 5 pounds?
15. If 3 barrels of cider cost 12, what cost 4 barrels ?
16. If 4 yards of cloth cost $28, what cost 7 yards ?
17. If 3 pounds cost 27 cents, what cost 8 pounds ?
18. If 5 barrels of flour cost 35, what cost 8 barrels ?
19. If 7 apples cost 28 cents, what cost 3 apples?
20. If 8 oranges are worth 24 apples, how many apples
are 3 oranges worth ?
21. If 4 pounds of cheese cost 36 cents, what will 3
pounds cost?
22. If 8 yards of cloth cost $56, what cost 7 yards ?
23. What sum of money must be divided among 11 men
that each shall receive 9 ?
24. A walks 5 miles, while B walks 3 : when A ha\
gone 35 miles, how far has B gone ?
ANALYSIS. In 35 there are 7 fives, and B walks as many
threes as A walks fives; that is, 7 times 3, or 21 miles.
25. Joseph and his father are husking corn : the father
can husk 7 rows while Joseph husks 3 : how many rows
will Joseph husk while his father husko 42 ?
26. Charles can earn 9 while Mary earns 4 : how
many S's will Charles earn while Mary earns 28 ?
27. If 6 horses eat 12 bushels of oats in a week, how
many will 10 horses eat in the same time ?
28. If 5 horses eat 16 bushels in 2 weeks, how long
would it take them to eat 56 bushels ?
29. If 6 apples are worth 18 cents, how many apples
must be given for 5 oranges worth 6 cents each ?
30. How many horses can eat in 9 days, the same
amount of hay that 12 horses eat in 6 days?
MISCELLANEOUS REVIEW. 81
31. If 5 men earn $30 in 3 days, how much will 2
men earn in the same time ? How much will 2 men earn
in I day?
LESSON II.
1. How many times 6 in 3-fifths of 40?
2. How many times 3-sevenths of 14 in 54 ?
3. How many times 5-sixths of 12, in 4-ninths of 72?
4. How many times 3-fifths of 20, in twice that num-
ber of which 14 is 7-ninths ?
5. If 3-eighths of a tun of hay cost $9, what will
5-sixths of a tun cost ?
6. If a man can earn $7 in 1 -fifth of a month, how
many dollars can he earn in 1 month ?
7. If a man can earn $42 in 1 month, how many
dollars can he earn in 1 -sixth of a month?
8. If a hogshead of sugar is worth $96, what are
7-eighths worth?
9. Five men paid $20 for a horse : what part of $20
did 2 men pay ?
10. If $7 will buy 56 yards of muslin, how many
yards will $4 buy ?
11. If 3 men can do a job of work in 16 days, in how
many days can 4 men do it ?
12. If 3 men spend $12 in 1 week, at the same, rate,
how many would 2 men spend in 6 weeks ?
13. If 6 men do a piece of work in 7 days, in how
many days can 3 men do it ?
14. If 5 men do a piece of work in 8 days, in how
many days can 4 men do a job twice as large ?
15. If 6 men do a job in 5 days, in how many days
can 2 men do a job half as large?
16. James had 16 apples : he kept 1-fourth himself
and divided the remainder equally among 3 of his com-
panions : how many did each receive ?
82 RAY'S INTELLECTUAL ARITHMETIC.
-**;
17. Three-fourths of 24, increased by 2-thirds of 12,
are equal to how many ?
18. Five-sixths of 24, diminished by 3-fourths of 20,
equal how many ?
19. One-half, and 2-thirds, and 3-fourths of 12 are
how many ?
20. Two-thirds of 12, less 1-half of 12, are 2-fifths of
what number?
21. From 10 take 3-fifths of itself; add to the remain-
der its 1-half: what is the result?
22. Thomas had 28 cents : he gave* 2-sevenths to his
sister, and 2-fifths of the remainder to his brother : how
many more did he give, than he had left?
23. James had 35 marbles : he gave to Thomas
3-sevenths, to Charles 2-fifths : to which did he give the
most, and how many ? What number had he left ?
24. Thomas had 28 : he kept 2-sevenths, and divided
the remainder equally among his 4 brothers: how many
dollars did each receive ?
25. A grocer had 14 barrels of flour : he sold 4-sev-
enths at S3 a barrel, and the remainder at 5 a barrel :
what did it amount to ?
26. Bought 15 yards of cloth, at 82 a yard : I sold
1 -third at 4 a yard, 2-fifths at 83 a yard, and the re,
mainder at 85 a yard : how much did I gain ?
27. Bought 10 yards of cloth for 90, and sold 2-fifths
of it for 840 : how much a yard did I gain ?
28. Two men travel the same direction : A is 40 miles
ahead of B ; but B travels 23 miles a day, and A 18 :
in how many days will B overtake A ?
29. A hare is 90 yards in advance of a hound : the
hound goes 10 yards in a minute, and the hare 7 : in how
many minutes will the hound overtake the hare? How
far will each run ?
30 .If a hound runs 7 rods while a hare runs 4,
far will the hare run while the hound runs 35 rods ?
MISCELLANEOUS KEVIEW. 83
31. One man is pursuing another, who is 4 miles in
advance : he travels 3 miles in pursuit, while the other
advances 1 : how much aoes he gain in going 3 miles ?
How far must he travel to overtake him ?
32. C and D travel in tne same direction : C is 15 miles
ahead of D ; but D travels 5 miles an hour, and C only
2 : in how many hours wnl L overtake C ? How far will
D have traveled ?
33. A cistern containing 2 gallons, is filled by a pipe
at the rate of 8 gallons an hour, and emptied by a pipe
at the rate of 5 gallons an hour : if both pipes are open,
how many gallons will remain in the cistern each hour?
How long will the cistern be in filling ?
34 .A cistern containing 36 gallons has 2 pipes ; by
the first it receives 6 gallons an hour, and by the second
it discharges 9 gallons an hour : if both pipes are left
open, how long will it take to empty the cistern ?
35. My pants cost $8, which were 2-fifths the cost of
my coat ; my vest cost 1-haif as much as my pants : what
was the cost of the whole ?
LESSON III.
1. What are the divlbors of a number called ? Ans.
Its factors.
2. What is the divisor of a number ? Ans. Any
number which is contained in it an exact number of times;
that is, without a remainder.
3. What is a multiple of a number ? Ans. Any
product of which that number is a factor.
4. Twelve can be divided by 2, 3, 4, and 6 : what are
the factors of 12 ? of 24 ?
5. What 2 numbers multiplied together, will make
28 ? What are the factors of 28 ?
6. What numbers w'll divide 22? What are th*
factors of 22 ?
84 RAY'S INTELLECTUAL ARITHMETIC.
7. Six is a factor of 24; what number is a multipk
of 6 ? of 12 ?
8. Seven is a factor of 21 ? What number is a mul-
tiple of 7 ? of 3 ?
9. What three numbers are multiples of 7 ? of 8 ?
of 9? of 10?
10. What numbers will divide 18? 27? 33? 36?
39? 48? 49? 63? 64? 65? 72? 75?
11. What number will divide both 10 and 15? 12
and 16 ? 20 and 32 ? 64 and 80 ? 72 and 96 ?
12. What is the greatest number less than 12, that will
divide 12 ? Less than 16 that will divide 16 ? Less than
20 that will divide 20 ?
13. What i? the greatest number that will divide both
6 and 10 ? 12 and 18 ? 20 and 32 ? 24 and 36 ? 42
and 48 ? 56 and 72 ?
14. What numbers can be divided by 9 ? 12 ? 8 ?
7? 16? 14? 15?
15. What is the least number that can be divided by
2 and 3? 4 and 7 ? 2 and 9 ? 7 and 21 ? 10 and 12?
12 and 20? *
The answers to the 13th question are called the Greatest
Common Divisors ; to the 15th question, the Least Common
Multiples or Dividends.
SECTION XIV. FRACTIONS.
LESSON I.
A single or whole thing of any kind, is called a unit
one ; as 1 yard, 1 dollar, 1 mile, 1, &c.
1. The above line represents a yard of tape : it is
called 1 yard.
FRACTIONS. 85
2. If you divide it into two equal parts, one of the
$arts is called one-half of a yard.
one-half. one-half.
One part is represented thus, |, and is read one-half.
3. If the yard of tape is divided into three equal
parts, one of the parts is called one-third; two parts are
called two-thirds.
one-third. one-third. one-third.
One part is represented thus, ^, and is read one-third;
2 parts are represented thus, |, read two-thirds.
4. If the yard of tape is divided into four equal parts,
one of the parts is called one-fourth; two parts two-fourths;
3 parts three-fourths.
one-fourth. one-fourth. one-fourth. one-fourth.
One part is represented thus, |, and is read one-fourth ;
2 parts are represented thus, |, read two-fourths; 3 parts
thus, |, read three-fourths.
5. If the yard of tape is divided into five equal parts,
one of the parts is called one-fifth; two parts two-fifths;
three parts three-fifths.
A FRACTION is a part of a unit, or one. In every
fraction the unit is divided into equal parts. The lower
number shows into how many equal parts the unit is
divided : the upper number, how many parts are tak^n,
2d Bk. 6
86 RATS INTELLECTUAL ARITHMETIC.
The lower number is called the DENOMINATOR, because
it denominates or names the parts : the upper number the
NUMERATOR, because it numbers the parts.
In reading fractions, that is, in expressing them bj '
words, read the Numerator first j the Denominator last.
6. If you divide a yard into 6 equal parts, whai
figures will express 2 parts ? What 5 parts ?
7. If you take away 2 of the 6 parts, what fraction
will express the remainder ?
8. How can you express in figures 3-sevenths of 1 ?
9. In the fraction f , into how many parts is the unit
(one) divided, and how many are represented ?
10. Read the following Fractions:
11111111 i i i i i i i
~g> 3> o? 6> 7) 6' ? To> TT> 72> 7.3? 74> To? 76? T7>
i I, I, I. 4, I i , T 3 j> T 9 T, IS. T 9 a> if, M> & IS-
11. What does the fraction f represent? Into how
many equal parts is the unit (one) divided?
12. What do you understand by T 9 T ? j-f ? |f ?
13. Divide an apple into 10 equal parts, and take away
3 parts : what fraction will express the remainder ?
14. What fraction will represent 1-half of 1 ? 1-half
of 3 ? 1-third of 5 ?
15. Which is the greater fraction, 4 or 4 ? - or j ?
ior? |or|?
16. What is a fraction ? What does the denominator
of a fraction show ? What does the numerator show ?
17. How many apples in 2-halves of an apple ? In
3-thirds ? In 4-fourths ? In 5 -fifths ?
18. When the numerator of a fraction is equal to the
denominator, as |, |, |, &c., what is its value ? Ans. 1.
19. Are 3-halves of an apple more or less than a whole
apple ? When the numerator is greater than the denom-
inator, is the value of the fraction greater or less than 1 ?
FKACTIONS, 7
20. If jker denominators of fractions be increased, do
the fractions become greater or less ? Why ?
21. If the numerators be increased, do the fractions
become greater or less ? Why ?
22. Such fractions as , |, |, &c., being either equal
to, or greater than 1, are called improper fractions; those
less than 1, as ^, |, &c., proper fractions.
23. Whole numbers and fractions joined together, as
14, 2|, 5j, 6|, 10|, &c., are called mixed numbers.
SECTION XV.
LESSON I.
1. How do you find the number of halves in any
number ? Why ?
2. In 1, how many halves ? In U ? In 2 ? In 2^ ?
In 3 ? In 3J ? In 4 ? 4 ' ? 5 ? 6 ? 7 ? 7i ?
3. How do you find how many thirds there are in
any number ? Why ?
4. How many thirds in 1 ? In 1| ? In If ? In 2 ?
In 21? In 23? 3? 31? 3 ? 4? 4 4 ? 5? 6?
O O DO O
5. How do you find how many fourths there are in
any number ? Why ?
6. How many fourths in 1 apple? In 1 and 1-fourth?
In 1 and 2-fourths ? In 1 and 3-fourths ? In 2 ? In 2
and 3-fourths ? In 3 ? In 4 and 3-fourths ?
7. How find how many fifths are in any number ?
8. How many fifths in 1 ? In H ? In If? In If ?
In 11 ? In 2 ? In 2i ? In 2| ? In 2| ? In 2| ? ' IB
B? "inSf? In3?In4f? In 54?
9. In 4 and 2-sevenths how many sevenths ?
10. In 5 and 3 -eighths how many eighths ?
Jl. In 4 and 7-ninths how many ninths?
88 RAY'S INTELLECTUAL ARITHMETIC
12. In 5 and 3-tenths how many tenths ?
13- In 12 and 1-third how many thirds?
14 In 11 and 1-half how many halves ?
15- In 6 and 3-fifths how many fifths ?
16* In 7 and 3-fourths how many fourths?
17. In 6 and 1-sixth ^ how many sixths ?
18. In 5 and 7-eighths how many eighths ?
19. In 11 and 4-sevenths how many sevenths ? J
20. In 7 and 9-tenths how many tenths?
LESSON II.
1. James's brother gave him 3-halves of an apple:
how many apples did he give him ? Ans. One apple
and 1-half.
2. In 3 halves how many ones ?
ANAL. Since 2-halves make 1, in 3-halves are as many ones
as 2-halves are contained times in 3-halves; 2-halves in
3-halves, one and one-half times. Ans. 1 and 1-half.
3. John's father gave him 4 half - dollars : how many
dollars did he give him ?
4. How many ones in 4-halves? in 5-halves? in
6-halves ? in 7 -halves ? in 8-halves ? in 9-halves ?
5. How do you find how many ones there are in any
number of halves?
6. If you divide an orange into 3 equal parts, what is
1 part called ? How many thirds in 1 orange ?
7. If 1 orange make 3-thirds, how many oranges would
there be in 3-thirds of an orange? in 4-thirds of an
orange? in 5 -thirds? in 6 -thirds?
8. How many ones in 3-thirds? in 4-thirds? in
5-thirds ? in 6-thirds ? in 7- thirds ? in 8-thirds ?
9. How do you find how many ones there are in any
number of thirds ?
FRACTIONS. 89
10. In 4-fourths of a dollar how many dollars ?
In 5-fourths? In 6-fourths ?
In 7-fourths? In 8-fourths ?
In 9-fourths? In 11-fourths?
In 13-fourths ? In 15-fourths ?
In 18-fourths ? In 19-fourths ?
11. How many ones in 4-fourths ? in 5-fourths ? in
6-fourths? in 7-fourths? in 8 fourths? in 9-fourths?
11-fourths? 13-fourths? 15-fourths? 19-fourths?
12. How do you find how many ones there are in any
number of fourths ?
13. In 5 -fifths of an orange, how many oranges ?
In 6-fifths? In 7-fifths?
In 8-fifths? In 9-fifths?
In 10-fifths ? In 11-fifths ?
Inl3-fifths? Inl5-fifths?
Inl7-fifths? In 18-fifths?
14. How many ones in 5-fifths ? 6-fifths ? 7-fifths ?
8-fifths? 9-fifths? 10-fifths? 11-fifths? 13-fifths?
15. How do you find how many times 1 there are in
any number of fifths ?
16. How many times 1, that is, how many whole ones,
in 23 -sixths ? Ans. Three and 5 -sixths.
17. In 30-sevenths are how many times 1 ?
18. In 35-sevenths ? 22. In 23-halves ?
19. In46-ninths? 23. In 3 3 -fifth s ?
20. In 53-tenths ? 24. In 31-fourths ?
21. In37-thirds? 25. In37-sixths?
26. In 47-eighths are how many whole ones ?
27. In 81-sevenths ? 30. In75-tenths?
28. In79-tenths? 31. In 89-elevenths?
29. In 53-ninths ? 32. In 93-twelfths ?
90
KAY'S INTELLECTUAL ARITHMETIC.
SECTION XVI.
LESSON I.
ILLUSTRATION. The first line represents a yard of
tape divided into two equal parts ; the second, a yard of
tape divided into four equal parts.
one-half. one-lialf.
one-fourth. one-fourth. one-fourth. one-fourth.
One-half Q), is equal to two-fourths (f).
1. If I give to Mary 1-half of an orange, and to Jane
1 -fourth, how much more will Mary have than Jane?
what part will be left ?
2. How much greater is 1-half than 1 -fourth ? How
much are 1-half and 1 -fourth?
3. James divided a melon, giving to his sister 1-half,
and to his brother 1 -fourth : what part did he give away ?
4. Thomas gave 3-fourths of a dollar for a geography,
and 1-half of a dollar for an arithmetic and slate : how
much did he give for both ?
5. How much are 1-half and 3-fourths?
ILLUSTRATION. The first line represents a yard of
ribbon divided into 2 equal parts; the second, a yard
divided into six equal parts.
one-half. one-half.
I -sixth. 1 -sixth. 1- sixth. I-sixth. 1-sixth. 1-sixth.
6. One-half is how many sixths? 1-third is how
many sixths ? 2-thirds are how many sixths ?
FRACTIONS. 91
7. James received 1-half of an orange, and Charles
I-third : how much more had James than Charles ?
J3. How much are 1-half and 1 -third?
9. If 1 -third is 2-sixths, how many sixths are there
in 2-thirds ?
10. A yard of flannel costs half a dollar : a yard of
sloth 2-thirds of a dollar : how much do both cost ?
11. James bought 2 melons; he gave to Lucy half of
the first ; to Jane 2-thirds of the second : what part of a
melon had Jane more than Lucy ?
12. How much greater are 2-thirds than 1-half?
13. After taking away 1-half and 1 -third of an apple,
what part will be left ?
14. I wish to divide an orange, and give to Mary 1-half,
to Jane 1 -fourth, and to William 1-eighth: how must 1
divide it ? how many eighths will each have ?
15. Two -fourths are how many eighths ? 3 -fourths are
liow many eighths ?
16. One-fifth is how many tenths? 2-fifths are how
many tenths ? 3-fifths ? 4-fifths ?
17. Thomas wishes to divide an orange, and give Ann
1-half, and Lucy 2-fifths: how must he divide it? what
part will he have left ?
18. How much are 1-half and 2-fifths?
19. One-third is how many twelfths? 1-fourth is how
many twelfths? How many twelfths in 2-thirds? in
2-fourths ? in 3-fourths ?
20. A farmer sows 1-half of a field in wheat, 1-third
in rye ; the rest in barley : how many twelfths are in
wheat ? how many in rye ? how many in barley ?
21. How many twelfths in 1-fourth? in 1-fourth and
1-third?
22. David bought a pound of figs: he gave 1-third of
them to his mother, 1-fourth to his sister, and 1 -sixth to
his brother : what part had he left ?
2 RAY'S INTELLECTUAL ARITHMETIC.
23. How many eighths are in 1-half? in 3-fourths?
24. How many tenths are in 1-half? in 1 -fifth?
in2-fifths? in 3 -fifths ?
25. How many fifteenths in 1-third? in 1-fifth?
in 3-fifths ? in 4-fifths ?
26. How many twentieths in 1-half? in 1 -fourth ?
in 1-fifth ? in 3-fourths ? in 3-fifths ?
27. Reduce -f and f to twelfths : ^ and f to fifteenths :
also, i and | to tenths.
28. Reduce f and f to twentieths : ^, |, and 1- to twen-
tieths.
2& Reduce | and ^ to fourteenths: ^ and f to eight-
eenths.
30. Reduce f, f, and | to twenty-fourths : | and T 3 to
fortieths.
REMARK. When two or more fractions have the same de-
nominator, they are said to have a common denominator;
thus, | and | have 6 for a common denominator.
When the denominators of two fractions are not the same, a
common denominator may be found by multiplying the denomi-
nators together.
Find the Common Denominator
31. Of and f .
1 and f
1 and |.
32. Of | and f .
1 an <* f
^ and jl.
33. Of | and |.
| and |.
i and 4.
34. Of f and f .
| and f .
| and f .
35, Of f and f
i and f .
| and |.
36. Of | and Jfc.
-|and/ T .
and ^
37. Of 1 and f
| and y^.
1 and |.
38. Of | and |.
1 and |.
1 and |.
39.Reduce i, |, f,
and j| to thirty-seconds.
40.Reduce 4, |, U
, and 4| to fif
fcy-fourtha.
FRACTIONS.
93
41.Reduce |, |,
42. Reduce -J, ^,
j, T 7 3 , and ^ to forty-eighths.
J, ^, , T V, and y^ to seventy-seconds,
LESSON II.
ILLUSTRATION. The first line represents a yard of
ribbon divided into 2 equal parts ; the second line, a
yard divided into 4 equal parts.
one-half. one-half.
one-fourth. one-fourth. one-fourth. one-fourth.
The first of the lines below represents a yard divided
into 3 equal parts ; the second, a yard divided into 6 equal
parts. From this it is seen that ^ is equal to f ; and
that | are equal to |.
one-third. one-third. one-third.
1-sixth. I-sixth. \-sixth. 1-sixth. 1-sixth. 1-sixth.
REMARK. The Numerator and Denominator are called the
terms of the fraction; when these are the smallest numbers of
-which the fraction admits, it is said to be in its lowest terms.
1. Reduce to its lowest terms. Divide loth terms ly
the greatest number (3), that will exactly divide loth.
2. Reduce | to its lowest terms. to its lowest
terms. | to its lowest terms.
3. Reduce to their lowest terms, |, .
4. Reduce to their lowest terms, j 2 ^, j 5 ^.
5. Reduce to their lowest terms, y 4 ^, T % -j%, -,.
6. Reduce to their lowest terms, y^, y%, y%, -f%, T 9 2, yf
7. Reduce to their lowest terms. ^, -f~, y%, y^, |f , yf .
94 RAY'S INTELLECTUAL ARITHMETIC.
8. Reduce to their lowest terms, T %, T 6 6 , T %, g, ft, ft.
9. Reduce to their lowest terms, T 2 g , T 4 g , 7 * g , T 8 g , ig, if.
10. Reduce to their lowest terms, 3 6 5 , o 3 T , ^, g, i, if.
11. Reduce to their lowest terms, ||, if, if, j, f {j, f |,
SECTION XVII.
LESSON I.
1. David divided an orange, giving to William 1 -fifth,
and to Sarah 2-fifths : how many fifths did he give away ?
how many fifths had he left ?
2. After taking from any thing 3-fifths of itself, what
part will be left?
3. John bought a quart of chestnuts : he gave 2-sixths
to Mary, and 3-sixths to Eliza : how many sixths did he
give away ? what part had he left ?
4. One-fifth, 2-fifths, and 7 -fifths of an orange, are
how many fifths ? how many oranges ?
5. How much are i, f, and J?
6. One -eighth, 3-eighths, and 7-eighths of a dollar
are how many eighths? how many dollars?
7. What is the sum of f , |, and | ?
8. Daniel's mother gave him $J, and his uncle $J:
how many sixths of a dollar did he receive from both ?
ANALYSIS. One-half is 3-sixths, and \-third is 2-sixths :
3-sixths and 2-sixths are 5-sixths. Ans. 5-sixths of a dollar.
The fractions are reduced to a common denominator before
adding, because you can not add things of different kinds.
This is the first step in adding or subtracting fractions.
9. Lucy divided an orange, giving to her sister 1 -third,
and to her brother 1 -sixth : how many sixths did she
give ? how many sixths did she have left ?
FRACTIONS. 95
10. James bought a lemon, and gave to Lucy 1 -eighth,
and to Susan 1 -fourth: how many eighths did he give?
what part did he retain ?
11. William cut a pine-apple, and gave to Mary
1 -third, and to Eliza 1-ninth: how many ninths did he
give to both ? what part did he retain ?
12. Thomas bought a copy-book for fty 1 ^, and a reader
for $J : how many tenths of a dollar did they both cost?
13. David gave 1 -fourth of a melon to Eliza, 1 -third
to his mother, and kept the remainder : how many twelfths
did he give ? what part did he retain ?
14. I bought 1^- yards at one store, and 2J yards at
another : how many yards did I purchase ?
15. I planted 2J acres of ground in corn ; 8| acres in
oats : how many acres were in both pieces ?
16. John bought a knife for $J, a slate for $J, and a
book for $f : how much did the whole cost ?
17. Add A and f. and f . f and |.
18. Add -| and f . and i. \ and \.
19. Add i and 4. f and If. | and f .
20. Add | and f . f and f . f and f .
21. Add 1| and 2i. 1, 1, |, and f
22. Add 3f and 4|. f , |, |, and .
23. Add 4f and 5f 1', 2|, |, and6 T %.
24. Add 5| and 4|. |, 3j, 4, and 5 f .
LESSON II.
1. Mary had 3-fourths of an orange ; she gave her
uister 1 -fourth : how many fourths had she left?
2. If you take 1-fifth from 3-fifths, what will be left ?
3. I bought 5-sixths of a quart of nuts, and gave
3-sixths to my sister : how many sixths had I left ?
4. One is how many sixths ? If you take 5-sixths
from 1, what will be left?
96 RAY'S INTELLECTUAL ARITHMETIC.
5. Two is how many fifths? If you take 3-fifths
from 2, what will be left ?
6. If you take i from f , what will be left ?
7. | from f? ' 10. T % from Jfc?
8. | from I? 11. -| from 1?
9. from I ? 12. f from 2 ?
13. Thomas divided an orange, giving his brother
1-half, and his sister 1 -third : how many sixths did each
receive? how much more the brother than the sister?
14. If a bushel of wheat cost $J, and of corn $J, how
much will the wheat cost more than the corn ?
