problem stringlengths 17 6.49k | generated_solution stringlengths 173 34k | expected_answer stringlengths 0 164 | problem_source stringclasses 4
values |
|---|---|---|---|
In the given problem, we are dealing with a two-row triangle made up of unit rods and connectors. To create a similar problem, we can modify the triangle's structure or the pieces used to build it.
Here's a new problem:
A three-row pyramid is constructed using a total of 28 pieces: 18 unit blocks and 10 connectors, a... | We are given a **three-row pyramid** made of:
- **18 unit blocks**
- **10 connectors**
We are to find the **total number of pieces** (unit blocks + connectors) needed to build a **six-row pyramid**.
---
### Step 1: Understand the Pattern
Let’s analyze the given data:
- **Three-row pyramid**:
- Unit blocks: 18
... | 91 | augmented_math |
Factor the following expression: $55z^{17}+121z^{34}$. | We are asked to **factor** the expression:
$$
55z^{17} + 121z^{34}
$$
---
### Step 1: Identify the **greatest common factor (GCF)**
Look at the coefficients and the powers of $ z $:
- Coefficients: 55 and 121
- Powers of $ z $: $ z^{17} $ and $ z^{34} $
#### GCF of coefficients:
- 55 = 5 × 11
- 121 = 11²
- So... | 11z^{17}(5+11z^{17}) | math |
The scores on a 30-point quiz were distributed among five students according to the bar chart below.
Bar Chart:
5 students
1st 0 - 4 pts: 1 student
2nd 5 - 8 pts: 1 student
3rd 9 - 12 pts: 1 student
4th 13 - 16 pts: 1 student
5th 17 - 20 pts: 1 student
A curious teacher noticed that if she doubled the score of every... | We are given a bar chart of quiz scores for 5 students, with each student scoring in a different range:
- 1 student scored **0–4 points**
- 1 student scored **5–8 points**
- 1 student scored **9–12 points**
- 1 student scored **13–16 points**
- 1 student scored **17–20 points**
Each range contains **1 student**, and ... | 0 | augmented_math |
Let $z$ be a complex number such that $|z| = 13.$ Find $z \times \overline{z}.$
Another problem:
Let $z$ be a complex number such that $|z| = 13.$ Find $z \times z.$ | We are given that $ |z| = 13 $, and we are asked to find two expressions:
1. $ z \times \overline{z} $
2. $ z \times z $
---
### **Problem 1: Find $ z \times \overline{z} $**
Let’s recall a key identity in complex numbers:
$$
z \times \overline{z} = |z|^2
$$
This is because if $ z = a + bi $, then $ \overline{z} ... | 169 | augmented_math |
What is the smallest integer value of $y$ for which $3-2y<-11$? | We are given the inequality:
$$
3 - 2y < -11
$$
### Step 1: Subtract 3 from both sides
$$
-2y < -11 - 3
$$
$$
-2y < -14
$$
### Step 2: Divide both sides by -2 (remember to **reverse the inequality** when dividing by a negative number)
$$
y > \frac{-14}{-2}
$$
$$
y > 7
$$
### Step 3: Find the **smallest integer valu... | 8 | augmented_math |
Let $p$ be a prime number such that $p^2-p-1$ is also prime. Let $T = \sum \limits_{i=0}^{p-1} i! \cdot (p-1-i)!$. What is the remainder when $T$ is divided by $p^2-p-1$? | We are given a prime number $ p $ such that $ p^2 - p - 1 $ is also prime. We are to compute the value of the sum:
$$
T = \sum_{i=0}^{p-1} i! \cdot (p-1 - i)!
$$
and find the remainder when $ T $ is divided by $ p^2 - p - 1 $, i.e., compute:
$$
T \mod (p^2 - p - 1)
$$
---
### Step 1: Understand the structure of th... | 1 | augmented_math |
In a square piece of paper, two congruent right triangles are removed from adjacent corners to create a trapezoid. If the side length of the square is 24 units and one of the acute angles of each triangle is $30^\circ$, what is the area of the remaining trapezoid, in square units? | We are given:
- A **square** with side length **24 units**.
- Two **congruent right triangles** are removed from **adjacent corners** of the square.
- Each triangle has an **acute angle of $30^\circ$**.
- The goal is to find the **area of the remaining trapezoid**.
---
### Step 1: Understand the Geometry
Let’s cons... | 576 - 144\sqrt{3} | augmented_math |
Expand the expression $\ds\left(\frac{1}{x}+\frac{1}{y}\right)\left(\frac{1}{x}-\frac{1}{y}\right)$. | We are given the expression:
$$
\left(\frac{1}{x} + \frac{1}{y}\right)\left(\frac{1}{x} - \frac{1}{y}\right)
$$
This is a **difference of squares** in the form:
$$
(a + b)(a - b) = a^2 - b^2
$$
Let’s identify $ a $ and $ b $:
- $ a = \frac{1}{x} $
- $ b = \frac{1}{y} $
Now apply the formula:
$$
\left(\frac{1}{x}... | \frac{1}{x^2} - \frac{1}{y^2} | augmented_math |
What is the number of square centimeters in the area of a circle with radius 7 centimeters? Express your answer in terms of $\pi$. | To find the **area of a circle**, we use the formula:
$$
A = \pi r^2
$$
Where:
- $ A $ is the area
- $ r $ is the radius of the circle
- $ \pi $ is a constant (approximately 3.14159...)
---
### Given:
- Radius $ r = 