15. If you take \ from f, what will remain ?
16. Joseph bought a quart of chestnuts, and gave 1-half
of them to his mother, and 1-sixth to his sister : how
many sixths did he give his mother more than his sister ?
17. Take \ and \ from 1, what will be left?
18. Jane divided an orange, giving to her sister
3-eighths, and to her brother 1 -fourth : to which did she
give the most ? what part of the orange was left ?
19. A man having 72 miles to travel, went 1 -third the
distance the first day, 2-ninths the second, the remainder
the third day : how many ninths of the distance did he
travel further on the first day, than on the second ? what
part did he travel the last day ?
20. If you take f from \, what will be left? If you
take | and -J from 1, what will be left?
21. If you take 1 from ^, what will be left?
ANALYSIS. \-fiftli is 2-tenths, and l-half is 5-tenths;
2-tenths from 5-tenths leave 3-tenths. Ans. 3-tenths.
What will be left if you take
om J ? I from | ? 1 fr
om ? f from ? fr
24. * from | ? 4 from f ? I from f ?
o 4 o 55 O
22. | from J ? I from | ? 1 from \ ?
23. 1 from ? from ? from f ?
FRACTIONS. 97
25. I from f ? J> from f ? ^ from -| ?
26. from f ? f from | ? f from | ?
27. J from f ? | from f ? * from 2 ?
LESSON III.
1. Mary divided a quart of pecans, giving Ann 1 -third,
and Jane 1-fourth of them : what part had she left?
2. A farmer had 1^ bushels of wheat : he gave to 1
poor man J of a bushel, and to another | of a bushel :
how much wheat was left ?
3. James had ^ of a pound of raisins : he gave to
his brother i of a pound, and to his sister \ of a pound :
how much had he left ?
4. If from 3^ bushels of corn, 1^ bushels be taken, how
much will there be left?
5. A lady bought 3| yards of muslin at one store,
and 2\ yards at another : after using 14 yards, how much
had she left ?
6. William's father gave him $| : he gave to a poor
person $|, for apples S-Jg, and for a book $| : what part
of a dollar had he left ?
7. James's mother gave him a book : he read the first
day|, ^he second | , the third ^, and the fourth the re-
mainder : what part did he read the fourth day ?
8. A farmer has a flock of 84 sheep in 4 fields : the
first contains i, the second J, and the third \ of them :
what part does the fourth field contain ?
9. Daniel spends -A of his time in sleep, \ of it at
school, y 1 ^ in reading, and ^ in learning music: what
part of his time is not employed?
10. A pole is standing in a pond; ^ of it is in the air,
pud | in the water : what part is in the earth ?
11. A student devotes | of his time to sleep, -| % to
Btudy, 75^ to reading, ^ to exercise, and T V to deeds of
charity 7 what part of his time is unemployed ?
98 RAY'S INTELLECTUAL ARITHMETIC.
12. One-third of an orchard is apple trees, ^ pear trees,
| plum trees, 7 V quince trees, and the remainder peach
trees : if the orchard contain 96 trees, how many are
there of each kind ?
13. After spending ^ and -J- of my money, and los-
ing jV, I had 8 remaining : how much had I at first?
14 .The difference between | and i of my money
is $3 : how much have I ?
15 .1 ate f of my peaches, gave away 10, and have 10
left : how many had I at first ?
SECTION XVIII.
LESSON I.
1. A father gave each of his two sons half a dollar:
how much did he give them both ?
2. A mother gave each of her 3 children half an
orange : how many half oranges did it take ? how many
oranges ? why ?
3. What are 3 times 1-half? 4 times 1-half?
4. John fed 5 horses, giving to each half a peck of
oats : how many half pecks did it take ? how many pecks ?
why?
5. What are 5 times 1-half? 6 times 1-half?
6. What are 8 times 1-half? 9 times 1-half?
7. James gave 1 -third of an orange to each of his
sisters : how much did he give to both ? why ?
8. If a man can eat 1 -third of a pound of meat in 1
day, how much can he eat in 3 days ? why ?
9. What are 4 times 1 -third ? 5-times 1 -third ?
10. John gave 2-thirds of a pine-apple to each of his 2
brothers : how many thirds did he give to both ? How
many pine-apples did it take ? Ans. 1 and 1-third.
FRACTIONS.
99
11. What are 4 times 2-thirds? 5 times 2-thirds?
12. Thomas gave 1 -fourth of an apple to each of his 3
playmates : how many fourths did it take ? why ?
13. If 1 bushel of oats cost 1 -fourth of a dollar, how
much will 4 bushels cost ? why ?
14. Charles gave 3-fourths of a pint of chestnuts to
each of his 2 brothers : how many fourths of a pint did
it take ? how many pints ? why ?
15. Mary gave 3-fourths of an orange to each of her
three brothers : how many fourths of an orange did it
take ? how many oranges ? why ?
16. What are 5 times 3-fourths ?
ANALYSIS. 5 times 3-fourths are I5-fourths ; 15-fourths
are as many ones as 4-fourths are contained times in
15-fourths ; 4-fourths in 15-fourths, 3 and 3-fourths times.
Ans. 3 and 3-fourths.
17. What are 6 times | ?
18. What are 3 times | ?
19. What are 3 times 4 ?
20. What are 5 times ?
21. What are 2 times ^ ?
What are 8 times f ?
What are 5 times X ?
22.
23.
24.
25.
26.
What are 7 times f ?
What are 3 times
What are 7 times
3?
I?
7
times
|
?
8
times
<i
?
5
times
2
?
6
times
5
?
4
times
?
5
times
4
?
2
times
|
?
4
times
F
?
4
times
~y
?
5
times
|
V
6
times
5
?
7
times
?
8
times
!
9
9
times
3
?
6
times
7
?
8
times
-|-
?
4
times
f
?
6
times
8
7
?
8
times
4
?
9
times
|
?
4
times
fr
-?
7
times
rt
.!
27. What are 3 times T 3 ? 4 times
LESSON II.
1. William gave an orange and a half to each of
his 2 sisters : how many oranges did it take ?
2. What are 2 times 1 and 1-half ?
3. What are 3 times H ? 4 times 11 ? 5 times l-.J ?
100 BAY'S INTELLECTUAL ARITHMETIC.
AXAL. 3 times I are 3, and 3 times l-half are 3-halves,
equal 1 and l-half; this added to 3 makes 4 and l-half.
4. If 1 bushel of wheat cost $1 and 1 -third of a dol-
lar, what will 2 bushels cost ?
5. How many are 3 times lJ ? 2 times 2f ?
6. How many are 3 times 3| ? 4 times 4^ ?
7. How many are 5 times 2| ? 6 times 3| ?
8. How many are 8 times 3| ? 9 times 4| ?
9. If 1 bushel of barley cost $1 and 1-fourth of a
dollar, what will 3 bushels cost ? 4 bushels ?
10. How many are 5 times 1| ? 6 times 1| ?
11. How many are 2 times 1| ? 3 times 2| ?
12. How many are 4 times 3i ? 5 times 3| ?
13. How many are 6 times 3| ? 8 times 3| ?
14. How many are 7 times 2| ? 9 times 2| ?
15. How many are 10 times 1| ? 10 times 3^ ?
16. How many are 11 times 2| ? 12 times 3| ?
17. If a family consume 3 and 1 -fifth barrels of flour
io one month, how much will they require for 3 months ?
18. How many are 4 times 3| ? 5 times 3| ?
19. How many are 2 times 6f ? 3 times 2| ?
20. How many are 6 times 4i ? 6 times 3| ?
21. How many are 7 times 4| ? 8 times 3| ?
22. How many are 9 times If ? 9 times 3 \ ?
SECTION XIX.
LESSON I.
1. If you divide an apple into two equal parts, >yhat
I part called? what is l-half of 1 ?
FRACTIONS. 101
2. If you divide 3 apples between 2 boys, how many
apples will each have ? how will you divide them ? what
is 1-half of 3?
ANALYSIS. l-half of I is l-half, and l-half of L is 3
times l-half of 1, which are 3-halves ; 3-halves are as many
ones as 2-halves are contained times in %-halves; 2-halves in
Sj 1 and l-half times. Ans. 1 and l-half
3. John bought 5 oranges, and divided them between
his 2 sisters : how many oranges did each have ?
4. What is l-half of 6? f of 7? Jof8? Jof9?
of 11? of 13?
5. If cloth is SI a yard, what must be paid for 1 -third
of a yard ? what is 1 -third of 1 ?
6. James has 2 oranges to divide among his 3 sisters :
how must he divide them, and what part of an orange
will each have? what is 1 -third of 2?
7. If 3 bushels of pears cost $4, how much is that a
bushel ? what is 1 -third of 4?
ANALYSIS. l-third of 4 is the same as 4 times l-third of
1, which are 4-thirds ; 4-thirds are as many ones as 3-thirds
are contained times in 4-thirds: 3-thirds w 4-thirds 1 and
l-third times. Ans. 1 and l-lhird.
8. A carpenter receives ?5 for 3 days' work: how
much. is that a day? what is l-third of 5?
9. What is l-third of 6? J of 7 ? J of 8 ? Jof9?
of 10? of 11? of 13?
10. What is l-third of 16? of 17? of 18? of 19?
of 20 ? of 21 ? of 22 ?
11. What is l-third of 23? of 24? of 25 ? of 26?
of 27? of 28? of 31?
12. If 4 bushels of oats cost $1, what part of a doL
lar will 1 bushel cost? what is 1 -fourth of 1 ?
13. A mother divided 3 pears equally among her 4
children, what part of a pear did each receive ?
2d Bk. 7
102 BAY ; S INTELLECTUAL ARITHMETIC.
14. What isl-fourthof 4? Jof5? Jof6? of 7?
of 8? of 9? of 10? of 11?
15. What is 1 -fourth of 12 ? of 13 ? of 15 ? of 17 ?
of 19? of 23?
16. What is 1 -fourth of 25 ? of 26 ? of 27 ? of 28 ?
of 29 ? of 30 ? of 31 ?
17. If a melon cost 5 cents, what part of a melon can
you buy for 1 cent? what is 1 -fifth of 1 ?
18. James had 2 pine-apples, and divided them among
5 of his companions : what part of a pine-apple did each
have ? what is 1 -fifth of 2 ?
19. What is 1-fifth of 3? of4? i of 5 ? of 6?
of 7? of 8.? of 9? of 10? of 11? of 12?
20. What is 1-sixth of 1 ? J of 2 ? I of 3 ? of 4 ?
of 5? of 6? of 7? of 8? of 10? of 15?
21. What is 1-ninth of 2? ' of 4? ^of7? of 8?
of 9? of 12? of 15? of 17? of 19?
22. How do you find one-half of any thing? one-
third? one-fourth? one-fifth? one-eighth?
LESSEN II.
1. If 3 bushels of wheat cost $1, what part of $1
will 1 bushel cost ? what part will 2 bushels cost ?
2. What is 1 -third of 1 ? what are 2-thirds of 1 ?
3. If 3 bushels of wheat cost $2, what part of 81
will 1 bushel cost ? what part will 2 bushels cost ?
ANAL. 1 bushel will cost l-third of $2, which is 2-thirds;
and 2 bushels will cost twice as much as 1 bushel; that is>
2 times 2-thirds, which are 4-thirds of $1, equal $1 and
l-third of a dollar. Ans. $1 and l-third.
4. If 3 barrels of cider cost $4, what part of a dol-
lar will 1 barrel cost ? what part will 2 barrels cost ?
5. What is l-third of 4 ? 2-thirds of 4 ?
6. What is l-third of 5 ? 2-thirds of 5 ?
FRACTIONS. 103
7. What is 1-third of 6? 2-thirds of 6?
8. What is 1-third of 7 ? 2-thirds of 7 ?
9. What is 1-third of 8 ? 2-thirds of 8?
10. What is 1-third of 9 ? 2-thirds of 9 ?
11. What is 1-third of 10 ? ' 2-thirds of 10 ?
12. If 4 barrels of apples cost $3, what part of a dol-
lar will 1 barrel cost ? 2 barrels ? 3 barrels ?
, 13. If 5 apples cost 2 cents, what part of a cent will
1 apple cost ? 2 apples ? 3 apples ? 4 apples ?
14. What are 2-fifths of 13 ? 3-fifths of 13 k
15. What is 1-sixth of 7 ? 5-sixths of 7 ?
16. What are 3-sixths of 10 ? 4-sixths of 10?
17. What is 1-seventh of 9 ? 3-sevenths of 9 ?
18. What are 2-sevenths of 11 ? 3-sevenths of 12 ?
19. What is 1-eighth of 13 ? 3-eighths of 13?
20. What are 5-thirds of 4 ? 7-thirds of 5 ?
21. What are 7-eighths of 10 ? 9-eighths of 3 ?
22. What are 3-tenths of 7 ? 7-tenths of 3 ?
23. What are 5-ninths of 11 ? 9-fifths of 7 ?
24. Which is the greater 4 of 2. or * of 1 ? i of 3.
or f of 1 ?
LESSON III.
1. James bought 3 lemons for 7 cents: how much
was that apiece?
2. William bought 5 quarts of chestnuts for 18 cents :
at that rate, what was the cost of 2 quarts ?
3. If 7 pounds of cheese sell for 40 cents, how much
should 5 pounds sell for?
4. Bought 6 yards of muslin ; gave my mother 3-fifths,
and kept the remainder : how many yards had each ?
5. If a quantity of provisions serve 2 men 7 days,
how long will it last 1 man ? how long 3 men ?
104 RAY'S INTELLECTUAL ARITHMETIC.
6. If 4 men perform a job of work in 8 days, how
long will it require 5 men ?
ANALYSIS. It will take 1 man 4 times as long as 4 men ;
*,nd 5 men \-fifth as long as 1 man.
7. If a barrel of cider will last 5 men 8 days, how
long will it last 3 men ?
8. If 1 barrel of flour serve 8 persons 10 days, how
long will it last 11 persons?
9. If 7 men can do a piece of work in 5 days, how
long will it require 8 men ?
10. If 2 men build a wall in 12 days, how long will
it take 7 men ?
11. If it requires 11 days, of 8 hours each, to do a
job, how many will be required of 10 hours each?
12. A man paid 37 cents for riding 8 miles : at the
same rate, what will it cost to ride 11 miles?
13. Two pipes of a certain size will empty a cistern
in 17 minutes : in what time will 3 pipes empty it ?
14. If 18 bushels of oats last 5 horses 1 week, how
many bushels will 7 horses require ?
15. If a laborer receive 5 bushels of wheat for 7 days'
work, how much should he receive for 11 days?
16. If a carpenter earn $8 in 5 days, how much will
he earn in 9 days ?
17. If 3 yards of cloth cost $13, what cost 8 yards?
18. A pole, 18 feet long, is 2-sevenths in the earth,
the rest in the air : what is the length of each part ?
19. Three men, A, B and C, found a bag containing
815: A got 2-ninths, B 1-third, and C the remainder:
what was the share of each ?
20. If J a yard of cloth cost 4, what cost 6 yards ?
21. If 3 ounces cost 36 cents, what should be charged
for of an ounce ?
22. If | of a number equal 18, what is the | of it?
FRACTIONS. ,05
23. How much must a man earn a day to receive $72
for 8 weeks, 6 days to a week ?
SECTION XX.
LESSON I.
ILLUSTRATION. If 1-half of a yard of tape be divided
into 2 equal parts, one of the parts is 1-half of 1-half;
that is, 1 -fourth of the whole yard.
one-half. one-half.
one-fourth. one-fourth. one-fourth. one-fourth.
If 1 -third of a yard be divided into two equal parts,
one of the parts is 1 -sixth of the whole yard; that is,
1-half of 1-third, is 1-sixth.
one-third. one-third. one-third.
l-sixth. 1-sixth. 1-sixth. 1-sixth. 1-sixth. 1-sixth.
1. Mary having 1-half of an orange, gave her brother
1 -half of what she had : what part of a whole orange did
she give him ?
2. James divided 1-third of an apple equally between
his 2 brothers ; what part did each receive ?
3. What is J of |? i of ?
ANALYSIS. 1-half of 1-third is the fraction obtained by
dividing a unit into 3 equal parts, and one of these parts
into 2 equal parts ; but if a unit (one) be thus divided, there
will be 6 parts, and each part will be 1-sixth of the whole;
therefore, 1-half of 1-third is 1-sixth. <
106 RAY'S INTELLECTUAL ARITHMETIC.
4. Thomas divided 1-half of a lemon equally between
his 3 sisters : what part did each receive ?
5. If 1 -fourth of an orange be divided into 2 equal
parts, what is 1 of the parts called ?
6. What is J of J ? J of J ?
7. If 1- third of an apple be cut into 3 equal parts,
what part of the apple will each piece be ?
8. If each half of an apple be divided into 5 equal
parts, how many parts will there be ? what is 1 part
called ? what is 4 of ?
9. If you divide an orange into 4 equal parts, and cut
each part into 3 pieces, what is 1 piece called ?
What single Fraction equals
10. I of
11. i of
12. 4 of
of |?
on?
? 4 of 4? 4 of 4? 4 of A?
4 * 5 ' 3 A g ' 5 U1 5 ' 3 UI 9
is. ion? i f i? iu? i f 4?
14. Thomas has 2-thirds of an apple, and wishes t<?
give his brother 1-half of what he has: what part of the
whole apple must he give him ?
15. What is J of J ? J of ?
ANALYSIS. I-half of 2-thirds is 2 times l-half of 1-third;
\-half of l-third is l-sixth; 2 times l-sixth are 2-sixths.
16. Daniel has 3-fifths of a melon to divide equally
between his brother and sister : how must he divide it
and what part of the whole will each receive ?
17. What is i of | ? i of | ? i of | ?
18. What is | of | ? 1 of ? i of | ?
19. What is I of 4 ? 4 of g ? -i of ?
20. What is 4 of | ? of 3 ? of f ?
21. Edward has 4-fifths of a melon, and wishes to give
his sister 2-thirds of what he has : what part of the whole
melon will she receive ?
FRACTIONS. 107
22. What'is of T % ? | of f ? f of f ?
ANALYSIS. 2-thirds of 4-jfifths are 2 times 1 -third of
trfifths, and l-third of 4-ffths is 4 times l-third of l-Jifth.
23. What are f of f ? | of f ? | of f ?
24. What are | of jj ? | of f ? | of | ?
25. What are f of f ? | of 4 ? | of T y
26. What are f of f ? f of _5_ ? f of f ?
27. If 1 yard of cloth is worth 2J bushels of wheat,
what is l-half of a yard worth ?
ANALYSIS. 2 and l-half are 5-halves, and l-half of 5-halves
is 5 times l-half of l-half , or 5-fourths, equal 1 and l-fourth.
Ans. 1 and l-fourth bushels.
What single Fraction will represent
28. i of 2-i ? i of 11 ? J of If ? | of 2?
29. A of 2| ? i of 3i ? i of 4| ? i of 51 ?
30. f of U ? | of 1| ? f of LI ? | of 1| ?
31. | of 21? | of 4| ? | of 2|? I of 3?
LESSON II.
1. A person owning | of a ship, sold | of his share :
what part of the whole ship did he sell ?
2. If 3 yards of cloth cost $2|, what cost 2 yards?
ANALYSIS. 1 yard will cost l-third as much as 3 yards,
and 2 yards will cost 2 times as much as 1 yard; that is,
2-thirds as much as 3 yards.
3. If 2 yards of cloth cost $1|, what cost 3 yards?
4. If 3 yards of cloth cost $5^, what part of that
sum will 2 yards cost?
5. If 5 gallons of molasses cost $3, what part of
that sum will 3 gallons cost ?
108 KAY'S INTELLECTUAL ARITHMETIC.
6. If 7 pounds of sugar cost l-i, what cost 4 pounds ?
7. If 8 pounds of butter cost $1J , what cost 7 pounds ?
8. If 7 yards of cloth cost $5|, what will be the cost
of 3 yards ? of 4 yards ?
9. If 3 barrels of cider cost $4|, what part of that
sum will 5 barrels cost?
10. If 5 gallons of oil cost $2|, what part of that sum
will 7 gallons cost ?
11. If 2 gallons of molasses cost $1|, what will be the
cost of 3 gallons ?
12. If 3 bottles of wine cost $2|, what will be the cost
of 8 bottles? of 10 bottles?
13. If 2 men do a job of work in 3| days, how many
days will it take 1 man ? 3 men ?
14. A man can do a job in 3| days, of 10 hours each:
how many days will it take of 7 hours each ?
15. If a man can do a piece of work in 15| days,
working 5 hours a day, how many days will it take,
working 8 hours a day ?
SECTION XXI.
LESSON I.
1. A boy had 2-thirds of an orange, which he divided
equally between his two sisters : what part of an orange
did each receive ?
ANALYSIS. Since 2 apples divided by 2 give 1 apple, and
$2 divided by 2 give $1, it follows that 2-thirds divided by 2
must give l-third. Each, therefore, must have received l-third
of an orange.
2. How often is 3 contained in 3-fourths ?
3. If 2 and 2-thirds dollars be divided among 4 men,
what is each man's share ?
FRACTIONS. 109
4. If 2f be divided by 7, what will be the result ?
5. How many times 6, in 3 and 3-fifths ?
6. How many thirds of 9, are contained in 3| ?
7. Divide 6 and 3-fourths cents among 9 boys.
8. A man having lOf acres of land, divided it equally
between his 5 children : how much did each receive ?
9. If 4f be divided by 11, what will be the quotient?
10. If $| be equally divided among 3 men, what part of
a dollar will each get ?
11. How divide a fraction by any whole number ?
LESSON II.
1. If 1 apple cost i a cent, how many will 1 cent buy?
why ? Ans. Because there are 2 half -cents in 1 cent.
2. If 1 pear cost 1- third of a cent, how many pears
can you buy for 1 cent? why?
3. When 1 bushel of wheat costs &|, how many bushels
can you buy for $H ?
4. How often is ^ contained in 2i or f ?
ANALYSIS. One-half is contained in Fiw-halves, as often
as 1 is contained times in 5 ; that is, 5 times : for, if 1 apple
is contained in 5 apples 5 times ; 1 cent in 5 cents, 5 times;
\-half must be contained in 5-halves, 5 times.
5. If 1 lemon cost half a cent, how many lemons can
be bought for 5 cents ?
6. If 1 apple cost 1 -third of a cent, how many apples
can John buy with 4 and 1 -third cents?
7. If 1 peach cost 2-thirds of a cent, how many
peaches can you purchase for 4 and 1 -third cents?
ANALYSIS. 4 and l-third are 13-thirds ; and Two-thirds are
contained in THiRTEEN-rffaVds as often as 2 is contained in 13,
that is, 6 and \-half times. Ans. 6^ peaches-
HO RAY'S INTELLECTUAL ARITHMETIC.
8. I distributed 2f bushels of wheat among a number
of poor persons, giving to each, 2-thirds of a bushel :
now many persons were there ?
9. At | a yard, how many yards of calico can be
purchased for &3f ?
10. At $| a yard, how many yards of , cloth can be
purchased for S3 ] ?
11. If a lemon cost 3-fourths of a cent, how many
can be purchased for 3| cents ? for 5 cents ?
12. One bushel of rye is worth 3-fourths of a bushel
of wheat: how many bushels of rye can be bought
with 4f bushels of wheat ? with 8| bushels ?
13. At 1 -fifth of a cent each, how many cherries can
Mary purchase with 3 cents ? why ?
14. At $? a gallon, how many gallons of vinegar can
you buy for ?2f ? for
15. How often is f contained in 2| ? in 4f ? in 61 ?
16. How often is J contained in 3| ? in 51 ? in 4| ?
17. How often is f contained in 1 ? in 3f ? in
18. How often is f contained in 3 ? in 4| ? in
19. At $| a yard, how much cloth can be had for
ANALYSIS. As many yards as l-third of a dollar, the price
of 1 yard, is contained times in I -half of a dollar; l-half is
equal to 3-sixths, and l-third to 2-sixths : 2-sixth.s in 3-sixths,
1 and l-half times. Ans. 1 and l-half yards.
20. At $^ a yard, how much gingham can be pur-
chased for $| ?
21. How often is i contained in | ?
22. If 1 yard of calico cost $A, how much can be pur-
chased for $| ?
23. How often is i contained in |?
NOTE. Reduce the fractions to a common denominator, then
divide their numerators.
FRACTIONS. 1 1 1
24. How often is ^ contained in | ? in f ? in { ?
25. How often is -| contained in f ? in f ? in }| ?
26. How often is f contained in f ? in | ? in i| ?
27. How often is | contained in f ? in | ? in T 9 ?
28. How often is U contained in | ? in | ? in 2| ?
29. How often is 2\ contained in ? in | ?. in 3| ?
30. How often is 3| contained in | ? in f ? in 5| ?
SECTION XXII.
LESSON I.
1. James gave his brother 1 apple and 1-half, which
Was 1-half of what he had : how many had he ?
2. If 1-third of a yard cost $l, what cost a yard?
3. If a man travel 1 and 1-third miles in 1- ourth of
an hour, how far can he travel in 1 hour ?
4. Daniel bought 1 -fifth of an orange for 3 J cents :
at that rate, what will a whole one cost ?
5. If 1 -fourth of a yard of tape cost 3 and 2-thirds
cents, what will 1 yard cost ?
ANALYSTS. 3 and 2-thirds are l-fourth of 4 times 3 and
2-thirds ; 4 times 2-thirds are S-thirds, equal 2 and 2-thirds ;
4 times 3 are 12, which added to 2 and 2-thirds^ make 14
und 2-thirds. Ans. 14 and 2-thirds.
6. 5 and 1-third are 1-half of what number?
7. 7 and 3-fourths are 1-third of what number ?
8. 5 and 3-eighths are 1 -fifth of what number?
9. 3 and 4-sevenths are 1-seventh of what number?
10. 4 and 4-fifths are 1-six.th of what number?
11. 9 and 2-thirds are 1 -eighth of what number?
112 RAY'S INTELLECTUAL ARITHMETIC.
12. If 2-thirds of a yard of ribbon cost 3 cents, what
will 1-third cost? If 1 -third of a yard cost 1 and 1-half
cents, what will a yard cost ?
13. One and 1-half is 1-third of what number ? Three
is 2-thirds of what number ?
14. If 2-thirds of a barrel of flour cost $5, what cost
1-third of a barrel ? If 1-third of a barrel cost $2J-, what
will a whole barrel cost ?
15. Five is 2 times what number ?
ANALYSIS. 5 is 2 times l-half of 5; 1-half of 5 is 2 and
1-half; therefore, 5 is 2 times 2 and 1-half
16. Five is 2-thirds of what number?
ANALYSIS. 5 is 2-thirds of that number of which 1-halJ-
of 5 is 1-third: 1-half of 5 is 2 and 1-half and 2 and 1-half
are 1-third of 3 times 2 and 1-half which are 7 and 1-half;
therefore, 5 is 2-thirds 0/7 and 1-half
17. If 3-fourths of a yard of velvet cost $4, what will
1 -fourth of a yard cost? If 1 -fourth cost $1|, what will
a whole yard cost ?
18. Four is 3-fourths of what number ?
19. If 2-fifths of a quart of chestnuts cost 3 cents,
what will 1 -fifth cost? If 1 -fifth of a quart cost 1 and
1-half cents, what will a whole quart cost?
20. Three is 2-fifths of what number ?
21. If 4-fifths of a yard cost $7, what will 1-fifth cost?
If 1-fifth of a yard cost $1|, what cost a yard?
22. Seven is 4-fifths of what number ?
23. If 3-eighths of a melon cost 4 cenfs, what will
1-eighth cost? what will a whole melon cost ?
24. If 3-fifths of a pound of sugar cost 10 cents, what
will 1-fifth cost ? what will 1 pound cost?
25. If 5-sixths of a barrel of beef cost $6, what will
1 -sixth cost? what will a barrel cost?
FRACTIONS. 113
26. Fifteen is 2-ninths of what number ?
27. Fourteen is 3-eighths of what number ?
28. Thirteen is 3-fourths of what number ?
29. Seventeen is 8-ninths of what number ?
30. If 1 yard and a half, that is, 3 half yards of rib-
bon cost 6 cents, what will a yard cost ?
31. Six is 3-halves of what number ?
ANALYSIS. Of that number of which l-third of 6 is
l-half; l-third of 6 is 2 ; 2 is \-lialf of 2 times 2, which are
4; therefore, 6 is 3-halves of 4.
32. If 1 yard and l-third of a yard, that is, 4-thirds
of a yard of cloth, cost $3, what will a yard cost ?
33. If a man travel 9 miles in 1 hour and 2-sevenths
of an hour, how far will he travel in 1 hour ?
34. By selling a watch for $18, I gained 1 -fourth of
what it cost me : how much did it cost ?
35. A grocer, by selling a lot of flour for $25, gained
1 -fifth of what it cost him: what was the cost? how
much did he gain ?
36. If a man pay $6 for 1 and l-third yards of cloth,
what is the cost of 1 yard ?
37. If a man receive $10 for 2 and 2-thirds days'
work, how much is that per day ?
38. If 2-thirds of a yard cost $4|, what cost a yard ?
39. Four and 6-sevenths, are 2-thirds of what ?
ANALYSIS Of that number of which l-half of 4 and 6-se&
#nths is l-third.
40. Four and 2-thirds .are 2-fifths of what ?
41. Three and 3-fourths are 5-sixths of what?
42. One and 3-fifths are 3-fifths of what?
43. Three and 2-thirds are 3-fourths of what ?
44. Four and 4-fifths are 6-halves of what ?
114 RAY'S INTELLECTUAL ARITHMETIC.
LESSON II.
1. A man gave to some poor persons $3, which was
2-fifths of his money : how much had he left ?
2. A pole stands 5 -sevenths in the air, and 5 feet in
the ground : how long is the pole ?
3. A man spent 2-fifths of his money, and had $10
left : how much did he spend ?
4. At 3 yards for 2 cents, how many yards of tape
can be bought for 7 cents ?
5. At 5 lemons for 3 cents, how many lemons can be
bought for 12 cents ?
6. If 4-fifths of a yard of calico cost 8 cents, how
many yards can be purchased for 25 cents ?
7c If 1 and 1 -fourth tuns of hay cost $8, what is the
price of 1 tun ?
8. If 6-sevenths of a yard of cloth cost $4, how many
yards can be purchased for 12 ?
9. A jockey, by selling a horse for $45, gained
1 -eighth of the cost: what was the cost?
10. If 1 and 1 -third yards of cloth cost $5, how much
can be purchased for $12 ?
11. By selling 5 yards of cloth for $12, I gained
1 -third of the cost: what did I pay per yard ?
12. By selling 7 yards of cloth for $21, I made
2-fifths of the cost : I paid for it with wheat, at $| per
bushel : how many bushels did I give ?
13. If 3-fifths of an apple cost 2-thirds of a cent, what
will 3 apples cost ?
14. What will be the cost of 11 yards of cloth, if 5
and 1-half yards cost $4| ?
15. By selling cloth at $8 a yard, 1-fifth of the cost
was lost : what part would have been gained, if the clotb
had been sold for $11 ?
TABLES. 115
SECTION XXIIL TABLES.
UNITED STATES MONEY.
10 mills, marked m., . . make 1 cent, . . marked ct.
10 cents " 1 dime, . " d.
10 dimes or 100 cents. " 1 dollar, . " $.
10 dollars " 1 eagle, . " E.
1. Repeat the table of United States Money.
2. How many cents in a half dime ? in a quarter of
a dollar ? in a half dollar ?
3. In 1 cent how many mills ? in 2 ? in 3 ?
4. In 1 dime how many cents ? in 2 ? in 4 ?
5. In 1 dollar how many cents ? in 2 ? in 4 ?
6. In 20 mills how many cents? in 30? in 50?
7. In 20 cents how many dimes ? in 25 ? in 30 ?
8. At 20 cents a yard, what will 3 and 3-fourths
yards of calico cost ? how many dimes ?
9. At 25 cents a yard, what will 3 and 1 -third yards
r}f muslin cost ? how many dimes ?
ENGLISH MONEY.
4 farthings (far.) . make 1 penny, . . marked d.
12 pence " 1 shilling, . " s.
20 shillings . . . . " 1 pound, . . .
1. Repeat the table of English Money.
2. In 1 penny how many farthings ?in2?3? 5?
3. In 1 shilling how many pence ? in 2? 3? 4?
4. How many shillings in 1 pound ? in 2? 3? 4?
5. How many pence in 8 farthings ? in 12? 16?
6. How many shillings in 24 pence ? in 36 ? 56 ?
7. How many pounds in 40 shillings ? in 60 ? 72 ?
U6 RAY'S INTELLECTUAL ARITHMETIC.
TROY WEIGHT.
24 grains (gr.) make 1 pennyweight, . marked pwt.
20 pennyweights " 1 ounce, .... " oz.
12 ounces ... " 1 pound,. ... " Ib.
1. Repeat the table of Troy Weight,
2. In 1 pennyweight how many grains ? in 2? 3?
3. In 1 ounce how many pennyweights ? in 2 ? 3 ?
4. In 1 pound how many ounces ? in 3 ? 5 ?
5. In 24 oz. how many Ib.? in 36 ? 48? 28? 56?
APOTHECARIES WEIGHT.
20 grains (gr.) . . make 1 scruple, . . marked 3.
3 scruples. ... " 1 dram, ... " 3-
8 drams .... " 1 ounce, ... " 5-
12 ounces .... " 1 pound, ... " Ib.
1. Repeat the table of Apothecaries Weight.
2. In 1 scruple how many grains ? in 2 ? 3 ?
3. In 1 dram how many scruples ? in 3 ? 7 ?
4. In 1 ounce how many drams ? in 2 ? 3 ?
5. In 1 pound how many ounces? in 3 ? 7?
6. In 15 oz. how many Ib. ? in 24 ? in 35 ? 48 ? 50 ?
AVOIRDUPOIS WEIGHT.
16 drams (dr.). . make 1 ounce, . . marked oz.
16 ounces, . . . . " 1 pound, . . " Ib.
25 pounds, . . . , " 1 quarter, . " qr.
4 quarters or 100 Ib. " 1 hundred weight, cwt.
BO hundred weight, " 1 tun, ... " T.
1. Repeat the table of Avoirdupois Weight.
2. In 2 pounds how many ounces? in 3? in 4?
TABLES.
3. In 2 quarters how many pounds ? in 3 ? in 4 ?
4. In 2 tuns how many cwt. ? in 3 ? in 4 ? in 5 ?
5. At 2 cents an ounce, what will 2 Ib. cost?
6. At 2 dollars a quarter, what will 2 cwt. cost?
DRY MEASURE.
2 pints (pt.) . . . make 1 quart, . . . marked qt.
8 quarts " 1 peck, ... " pk.
4 pecks " 1 bushel, . . " bu.
1. Repeat the table of Dry Measure.
2. In 1 quart how many pints ? in 2? in 3? 5?
3. In 1 peck how many quarts ? in 2 ? jn 4 ? 7 ?
4. In 1 bushel how many pecks ?in2? in 3? 5?
5. In 2 pints how many quarts ?in5? in 8? 9?
6. In 8 quarts how many pecks ? in 16 ? 24 ? 32 ?
7. In 4 pecks how many bushels ? in 12 ? in 20 ?
8. At 5 cents a pt., what will 2 qt. of beans cost ?
9. At 6 cents a qt., what will 1 peck of corn cost?
10. At 3 cts. a qt., what cost 3 and 3-fourths pk. of salt?
LIQUID OR WINE MEASURE.
marked pt.
" qt.
" gal.
bl.
hhd.
" T.
1. Repeat the table of Wine Measure.
2. In 1 pint how many gills ? in 2 ? in 3 ? in 5 ?
3. In 2 quarts how many pints ?in3? 5? 6? 8?
2d Bk. 8
4 gills (gi.) . . . i
2 pints
make 1 pint, . . .
u 1 auart.
4 quarts ....
31-| gallons ....
63 gallons ....
4 hogsheads. . .
L VJ 1*1* M. Vj
" 1 gallon, . .
" 1 barrel, . .
" 1 hogshead, .
" 1 tun, . . .
118 RAY'S INTELLECTUAL ARITHMETIC.
4. In 1 gal. how many quarts? in 2 ? 3 ? 7 ? 10?
5. In 1 tun how many hogsheads ? in 3? in 5? 8?
6. In 4 gi. how many pints ? in 7 ? in 8 ? in 9 ?
7. In 3 pt. how many quarts ? in 6? in 8? 11?
8. In 4 qt. how many gallons ? in 7 ? in 11 ? 12 ?
9. At 12 cents for one pint, what will 1 and 3-fourtha
gallons of Port wine cost ?
LONG MEASURE.
12 inches (in.) . . make 1 foot, . . . marked ft.
3 feet " 1 yard, . . " yd.
5 yards or 16J ft. " 1 rod (or pole), " rd.
40 rods or 220 yd. " 1 furlong, . " fur.
8 furlongs ... " 1 mile,. . . " mi.
1. Eepeat the table of Long Measure.
2. In 2 feet how many inches ? in 3 ? in 5 ? in 8 ?
3. In 2 yards how many feet ? in 3 ? in 4 ? in 5 ?
4. In 2 rods how many yards ? in 3 ? in 4 ? in 5 ?
5. In 1 furlong how many rods ? how many yards ?
6. In 1 mile how many furlongs ? in 3? 4? 5?
7. How many inches in 8 and 5-sixths feet?
8. In 2 rods how many feet ? in 3 rods ? 4 rods ?
5 rods ? 6 rods ?
SQUARE MEASURE.
144 square inches . make 1 square foot, marked sq. ft.
9 square feet . . " 1 square yard,. " sq. yd.
30J square yards . " 1 square rod, . " sq. rd.
40 square rods . . " 1 rood, . . . . " R.
4 roods .... " 1 acre, . . . . " A.
640 acres " 1 square mile, . " sq. mi
TABLES. 119
1. Repeat the table of Square Measure.
2. How many square feet in 3 square yards ? in 5 ?
3. How many roods in 4 acres ? in 6 ? in 8 ? in 10 ?
CLOTH MEASURE.
4 nails (na.) . . make 1 quarter, . . marked qr.
4 quarters .
u
1 yard,. . . .
u
yd.
3 quarters .
(I
1 ell Flemish,.
u
E.F1.
5 quarters .
u
1 ell English, .
If
E.En.
6 quarters .
a
1 ell French, .
a
E.Fr.
1. Repeat the table of Cloth Measure.
2. How many quarters in 2 yards ? in 3? 4? 5?
3. How many quarters in 1 ell Flemish ? in 4 ? 5 ?
4. How many quarters in 1 ell English ? in 3 ? 4 ?
5. How many quarters in 1 ell French ? in 2 ? 4 ?
6. How much longer is 1 ell French than 1 ell Eng-
lish? than 1 ell Flemish?
7. There are 36 inches in 1 yard : how many inches
are there in 1 quarter? in 1 nail?
TIME MEASURE.
60 seconds (sec.) make 1 minute, marked min.
60 minutes " 1 hour, . " hr.
24 hours " 1 day, . . da.
365 days 6 hours (365J days) " 1 year, . " yr.
100 years " 1 century, " cen.
7 days " 1 week, . u wk.
4 weeks " 1 month, " mon
12 calendar months .... " 1 year, . " yr.
NOTE. One Solar year contains 365 days, 6 hours, 48 minute*
and 48- seconds, or 365 \ days nearly.
120 BAY'S INTELLECTUAL ARITHMETIC.
The following table shows the names of the different
months of the year, and the number of days embraced
in each.
January, 1st mon., 31 da.
February, 2d " 28
March, 3d 31
April, 4th 30
May, 5th 31
June, 6th 30
July, 7th mon., 31 da.
August, 8th " 31
September, 9th " 30 "
October, 10th " 31
November, llth " 30 "
December, 12th " 31 "
The number of days in each month of the year may
be retained in the mind by committing the following lines
to memory :
Thirty days has September,
April, June, and November;
Other months have thirty-on-e,
Except the second month alone;
To this we twenty-eight assign,
Till leap-year gives it twenty-nine.
1. Repeat the table. "What make minutes ? days ?
2. In 14 days how many weeks ? in 21 ? in 28 ?
3. How many minutes in 1 hour and a quarter ?
4. There are 12 months in 1 year : what part of a
year is 1 month ? what part are 2 months ? 3 months ?
i months? 5? 6? 8? 9? 10?
5. If there are 30 days in 1 month, what part of a
month are 3 days? what part are 5 days? 6 days?,
10 days? 12 days? 15 days? 18 days? 20 days?
84 days? 27 days?
GENERAL REVIEW.
APPLICATIONS OF MENTAL ARITHMETIC,
FOE ADVANCED CLASSES.
SECTION XXIV.
THM preceding sections contain all the elementary forms of
Anatysiw. Those who are properly acquainted with them, will
find bnt little difficulty in these applications.
LESSON I. ADDITION.
SUGGESTION. Where the numbers are large, it is better to add
by 10' s or 100's, as the case may be.
To add 25 ai*d 35 : 20 and 80 are 50 ; 5 and 5 are 10 ; 50 and
10 are 60. Or, 5 and 30 are 55; 55 and 5 are 60.
OP THE SIGNS.
The sign +, called plus, means more. The numbers
between which it is placed are to be added.
Thus, 2+4, (2 plus 4), shows that 2 and 4 are to be
added together.
The sign of equality, =, shows that the number of
units on the right and left of it are equal to each other.
Thus, 2 + 4 = 6, means that 2 added to 4 equal 6 j and
is read, 2 and 4 are 6, or 2 plus 4 equal 6.
EXAMPLES. 4 + 2 = how many ?
3 + 5 = how many ?
2 + 3 + 4 = how many ?
1 + 3 + 5 + 7 = how many ?
$4 + $3 + $1 + $2 = how many $'s ?
(121)
122 KAY'S INTELLECTUAL ARITHMETIC.
1. What is the sum of 4+ 5+ 6+7?
2. What is the sum of 9 + 10 + 11 + 8 ?
3. What is the sum of 20 + 12 + 9 + 11 ?
4. What is the sum of 24 + 20 + 12 + 30 ?
5. What is the sum of 35 + 40 + 15 + 20 ?
6. What is the sum of 50 + 60 + 70 + 80 ?
7. What is the sum of 54 + 20 + 13 + 12 ?
8. What is the sum of $21+$16+$13+$20?
9. Bought at one time 33 gal. of oil, at another 20,
at another 40, at another 50, and at another 62 : how
many gal. did I buy ?
10. A lady paid 23 for a dress, $18 for a shawl, and
69 for a bonnet : how much did she spend ?
11. I owe A 850, B $75, C $40, and D $20: how
much money do I owe ?
12. I collected of one man $110, of another $90, of
another $75, and of another $50 : how much in all ?
13. Thirty-two plus 16 + 20 + 21 + 18 = what ?
14. Fifty-nine plus 21 + 32 + 15 + 11 = what?
LESSON II. SUBTRACTION.
The sign , called minus, means less. Placed between
two numbers, it shows that the one on the right, is to be
taken (subtracted), from the one on the left.
Thus, 4 2=2; read> 4 minus (less) 2 equal 2.
EXAMPLES. 5 2 = how many ?
3 + 9 8 = how many ?
1 + 5 + 9 10= how many ?
2 + 3 + 4+8 11= how many ?
$5 + $9 + $7 + $4 $12 = how many $'s ?
RE VIEW. MULTIPLICATION. 1 23
1. What number equals 75 less 40 ? 160 120 ?
100 45? 110 90? 120 95?
2. A boy having 75 ct, purchased 55 ct. worth of
goods : how much change did he receive ?
3. Having $92, I purchased a watch for $73: how
much had I left?
4. Bought a horse for $110, and sold him for $145:
how much did I make?
5. George bought candles for 25 ct., soap for 10 ct.,
sugar for 35 ct., and starch for 3 ct. : he gave $1, and
received 30 ct. change : was this correct ?
6. A boy had $5, from which he took at one time $1
and 50 ct. ; at another, 40 ct. ; at another, $1 and 10 ct.:
how much had he left ?
7. Ten plus 22 + 19 less 8 + 3 + 9 + 6=what ?
8. 42 + 19 + 13 + 15 12 17 20 4=?
9. $37 + $33 + $45 + $25 $35 $20 $40= ?
10. $125 + $140 + $20 $100 $50 $8 $30= ?
11. $160 + $80 + 130 $210 $30 $10 $40==?
LESSON III. MULTIPLICATION.
The sign X, denotes multiplication, read, multiplied
ly. When placed between two numbers, it shows that
ihey are to be multiplied together.
Thus, 3x5, means 3 multiplied by 5.
EXAMPLES. 3X5 = how many ?
2X5X7 = how many ?
2X4X6X3 = how many ?
$8 + $12 $7 X 10 = how many $'s ?
124 RAY'S INTELLECTUAL ARITHMETIC.
1. What is the product of 2+6X2? 25X2? 16X3?
20X5? 22X6? 40X5? 38X3? 60X4? 45x5?
24X6? 53X9? 65X8?
2. What is the product of 14x6? 4X7X5?
5X6x7? 9X10X5? 6x8X5?
3. What will be the cost of 5 yd. of cloth, at $2
and 50 ct. a yd. ?
4. A man traveling at the rate of 5 mi. an hr., meets
a stage going at the rate of 9 mi. an hr. : how far from
the man will the stage be in 10 hr. ?
5. What cost 9 boxes, at 7 dimes each ?
6. What cost 8 Ib. and 4 oz. of sugar at 12 ct. a Ib. ?
7. What cost 75 ft. of lumber, at 3 ct. a ft. ?
8. If 1 A. of land produce 85 bu. of corn, how many
bu. will 11 A. produce ?
9. Bought 15 Ib. of coffee at 10 ct. a Ib., and 13 Ib.
at 9 ct. a Ib, : what did the whole cost ?
10. Henry has 19 ct ; George 3 times as many, lack-
ing 10 : how many have beth?
11. How many yd. in 3 bales of cloth, each containing
6 pieces of 35 yd. each ?
LESSON IV. DIVISION.
The sign -7-, is read, divided ~by. When placed be-
tween two numbers, it shows that the first is to be di-
vided by the second.
Thus, 6-7-2, means that 6 is to be divided by 2.
EXAMPLES. 6-
4X5
4 + 10X 2
$3 + $12 $5X4
2 = how many ?
2 = how many ?
7 = how many ?
8 = how many $'
REVIEW. DIVISION. 1 25
1. How often is 3 contained in 48 ? in 51 ? in 60 ?
In 75 ? in 81 ? in 90 ? in 144 ?
2. Divide 125 by 5; multiply the result by 10; the**
divide by 2 : what is the last quotient ?
3. Multiply 14 by 20, and divide the product by 7*
4. What does 12 X 13 -=- 2 -f- 6 = ?
5. What does 15 X 12 -*- 3 -f- 12 3 == ?
6. What does 27 + 9 -f- 12 + 20 17 -f-2 X 5 =?
7. What is 1-half of 28 + 1-third of 72 ?
8. What are 9-fifteenths of 120 + of 60 ?
9o If a boat sail 48 mi. in 12 hr., how far will she
sail in 6 hr. ?
10. At 15 ct. a lb., what quantity of beef can be pur-
chased for $6 ?
11. Three men bought a horse for $90 : after keeping
him 6 wk.j at $3 a wk., they sold him for $99 : what did
each man lose ?
12. Seven multiplied by 9, divided by 3, 2 added, 13
subtracted, and divided by 5, will = what ?
13. Add 10 to 12 X 3-7-6 + 5-7-8+10 X 5-7-6-^
2 = what ?
14. Seventeen + 6 8 -r- 3 X 8 6 + 14 -r- 4 X 8~
8+12 --10 + 6 + 7 12X 6= what?
15. What number added to itself will give a sum equal
to 14?
Explanation. If a number be added to itself, the sum will
be 2 times the number : 14, then, is 2 times what number ?
16. What number added to itself 3 times, will make 32 ?
17. Divide 16 into 2 parts, so that the second pari
will be 3 times the first.
Explanation. The sum of the parts (16), will be 4 time* the
first part
126 RAY'S INTELLECTUAL ARITHMETIC.
18. Divide 48 into 2 such parts that the second shall
be 7 times the first. 48 is 8 times what number ?
19. Divide 24 into 3 parts, so that the second shall
be 2 times the first, and the third 3 times.
20. A boy being asked the number of ct. he had, re-
plied : " Five times the number I have, is just 40 less
than 10 times the number :" how many had he ?
21. Find a number which, being multiplied by 2 -7- 8
X 3 9-f-3X 3 + 11 X 3 1-5-10, equals 5.
Explanation. Begin with the last number mentioned, and re-
verse every operation indicated by the signs : thus, 5 X 10 = 50 ;
60+1 = 51; 51 -J- 3 = 17, &c.
22. What does 12X5 + 3-r-7+ll-f-5 1 + 10
+ 14 .^3 + 19 + 8 9 + 17 + 8=?
23. What does 13 + 27 + 14 + 10 -*- 8 + 21 + 13 -*-
7 + 14 + 20 + 23 + 3 -f- 11 = ?
24. What does 19 + 2 13x6-f-4 + 7 12X5
--7 + 15 11 X 8 --12 + 15 14=?
LESSON V. PRINCIPLES.
1. When 10 was taken from a number, only 2-thirds
of the number remained : what was the number?
2. The sum of two numbers is 12 ; if 6 be added to
the sum, the result will be twice the greater number : what
are the numbers?
3. The sum of two numbers is 16 more than their
difference : if their difference be 4, and 8 one of the num-
bers, what is the other number ?
4. The sum of two numbers diminished by the less
gives 15 : if 10 is the less number, what is their difference ?
REVIEW. PRINCIPLES. 127
5. If 6 be taken from the difference of two numbers,
the remainder will be 2 : if 4 is one of the numbers,
what is the other ?
6. If 10 be added to the difference of two numbers,
the sum will be 6 more than the greater number, which
is 19 : what is the less number ?
7. If 10 be taken from the sum of two numbers, of
which 5 is one, there will be 8 left: what is the other
number ?
8. By what part of 6 must 4 be X to = f of 20 ?'
9. What number -=- 8 will give 13 for a quotient ?
10. What number X 12, will give 156 for a product ?
11. If 15 be multiplied by some number and 20 added
to the product, the sum will be 200 : what is the multi-
plier ?
12. A certain number X 2, gives a result as much less
than 20 as the number is greater than 7 ; but when it is
subtracted from 11, it leaves the same remainder as 5
from 7 : what is the number ? why ?
13. Six is contained in a certain number 12 times, with
a remainder of 5 : what is the number ?
14. If 12 be added to a certain number, 7 will be con-
tained 9 times in the sum, with a remainder of 1 : what
Is the number?
15. If 13 be taken from a certain number, 8 will be
contained 10 times in the difference, with a remainder of
3: what is the number ?
16. When the divisor of 132 was increased by 6, the
quotient was found to be 11 : what was the divisor?
17. When the divisor of 72 was multiplied by 2, the
quotient was 9 : what was the divisor ?
18. When the divisor of 84 was divided by 3, the quo-
tient was 4 : what was the divisor ?
128 RAY'S INTELLECTUAL ARITHMETIC.
19. If 1 be added to the number of times a certain
number is contained in 60, the result will be 11 : what is
that number ?
20. Twice the greater of two numbers, 2 = their
sum, which is 20 : what are the numbers ?
21. A boy received of his father 3 ct. ; of his mother
twice as many less 1 ; if he had received from his father
5 ct. more, his father would have given him 4 times as
many as his sister : how many ct. did he receive ?
SECTION XXV. QUESTIONS.
LESSON I.
1. If I of a yd. of cloth cost $2, what cost ^ of a yd. ?
2. If | of a yd. of cloth cost $5, what cost | of a yd. ?
ANALYSIS. \-tliird of a yd. will cost \-Tialf as much as
2-thirds ; and 3-thirds, or a whole yd., will cost 3 times as
much as I-third ; and,
One-fourth of a yd. will cost l-fourth as much as 1 yd.,
and ^-fourths, 3 times as much as l-fourth.
3. If | of a bl. of flour cost 3, what cost f of a bl. ?
4. If 4 of a yd. of muslin cost 24 ct., what will y\ of
a yd. cost ?
5. If | of a tun of hay cost $15, what will one-half a
tun cost?
6. If | of an orchard contain 30 fruit trees, how many
trees are there in T 7 ^ of it ?
7. If 1| yd. of cloth cost $14, what cost 2^ yd. ?
8. If 14, bl. of flour cost $5J, what cost 2J bl. ?
9. If 3J Ib. of cheese cost 20 ct., what cost 2| Ib. ?
RE VIE W. QUESTIONS. 1 29
10. A traveled 30 mi. in 3f hr. : at that rate, how far
can he travel in 7J hr. ?
11. If a man earn $1J in 10 hr., how much can he
earn in 11 hr. ?
12. A can earn $9? in 6 da., of 8 hr. each : how much
tan he earn in 7 da., of 9 hr. each ?
13. If 5| bu. of wheat cost $94, what cost 3| bu. ?
14. If 8J is f of a number, what is f of it?
15. If 3^ is 2 times some number, what is 2J times
that number ?
16. If | of a bl. of flour cost $4i, what cost f of a bl.?
17. If f of a yd. of lace cost $f , what cost | of a yd. ?
18. If the wages of 3 men for 5 da, is $30, what will
be the wages of 4 men for 7 da. ?
SUGGESTION. First find the wages of one man for one day.
19. If 6 persons spend $36 in 8 da., how much, at that
rate, would 5 men spend in 12 da. ?
20. If 3 men can build 12 rd. of wall in 8 da., how
many rd. can 5 men build in 3 da. ?
21. If 6 horses eat 36 bu. of oats in 10 da., how many
bu. will 5 horses eat in 9 da. ?
22. If 5 oxen eat 2 A. of grass in 6 da., in how many
da. will 12 oxen eat 8 A. ?
23. If a family of 8 persons spend $400 in 5 mon.,
how much would maintain them 8 mon., if 3 more per-
sons were added ?
ANALYSIS. $400 for 5 mon. is $80 for 1 mon. : if 8 per-
sons spend $80 in 1 mon., I person spends $10 in I mon.
Hence, 1 1 persons spend $1 10 in 1 mow., and $880 in 8 mon.
24. If 10 oxen can be kept on 5 A. for 3 mon., how
many sheep can be kept on 15 A. for 5 mon., if 7 sheep
eat as much as 1 ox ?
130 RAY'S INTELLECTUAL ARITHMETIC.
LESSON II.
1. If 5 men can do a piece of work in 18 da., how
many men will do it in 9 da. ?
2. If 8 men can do a piece of work in 15 da., how
many men can do it in 12 da. ?
3. If 9 pipes fill a cistern in 2^-hr., in what time
will 5 such pipes fill it ?
4. If 5 men do a piece of work in 6 da., how many
can do a piece twice as large, in 1 -fifth the time?
5. If 8 men can do a piece of work in 5 da., in what
time can 5 men do it ?
6. If 6 men can do a piece of work in 5 da., in what
time can they do it, if they receive the assistance of 3
additional men when the work is half completed ?
7. If 7 men can do a piece of work in 4 da., in what
time can it be done, if 3 of the men leave when the work
is half completed ?
'8. If 20 Ib. of flour afford 8 five ct. loaves, how many
one ct. loaves will it furnish ? how many four ct. loaves ?
how many ten ct. loaves ?
9. If 10 Ib. of flour afford 6 five ct. loaves, how
many 3 ct. loaves will it furnish ?
ANALYSIS. It will afford 5 times as many 1 ct. loaves as 5
ct. loaves : l-third as many 3 ct. loaves as 1 ct. loaves.
10. If a sack of flour make 20 three ct. loaves, how
many 4 ct. loaves will it make ? 5 ct. loaves ?
11. If the 5 ct. loaf weigh 8 oz. when flour is $3 a
bl., what should it weigh when flour is 1 a bl. ? what if
flour is 2 a bl. ? $4 a bl. ?
12. A 4 ct. loaf weighs 10 oz. when flour is $6 a bl. ;
what will it weigh when flour is $5 a bl. ?
RE VIE W. QUESTIONS. 1 3 1
ANALYSIS. If flour were $1 a bl, it ought to weigh 6 times
as much as when flour is $6 a bl. ; that is, 60 oz. : and.
When flour is $5 a bl., it ought to weigh J as much as when
it is $1 a bl., that is, 12 oz.
13. If the 3 ct. loaf weigh 7 oz. when flour is $3J a bl.,
what ought it to weigh when flour is $2J a bl. ?
14. If 6 men can mow a field in 5J da., how muck
time would be saved by employing 4 more men ?
LESSON III.
1. A and B hired a pasture for $45 : A pastured 4
cows, and B 5 cows : what should each pay ?
ANALYSIS. They together pastured 9 cows, of which ^ were
As, and f J3's; hence, A should pay | of $45, which are
$20; and B f of $45, which are $25.
2. William had 3 ct., Thomas 4 ct., and John 5 ct. ;
they bought 36 peaches : what was the share of each ?
3. Two men paid $3 for 7^ dozen oysters : the first
paid $2, and the second $1 : how many should each have ?
4. A and B bought a horse for $40; A paid $25,
and B the rest : they sold him for $56 : what should each
receive ?
5. A boat worth $860, of which ^ belonged to A, J to
* B, and the rest to C, was entirely lost : what loss will
each sustain, it having been insured for $500 ?
6. Two boys bought a silver watch for $7 : the first
paid $2^ ; the second, $4A ; they sold it for $21 : what
was each one's share ?
7. A man failing, paid 80 ct. on each dollar of his
indebtedness : what did I receive, if he owed me $60 ?
ANALYSIS. 80 ct. are f of $1 ; he therefore paid me | of
">,or$48.
132 KAY'S INTELLECTUAL ARITHMETIC
8. A grocer failing, pays 60 ct. on the dollar : what
mil B receive to whom he owes $25 ?
9. A trader failing, pays only 15 ct. on the dollar:
what will C receive to whom he owes $80 ?
10. A and B rent a pasture for $25 : A puts in 27
oxen, and B 180 sheep : what should each pay, supposing
an ox to eat as much as 10 sheep ?
11. A and B rent a pasture for $60 : A puts in 14
horses, and B 15 cows : what should each pay, if 2 horses
eat as much as 3 cows ?
12. A and B rent a pasture for $75 : A puts in 8
horses ; B 15 oxen and 120 sheep : what should each pay,
if a horse eat as much as 20 sheep, and 2 horses as much
as 3 oxen ?
13. A and B rent a pasture for $35 ; A puts in 4 horses
2 wk. ; B, 3 horses 4 wk. : what ought each to pay ?
ANALYSIS. 4 horses for 2 wk. = 1 horse for 8 wk. ; and
3 horses for 4 wk. 1 horse for 12 wk. : 8 wk. and 12 wk.
are 20 wk ; hence,
A must pay ^ or f of the rent, = $l4; and B , or f of
the rent, = $21.
14. C and D join their stocks in trade ; puts in $50
for 4 mon., and D $60 for 5 mon. : they gain $45 : what
is the share of each?
15. Two masons, A and B, built a wall for $81 ; A
sent 3 men for 4 da., and B 5 men for 3 da. : what ought
each to receive ?
16. A and B traded in company ; A put in $2 as often
as B put in $3 ; A's money was employed 5 mon., and
B's 4 mon. ; they gained $55 : what was each man's share ?
17. E and F rented a field for $27 ; E put in 4 horses
for 5 mon., and F 10 cows for 6 mon. : what ought each
to pay, if 2 horses eat as much as 3 cows ?
EEVIEW. QUESTIONS. 133
LESSON IV.
1. Divide 20 apples between A and B, so that A may
as often as B gets 3.
ANALYSIS. Of each 5 apples, A must get 2 and B 3J
In 20 apples there are 4 times 5 apples ; hence,
A must get 4 times 2 apples, or 8 apples; and B 4 times 3
zpples, or 12 apples.
2. Divide 28 ct. between John and James, so that
John may get 3 as often as James gets 4.
3. Divide 45 ct. between A, B, and C, so that A may
get 4 ct. as often as B gets 3, and gets 2.
4. In an orchard of 96 trees, there are 5 apple-trees
for 3 peach-trees : how many of each kind ?
5. On a farm there are 60 animals horses, cows, and
sheep ; for each horse there are 3 cows, and for each cow
there are 2 sheep : how many animals of each kind ?
6. A school of 35 pupils has 2 boys for 3 girls : how
many of each in the school ?
7. What number is that which being added to 3 times
itself will make 48 ?
8. Divide 42 plums between A, B, and C, so that B
may get twice, and C three times as many as A.
9. Mary has 25 yd. of ribbon : she wishes to divide
it into two parts, so that one shall be 4 times the length
of the other : what will be the length of each part 1
10. Divide 35 cherries between Emma, Agnes, and
Sarah, so that Agnes shall have twice as many as Emma,
and Sarah twice as many as Agnes.
11. Divide 28 into two parts, so that one shall be 3
fcimes a certain number, and the other 4 times.
2d Bk. 9
134 RAY'S INTELLECTUAL ARITHMETIC.
LESSON V.
1. What part of 8 is 2? what part is 4? is 1 ?
2. How many times does 10 contain 2 ? 2 is what
part of 10?
3. Twelve is how many times 2 ? 2 is what part of
12 ? what is the ratio of 2 to 12 ?
EXPLANATION. When two numbers are compared, to
see how many times greater one is than the other, what
do you find? Ans. The ratio of the numbers.
How do you find the ratio or relation of two numbers?
Ans. Divide the second number by the first.
4. How many times does 18 contain 9 ? what is the
ratio of 9 to 18?
5. What is the ratio of 12 to 36 ? 9 to 45 ? 11 to 66?
13 to 52 ? 2 to 1 ? 4 to 3 ?
6. What is the ratio of 2i to 5 ? 6] to 12-^ ? \ to \ ?
of \ to ? | to i ? i to f ?
is b o O xv o
7. If the ratio of two numbers is 5, and 6 is the less
number, what is the greater ?
8. The ratio of 7 to 21 is equal to the ratio of some
number to 36 : what is the number ?
9. Five less than the ratio of 2 to 20 is | of the ratio
of 2 to what ?
10. The ratio of 2 to 18, plus 3, is 7 less than the
ratio of 2 to what?
11. The ratio of 9 to 27, increased by 5, is equal to
the ratio of 2 i to what?
12. Divide 25 ct. between John and George, so that
their shares shall be in the ratio of 2 to 3.
Explanation. When two numbers are in the ratio of 2 to 3,
one will contain 2 as often as the other contains 3.
REVIEW. QUESTIONS. 135
13. Divide the number 48 into two parts that shall be
in the ratio of 5 to 7.
14. Divide the number 60 into three parts that shall
be to each other as 3, 4, and 5.
15. Divide the number 70 into 4 parts that shall be
to each other as 1, 2, 3, and 4.
16. Divide the number 22 into two parts that shall be
to each other as 2| to 3.
ANALYSIS. 2^ and 3 are 5J units. The first part will
therefore be as many times 2^, and the second as many times
3, as 5J are contained times in 22.
17. Divide 16 apples between Henry and Oliver, so
that their shares shall be in the ratio of 1J to 2J.
18. Divide the number 39 into three parts that shall
be to each other as ^, J-, and J.
19. Divide 14 ct. between A and B, so that B may
have 1J times as many as A.
ANALYSIS. As often as A gets 1 ct., B gets 1J; that is, of
each 2J ct. A gets 1, and B \\ct. ; Jience,
A gets as many times 1 ct. and B as many times 1| ct.,
as 2J ct. are contained times in 14 ct. Ans. A 6 ct., B 8 ct.
20. John and James together have 33 marbles; James
has If times as many as John : how many has each ?
21. William's age is 1| times Frank's age; the sum of
their ages is 32 yr. : what is the age of each ?
22. A basket contains 30 apples : the number of those
which are sound, is 2J times the number of those not
sound : how many are there of each ?
23. Two men built 27 ft. of wall : how much did eacli
build, if one built J as much as the other ?
24. A, B, and C, have $42 ; B has half as many as
A, and C half as many as B : how many has each ?
136 RAY'S INTELLECTUAL ARITHMETIC.
25. On a, farm there are 104 animals hogs, sheep, and
cows ; there are f as many sheep as hogs, and | as many
cows as sheep : how many are there of each ?
LESSON VI.
1. Divide 15 ct. between A and B, so that B may
have 3 more than A.
ANALYSIS. Reserving 3 ct. for the number that B receives
more than A, there, are 12 ct. left', dividing these equally, A
will get 6 <tf. ; and B, 6 ct. plus 3 ct. reserved, or 9 ct.
2. Thomas has 5 apples more than James, and they
both together have 19: how many has each?
3. The sum of two numbers is 31, and the greater
exceeds the less by 7 : what are the numbers ?
4. Thomas and James each had the same number of
ct., when Thomas found 8 ct. more ; they then had to-
gether 32 ct. : how many had each ?
5. Thomas and William each bought the same number
of peaches ; after Thomas ate 4, and William 6, they both
together had 20 left : how many had each remaining ?
6. Mary bought twice as many cherries as Sarah ; and
after Mary ate 7, and Sarah 5, they had only 24 left:
how many had each left ?
7. If 5 be added to the treble of a certain number,
the sum will be 50 : what is the number ?
8. If | of a certain number be increased by 10, the
gum will be 31 : what is the number ?
9. If | of a number be diminished by 7, the remain-
der will be 21 : what is the number ?
10. James is 4 yr. older than Henry, and Henry is
E yr. younger than Oliver j the sum of their ages is 37 yr. :
what is the age of each ?
REVIEW. QUESTIONS. 137
11. Mary has 8 ct. more than Jane, and Sarah 3 less
than Mary ; they all have 43 ct. : how many has each ?
12. The sum of the ages of Mary and Frank is 42 yr.;
Mary is twice as old as Frank, less 3 yr. : what is the
age of each ?
13. I bought a watch, a chain, and a ring, for $62 ;
the chain cost $5 less than the ring, and the watch $12
more than the chain : what did I pay for each ?
14. Thirty ct. are 6 ct. less than of ^ of my money :
how much have I ?
15. John has twice as much money as James, + $3 ;
Frank has as much as John and James, + $7 ; together
they have $55 : how much has each ?
LESSON VII.
1. Divide the number 15 into two parts, so that the
less part may be | of the greater.
ANALYSIS. The greater part being 3-thirds, and the less,
2-thirds, their sum will be 5-thirds : 15, then, is 5-thirds of
what number ?
2. Thomas and John have $60 to pay ; John has | as
much to pay as Thomas : what must each pay ?
3. The time past from noon is equal to half the time
to midnight : what o'clock is it ?
4. The time elapsed since noon is | of the time to
midnight : what is the hour ?
5. I had 56 mi. to travel in 2 da. ; the second da., I
went | as far as the first : how far did I travel each da. ?
6. Divide 100 into two such parts, that If of the first
less 8 will = the second.
7. Divide the number 45 into three such parts, that
the second shall be J, and the third J of the first part.
The sum of all the parts will be nine-fourths of the first part
138 RAY'S INTELLECTUAL AKITHMETIC.
8. A, B, and C, together have 40 ct. ; B has f as
many as A, and C f as many as B : how many has each ?
9. A tree 70 ft. long was broken into 3 pieces; the
middle part was | of the top part ; the lower part was |
of the middle part : what was the length of each ?
10. I bought a hat, coat, and vest, for 34 ; the hat
cost | of the price of the coat, and the vest f the price
of the hat : what was the cost of each ?
11. In a field containing 55 sheep and cows, A of the
cows = of the sheep : how many are there of each ?
12. The sum of two numbers is 100 ; and A of the less
equals | of the greater : what are the numbers ?
13. One-fourth of Mary's age = J of Sarah's, and the
sum of their ages is 14 yr. : what the age of each ?
14. Divide 38 ct. between A and B, so that f of A's
share may be equal to f of B's.
ANALYSIS. If f of As share = f of B's, then i of As
share = ^ of | ; that is. T 3 Q of It's ; and the whole of A's =
3 times T 3 , that is, -^ of B's ; and,
Hence, {g of B's + -^ of B's = i of B's = 38 ct. : 38 ct,
then, are || of what number ?
15. Divide the number 51 into two such parts, that | of
the first will equal | of the second.
16. In an orchard of 65 apple and peach-trees, f of the
apple-trees = 4 O f the peach-trees : how many are there
of each ?
17. From C to D is 66 mi. ; A left C at the same time
B left D ; when they met, f of the distance A had trav-
eled =g of the distance B had traveled: how much
farther did B travel than A ?
18. The time past noon, + 3 hr., is equal to of the
time to midnight : what is the hour ?
REVIEW. QUESTIONS. 139
19. What is the hour in the afternoon, when the time
past noon is equal to J of the time past midnight ?
Explanation. Since the time past noon is one-fifth of the whole
time from midnight, the time from midnight to noon, which is 12
hr., must equal the remaining four-fifths of the time.
20. What is the hour in the afternoon, when the time
past noon is | of the time past midnight ?
21. What is the hour of the day, when ^ of the time
past noon is ^ of the time past midnight ?
LESSON VIII.
1. What number is that to which, if its half be added,
the sum will be 15 ?
ANALYSIS. The number + its ^ = f of the number. Now }
if | 15, J5, and f , or the whole number, = 10.
2. What number is that to which if its f be added,
the sum will be 20?
3. If to Mary's age its f be added, the sum will be
21 yr. : what is her age ?
4. What number is that which being doubled, and in-
creased by its |, the sum will be 52 ?
5. What number is that which being doubled, and
diminished by its ^, the remainder will be 40 ?
6. What number is that which being trebled, and
diminished by its f , the remainder will be 48 ?
7. If to David's age you add its ^ and its |, the sum
will be 26 : what is his age ?
8. If to Sarah's age you add its |, its J-, + 10 yr.,
the sum will be twice her age : how old is she ?
9. Thomas spent | of his money, and had 30 ct. left ;
how much had he at first ?
140 RAY'S INTELLECTUAL ARITHMETIC.
10. If to a certain number you add its J, its | + 27,
the number will be trebled : what is the number ?
11. A father is 40 yr. older than his son ; the son's
age is T 3 T of the father's age : what is the age of each ?
12. If to Susan's age you add its 4 + 18 yr., the sum
will be 3 times her age : how old is she ?
13. A piece of flannel, losing | of its length by shrink-
age, measured 28 yd. : what was its length ?
14. Tha distance from A to B is ^ the distance from
C to D, and f of the distance from A to B, + 20 mi., =
the distance from C to D : what is the distance from A
to B, and from C to D ?
15. My age + its ^, its i, and its f = 94 : what is
my age ?
LESSON IX.
1. If A can do a piece of work in 2 da., what part
of it can he do in 1 da. ?
2. A can drink a keg of cider in 4 da. : what part
of it can he drink in 1 da. ?
3. B can do a piece of work in 4 a da. : how many
times the work can he do in 1 da. ?
4. C can mow a certain lot in | of a da. : how many
such lots can he mow in a da. ?
5. A can mow a certain field in 2^ da. : what part of
it can he mow in 1 da. ?
ANALYSIS. If he can mow the field in 2^ da., he could mow
twice the field in 5 da., and of twice the field, that is, f of
ihe field in 1 da.
6. B can dig a trench in 3^- da. : what part of it can
he dig in 1 da, ?
7. C can walk from Cincinnati to Dayton in 3^ da. :
what part of the distance can he walk in 2 da. ?
RE VIEW. QUESTIONS. 141
8. A can do -J of a piece of work in 1 da., and B J of
It : what part of the work can both do in a da. ?
9. A can do 4, B -j, and C I of a piece of work in 1
da. : what part of it can they all do in a da. ?
10. If A can do a piece of work in 2 da., and B in 3 da. :
in what time can they both together do it ?
ANALYSIS. If A does it in 2 da., he can do J of it in 1 da. ;
and if B does it in 3 da., he can do i of it in 1 da. ; hence,
Both can do | + \ f in 1 da. : it will require them both
as many da. as | are contained times in 1, (the whole work).
Ana.
11. A can dig a trench in 6 da., and B in 12 da. : in
what time can they both together do it ?
12. C alone can do a piece of work in 5 da., and B
in 7 da. : in what time can both do it ?
13. A can do a piece of work in 2 da., B in 3 da., and
C in 6 da. : in what time can all three do it ?
14. A cistern has 3 pipes; the first will empty it in
3 hr., the second in 5 hr., and the third in 6 hr. : in what
time will all three empty it ?
15. A and B mow a field in 4 da.; B can mow it alone
in 12 da. : in what time can A mow it?
Explanation. Both mow one-fourth in 1 da., and B one-twelfth
in 1 da.; therefore, A can mow one-fourth less one-twelfth, which
is one-sixth in 1 da. ; hence, he can mow it in 6 da.
16. A man and his wife can drink a keg of beer in 12
da. ; when the man is away, it lasts the woman 30 da. :
in what time can the man drink it alone ?
17. Three men, A, B, and C, can together reap a field
of wheat in 4 da. ; A can reap it alone in 8 da., and B
in 12 da. : in what time can C reap it ?
18. A can do a piece of work in J a da., and B in J of
a da. : how long will it take both to do it ?
142 RAY'S INTELLECTUAL ARITHMETIC.
ANALYSIS. If A does the work in J of a da., he can do 2
times the work in I da. ; and
If B does the work in\ of a da., he can do 3 times the work
in I da. ; hence,
They both together can do 5 times the work in 1 da., or the
whole work in of a da.
19. A cistern has two pipes ; by the 1st it may be
emptied in * of an hr., and by the 2nd in 1 of an hr. :
in what time will it be emptied by both together ?
20. A can alone dig a cellar in 2 \ da., and B in 3J da. :
in what time can they both together dig it ?
Explanation. A digs 2-fifths in 1 da., and B 3-tenths in 1 da. ;
hence, they both dig 7-tenths in 1 da.
21. C can reap a field of wheat in 5 da., and D can
reap it in 3J da, : in what time can both reap it ?
22. A can do a piece of work in 3 da,, and B a piece
3 times as large in 7 da. : in what time can they together
do a piece 5 times as large as the piece done in 3 da. ?
23. A cistern of 100 gal., is emptied in 20 min. by 3
pipes ; the 1st discharges \ a gal. in a min. ; the 2nd, \\
gal. : how much does the 3rd discharge ?
LESSON X.
1. Two-thirds of 14 is f of what number?
ANALYSIS. 1 \ is f ; f of f is twice i of f ; J of f is f , and
twice | are |; if i is % of some number, of f, or is %of
the number ; | is % of seven times |, which are M.
2. | of 5| is | of what number ?
3. 4 f 4f is j^ of what number ?
4. f of 5| is T 7 n of what number ?
5. | of 2| is \ of how many times 2 ?
REVIEW. QUESTIONS. 143
6. Three-fifths of li is f of how many times 4?
7. Three-fourths of 3J- is j of how many times 3?
8. John has 10 marbles, and f- of what John has is
the T 8 7 of what James has : how many has James ?
9. Jane received f of 60 plums ; she gave away | of
her | : how many were left ?
10. James has a given distance to travel ; after going
35 mi., there remains f of the distance : when he has
gone I of the remainder, how far must he then go ?
11. A horse cost $40 ; f of the price of the horse = f of
the price of the cart : what did the cart cost ?
12. B's coat cost $27, and his hat $8; of the cost
of the coat + f that of the hat, = f of the cost of his
watch : what did the watch cost ?
13. Mary lost f of her plums ; she gave f of the re-
mainder to Sarah, and had 6 plums left : how many had
she at first ?
14. John has 12 ct. ; f of his money = J of of
William's money : how much has William ?
15. From A to B is 36 mi. ; f of this is f of the dis-
tance from C to D : what is the distance from C to D ?
16. On counting their money, it was found that A
had 12 ct. more than B ; and that ^ of B's = f of A's :
how much had each ?
17. In an orchard, J are apple-trees, J are pear-trees,
T V 2 are plum-trees, and the remainder, which is 32, cherry-
trees : how many trees are thm*e of each kind ?
18. In an orchard of apple and pear-trees, the latter
are f of the whole ; the apple-trees are 25 more than the
pear-trees : how many are there of each ?
19. In an orchard of apple, plum, and cherry-trees, 69
in all, the plum-trees = J of the apple-trees, and the
cherry-trees = J of the apple-trees + i of the plum :
how many trees are there of each kind ?
144 RAY'S INTELLECTUAL ARITHMETIC.
20. The age of Jane is | of the age of Sarah, and | of
both their ages is | of the age of Mary, which is 12 yr. ;
what are the ages of Jane and Sarah ?
21. How many times T 3 T of 55, is twice that number
of which | of 30 is | ?
22. John's money is f of Charles's ; and | of John's
+ 33 = Charles's : how much has each ?
LESSON XI.
1. A hare takes 4 leaps while a hound takes 3 ; 2 of
the hound's leaps = 3 of the hare's : how many leaps
must the hound take to gain the length of a hare's leap
on the hare ?
ANALYSIS. Since 2 of the hound's leaps 3 of the hare's,
1 of the hound's leaps = l^of the hares, and 3 of the hound's
leaps 4^ of the hares; hence.
In taking 3 leaps the hound gains % the length of a hare's
leap on the hare; therefore,
The hound must take 6 leaps to gain 1 leap on the hare.
2. Henry takes 6 steps while John takes 5 ; but 4 of
John's steps = 5 of Henry's : how many steps must John
take, to gain one of Henry's steps on him?
3. Henry is 30 steps before John, but John takes 7
steps while Henry takes 5 : supposing the length of their
steps to be equal, how many steps must John take to
overtake Henry ?
4. A hare is 10 leaps before a hound, and takes 4 leaps
while the hound takes 3 ; but 2 of the hound's leaps = 3 of
the hare's : how many leaps must the hound take to
catch the hare ?
ANALYSIS. Since the hound takes 3 leaps while the hare
takes 4, the hound will take 1 leap while the hare takes 1|
leaps, and 2 leaps, while the hare takes 2| ; but,
RE VIE W. QUESTIONS. 1 45
By the second condition, 2 of the hounds leaps 3 of the
hare's ; therefore,
In making 2 leaps the hound gains | of a hards leap on the
hare ; that is, 3 leaps 2f leaps ; hence,
To gain 10 of the hares leaps, the hound must make as many
times 2 leaps as is contained times in 1 0, that is, 60 leaps.
5. A hare is 100 leaps before a hound, and takes 5
leaps while the hound takes 3 ; but 3 leaps of the hound
= 10 of the hare : how many leaps must the hound take
to catch the hare ?
6. N is 35 steps ahead of M, and takes 7 steps while
M takes 5 ; but 4 of M's steps = 7 of N's : how many
steps must M take to overtake N ? How many more will
N have made, when he is overtaken ?
7/* A hare is 8 leaps before a hound, and takes 3 leaps
while the hound takes 2 ; but 2 of the hound's leaps are
equal to 3 of the hare's : will the hound catch the hare,
and if not, why ?
8. A fish's head is 6 in. long ; its tail is as long as
the head and half of its body ; and the body is as long
as both head and tail : what is the length of the fish ?
ANALYSIS. Since the tail is as long as the head and half
the body, the tail is 6 in. + ^ the body ; but,
The body = the head and tail ; therefore, the body = 6 in. +
6 in. + ^ the body ; that is,
The body = 12 in. + i the body ; therefore, J the body is 12
in., and the body is 24 in. long.
The tail 6 in. + J of 24 in. = 18 in. Therefore, the whole
length is 6 in. + 24 in. + 18 in. = 48 in.
9. A trout's head is 4 in. long ; its tail is as long as
its head and | of its body ; the body is as long as its
head and tail : what is its length ?
10. A has 10 ct. ; C has as many as A, + 1 as many
as B ; B has as many as both A and C : how many ct.
have B and C each ?
146 RAY'S INTELLECTUAL ARITHMETIC.
11. A man bought a sheep, cow, and horse ; the sheep
cost $8; the cow, as much as the sheep, and 1 as much
as the horse ; and the horse cost twice as much as both
the sheep and cow : what did each cost ?
12. The head of a fish weighs 8 Ib. ; the tail weighs 3 Ib.
more than the head and half the body ; and the body
weighs as much as both the head and tail : what is the
weight of the fish ?
13. B is pursuing A, who is some distance in advance ;
B goes 4 steps, while A goes 5, but 3 of A's steps = 2
of B's ; B goes 30 ft. before overtaking A : how many ft.
is A in advance of B ? Ans. 5 ft.
LESSON XII.
1. A gentleman meeting some beggars, found that if
he gave each of them 3 ct., he would have 12 ct. left, but
if he gave each of them 5 ct., he would not have money
enough by 8 ct. : how many beggars were there ?
ANALYSIS. Each beggar will receive 2 ct. more when there
are 5 ct. given to each, than when there are 3 ct. ; but,
Since the money to be distributed is 12 ct. more than 3 ct.
for each beggar, and 5 ct. for each beggar is S ct. more than
the money, it will, therefore, require 20 ct. more to give 5 ct. to
each, than to give 3 ct. ; hence,
As each beggar gets 2 ct. more, it will take as many beggars
to get 20 ct. more, as 2 ct. are contained times in 20 c, which
are 10. Ans. 10 beggars.
2. A father wishes to distribute some peaches among
his children ; if he gives each of them 2 peaches, he will
have 9 left ; but if he gives each 4 peaches, he will have
3 left : how many children has he ?
Explanation. The difference between having 3 peaches left ay
9 left, is 6.
RE VIE W. QUESTIONS. 1 47
3. Mary wishes to divide some cherries among her
playmates ; she finds that if she gives each of them 5, she
will have 21 left; but if she gives each 8, she will have
none left : what is the number of her playmates ?
4. A lady wished to buy a certain number of yd of
silk for a dress ; if she paid $1 a yd., she would have
$5 left ; but if she paid $H per yd., it would take all
her money : how many yd. did she want ?
5. To buy a certain number of oranges at 8 ct. each,
requires 6 ct. more than all the money James has ; but
if he buys the same number of lemons, at 3 ct. each, he
will have 29 ct. left : how much money has he ?
6. There are two pieces of muslin, each containing the
same number of yd. ; to buy the first at 12^ ct. a yd.,
requires 40 ct. more than to buy the second at 10 ct. a yd. :
how many yd. in each ?
7. Five times a certain number is 16 more than 3
times the same number : what is the number ?
ANALYSIS. 5 times any number less 3 times the same num-
ber, is 2 times the number; therefore, 2 times the number is 16.
' 8. Thomas's age is three times that of James, and the
difference of their ages is 10 yr. : what is the age of each ?
9. The age of A is 5 times the age of B ; and the
age of B is twice the age of C ; A is 45 yr. older than
C : what is the age of each ?
10. A has ^- as much money as B ; B has | as much
as C ; C has $15 more than A : how much money has each?
11. A farmer's sheep are in 3 fields; the second con-
tains 4 times as many as the first; the third 3 times aa
many as the second, and 70 more than both the first and
second : how many sheep are there in each field ?
12. The age of A is the age of B ; twice the age of
A is | the age of C; C is 20 yr. older than B : what is
the age of each ?
148 KAY'S INTELLECTUAL ARITHMETIC.
13. A father, who had as many sons as daughters,
divided 818 among them, giving to each daughter 82 {
and to each son 1 : how many children had he ?
14. A man agreed to pay a laborer 82 for every da.
he worked ; and the laborer, for every da. he was idle,
was to forfeit 81 ; at the expiration of 20 da., the laborer
received 825 : how many da. was he idle ?
ANALYSIS. Had he worked every da. he would have received,
at the expiration of the 20 da,, $.40; but as he received only
$25, he lost $15 by being idle.
Each da. he was idle he received $3 less than if he had
worked, $2 wages, and the $1 forfeited; hence,
He was idle as many da. as $3 are contained times in $15,
that is, 5 da.
15. James was hired for 30 da. ; for every da. he
worked, he was to receive 30 ct., and for every day he
Was idle, he was to pay 20 ct. ; at the end of the time,
he received 85 : how many da. did he work ?
16. When A and B entered school, the age of A was 3
times that of B ; but in 5 yr., A's age was only twice
B's : what were their ages at first ?
ANALYSIS. Since the age of A is three times that of B, and
when 5 yr. are added to each, the age of A is only twice that
of B, therefore,
Three times the age of B, (which is the age of A,) in-
creased by 5 yr. = twice the age of B and 5 yr ; that is, =
twice the age of B, and twice 5 yr. ; hence,
The age of B must be 5yr.; and that of A, 15 yr.
17. Four yr. ago, the age of A was 3 times the age
of B ; but 4 yr. hence, it will be only twice his age : what
were their ages 4 yr. ago ?
18. B's age is twice A's ; in 10 yr., A's will be f of
B's : what are their ages ?
REVIEW. QUESTIONS. 149
19. A person has 2 watches, and a chain worth $10 ;
the first watch and chain are worth half as much as the
second; the chain and second watch, are worth 3 times
as much as the first : what is the value of each ?
ANALYSIS, Since the 1st watch and chain are together
worth 1-halfofthe 2nd, the 2nd must be worth twice the 1st +
twice the chain. But,
The 2nd watch and chain = 3 times the first; or the 2nd
watch = 3 times the 1st the chain; hence,
Three times the 1st the chain 2 times the first + 2 times
the chain; or the 1st = 3 times the chain, or $30; and
the 2nd = $80.
20. A person has 2 horses, and a saddle worth $12 :
the 1st horse and saddle are worth ^ of the 2nd horse ;
but the 2nd horse and saddle are worth four times the
1st horse : what is the value of each horse ?
21. If a herring and a half cost 2 pence and a half,
how many can you buy for 9 pence ?
LESSON XIIL
1. If 12 peaches are worth 84 apples, and 8 apples
24 plums, how many plums shall I give for 5 peaches ?
2. If 1 ox is worth 8 sheep, and 3 oxen are worth
2 horses, what is the value of 1 horse, if a sheep is
worth $5 ?
3. A walks 10 mi. in 1-| da., and B 8 mi. in If da. :
how far will B travel while A is traveling 20 mi. ?
4. A bought a number of apples at 2 for 3 ct., and
as many more at 2 for 5 ct. ; he sold them at the rate
of 3 for 7 ct. : how much per dozen did he gain ?
5. C bought a number of eggs at 2 ct. each, and twice
as many at 3 ct. each ; he sold them at the rate of 3 for
10 ct. : how much per dozen did he gain ?
2d Bk. 10
150 AAY'S INTELLECTUAL ARITHMETIC.
If he had sold them at the rate of 4 for 10 ct., how
much per dozen would he have gained or lost ?
6. Bought a number of pears at 2 for 1 ct., and as
many more at 4 for 1 ct. ; by selling 5 for 3 ct., I gained
18 ct. : how many pears did I buy ?
7. A poulterer bought a number of ducks, at the rate
of 6 for 81, and twice as many chickens, at the rate of
8 for 81 ; by selling 2 chickens and 1 duck for $, he
gained $2^ : how many of each did he buy ?
8. If 3 men can perform a piece of work in 4 da.,
working 10 hr. a da., in how many da. can 8 men perform
the^same job, working 6 hr. a da. ?
9. Divide 32 peaches between Mary, James, and Lucy,
giving Mary 2, and Lucy 3 more than James.
10. If 10 gal. of water per hr. run into a vessel contain-
ing 15 gal., and 17 gal. run out in 2 hr., how long will
the vessel be in filling ?
11. A can do a piece of work in 4^ da., and A and B
together in 2| da. : in what time can B do it alone ?
12. A, B, and C, together, can do a piece of work in
5 da. ; A and B, in 8 da. ; and B and C, in 9 da. : in what
time can each of them do it alone ?
13. If 5 men or 7 women can do a piece of work in
35 da., in what time can 5 men and 7 women do it?
14. If 2 men and 4 women can do a piece of work in
28 da., in what time can 1 man and 1 woman do it, if a
woman does | of a man's work ?
15. A man and his wife consume a sack of meal in
15 da.; after living together 6 da., the woman alone con r
sumed the remainder in 30 da. : how long would a ?ack
last either of them alone ?
16. A man had 80 eggs, which he intended to sell as
follows: 36 at 3 for 4 ct., 24 at 4 for 3 ct., and the rest
at 10 for 17 ct. : but having mixed them, how must he
sell them per dozen to get the intended price ?
RE VIEW. QUESTIONS. 151
17. If | of James's money be increased by $6. the sum
will equal what Thomas has; both together have $34:
how much has each ?
18. If $5 be taken from the | of A's money, the re-
mainder will equal B's ; both together have $51 : how
much has each ?
19. The age of A is twice the age of B ; and f of B's
age + 44 years, = 2^ times the age of A : what is the
age of each ?
20. A has not $40 ; but if he had half as many more,
and $2^ besides, he would have $40 : how many has he ?
21. Two and a half times a number + 2 = 100: What
is the number ?
22. A farmer sold | of his sheep, but soon afterward
purchased 4 as many as he had left, when he had 65
sheep : how many sheep had he at first?
23. John had $50 in silver and gold j | of the silver,
increased by $10, is equal to 1| times the gold : what
amount has he of each ?
24. One-half of A's money, diminished by $3, is equal
to J- of B's, increased by $5, and both together they have
$56 : how much money has each ?
25. A started from C the same time that B started
from D ; when they met, | of the distance A traveled
= | of the distance B traveled ; from C to D is 86 mi. :
what was the distance each traveled ?
26. Two-thirds of A's money = f of B's, and | of
their difference is $15 : how much money has each ?
27. My watch and chain cost f as much as my watch ;
3 times the price of my chain + twice the price of my
watch = $100 : what did each cost?
28. Three towns, A, B, and C, are situated on the same
road ; the distance from A to B is 24 mi. ; and | of the
distance from A to B = f of the distance from B to C
how far is it from A to C?
152 KAY'S INTELLECTUAL ARITHMETIC.
29. A, B, and C, rent a pasture for 92 ; A puts in 4
horses for 2 mon., B 9 cows for 3 mon., and C 20 sheep
for 5 mon. : what should each pay, if 2 horses eat as much
as 3 cows, and 3 cows as much as 10 sheep ?
30. John bought 5 melons for 5 ct., and James 3
melons for 3 ct. ; they then joined Thomas, and each one
ate an equal part of the melons ; when Thomas left, he
gave them 8 ct. : how should this be divided ?
31. A person having 3 sons, A, B, and C, devised f of
his estate to A, ^ to B, and the remainder to C ; the dif-
ference of the legacies of A and C was 160 : what amount
did each receive ?
32. The age of A is f of the age of B ; and the sum
of their ages -f- half the age of B = twice the age of
A 2 yr. : what is the age of each ?
33. A and B together can do a job in 16 da. ; they
work 4 da., when A leaves, and B finishes the work in
36 da. more : in how many da. can each do it ?
34. Three persons, A, B, and C, are to share a certain
sum of money, of which A's part is $12, which is % of the
sum of the shares of B and C; and | of C's share is equal
to T 3 of the sum of the shares of A and B : what are the
shares of each ?
35. If 9 men mow a field in 12 da., how many men can
mow of it in of the time ?
36. Two men formed a partnership for 1 yr. ; the 1st
put in 100, and the 2nd, 200 : how much must the
first put in at the end of 6 mon., to entitle him to 1-half
of the profits ?
37. A and B had 24 ct. ; A said to B, Give me 2 of
your ct., and I shall have twice as many as you ;" B re-
plied, " Give me two of yours, and I shall have as many
as you :" how many had each?
38. A gentleman being asked his age, replied : " The
excess of f of 50 above *my age, is equal to the difference
between my ag~ and 10 yr. :" what was his age ?
PERCENTAGE. 153
39. If I sell my eggs at 6 ct. a dozen, I will lose 12 ct. ;
but if I sell them at 10 ct., I will gain 18 ct. : what did
they cost per dozen ?
40. If I sell my sugar at a certain price per lb., I will
lose $1, but if I increase the price 3 ct. per lb., I will
gain 50 ct. : how many lb. have I ?
41. If the labor of 1 man is equal to that of 2 women,
and the labor of 1 woman is equal to that of 3 boys, how
many men would it take to do in 1 da., what 12 boys are
a wk. in doing ?
42. If sugar worth 3i ct. a lb., be mixed in equal
quantities with sugar worth 6i ct., what will ^ a lb. of
the mixture be worth, and how many lb. must be given
for $1 ?
43. If | of the gain = 7 \ of the selling price, for how
much will 3| yd. of cloth be sold, that cost $4 a yd.?
44. When sugar is worth 7 ct. a lb., a package was sold
for 24 ct., gaining 3 ct. : for how much should a package
weighing twice as much be sold, to gain 5 ct., when sugar
costs 8 ct. a lb. ?
SECTION XXVI. PERCENTAGE.
LESSON I. GAIN AND LOSS.
Explanation. The terras Percentage and Per cent, mean a certain
number of parts out of each hundred parts ; that is, a certain
number of hundredths of the sum considered.
One per cent, of any number is 1-hundredth of that number,
that is, 1 of each 100 parts.
Thus, 1 per cent, of $100 is $1 : 1 per cent, of $200 i* $2 , of
$50, 50 cents: of $1, 1 cent.
1. What is 1 per cent, of $1 ? $2 ? $5 ?
2. What is 2 per cent, of $3 ? $4 ? $6 ?
154
RAY'S INTELLECTUAL ARITHMETIC.
3. What is 3 per cent, of $10 ? $20 ? $60 ?
4. What is 4 per cent, of $25 ? $45 ? $75 ?
5. What is 5 per cent, of $100 ? $300 ? $700 ?
6. What is 6 per cent, of $150 ? $250 ? $350 ?
7. What is 2^ per cent, of $100 ? $200 ? $500 ?
8. I bought a piece of cloth for $15, and in selling
it gained 5 per cent, of the cost : what did I gain ?
ANALYSIS. 5 per cent, of 1 is five \-JiundredtTis; 5 per cent,
of $1 is 5 ct. ; of $15 it will be 15 times 5 ctf., wfo'cA are
75 c Ansstfb cZ.
9. A grocer bought a bl. of sugar for $10, and in
selling it gained 10 per cent. : how much did he gain ?
10. A farmer having a flock of 40 sheep, lost 5 per
cent, of them : how many had he left ?
11. A flock of 50 sheep increases 10 per cent, in one
year : how many are then in the flock ?
12. A lady having $20, spent 10 per cent, for muslin,
and 10 per cent, of the remainder for calico : how much
did she pay for both ?
One per cent, of anything is T ^ part of it; two per
cent, is T |^ = J^ ; four per cent. T ^ = r, 1 -.
5 per cent. = ^
8| per cent. = j 1 ^
10 per cent. = -L
12^ per cent. = J
16| per cent. = 1
20 per cent, =
25 per cent. =
33| per cent. =
50 per cent. =
75 per cent. =
13. I paid 30 ct. per yd. for muslin : at what price
must I sell it, to make 10 per cent. ?
ANALYSIS. Ten per cent, being IQ-hundredths, or \-tenth,
I must add to the first cost \-tenih of itself: but,
PERCENTAGE. 155
One-tenth of3Qct. is 3 ct, and 30 ct. adaed to 3e. arc
33 ct Ans. 33 ct. per yd.
14. To make 12| per cent, profit, what must muslin be
sold at that cost 8 ct. per yd.? 25 ct. ?
15. To make 8| per cent, profit, what must sugar be
sold for that cost 6 ct. per Ib. ? 12 ct. ?
16. To make 25 per cent, profit, what must calico be
sold for that cost 12 ct. per yd.? 16 ct? 20 ct.? 35 ct?
LESSON II.
1. A merchant bought cloth at $5 per yd., and sold
it at $7 per yd. : what did he gain on a yd. ? how much
v per cent. ?
ANALYSIS. Since Tie bought at $5 and sold at $7 per yd.j
he gained $2 on every $5, that is,
He gained | of the first cost ; ^ of this is T ^j, and are
twice -$j which are T 4 ^, or 40 per cent.
2. James bought a melon for 4 ct., and sold it for 5 ct. :
what per cent, did he gain ?
3. An orange was bought for 5 ct., and sold for 4 ct. :
what was the per cent, of loss ?
4. Thomas bought a watch for $4, and sold it for $6 :
what per cent, did he gain ?
5. Henry bought a horse for $15, and sold it for $24 :
what per cent, did he gain ?
6. A keg of wine holding 5 gal., lost 6 qt by leak-
age : what was the loss per cent. ?
7. By selling citrons at 6 ct. each, John cleared 1 of
the first cost : what per cent, would he have cleared by
selling them at 8 ct. each ?
8. A merchant bought cloth at the rate of 6 yd. for
83, and sold it at the rate of 5 yd. for $4 : what per cent,
did he gain ?
156 RAY'S INTELLECTUAL ARITHMETIC.
9. Henry sold melons at 8 ct. each, and lost i of the
first cost : what per cent, would he have lost by selling
them at 3 for 25 ct. : what per cent would he have gained
by selling them at 2 for 25 ct. ?
10. James bought a lot of lemons, at the rate of 2 for
3 ct. ; but finding them damaged, he sold them at the rate
of 3 for 2 ct. : what per cent, did he lose ?
11. Sold a watch for $12, and gained 20 per cent.: what
was the first cost ?
ANALYSIS. 20 per cent, is -%$ or , hence the gain was equal
to \ of the cost; therefore,
The watch:*oldfor | + I = f of the cost 12 then is f of
what?
12. I sold a piece of cloth for S26, and gained 30 per
cent. : what did the cloth cost me ?
- 13. If there is a gain of 40 per cent, when muslin is
sold at 14 ct. a yd., what is the cost price ?
14. By selling a horse for 45, there was a gain of 12^
per cent. : what did the horse cost ?
15. Sold a horse for 145, and lost 10 per cent. : what
was the cost ?
16. Thomas sold a watch for 21, and gained 75 per
cent. : what did he pay for it ?
17. James sold 10 oranges for 40 ct., and gained 33^
per cent. : how much did each orange cost ?
18. "When an article is sold at | of its cost, what is the
gain per cent. ?
19. When an article is sold at | of its cost, what is the
loss per cent.? at f ? at T % ? at $% ?
LESSON III.
1. When the gain is 20 per cent., what part of the
cost is equal to the gain ? when it is 75 per cent. ?
PERCENTAGE. 157
2. When the gain is 100 per cent., what part of the
cost is equal to the gain ? when it is 150 per cent. ?
3. When the loss is 25 per cent., what part of the
cost is equal to the loss ? when it is 35 per cent. ?
4. What is the loss per cent, when the whole is lost?
What is the gain per cent, when the gain is three times
the cost?
5. When -i of the gain is equal to \ of the cost, what
is the gain per cent. ?
Explanation. If 1-third of the gain is equal to 1-fifth of the
cost, the whole gain is equal to 3-fifths of the cost, and 3-fifths
are 60-hundredths or 60 per cent.
6. A sold a watch, so that of the gain was equal
to 2 6 - of the cost : what did he gain per cent. ?
7. When | of the gain is equal to i| of the cost, what
is the gain per cent. ?
8. Sold a watch for $10, by which I gained 25 per
cent. : what per cent, would I have gained by selling it
for $12?
9. By selling muslin at 7 ct. per yd., there is a loss
of 124 per cent. : what will be the loss per cent, by sell-
ing it at 6 ct. per yd. ?
10. By selling my horse for $35, there was a loss
of 16| per cent. : what would have been the gain per cent,
by selling him for $63?
11. I bought a watch for $18, which was 20 per cent,
more than its value : I sold it at 10 per cent, less than its
value : what sum did I lose ?
12. A sold B a watch for $60, and gained 20 per cent. :
afterward B sold it and lost 20 per cent, on what it cost
him : how much did B lose more than A gained ?
13. A watchmaker sold 2 watches for $30 each : on
one he gained 25 per cent., and on the other he lost 25
per cent. : how much did he gain or lose by the sale ?
158 RAY'S INTELLECTUAL ARITHMETIC.
14. By selling 4 apples for 3 ct., a dealer gains 50 pei
cent. : what per cent, will he gain by selling them at the
rate of 5 for 4 ct. ?
15. Sold 5 lemons for 4 ct., and lost 20 per cent. : what
per cent, will I lose by selling 6 for 5 ct. ?
16. Two-thirds of 10 per cent, of 60, is i of what per
cent, of 40 ?
17. One-half of f of 50 per cent, of 120, is 10 less
than 20 per cent, of what ?
iv. INTEREST.
EXPLANATION. Interest is money paid for the use of money.
The Principal is the sum of money which is loaned.
The Amount is the principal and interest added together.
The Rate Per Cent, is so many cents paid on each dollar.
1. If the interest of $1 at 6 per cent, for 1 yr., is 6 ct.,
what will be the interest of $10? of $12? of $15?
of $20 ?
2. What is the interest of $2 for 3 yr., at 5 per cent.?
ANALYSIS. The interest of $1 for 1 yr. at 5 per cent., is 5
ct. ; and for $2 the interest is twice as much as for $1, which
is 2 times 5 ct., equal 1 ct. ; and,
For 3 yr. the interest is 3 times as much as for 1 yr., which
is 3 times 10 ct., equal 30 ct. Ans. 30 ct.
3. Find the interest of $5 for 2 yr., at 6 per cent.
4. Find the interest of $8 for 5 yr., at 5 per cent.
5. Find the interest of $20 for 3 yr., at 8 per cent.
6. Find the interest of $25 for 6 yr., at 4 per cent.
7. Find the interest of $40 for 4 yr., at 5 per cent.
8. Find the interest of $50 for 3 yr., at 6 per cent.
PERCENTAGE. 159
9. Find the interest of $60 for 2 yr., at 7 per cent.
10. Find the interest of $75 for 3 jr., at 4 per cent.
LESSON Y.
1. What the interest of $50 for 5 mon., at 6 per cent.?
ANALYSIS. For 5 mon. the interest will be 5 times as much
a#for\ mon.; and for 1 mon., \-twelfth as much as for a yr.
2. Find the interest of $60 for 4 mon., at 5 per cent.
3. Find the interest of $80 for 7 mon., at.6 ; per cent.
4. Find the interest of $40 for 9 mon., at 8 per cent.
5. Find the interest of $75 for 8 mon., at 9 per cent.
What is the interest
6. Of $120 for 6 mon. 15 da., at 5 per cent. ?
7. Of $150 for 10 mon. 10 da., at 4 per cent. ?
8. Of $45 for 11 mon. 23 da., at 8 per cent. ?
9. Of $200 for 4 mon. 24 da., at 6 per cent. ?
10. Of $480 for 9 mon. 18 da., at 5 per cent.?
11. Of $360 for 5 mon. 19 da., at 5 per cent.?
12. Of $144 for 8 mon. 25 da., at 4 per cent.?
13. Of $40 for 1 yr. 4 mon., at 6 per cent. ?
14. Of $60 for 2 yr. 3 mon., at 5 per cent. ?
15. Of $75 for 1 yr. 3 mon. 6 da., at 4 per cent. ?
What is the amount
16. Of $25 for 3 yr., at 4 per cent. ?
17. Of $40 for 2 yr., at 5 per cent. ?
18. Of $55 for 3 yr., at 8 per cent. ?
19. Of $30 for 1 yr. 4 mon., at 7 per cent. ?
160 RAY'S INTELLECTUAL ARITHMETIC.
20. Of $50 for 2 yr. 3 mon. 6 da., at 6 per cent. ?
21. Of 90 for 1 yr. 3 inon. 6 da., at 8 per cent. ?
LESSON VI.
1. The interest of a certain principal for 2 yr., at 6
per cent., is $3 : what is the principal ?
ANALYSIS. The interest of $1 for 2 yr., at 6 per cent., is
12 ct. ; and the principal must be as many times $1 as 12 ct.
are contained times in $3. Ans. $25.
2. The interest of a certain principal for 3 yr., at 4
per cent., is $6 : what is the principal ?
3. What principal at interest for 4 yr., at 5 per cefct.,
will produce $12 interest ?
4. What principal at interest for 5 yr., at 8 per cent.,
will produce $30 interest?
5. What principal at interest for 4 yr., at 7J per cent.,
will produce $42 interest ?
6. What principal at interest for 2 yr. 6 mon., at 6
per cent., will produce $36 interest?
7. What principal at interest for 3 yr. 4 mon., at 6
per cent., will produce $70 interest ?
8. A father wishes to place such a sum at interest
at 5 per cent., as will produce for his son an annual in-
come of $200 : what sum must he invest ?
LESSON VII.
1. What principal on interest for 2 yr., at 5 per cent.,
will amount to $55 ?
ANALYSIS. The amount of $1 for 2 yr. at 5 per cent., is
$1.10, and it will require as many times $1 to amount to $55
as $1.10 is contained times in $55.
PERCENTAGE. 161
What principal on interest,
2. At 6 per cent., for 3 yr., will amount to $236 ?
3. At 5 per cent., for 4 yr., will amount to $600 ?
4. At 10 per cent., for 5 yr., will amount to $375 ?
5. At 6 per cent., for 5 yr., will amount to $390 ?
6. The amount due on a note which has been on in-
terest 3 yr. 4 mon., at 6 per cent., is $30 : what is the
face of the note ?
7. Two-fifths of A's money on interest for 2 yr. 6
mon., at 8 per cent., is $60: what is his whole money ?
LESSON VIII.
1. In what time, at 6 per cent., will $50 give $10
interest ?
ANALYSIS. The interest of $50 for 1 yr., at 6 per cent, is
$3 ; and it will require $50 as many yr., to give $10 interest,
as $3 is contained times in $10, which is 3J.
In what time,
2. At 5 per cent., will $40 give $8 interest ?
3. At 8 per cent., will $75 give $15 interest?
4. At 10 per cent., will $60 give $16 interest?
5. At 5 per cent., will $14$ give $24 interest ?
6. At 6 per cent, will $25 give $10 interest?
7. In what time, at 4 f>er cent., will any given prin-
cipal double itself?
ANALYSIS. At 1 per cent, any given sum, as $100, will
double itself in 100 yr. ; and,
At 4 per cent, it will double itself in J of the time that it
will at 1 per cent : \ of lOOyr. is 25 yr. Ans. 25 yr.
162 KAY'S INTELLECTUAL ARITHMETIC.
8. In what time will any given principal double itself,
at 2 per cent. ? at 3 per cent. ? at 5 per cent. ? at 6 per
cent. ? at 7 per cent. ? at 8 ? at 10 ? at 12 ?
9. In what time will any given principal treble itself,
at 5 per cent. ?
10. In what time will any given principal treble itselfj
at 8 per cent. ? at 10 per cent. ?
LESSON IX.
1. At what per cent, will 200, in 2 yr., give $24 in.
terest ?
ANALYSIS. At I per cent, for 2 yr., $200 will give $4 in*
terest; and,
It will take as many times 1 per cent, for $200 to give $24
interest, as $4, the interest of $200 at 1 per cent., is contained
times in $24.
At what per cent.,
2. Will $50 in 5 yr., give $20 interest ?
3. Will $75 in 3 yr., give $11J interest ?
4. Will $300 in 3 yr., give $63 interest?
5. Will $300 in 2 yr. 3 mon., give $54 interest ?
6. Will $240 in 3 yr. 4 mon., give $56 interest?
7. Will $200 in 4 yr., amount to $240 ?
8. Will $150 in 3 yr. 8 mon., amount to $183 ?
j, 9. Will any given principal double itself in 20 yr. ?
ANALYSIS. Any principal, as $100, will double itself
in 1 yr., at 100 per cent., and in 20 yr., at ^ of 100 per
cent., or 5 per cent.
At what per cent.,
10. Will any given principal double itself in 12 yr. ?
PERCENTAGE. 163
11. Will any given principal double itself in 10 yr. ?
12. Will any given principal double itself in 8 yr. ?
in 5 yr. ? in 4 yr. ? in 2 yr. ?
LESSON X.
1. What principal, at 5 per cent, for 4 yr., will amount
to $72?
ANALYSIS. $1, at 5 per cent., for 4 yr., will amount to
$1.20; and,
The required principal will be as many times $1, as the
amount of $1 at the given rate for the given time, which is
$1.20,.& contained times in $72.
2. What principal, at 6 per cent., for 5 yr., will
amount to $520 ?
3. What principal, at 4 per cent., for 5 yr., will
amount to $30 ?
4. What principal, at 10 per cent., for 5 yr., will
amount to $750 ?
5. What principal, at 5 per cent., for 3 yr., will
amount to $345 ?
6. What principal, at 6 per cent., for 4 yr., will
amount to $496 ?
7. What is the present worth of $24, due 4 yr. hence >
reckoning interest at 5 per cent. ?
Explanation. The present worth is that principal of which $24
is the amount. The discount is the interest on the present worth.
8. What is the present worth of $65, due 5 yr. hence,
interest at 6 per cent. ? what the discount ?
9. What is the present worth of $55, due 5 yr. hence,
interest at 5 per cent. ? what the discount ?
10. A owes $77, payable 6 yr. 8 mon. hence : what will
he gain by paying it now, money worth 6 per cent. ?
164 RAY'S INTELLECTUAL ARITHMETIC,
LESSON XI.
1. At 6 per cent., for 4 yr. 2 mon., what part of the
principal is equal to the interest ?
2. At 5 per cent., for 5 yr., what part of the amount
is equal to the interest ?
3. When the interest for 2 yr. = 1 of the principal,
what is the rate per cent. ?
4. When the interest for 2 yr. 6 mon. == \ of the
principal, what is the rate per cent. ?
5. When the interest, at 10 per cent. = f of the prin-
cipal, what is the time ?
6. When 3 times the yearly interest = ^ of the prin-
cipal, what is the rate per cent. ?
7. When I of the interest for 2 yr. = ^ of the prin-
cipal, what is the rate per cent, ?
8. When f of the interest for 3 yr. = g % of the prin-
cipal, what is the rate per cent. ?
9. The interest for 8 mon. is ^ of the principal :
what is the interest of 200 for 1 yr. 4 m$n. ?
10. If the interest for 1 yr. 4 mon., is ^ of the prin-
cipal, what the interest of $100 for 1 yr. 8 mon. 18 da. ?
11. In what time will any principal at 5 per cent., give
the same interest as in 4 yr., at 10 per cent. ?
12. The interest of A's and B's money for 3| yr., at 5
per cent., is $40, and A's money is twice that of B's :
what sum has each ?
13. Twice A's money = 3 times B's ; and the interest
at 7 per cent, for 1| yr., of what they both have, is $49 :
how much money has each ?
14. One-half of A's money = f of B's ; and the inter-
est of | of A's and ^ of B's money, at 4 per cent, for
2 yr. 3 mon., is $18 : how much has each ?
THE END,
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YOUNO SINGER'S I ~\>l (SCHOOL MCM'C), 3 NO'S | 677.169 | 1 |
Description: This book is an introduction to linear algebra for pre-calculus students. It is a stand-alone unit in the sense that no prior knowledge of matrices is assumed. Students with experience in general mathematics, up to and including Algebra I, should be able to comprehend the material. However, most students have not had experience with the topics in the latter chapters, so the pace of the course should allow for the students to spend extra time with these chapters. | 677.169 | 1 |
Subject: Re: Calculus calculators
From: Bill Richter
Date: Tue, 23 Jan 2007 09:22:01 -0600
Steve, that's interesting:
I liked this:
An easy explanation is to assume that this is the result of a
slowly degenerating mathematics professor. I am not inclined to
look favorably upon that explanation.
Can you explain your thesis, though? That the problem is that the SAT
tests allows & encourages the use of calculators? I wouldn't be too
surprised if you were right, but you didn't explain why it would be. | 677.169 | 1 |
As with Proportionality & Similar Figures, Patterns
& Functions teaches students to see relationships
among different data points and how those relationships
can be used to solve problems. Students move from the
topic introduction — a series of hands-on activities
that encourage them to analyze patterns in data — to
real-life problems that help students build a conceptual
understanding of algebraic concepts. By presenting their
data as tables, graphs and equations, students learn
different ways to solve problems and share information.
And they can see which of the three representations
most effectively communicates the solution to any given
problem.
...
use real-life problems and experiments to gather
and display experimental data in graphs and tables.
They then analyze the resulting patterns to make
predictions and develop algebraic equations. They
gain the knowledge and confidence to teach middle
schoolers concepts that often are not introduced
until high school, such as linear functions, independent
and dependent variables, y-intercept, and slope.
Teachers also see how linear models are useful to
solve real-world problems that involve constant
rates of change, whether it's figuring which company
has the best buy for t-shirts or selecting the best
pledge plan for a charity walkathon.
...
discuss the experience of teaching the Patterns
& Functions lessons in their classrooms. They
create task-specific scoring guides to evaluate
student work, using a scale of 1 to 4. Then, they
work in small groups to develop high-quality, engaging
performance tasks that will deepen students' understanding
of the big ideas of Patterns & Functions.
On
the following pages, we offer agendas for two different
levels of professional development:
a
two-hour workshop, where teachers have an opportunity
to watch the program, do one lesson and briefly
discuss it
a
four-hour workshop, where teachers can watch the
program, pause the tape after each lesson, do
the lesson on their own and have extended discussions
with their colleagues
If at
all possible, we recommend that you make time for
the longer workshop. Even if you don't have time
to do all the lessons with your colleagues during
the workshop, it's essential that you do them on
your own between the Discovery and In Practice workshops. | 677.169 | 1 |
Using a scientific calculator
Free Course
Do you have a Casio fx-83 ES scientific calculator (or a compatible model) and want to learn how to use it? This free course, Using a scientific calculator, will help you to understand how to use the different facilities and functions and discover what a powerful tool this calculator can be!
After studying this course, you should be able to:
understand the basic functions on your calculator
understand which calculator functions are needed for a given problem
understand what may go wrong when entering calculations and know how to fix them
apply knowledge of calculator functions to a range of mathematical calculationsUsing a scientific calculator
Introduction
The course describes some of the main features of a scientific calculator and encourages you to use your calculator, both for everyday arithmetic and for more complicated calculations that use the function keys as well. Key sequences, which describe which keys to press, are included in all the activities, so you can try out the ideas straightaway.
Due to the wide range of scientific calculators available, for the purposes of this course we will be concentrating on the Casio fx-83ES model. Other calculators may function differently to the methods described within this course 3rd May 2012
Last updated on: Tuesday, 23rd | 677.169 | 1 |
9780195301ciples of Finance with Excel: Includes CD
Principles of Finance with Excel is the first textbook that comprehensively integrates Excel into the teaching and practice of finance. This book provides exceptional resources to the instructor and student, combining classroom-tested pedagogy with the full potential of Excel's powerful functions.
In today's business world, computation is done almost wholly in Excel. Excel's ability to combine graphics with computation and perform complex sensitivity analysis with ease provides potent insights into financial problems. Despite this, most finance texts rely heavily on hand-held calculators and ignore Excel. As a result, many students find that after they enter the professional environment, they have to relearn both finance and Excel.
Principles of Finance with Excel is ideal for undergraduate courses in introductory finance or as a reference for finance professionals. A Free In-Text CD for students contains electronic versions of all spreadsheets in the book. A Companion Website -- -- contains lecture notes, PowerPoint Slides, and a Test Bank for instructors | 677.169 | 1 |
(NIOS Syllabus) Class 10 NIOS Syllabus | Mathematics
RATIONALE Mathematics is an important discipline of learning at the secondary stage. It helps the learners in acquiring decision- making ability through its applications to real life both in familiar and unfamiliar situations. It predominately contributes to the development of precision, rational and analytical thinking, reasoning and scientific temper. One of the basic aims of teaching Mathematics at the Secondary stage is to inculcate the skill of quantification of experiences around the learner. Mathematics helps the learners to understand and solve the day to day life problems faced by them including those from trade, banking, sales tax and commission in transaction. It also helps them to acquire the skill of representing data in the form of tables/graphs and to draw conclusions from the same. The present curriculum in Mathematics includes the appreciation of the historical development of mathematical knowledge with special reference to the contribution of Indian mathematicians particularly in the introduction of zero, the decimal system of numeration in the international form (popularly known as Hindu – Arabic numerals ). The learners are encouraged to enhance their computational skills using Vedic Mathematics.
OBJECTIVES The main objectives of teaching Mathematics at the Secondary stage are to enable the learners to : · acquire knowledge and understanding of the terms, concepts, symbols, principles and processes. · acquire the skill of quantification of experiences around them. · acquire the skill of drawing geometrical figures, charts and graphs representing given data. · interpret tabular/graphical representation of the data. · articulate logically and use the same to prove results. · translate the word problems in the mathematical form and solve them. · appreciate the contribution of Indian mathematicians towards the development of the subject. · develop interest in Mathematics.
DESCRIPTION OF COURSE The present syllabus in Mathematics has been divided into six modules namely Algebra ,Commercial Mathematics ,Geometry, Mensuration ,Trigonometry and Statistics . The marks allotted , number of lessons and suggested study time for each module are as under :
Name of the module
Number of lessons
Study time ( in hours )
Marks
1. Algebra
8
50
26
2. Commercial Mathematics
4
35
15
3. Geometry
10
75
25
4. Mensuration
2
25
10
5. Trigonometry
2
20
12
6. Statistics
4
35
12
30
240
100
There will be three Tutor Marked Assignments (TMA's) to be attempted by the learner. The awards/grades of the best two TMA's will be reflected in the Mark sheet.
DETAILED DESCRIPTION OF EACH MODULE IS AS FOLLOWS :
Module 1 : Algebra Study time : 50 Hours Marks : 26
Scope and Approach : Algebra is generalized form of arithmetic. Here we would deal with unknowns in place of knowns as in arithmetic. These knowns are, in general, numbers. It may be recalled that the study of numbers begin with natural numbers without which we would not be able to count. The system of natural numbers is extended to rational number system. To be able to measure all lengths in terms of a given unit, the rational numbers have to be extended to real numbers. Exponents and indices would simplify repeated multiplication and their laws would be introduced. These would be used to write very large and very small numbers in the scientific notation.
Algebraic expressions and polynomials would be introduced with the help of four fundamental operations on unknowns. Equating two algebraic expressions or polynomials leads to equations. In the module a study of linear and quadratic equations would be taken up to solve problems of daily life.
The learners would be acquainted with different number patterns. One such pattern, namely Arithmetic Progression would be studied in details.
1.1 Number Systems –Review of natural numbers ,integers and rational numbers, rational numbers as terminating or non – terminating decimals. Introduction of irrational numbers as nonterminating and non – recurring decimals. –Rounding of rational numbers and irrational numbers. Real numbers. –Representation of irrational numbers such as 2 , 3 and 5 on the number line. –Operations on rational and irrational numbers.
1.3 Radicals( Surds ) –Meaning of a radical, index and radicand. Laws of radicals. Simplest form of a radical. –Rationalising a radical in the denominator. Simplification of expressions involving radicals.
1.4 Algebraic Expressions and Polynomials –Introduction to variables. Algebraic expressions and polynomials. Operations on algebraic expressions and polynomials. Degree of a polynomial. Value of an algebraic expression .
1.5 Special Products and Factorisation –Special products of the type ( a ± b )2 , (a + b)(a – b) , ( a ± b )3. –Application of these to calculate squares and cube of numbers. –Factorisation of the algebraic expressions. –Factorisation of expressions of the form a2 – b2, a3 ± b3 . –Factorisation of the polynomial of the form ax2 + bx + c ( a ¹ 0) by splitting the middle term. –H.C.F and L.C.M of two polynomials in one variable only by factorisation. –Rational expressions. Rational expression in the simplest form. –Operations on rational expressions.
1.6 Linear Equations –Linear equations in one variable and in two variables. Solution of a linear equation in one variable. –System of linear equations in two variables. Graph of a linear equation in two variables. –Solution of a system of linear equations in two variables ( graphical and algebraic methods). –Solving word problems involving linear equations in one or two variables.
1.8 Number Patterns -Recognition of number patterns. Arithmetic and Geometric progressions. nth term and sum to n terms of an Arithmetic Progression.
Module 2 : Commercial Mathematics Study time : 35 Hours Marks : 15
Scope and Approach : After passing Seco
ndary level examination ,some learners may work in banks, business, houses, insurance companies dealing with sales tax ,income tax , excise duty etc. Some other may enter business and ind ustry. Some may go for higher studies. All of them will need mathematics of finance. In any case ,every citizen has to deal with problems involving interest , investment , purchases etc. It is in this context ,the present module would be developed.
In this module , applications of compound interest in the form of rate of growth ( appreciation ) and depreciation(decay) will be dealt. In solving problems related to all the stated areas , the basic concepts of direct and inverse proportion (variation) ,and percentage are all pervading.
2.1 Ratio and Proportion Review of ratio and proportion. Application of direct and inverse proportion (variation).
2.3 Compound Interest Compound interest and its application to rate of growth and depreciation. (conversion periods not more than 4 )
2.4 Banking Concept of Banking. Types of accounts : (a) Saving (b) Fixed/term deposit Calculation of interest in saving account and on fixed deposit with not more than 4 conversion periods.
Module 3 : Geometry Study time : 75 Hours Marks : 25
Scope and Approach : Looking at the things around him , the learner sees the corners ,edges , top of a table , circular objects like rings or bangles and similar objects like photographs of different sizes made from the same negative which arouse his curiosity to know what they represent geometrically.
To satisfy the learners curiosity and to add to his knowledge about the above things, the lessons on Lines and Angles, congruent and similar triangles and circles will be introduced. Some of the important results dealing with above concepts would be verified experimentally while a few would be proved logically. Different types of quadrilaterals would also be introduced under the lessons on Quadrilaterals and Areas.
The learners would also be given practice to construct some geometrical figures using geometrical instruments. In order to strengthen graphing of linear equations , the basic concept of coordinate geometry has been introduced.
Note : Proofs of only " * " marked propositions and riders based on " * " marked propositions using unstarred propositions may be asked in the examination. However direct numerical problems based on unstarred propositions may also be asked in the examination.
3.1 Lines and Angles Basic geometrical concepts : point ,line ,plane,parallel lines and intersecting lines in a plane. Angles made by a transversal with two or more lines. –If a ray stands on a line, the sum of the two angles so formed is 180o. –If two lines intersect, then vertically opposite angles are equal. –If a transversal intersects two parallel lines then corresponding angles are equal. –If a transversal intersects two parallel lines then (a) alternate angles are equal (b) interior angles on the same side of the transversal are supplementary. –If a transversal intersects two lines in such a way that (a) alternate angles are equal ,then the two lines are parallel. (b) interior angles on the same side of the transversal are supplementary ,then the two lines are parallel. *Sum of the angles of a triangle is 180o. –An exterior angle of a triangle is equal to the sum of the interior opposite angles. –Concept of locus (daily life examples may be given) –The locus of a point equidistant from two given : (a) points (b) intersecting lines.
3.2 Congruence of Triangles –Concept of congruence through daily life examples . Congruent figures. –Criteria for congruence of two triangles namely : SSS,SAS,ASA,RHS *Angles opposite to equal sides of a triangle are equal. *Sides opposite to equal angles of a triangle are equal. *If two sides of a triangle are unequal ,then the longer side has the greater angle opposite to it. –In a triangle , the greater angle has the longer side opposite to it. –Sum of any two sides of a triangle is greater than the third side.
3.3 Concurrent Lines –Concept of concurrent lines. –Angle bisectors of a triangle pass through the same point. –Perpendicular bisectors of the sides of a triangle pass through the same point. –In a triangle the three altitudes pass through the same point. –Medians of a triangle pass through the same point which divides each of the medians in the ratio 2 : 1.
3.4 Quadrilaterals –Quadrilateral and its types. –Properties of special quadrilaterals viz. trapezium ,parallelogram ,rhombus , rectangle ,square. –In a triangle , the line segment joining the mid points of any two sides is parallel to the third side and is half of it. –The line drawn through the mid point of a side of a triangle parallel to another side bisects the third side. –If there are three or more parallel lines and the intercepts made by them on a transversal are equal, the corresponding intercepts on any other transversal are also equal. –A diagonal of a parallelogram divides it into two triangles of equal area. *Parallelograms on the same or equal bases and between the same parallels are equal in area. –Triangles on the same or equal bases and between the same parallels are equal in area. –Triangles on equal bases having equal areas have their corresponding altitudes equal.
3.5 Similarity of Triangles –Similar figures ,concept of similarity in geometry. Basic proportionality theorem and its converse. –If a line is drawn parallel to one side of a triangle , the other two sides are divided in the same ratio. –If a line divides any two sides of a triangle in the same ratio , it is parallel to the third side. –Criteria for similarity of triangles : AAA, SSS and SAS . –If a perpendicular is drawn from the vertex of the right angle of a triangle to its hypotenuse , the triangles on each side of the perpendicular are similar to the whole triangle and to each other. –The internal bisector of an angle of a triangle divides the opposite side in the ratio of the sides containing the angle. –Ratio of the areas of two similar triangles is equal to the ratio of the squares on their corresponding sides. *In a right triangle ,the square on the hypotenuse is equal to the sum of the squares on the other two sides (Baudhayan / Pythagoras theorem) In a triangle ,if the square on one side is equal to the sum of the squares on the remaining two sides ,the angle opposite to the first side is a right angle ( converse of Baudhayan /Pythagoras theorem)
3.6 Circles Definition of a circle and related concepts. Concept of concentric circle. Congruent circles : –Two circles are congruent if and only if they have equal radii. –Two arcs of a circle( or congruent circles) are congruent , if the angles subtended by them at the c
entre(s) are equal and its converse. –Two arcs of a circle( or congruent circles)are congruent ,if their corresponding chords are equal , and its converse. –Equal chords of a circle( or congruent circles) subtend equal angles at the centre(s) and conversely , if the angles subtended by the chords at the centre of a circle are equal , then the chords are equal. –Perpendicular drawn from the centre of a circle to a chord bisects the chord. –The line joining the centre of a circle to the mid point of a chord is perpendicular to the chord. –There is one and only one circle passing through three given non collinear points. –Equal chords of a circle (or of congruent circles) are equidistant from the centre (centres) and its converse.
3.7 Angles in a Circle and Cyclic Quadrilateral The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle. *Angles in the same segment of a circle are equal. Angle in a semi circle is a right angle. Concyclic points. *Sum of the opposite angles of a cyclic quadrilateral is 180o. If a pair of opposite angles of a quadrilateral is supplementary , then the quadrilateral is cyclic.
3.8 Secants , Tangents and their Properties Intersection of a line and a circle. Point of contact of a line and a circle. A tangent at any point of a circle is perpendicular to the radius through the point of contact. Tangents drawn from an external point to a circle are of equal length. If two chords AB and CD of a circle intersect at P (inside or outside the circle), then PA ´ PB = PC ´ PD If PAB is a secant to a circle intersecting the circle at A and B, and PT is a tangent to the circle at T, then PA ´ PB = PT2. If a chord is drawn through the point of contact of a tangent to a circle , then the angles which this chord makes with the given tangent are equal respectively to the angles formed by the chord in the corresponding alternate segments.
3.9 Constructions –Division of a line segment internally in a given ratio. –Construction of triangles with given data: (a) Construction of a triangle with given data : SSS , SAS , ASA , RHS (b) perimeter and base angles (c) its base , sum and difference of the other two sides and one base angle.(d) its two sides and a median corresponding to one of these sides. –Construction of parallelograms , rectangles, squares , rhombuses and trapeziums. –Constructions of quadrilaterals given : (a) four sides and a diagonal (b) three sides and both diagonals (c) two adjacent sides and three angles (d) three sides and two included angles (e) four sides and an angle –Construction of a triangle equal in area to a given quadrilateral. –Construction of tangents to a circle from a point (a) outside it (b) on it using the centre of the circle . –Construction of circumcircle and incircle of a triangle.
Scope and Approach : In this module an attempt would be made to answer the following questions arising in our daily life.
–How do you find the length of the barbed wire needed to enclose a rectangular kitchen garden ? –What is the cost of constructing two perpendicular concrete rectangular paths ? –What is the area of the four walls of a room with given dimensions ? –How much plywood is needed to be fixed on the top of a rectangular table ? –The formulae for areas of plane figures would be taught in the first lesson.
In the second lesson , the surface and volume of the different solids ( three dimensional figures ) would be taken up and formulae given. Their applications to daily life situations would then be taken up.
4.1 Area of Plane Figures –Rectilinear figures. Perimeter and area of a square , rectangle ,triangle, trapezium , quadrilateral , parallelogram and rhombus. –Area of a triangle using Hero's formula. Area of rectangular paths . –Simple problems based on the above. –Non rectilinear figures : Circumference and area of a circle. –Area and perimeter of a sector. –Area of circular paths. Simple problems based on the above.
4.2 Surface Area and Volume of Solids –Surface area and volume of a cube , cuboid , cylinder , cone , sphere and hemisphere. ( combination of two solids should be avoided ). –Area of four walls of a room.
Module 5 : Trigonometry Study time : 20 Hours Marks : 12
Scope and Approach : In astronomy one often encounters the problems of predicting the position and path of various heavenly bodies ,which in turn requires the way of finding the remaining sides and angles of a triangle provided some of its sides and angles are known. The solutions of these problems has also numerous applications to engineering and geographical surveys ,navigation etc. An attempt has been made in this module to solve these problems. It is done by using ratios of the sides of a right triangle with respect to its acute angle called trigonometric ratios. The module will enable the learners to find other trigonometric ratios provided one of them is known. It also enables the learners to establish well known identities and to solve problems based on trigonometric ratios and identities.
Measurement of accessible lengths and heights (e.g. height of a pillar, height of a house etc.) and inaccessible heights ( e.g. height of a hill top, height of a lamp post on the opposite bank of a river (without bridge),celestial objects etc. ) is a routine requirement. The learners will be able to distinguish between angles of elevation and depression and use trigonometric ratios for solving simple real life problems based on heights and distances , which do not involve more than two right triangles.
5.2 Trigonometric Ratios of Some Special Angles Trigonometric ratios of 30o,45o and 60o. (Results for trigonometric ratios of 30o,45o and 60o to be proved geometrically) Trigonometric ratios of complementary angles. Application of these trigonometric ratios for solving problems such as heights and distances( problems on heights and distances should not involve more than two right triangles)
Module 6 : Statistics Study time : 35 Hours Marks :12
Scope and Approach : Since ancient times, it has been the practice by the householders , shopkeepers , individuals etc to keep records of their receipts, expenditures and other resources. To make the learners acquainted with the methods of recording, condensing and culling out relevant information from the given data, the learners would be exposed to the lesson on Data and their Representation.
Everyday we come across data in the form of tables, g
raphs, charts etc on various aspects of economy, advertisements which are eye catching. In order to read and understand these, the learners would be introduced to the lesson on Graphical Representation of Data.
Sometimes we are required to describe data arithmetically like average age of a group median score of a group or modal collar size of a group. To be able to do this, the learners would be introduced to the lesson on Measures of Central Tendency. They would also be taught characteristics and limitation of these measures.
'It will rain today', 'India will win the match against England', are statements that involve the chance factor. The learners would be introduced to the study of elementary probability as measure of uncertainty, through games of chance- tossing a coin, throwing a die , drawing a card at random from a well shuffled pack etc.
6.2 Graphical Representation of Data –Drawing of Bar charts, Histograms and frequency polygons. –Reading and interpretation of Bar charts and Histograms. Reading and construction of graphs related to day to day activities ;temperature – time graph ,pressure – volume graph and velocity – time graph etc.
6.4 Introduction to Probability –Elementary idea of probability as a measure of chance of occurrence of an event ( for single event only ) Problems based on tossing a coin ,throwing a die, drawing a card from a well shuffled pack . | 677.169 | 1 |
Product Description
In this program we are going to raise the hood and show you how algebra works! With some things in algebra, it's just a step-by-step process to get to where you want to go and that's exactly what we are going to work on today, the mechanics of algebra! Topics Covered: Rule of Equality Solving for Solving Equations with Variables on Both Sides Absolute Value Simplifying Expressions Solving InequalitiesGrade Level: 8 - 12. 26 minutes.
DVD Playable in Bermuda, Canada, United States and U.S. territories. Please check if your equipment can play DVDs coded for this region. Learn more about DVDs and Videos | 677.169 | 1 |
A 10 minute tutorial for solving Math problems with Maxima
About 50,000 people read my article 3 awesome free Math programs. Chances are that at least some of them downloaded and installed Maxima. If you are one of them but are not acquainted with CAS (Computer Algebra System) software, Maxima may appear very complicated and difficult to use, even for the resolution of simple high school or calculus problems. This doesn't have to be the case though, whether you are looking for more math resources to use in your career or a student in an online bachelor's degree in math looking for homework help, Maxima is very friendly and this 10 minute tutorial will get you started right away. Once you've got the first steps down, you can always look up the specific function that you need, or learn more from Maxima's official manual. Alternatively, you can use the question mark followed by a string to obtain in-line documentation (e.g. ? integrate). This tutorial takes a practical approach, where simple examples are given to show you how to compute common tasks. Of course this is just the tip of the iceberg. Maxima is so much more than this, but scratching even just the surface should be enough to get you going. In the end you are only investing 10 minutes.
Maxima as a calculator
You can use Maxima as a fast and reliable calculator whose precision is arbitrary within the limits of your PC's hardware. Maxima expects you to enter one or more commands and expressions separated by a semicolon character (;), just like you would do in many programming languages.
(%i1) 9+7;
(%o1)
(%i2) -17*19;
(%o2)
(%i3) 10/2;
(%o3)
Maxima allows you to refer to the latest result through the % character, and to any previous input or output by its respective prompted %i (input) or %o (output). For example:
(%i4) % - 10;
(%o4)
(%i5) %o1 * 3;
(%o5)
For the sake of simplicity, from now on we will omit the numbered input and output prompts produced by Maxima's console, and indicate the output with a => sign. When the numerator and denominator are both integers, a reduced fraction or an integer value is returned. These can be evaluated in floating point by using the float function (or bfloat for big floating point numbers):
2D and 3D Plotting
Maxima enables us to plot 2D and 3D graphics, and even multiple functions in the same chart. The functions plot2d and plot3d are quite straightforward as you can see below. The second (and in the case of plot3d, the third) parameter, is just the range of values for x (and y) that define what portion of the chart gets plotted.
Limits
Differentiation
We can calculate higher order derivatives by passing the order as an optional number to the diff function:
diff(tan(x), x, 4);
=>
Integration
Maxima offers several types of integration. To symbolically solve indefinite integrals use integrate:
integrate(1/x, x);
=>
For definite integration, just specify the limits of integrations as the two last parameters:
integrate(x+2/(x -3), x, 0,1);
=>
integrate(%e^(-x^2),x,minf,inf);
=>
If the function integrate is unable to calculate an integral, you can do a numerical approximation through one of the methods available (e.g. romberg):
romberg(cos(sin(x+1)), x, 0, 1);
=> 0.57591750059682
Sums and Products
sum and product are two functions for summation and product calculation. The simpsum option simplifies the sum whenever possible. Notice how the product can be use to define your own version of the factorial function as well.
Series Expansions
Series expansions can be calculated through the taylor method (the last parameter specifies the depth), or through the method powerseries:
niceindices(powerseries(%e^x, x, 0));
=>
taylor(%e^x, x, 0, 5);
=>
The trunc method along with plot2d is used when taylor's output needs to be plotted (to deal with the in taylor's output):
plot2d([trunc(%), %e^x], [x,-5,5]);
I hope you'll find this useful and that it will help you get started with Maxima. CAS can be powerful tools and if you are willing to learn how to use them properly, you will soon discover that it was time well invested.
Sponsor's message: Math Better Explained is an insightful ebook and screencast series that will help you deeply understand fundamental mathematical concepts, and see math in a new light. Get it here.
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About The Author
Antonio started Math Blog more than a decade ago. He has a deep-rooted passion for mathematics and programming. He is part of the Emerging Technologies team in the Analytics group at IBM, a team that focuses on data science and big data.
116 Comments
nice infos, in my opinion these kind of software should be much more supported by universities, instead of (implicitly) encouraging the use of pirated software…
thanks! I was just trying to improve my Maxima skills…
The fancy calculus stuff doesn't translate. Most of the basic functions (sin, sqrt, log, etc) are available as Math.sin() and whatnot. Google for "javascript math" to get a full list of Javascript's math functions and constants.
I've recommended Maxima to many people. Most of these people rejected it and did not even bother to look it up because they never heard of the name or believe free software cannot be of good quality.
I would like to see a side-to-side comparison between Maxima and the commercial alternatives. I can imagine Mathematica and Maple having more features (never missed them though) and a more optimized solver.
Hi!The information was delivered in a simple way that it is very easy to understand in using the software which initially i found it very difficult to use Maxima when I explore it al by my own. But now you had help me to solve my probs.
It is indeed a fantastic piece of info for me especially at this moment in rushing my assignment. Thanks…
Hi
Thanks for the Tut. However I cant find information to help me. My problem is as follows:
I want to solve f in the following two functions, when I choose the following:
s=2400 and d = 2000. f is the required value, which is the focal point of a parabole.
with Mathematica, I can do wonderful things to turn a series of calculations into a semi-professional looking document. Can I do that with maxima? (or wmaxima, or …?) I've played with it (on windows) and it just seems like a 'calculator' I can solve it all, but it's not really made to give a nice formatted output. Seems. I claim no full knowledge, maybe I'm missing something.
This is a fantastic tutorial. Thanks to the author, I was able to discover the great power of this amazing program. Based on this page, I started to write a tutorial for Maxima in Greek.
fantastic!! This is a great getting started tutorial and showes the power of the program.
I teach at University of Colorado and really push the value and quality of open source software (which I use in my own research). Next time I teach numerical analysis I will definitly use Maxima!
I used to lust after Mathematica but it is outrageously expensive: $2,500. We are approaching the point where open source alternatives, like Maxima, are the way to go for heavy duty computer assisted symbolic and numerical mathematics. | 677.169 | 1 |
Education Announces Interactive E-books with ALEKS 360
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McGraw-Hill Education has introduced ALEKS 360, a mathematics solution that combines an artificial intelligence and personalized learning program with a fully integrated, interactive e-book package. ALEKS 360 delivers assessments of students' math knowledge, guiding them in the selection of appropriate new study material, and recording their progress toward mastery of course goals. Through adaptive questioning, ALEKS accurately assesses a student's knowledge state and delivers targeted instruction on the exact topics a student is most ready to learn.
The e-books featured within ALEKS 360 are interactive versions of their physical counterparts, which offer virtual features such as highlighting and note-taking capabilities, as well as access to multimedia assets such as images, video, and homework exercises. E-books are accessible from ALEKS Student Accounts and the ALEKS Instructor Module for convenient, direct access.
The initial e-books to be offered in ALEKS 360 include: Introductory Algebra, Second Edition, by Julie Miller and Molly O'Neill; Intermediate Algebra, Second Edition, by Miller and O'Neill; College Precalculus, Second Edition, by John W. Coburn; and College Algebra, Second Edition, by Coburn. | 677.169 | 1 |
Details about Problem Solving Through Recreational Mathematics:
Historically, many of the most important mathematical concepts arose from problems that were recreational in origin. This book takes advantage of that fact, using recreational mathematics — problems, puzzles and games — to teach students how to think critically. Encouraging active participation rather than just observation, the book focuses less on mathematical results than on how these results can be applied to thinking about problems and solving them. Each chapter contains a diverse array of problems in such areas as logic, number and graph theory, two-player games of strategy, solitaire games and puzzles, and much more. Sample problems (solved in the text) whet readers' appetites and motivate discussions; practice problems solidify their grasp of mathematical ideas; and exercises challenge them, fostering problem-solving ability. Appendixes contain information on basic algebraic techniques and mathematical inductions, and other helpful addenda include hints and solutions, plus answers to selected problems. An extensive appendix on probability is new to this Dover edition.
Back to top
Rent Problem Solving Through Recreational Mathematics 1st edition today, or search our site for other textbooks by Bonnie Averbach | 677.169 | 1 |
About this course
This distance learning course provides the information you will need to prepare for the AQA A-Level in Maths with Statistics. In this home study course, you will focus on four core topics of algebra, geometry, trigonometry and calculus, which make up two-thirds of the A-Level qualification. The remaining third is focused on the study of statistics, including estimation, probability and distributions. The course is optimized for students studying at home and includes full tutor support via email.
A-Level Maths with Statistics is a valuable complement to other A-Level courses with a statistical element, such as biology, sociology or psychology, and for those wishing to study these subjects at a higher level. A-Level Maths with Statistics is also applicable to many jobs and careers and is a well-respected qualification that can be used for career progression and further training whilst in employment.
Entry requirements
English reading and writing skills, and maths to at least GCSE grade C or equivalent are required. You will need to have general skills and knowledge base associated with a GCSE course or equivalent standard.
This specification is designed to:
develop the student's understanding of mathematics and mathematical processes in a way that promotes confidence and fosters enjoyment
develop abilities to reason logically and to recognise incorrect reasoning, to generalise and to construct mathematical proofs
extend their range of mathematical skills and techniques and use them in more difficult unstructured problems
use mathematics as an effective means of communication
acquire the skills needed to use technology such as calculators and computers effectively, to recognise when such use may be inappropriate and to be aware of limitations
develop an awareness of the relevance of mathematics to other fields of study, to the world of work and to society in general
On this course you will study six units:
AS Level
Unit 1 MPC1 Core 1
Unit 2 MPC2 Core 2
Unit 3 MS1B Statistics 1B
A2 Level
Unit 4 MPC3 Core 3
Unit 5 MPC4 Core 4
Unit 6 MS2B Statistics 2
Each unit has 1 written paper of 1 hour 30 minutes.
Course Content
AS Level
Unit 1 MPC1 Core 1
Co-ordinate Geometry
Quadratic functions
Differentiation
Integration
Unit 2 MPC2 Core 2
Algebra and Functions
Sequences and Series
Trigonometry
Exponentials and logarithms
Differentiation
Integration
Unit 3 MS1B Statistics 1B
Statistical Measures
Probability
Discrete Random Variables
Normal Distribution
Estimation
A2 Level
Unit 4 MPC3 Core 3
Algebra and Functions
Trigonometry
Exponentials and Logarithms
Differentiation
Integration
Numerical Methods
Unit 5 MPC4 Core 4
Algebra and Functions
Coordinate Geometry in the (x, y) plane
Sequences and Series
Trigonometry
Exponentials and Logarithms
Differentiation and Integration
Vectors
Unit 6 MS2B Statistics 2
Poisson distribution
Continuous random variables
The t-distribution
Hypothesis Testing
Chi-squared tests
AS +A2 = A Level in Maths with Statistics. Both AS and A2 level courses and examinations must be successfully completed to gain a full A Level.
AQA Specification 6360
Paper Based Version
The course comes to you as a paper-based pack delivered by courier
You will be given guidance through the Study Guide on the nuts and bolts of studying and submitting assignments
Postal assignments cannot be accepted without prior permission from the tutor
Online Version
Our online A-Level courses are fully digitised versions of the paper-courses, so you can study on any PC or smart device when connected to the internet.
As with the paper course, your online learning programme is completely flexible, so you can study at a pace that suits you.
All of our online course content is broken down into bite size chunks to make your learning more manageable and effective.
The course contains a number of assignments which your tutor will mark and give you valuable feedback on. We call these Tutor Marked Assignments (TMAs). You need only send the TMAs to your tutor for comment, not the self-assessment exercises which are also part of the course to help you gauge your progress.
Exams are taken at an AQA centre and we can provide an extensive list of centres for you. Please read our FAQs for further information
Our A Levels come with tutor support for up to 24 months and for this course support expires in June 2017 due to syllabus changes brought in by the Government.
You will have access to a tutor, via email, who will mark your work and guide you through the course to help you be ready for your examinations. In addition you will be supplied with a comprehensive Study Guide which will help you through the study and assessment process.
English Literature is a wonderful subject that can open windows into many different areas. A sound knowledge of English Literature is essential if you want to fully understand the history and philosophy of the nation. It can also help to enhance and enrich the reading experience, providing a multi-layered, deeper understanding of the texts we all love.
On this course you will study a variety of prose, poetry and drama, including well known names like Shakespeare, Oscar Wilde and Mary Shelley. All of the authors that appear in the A Level often feature heavily in university curriculums, making it an excellent preparation for higher education.
NBAs you progress through this engaging and interesting home study course you will have 24 hour access to your own specialist tutor via email. They will guide you through interesting and complex topics such as Soviet revolutionary history and British Parliamentary reform in a way that is accessible to students with little or no prior knowledge of the subject.An A Level in History is a great platform for those wishing to move on to higher education, but could be equally useful for those wising to bolster their CV, or simply learn something new. NBThis distance learning A-Level Citizenship course explores and debates the issues relevant to local and global citizenship, helping students to take an informed and effective role in society.
In this course, you will study online and at home to learn more about such concepts as identity, democracy, power and justice and you will be encouraged to approach different viewpoints and opinions critically, in order to gain a deeper understanding of contemporary debates surrounding modern citizenship. Closely based on the AQA A-Level Citizenship specification, the course encourages an active and participatory approach to citizenship, including the opportunity to conduct individual research in an area of personal interest.
So whether you seek a deeper knowledge of citizenship to prepare you for study at university level, or whether you simply want to become a more informed member of society, this home study course will give you a valuable overview of contemporary citizenship issues, allowing you to become an active and engaged global citizen.
Psychology is a highly respected A Level subject that is applicable across a wide range of professions and university courses. In this course students will learn about different psychological conditions, their causes and their treatments. Importantly, they will also learn about different kinds of research, data handling and analysis: knowledge and skills that can be transferred across any number of different job roles.
NB: Under the new A Level specifications, the AS Level is now a separate qualification, and does not count towards the full A Level. If students still wish to sit the AS exams our course does cover the content. However, we would always advise students to check with their chosen exam centre about the availability of exams.
In this home study course from UK Distance Learning and Publishing This distance learning course in A-Level Classical Civilisation allows students to study at home and online, providing much greater access to a subject rarely offered at comprehensive schools or sixth form colleges.
The OCR A-level in Classical Civilisation is an ideal complement to many other A-Levels and it can provide a useful basis for those wishing to study related subjects (such as Classics, Ancient History or Archaeology) at university level. Alternatively, it is also an engaging and deeply absorbing subject worthy of study in its own right, and interested students will find studying Classical Civilisation to be both intellectually stimulating and rewarding. | 677.169 | 1 |
text is known for its quality and quantity of exercises, examples with detailed solutions, interesting applications, and innovative resources.
Table of Contents
Prerequisites
Real Numbers
Exponents and Radicals
Polynomials and Factoring
Rational Expressions
The Cartesian Plane
Representing Data Graphically
Functions And Their Graphs
Introduction to Library of Functions
Graphs of Equations
Lines in the Plane
Functions
Graphs of Functions
Shifting, Reflecting, and Stretching Graphs
Combinations of Functions
Inverse Functions
Solving Equations And Inequalities
Linear Equations and Problem Solving
Solving Equations Graphically
Complex Numbers
Solving Quadratic Equations Algebraically
Solving Other Types of Equations Algebraically
Solving Inequalities Algebraically and Graphically
Linear Models and Scatter Plots
Cumulative Test: Chapters P-2
Progressive Summary: Chapters P-2
Polynomial And Rational Functions
Quadratic Functions
Polynomial Functions of Higher Degree
Real Zeros of Polynomial Functions
The Fundamental Theorem of Algebra
Rational Functions and Asymptotes
Graphs of Rational Functions
Quadratic Models
Cumulative Test: Chapters 1-3
Exponential And Logarithmic Functions
Exponential Functions and Their Graphs
Logarithmic Functions and Their Graphs
Properties of Logarithms
Solving Exponential and Logarithmic Equations
Exponential and Logarithmic Models
Nonlinear Models
Cumulative Test: Chapters 3-4
Progressive Summary: Chapters P-4
Trigonometric Functions
Angles and Their Measure
Right Triangle Trigonometry
Trigonometric Functions of Any Angle
Graphs of Sine and Cosine Functions
Graphs of Other Trigonometric Functions
Inverse Trigonometric Functions
Applications and Models
Library of Parent Functions Review
Analytic Trigonometry
Using Fundamental Identities
Verifying Trigonometric Identities
Solving Trigonometric Equations
Sum and Difference Formulas
Multiple-Angle and Product-to-Sum Formulas
Additional Topics In Trigonometry
Law of Sines
Law of Cosines
Vectors in the Plane
Vectors and Dot Products
Trigonometric Form of a Complex Number
Cumulative Test: Chapters 5-7
Progressive Summary: Chapters P-7
Linear Systems And Matrices
Solving Systems of Equations
Systems of Linear Equations in Two Variables
Multivariable Linear Systems
Matrices and Systems of Equations
Operations with Matrices
The Inverse of a Square Matrix
The Determinant of a Square Matrix
Applications of Matrices and Determinants
Sequences, Series, And Probability
Sequences and Series
Arithmetic Sequences and Partial Sums
Geometric Sequences and Series
The Binomial Theorem
Counting Principles
Probability
Topics In Analytic Geometry
Circles and Parabolas
Ellipses
Hyperbolas
Parametric Equations
Polar Coordinates
Graphs of Polar Equations
Polar Equations of Conics
Cumulative Test: Chapters 8-10
Progressive Summary: Ch P-10
Technology Support Guide
Concepts In Statistics (Web only)
Measures of Central Tendency and Dispersion
Least Squares Regression
Variation (Web only)
Solving Linear Equations And Inequalities (Web only)
Systems Of Inequalities (Web only)
Solving Systems of Inequalities
Linear Programming
Mathematical Induction (Web only)
Table of Contents provided by Publisher. All Rights Reserved.
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Reading to learn technical material takes some work and
practice. Invest the time to get the most from your textbook. You will reap the
benefits as the semester progresses, and in future courses: the "art" of
technical reading should become second nature to you.
Here are some suggestions we have compiled
for reading a math textbook:
**LEARN BY DOING: Mathematics is not a
spectator sport. We learn
mathematics by participating, so "participate" while you read. When
you read your textbook, you should do so with a paper and pencil.
* Take notes on
definitions, theorems or key concepts in your notebook. You should try to state
the material that you read in your own words. If you encounter an unfamiliar
term, look it up and make a note of it.
For definitions,
find examples of the defined objects and examples of objects that do not fit
the definition.
Figure out why
each piece of a theorem is necessary or sufficient.
* When you come to
an example or theorem, work through it carefully step by step. Try to
understand and follow how the author is progressing through it. After reading
it, cover it up and try to work through the details on your own. Authors often
omit steps. Fill in the gaps to deepen your understanding of the material.
* When you can work
through an example, try to think of other examples that would fit the idea
being discussed. Think of other relevant problems and try to solve them.
* Make a note
of those things you do not understand and discuss them with your study group,
in class, via email or during office hours.
* Discuss the text
with other students. Even a short discussion of a concept or example may help
deepen your understanding.
* Ask questions as
you read: Why are the topics presented in this order? What may be a better
order? What's coming up next? Does this make sense?Is this a sound argument? If something does
not make sense to you, explore it further.You may find mistakes.Keep
track of mistakes/typos you find.
**SLOW DOWN!! Math is dense.
The flow of a math book is not like the flow of a novel. Reading a mathematics
textbook requires slow and careful reading of each word. A typical novel might
be read at the rate of a page a minute. Expect to spend 30-60 minutes
working through the few selected pages for each reading assignment thoroughly
for the first time.
* Every word
counts. Writers of math texts believe that extra words and repetition get in
the way of clarity, so there is little chance of picking up missed information
from reading on.
* Understand each
sentence before you go on. Expect to re-read, and then to re-read again. It may
take several passes through a section before you start to absorb the material.
* Study diagrams
and other kinds of illustrative material.
* Read when you are
relatively alert.
**DON'T GET DISCOURAGED. Even after you follow all the suggestions, you
probably will not completely understand everything in the section, but the
class meetings will be much more meaningful if you spend time to understand
while you read.
When you encounter a new topic that is frustrating
you, try to remember that previous topics were also difficult at first, but
that there is great satisfaction in learning and mastering a concept.
With hard work, you'll be able to gain that same sense of satisfaction with the
material in this course.
**GIVE IT A RE-RUN! After we have discussed a section in class, go back
and re-read the section -- Many points will be much clearer.Read the chapter over, soon after class. This
second reading will help you store the information you've learned in your
long-term memory.
**Print these suggestions and use them as
a bookmark. Reread them as needed.
If all of this seems like too
much work consider that it will take nearly as much work to fail. If it takes
only a little more work to succeed, then take the time to succeed! | 677.169 | 1 |
Classroom Activity
This problem focuses on the MMACS flight controller. Students learn about one of his/her duties in monitoring the Auxiliary Power Units of the space shuttle. Students will apply various calculus concepts including an application of related rates. The focus is on interpretation of the derivative as a rate of change.
Students will
analyze graphs to determine the rate of change at specific points; and
use the chain rule to determine the rate of change of two or more variables that are changing with respect to time. | 677.169 | 1 |
An NSF-sponsored project designed to help secondary school and college teachers of mathematics bring contemporary topics in mathematics (chaos, fractals, dynamics) into the classroom, and to show them how to use technology effectively in this process. There are several interactive papers available designed to help teachers understand the mathematics behind such topics as iterated function systems (the chaos game) and the Mandelbrot and Julia sets, with some JAVA applets for chaos and fractals. | 677.169 | 1 |
With an emphasis on relevant applications to motivate students, these lesson plans bridge the gap between classic Algebra 2 and real life. Cotant's goal is "to give high school math teachers a ready-to-use lesson plan as well as help students with their Algebra 2 homework and understanding of various Algebra 2 concepts." Sample a free lesson on quadratics and square roots; register and pay for more. | 677.169 | 1 |
1 Operations on Whole Numbers1.1 The Decimal Place-Value System1.2 Addition1.3 Subtraction1.4 Rounding, Estimation, and Order1.5 Multiplication1.6 Division1.7 Exponential Notation and the Order of Operations2 Multiplying and Dividing Fractions2.1 Prime Numbers and Divisibility2.2 Factoring Whole Numbers2.3 Fraction Basics2.4 Simplifying Fractions 2.5 Multiplying Fractions2.6 Dividing Fractions3 Adding and Subtracting Fractions3.1 Adding and Subtracting Fractions with Like Denominators3.2 Common Multiples3.3 Adding and Subtracting Fractions with Unlike Denominators3.4 Adding and Subtracting Mixed Numbers3.5 Order of Operations with Fractions3.6 Estimation Applications4 Decimals4.1 Place Value and Rounding4.2 Converting Between Fractions and Decimals4.3 Adding and Subtracting Decimals4.4 Multiplying Decimals4.5 Dividing Decimals5 Ratios and Proportions5.1 Ratios5.2 Rates and Unit Pricing5.3 Proportions5.4 Solving Proportions6 Percents6.1 Writing Percents as Fractions and Decimals6.2 Writing Decimals and Fractions as Percents6.3 Identifying the Parts of a Percent Problem6.4 Solving Percent Problems 7 Measurement7.1 The Units of the English System7.2 Metric Units of Length7.3 Metric Units of Weight and Volume7.4 Converting Between the English and Metric Systems8 Geometry8.1 Area and Circumference8.2 Lines and Angles8.3 Triangles8.4 Square Roots and the Pythagorean Theorem9 Data Analysis and Statistics9.1 Means, Medians, and Modes9.2 Tables, Pictographs, and Bar Graphs9.3 Line Graphs and Predictions9.4 Creating Bar Graphs and Pie Charts9.5 Describing and Summarizing Data Sets10 The Real Number System10.1 Real Numbers and Order10.2 Adding Real Numbers10.3 Subtracting Real Numbers10.4 Multiplying Real Numbers10.5 Dividing Real Numbers and the Order of Operations11 An Introduction to Algebra11.1 From Arithmetic to Algebra11.2 Evaluating Algebraic Expressions11.3 Adding and Subtracting Algebraic Expressions11.4 Using the Addition Property to Solve an Equation11.5 Using the Multiplication Property to Solve an Equation11.6 Combining the Properties to Solve Equations
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Rent Combo: Hutchinson's Basic Math Skills with Geometry with MathZone Access Card 8th edition today, or search our site for other textbooks by Stefan Baratto. Every textbook comes with a 21-day "Any Reason" guarantee. Published by McGraw-Hill. | 677.169 | 1 |
Mathematics Formula Reference
Math Formula Reference is a simple application that helps you access a vast collection of math formulas, with thousands of
mathematical equations.
This app can help you as a quick reference (cheat sheet) so you can access math formulas whenever and wherever you want.
This math formulary app is one of the most comprehensive of its | 677.169 | 1 |
Winthrop Harbor ChemistryEdward P.
...In algebra 2 students are introduced to polynomials. The rules for basic arithmetic operations on polynomials (addition, subtraction, multiplication, division and exponentiation) are examined. Rational expressions (quotients of two polynomials) are studied along with the rules for basic arithmetic operations on rational expression | 677.169 | 1 |
97803140126Mathematics for Electronics
Algebra-trigonometry based book bridges the gap between math and technology. Open workbook format includes tear-out pages so that assigned work can be collected. Self-paced examples with solutions and exercises assist students with difficult concepts and allow for self-study. Applications are relevant to students' interest in electronics. Includes chapter on digital mathematics for computer technicians.
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ISBN-10/ISBN-13: 1886363110 / 978-1886363113 / A Complete List of British and Colonial Law Reports and Legal Periodicals: Arranged in Alphabetical and in Chronological Order With Bibliographical Notes With a Check List of Canadian Statutes / 1886363153 / 978-1886363151 / Blackstone's Commentaries: With Notes of Reference to the Constitution and Laws, of the Federal Government of the United States, and of the Commonwealth of Virginia : In Five / St. George Tucker, William Blackstone 188636348X / 978-1886363489 / Slavery in the Courtroom: An Annotated Bibliography of American Cases / Paul Finkelman 1886363501 / 978-1886363502 / A Sketch of English Legal History / Frederic William Maitland, F. C. Montague 1886363552 / 978-1886363557 / Citizenship Sovereignty / John S. Wright, John Holmes Agnew 1886363668 / 978-1886363663 / Commentaries on the Constitution 1790-1860 (Studies in History, Economics and Public Law, No. 575.) / Elizabeth Kelley Bauer 1886363714 / 978-1886363717 / A Familiar Exposition of the Constitution of the United States: Containing a Brief Commentary on Every Clause, Explaining the True Nature, Reasons, and ... Designed for the Use of School Libraries and / Joseph Story 1886363838 / 978-1886363830 / A System of Penal Law for the State of Louisiana: Consisting of a Code of Crimes and Punishment, a Code of Procedure, a Code of Evidence, a Code of Reform and Prison Discipline, a Book of Definitions / Edward Livingston 1886363854 / 978-1886363854 / The Cyclopedic Law Dictionary: Comprising the Terms and Phrases of American Jurisprudence, Including Ancient and Modern Common Law, International Law, and Numerous Select Titles / Walter A. Shumaker, George Foster Longsdorf, James Christopher Cahill 1886363862 / 978-1886363861 / Biographia Juridica: A Biographical Dictionary of the Judges of England from the Conquest to the Present Time, 1066-1870 / Edward Foss 1886365121 / 978-1886365124 / Anthologie De LA Poesie Fraucaice Du 16 Siecle (Rookwood Texts) / 1886360081 / 978-1886360082 / Burning Point / Dennis N. Hinkle 188636009X / 978-1886360099 / What the Bible Really Says About Homosexuality / Daniel A. Helminiak 1886360103 / 978-1886360105 / Our Tribe: Queer Folks, God, Jesus & the Bible (Millennium Edition) / Nancy Wilson 188636219X / 978-1886362192 / Electric Furnace Conference 1997 (Electric Furnace Conference//Proceedings) / 1886362513 / 978-1886362512 / J. K. Brimacombe: Reflections And Perspectives / J. K. Brimacombe 1886363781 / 978-1886363786 / The Supreme Court and the Constitution / Charles Austin Beard 1886363846 / 978-1886363847 / The Courts of the State of New York: Their History, Development and Jurisdiction / Henry Wilson Scott 1886363900 / 978-1886363908 / The Lawyer in Literature / John Marshall Gest 1886364001 / 978-1886364004 / I Made It to Broadway - UMW / Eloise Marx 1886365032 / 978-1886365032 / Pistoles/Paroles: Money and Language in Seventeenth-Century French Comedy (Emf Monographs) / Helen L. Harrison 1886365113 / 978-1886365117 / Science and Humanism in the French Enlightenment (Emf Critiques) / Aram Vartanian 1886365180 / 978-1886365186 / Emf: Studies in Early Modern France : Rethinking Cultural Studies 2 : Exemplary Essays (EMF Studies in Early Modern France) / 1886365504 / 978-1886365506 / Theatrum Mundi: Studies in Honor of Ronald W. Tobin / 1886362335 / 978-1886362338 / Steelmaking Conferences (Steelmaking Conference//Proceedings) / 1886362378 / 978-1886362376 / Mechanical Working and Steel Processing Conference 1999 (Mechanical Working and Steel Processing Conference//Mechanical Working and Steel Processing) / 1886363250 / 978-1886363250 / The Struggle for Law / Rudolf Von Jhering 1886363285 / 978-1886363281 / Great Jurists of the World (Continental Legal History Series) /79 / 978-1886363571 / Roman Canon Law in the Church of England: Six Essays / Frederic William Maitland | 677.169 | 1 |
Preface
The mathematical sciences have enormous range and diversity. They have demonstrated the consistent ability to renew themselves through synthesis of preceding work and infusion of new ideas, some of which originate through the application of mathematics in other disciplines. This process of rejuvenation and evolution is indispensable for discovery at the frontiers of the mathematical sciences.
The essential role of the mathematical sciences in almost all aspects of the scientific, engineering, and educational enterprise has become increasingly apparent. The mathematical sciences have enabled extraordinary advances in every area of science, engineering and technology, yielding new analytical and experimental tools that address a broad range of scientific and technological challenges previously considered intractable. With this greatly expanded capacity for discovery and its subsequent applications to meet societal needs has come a dramatic demand for new mathematical techniques and capabilities that will ensure the continued growth of our nation's scientific and technological capacity. To enable progress in information technology and the biomedical sciences, participation of the mathematical sciences is indispensable. To respond to this demand, substantial progress in the development of new fields of fundamental mathematics is required. Further, in light of our increasing reliance on science, engineering and technology to sustain economic growth and improve the national quality of life, there is a growing need for improved education and training in mathematics and statistics, both for the scientific and technical workforce and for the population at large.
The Division of Mathematical Sciences at NSF organized a June 26-27, 2000 workshop to outline some of the exciting opportunities. Contributions were submitted prior to the workshop, and formed the basis of the workshop discussions. Together with the summary article "Mathematics—The Science of Patterns and Algorithms," these documents are an outstanding contribution to DMS's planning process which will permit the realization of some of these opportunities.
We are greatly indebted to the contributors for their vision and effort. We are pleased to be able to share this vision and support the mathematical sciences community in the realization of the exciting research and educational opportunities that will present themselves during the next decade. | 677.169 | 1 |
Elementary AlgebraAlgebra can be like a foreign language. But one text delivers an interpretation you can fully understand. Building a conceptual foundation in the "language of algebra," ELEMENTARY ALGEBRA, 4e provides an integrated learning process that helps you expand your reasoning abilities as it teaches you how to read, write, and think mathematically. Packed with real-life applications of math, it blends instructional approaches that include vocabulary, practice, and well-defined pedagogy with an emphasis on reasoning, modeling, communication, and technology skills. The authors' five-step problem-solving approach makes learning easy. More student-friendly than ever, the text offers a rich collection of student learning tools, including Enhanced WebAssign online learning system. With ELEMENTARY ALGEBRA, 4e, algebra makes sense! | 677.169 | 1 |
Find a Deptford Township, NJ can guide, but students must do the work.Introduction to the basics of symbolic representation and manipulation of variables. Liberal use of concrete examples. May require review of arithmetic concepts, including fractions and decimals. | 677.169 | 1 |
Description
Written in a relaxed and conversational style, this book addresses the need for a complete and mathematically sound presentation of algebra at the college level. It is appropriate for a core curriculum college algebra course, and also contains all the material necessary to prepare students for more advanced classes such as calculus and statistics.
This book was written using Mathematica 5.2 but is also compatible with Mathematica 6. Related Topics | 677.169 | 1 |
IAI GECC Course Description: M1901
M1 901: Quantitative Literacy (3-4 Semester Credits)
Develops conceptual understanding, problem-solving, decision-making and analytic skills dealing with quantities and their magnitudes and interrelationships, using calculators and personal computers as tools. Includes: representing and analyzing data through such statistical measures as central tendency, dispersion, normal and chi-square distributions, and correlation and regression to test hypotheses (maximum of one-third of course); using logical statements and arguments in a real-world context; estimating, approximating and judging the reasonableness of answers; graphing and using polynomial functions and systems of equations and inequalities in the interpretation and solutions of problems; and selecting and using appropriate approaches and tools in formulating and solving real-world problems. Prerequisite: The GECC Mathematics panel will support the pilot of the co-requisite model from 2016 to 2019. During the 3 year co-requisite pilot, the GECC Mathematics panel will allow the co-requisite to serve as the prerequisite for M1901, M1902, M1904, and M1907. The panel has compared the IAI GECC M1 901 descriptor against the AP Calculus AB and BC exams and determined there is not a match. Feb 2016
Former Prerequisite: Intermediate Algebra or PMGE and Geometry, both with a grade of "C" or better. | 677.169 | 1 |
Browse related Subjects ...
Read More can gain expertise and ease in all the algebra concepts that often stump students. How? Each lesson gives one small part of the bigger algebra problem, so that every day students build upon what was learned the day before. Fun factoids, catchy memory hooks, and valuable shortcuts make sure that each algebra concept becomes ingrained. With "Junior Skill Builders: Algebra in 15 Minutes a Day", before you know it, a struggling student becomes an algebra pro-one step at a time. In just 15 minutes a day, students master both pre-algebra and algebra, including: fractions, multiplication, division, and other basic math; translating words into variable expressions; linear equations; real numbers; numerical coefficients; inequalities and absolute values; systems of linear equations; powers, exponents, and polynomials; quadratic equations and factoring; rational numbers and proportions; and, much more! In addition to all the essential practice that kids need to ace classroom tests, pop quizzes, class participation, and standardized exams, "Junior Skill Builders: Algebra in 15 Minutes a Day" provides parents with an easy and accessible way to help their children excel | 677.169 | 1 |
Perth Amboy Algebra 1This is how all math works! Math is a system that makes sense, and once you understand it, homework, quizzes, and tests are easy! Greetings!Sheryl L | 677.169 | 1 |
Site owners
Calculators for our Courses
In all of our mathematics courses, calculators and computer software,
when used, are used as tools to support the curriculum. None of our
courses are particularly focused on teaching you to use these tools. We
specifically use tools that you can learn very quickly, so that our
classes and instruction are focused on teaching you the concepts and
then on using the tools to do more interesting and more complex problems
than you could do by hand.
The tools we use are scientific (or business) calculators, graphing
calculators, and computer software. This document has information to
help you to understand how these will be used in the various courses and
how you can use them at school if you wish or acquire them to use at
home.
What is used in which courses?
In almost all mathematics courses (except perhaps the most basic
developmental courses), students are expected to use a scientific or
business calculator to do messy computations. While some ACC campuses
have these available for short-term loan in the LRS or the Testing
Center, generally it will be best for you to have one of your own. Most
students already have one. Also, they can be purchased for as little as
$10. If you do purchase one, hold onto the manual that comes with it.
You will probably find that useful. See below for some advice on how to
buy one.
In the main calculus courses, substantial use is made of graphing
technology. At this level, most students find it inconvenient to only be
able to use it at school and buy what is needed to be able to use it at
home. However, most of the ACC campuses do have a some graphing
calculators that students can check out of the LRS for two hours at a
time and computer labs in which students can use graphing software, so
it is not necessary for students to purchase anything.
In the algebra and trig courses instructors are encouraged to bring
classroom sets of calculators to class occasionally and to help students
become familiar with and comfortable using these as tools. Students
should not be pressured to purchase anything, although students may be
asked to use the calculators and computers available at ACC to do some
work outside of class.
In the higher-level courses (past Calculus II), most instructors will
use some computer software. There should be plenty of access to
computer labs at ACC for all students in these courses. Also, we make
efforts to identify public-domain or very low-cost software so that
students who wish to use it at home can do so economically.
What kind of scientific or business calculator is best?
Notice that a scientific calculator is one that does more than just
the basic four arithmetic functions. In particular, it must do
exponential and logarithmic functions. Students who plan to take
technical math, trigonometry, or scientific calculus should have a
scientific calculator, with trig functions. Students who plan to take
statistics will find that a calculator that does two-variable statistics
is useful. About half of the scientific calculators on the market will
do this, and they don't particularly cost more than the others.
Students often find the kind with two-line displays easier to use
than those with one-line displays. With a two-line display, you can see
the formula you are typing in as you type it. Typically the price
difference is only a dollar or two. And, when you see the numerical
answer, the formula is still visible.
See the scientific calculator page from these companies. Look at the
difference in appearance between the two-line display calculators and
the single-line calculators. Texas Instruments | Casio | Sharp
(If these links are no longer good, just do a web search for each of
these companies, and, under "Products" look for "Scientific
Calculators.")
What kind of graphing calculator should you buy?
We can't afford (in either money or time) to provide and learn about
all the types of calculators. Our instructors know how to help you with
some of the Texas Instruments
calculators. There are several different versions of the TI-83 and
TI-84 for sale (see below) and our instructors can help you learn to do
the required things on any of these. You can get some help with other
models, such as the TI-86 or the older TI-85 or TI-82. Some, but not
all, of the instructors are able to help with the TI-89 family of
calculators and/or the CAS functionality of the TI-Nspire.
As far as working in our ACC classes, any of the various TI-83 or
TI-84 options will be acceptable. The following information summarizes
the difference between the various TI-83 and TI-84 options.
The TI-83 has the least amount of memory, and does not contain Flash™ Memory; therefore, no APPS button.
The TI-83 Plus and TI-84 Plus both have Flash Memory.
The TI-83 Plus allows you to graph and compare functions, as
well as perform data plotting and analysis. Its FLASH™ ROM memory
allows you to update and add software applications (Apps).
The TI-84 Plus graphing calculator offers three times the
memory, more than twice the speed and a higher contrast screen than the
TI-83 Plus model. It's keystroke-for-keystroke compatible to the TI-83
Plus.
Cost ranges from $99 - $150.
The TI-89 Titaniumn works differently. It. is equipped with
Computer Algebra System (CAS) which contains symbolic manipulation,
polar graphing, sequence graphing, differential equation graphing and
programming. Most ACC instructors will not allow you to use
calculators which will do symbolic manipulation, like TI-89s, on tests
even if they will allow graphing calculators, such as TI-83s or TI-84s.
The TI-Nspire has two different keyboards which allow it to be used as essentially a TI-84 Plus or as a TI-89 Titanium.
Other companies, such as Casio and Sharp, have calculators with
similar capabilities to those previously mentioned. If you use one of
these, our instructors won't be able to help you very much. However,
many ACC mathematics students have used these and been successful.
Hewlett-Packard calculators are highly respected and are widely used
by engineers. In the calculator world, HP is most notable for having
pioneered the use of Reverse Polish Notation (RPN) in calculators. These
worked differently from other calculators and were confusing to most
students. These days, the RPN calculators are mostly in the
more-expensive range of calculators. You should not be using one of
these RPN calculators unless you have already learned to use it well. Do
not expect any ACC instructors to help you with it, as we do not
provide any such calculators for the instructors to use. | 677.169 | 1 |
Designed to prepare students for success in the sciences by providing them with appropriate mathematics and quantitative reasoning skills. Course topics include measurement and estimation, growth and decay phenomena, scaling transformations, and an introduction to probability and statistics.
121 – Calculus I (3)
First course in calculus. Includes functions, limits, derivatives, and applications. May include some proofs.
122 – Calculus II (3)
Prerequisite: MATH 121. Includes antiderivatives, definite integrals and their applications, the fundamental theorem of calculus, derivatives and integrals of inverse functions, and techniques of integration. (Prospective mathematics majors should take this course during their freshman year.)
200 – Introduction to Statistics (3)
First course in statistical methods. Includes descriptive and inferential techniques and probability, with examples from diverse fields. Topics vary with instructor and may also include sampling methods, regression analysis, and computer applications.
201 – Introduction to Discrete Mathematics (3)
Designed to prepare prospective mathematics majors for advanced study in the field by introducing them to a higher level of mathematical abstraction. Topics include sets and logic, functions and relations, methods of mathematical proof including mathematical induction, and elementary counting techniques. (Prospective mathematics majors should take this course during their freshman year.)
204 – Mathematical Concepts and Methods I (4)
Prerequisite: EDUC 203. Mathematical concepts and methods of teaching for the elementary school. Topics include number systems and their properties, problem solving, and topics in number theory. Course intended for students certifying to teach grades PreK-6. Significant field experience required. (3 lecture credits, 1 practicum credit).
The history of mathematics begins with the early numbering systems and mathematics of the Egyptians and the Babylonians. The course then turns to the Greeks and their emphasis on logical deduction and geometry. The Arabs develop algebra in the Middle Ages, and calculus is created during the Age of Reason. The development of individual branches of mathematics then is studied (probability, number theory, non-Euclidean geometry, set theory, and topology). The course ends with the Computer Age and implications for the future.
Prerequisite: MATH 122. Includes vectors in two- and three-dimensional space, vectorvalued functions, functions of several variables, partial derivatives, multiple integrals, and line integrals.
280 – Statistical Methods (3)
Prerequisite: MATH 200. Second course in statistical methods. Includes one-way and higher ANOVA, multiple regression, categorical data analysis, and nonparametric methods with examples from diverse fields. Topics vary with instructor and may also include time series and survival analysis.
300 – Linear Algebra (3)
Prerequisites: MATH 122 and MATH 201. CPSC 125 may be used as a substitute for MATH 201 with department approval. An introduction to linear algebra. Usually includes matrix algebra, systems of equations, vector spaces, inner product spaces, linear transformations, and eigenspaces.
Prerequisite: MATH 201. CPSC 125 may be used as a substitute for MATH 201 with department approval. An elementary, theoretical study of the properties of the integers.
325 – Discrete Mathematics (3)
Prerequisite: MATH 201. CPSC 125 may be used as a substitute for MATH 201 with department approval. Includes topics such as discrete probability, graph theory, recurrence relations, topics from number theory, semigroups, formal languages and grammars, finite automata, Turing machines, and coding theory.
Prerequisite: MATH 223 and either MATH 300 or 312. Mathematics 351 introduces the theory and applications of the basic computational techniques of numerical approximation. Topics include an introduction to computer programming and algorithms, root finding, interpolation, polynomial approximation, numerical differentiation and integration, and numerical linear algebra. Mathematics 352 expands on the basic approximation techniques to include scientific computing. Topics include methods of simulation, initial value problems and boundary value problems for ordinary/partial differential equations, and applications in science and engineering. Only in sequence.
Prerequisite: MATH 300. Axiomatic development of various geometries including modern Euclidean and non-Euclidean geometry, finite geometries, hyperbolic geometry, and elliptic geometry. Topics could also include convexity, transformational geometry, projective geometry, and constructability.
381, 382 – Probability and Statistical Inference (3, 3)
Prerequisite: MATH 223. An introduction to probability theory and calculus-based statistics including probability distributions of discrete and continuous random variables, functions of random variables, methods of estimation, and statistical inference. Only in sequence.
Prerequisite: MATH 300 and at least one other 300- or 400-level mathematics course. Mathematical systems including groups, rings, fields, and vector spaces. Only in sequence.
441 – Topology (3)
Prerequisite: MATH 300 and at least one other 300- or 400-level mathematics course. Includes topics from point-set topology such as continuity, connectedness, compactness, and product and quotient constructions.
461 – Topics in Mathematics (3)
Prerequisite: Course dependent. Topics such as partial differential equations, optimization, Fourier series, ring theory, cryptology, algebraic number theory, coding theory, and modeling. May be taken up to three times for credit.
471, 472 – Real Analysis (3, 3)
Prerequisites: MATH 223, 300, and at least one other 300- or 400-level mathematics course. A rigorous, real analysis approach to the theory of calculus. Only in sequence. | 677.169 | 1 |
Key Skills
So here's the thing: this is a math class. That means one thing: there will be numbers. Okay, two things. There will be a lot of building up from the stuff we learned since…well…kindergarten.
There will be a lot of adding, subtracting, multiplying, and dividing. And yes, PEMDAS is still a thing.
We'll still be asked to solve for x. In addition to this, we will also be asked to solve for a bunch of other variables. There will even be several variables linked to several equations. We'll learn how to solve for them all.
We'll also use what we've learned about graphing and jump right in to more interesting problems. Join Shmoop, and get fired up. | 677.169 | 1 |
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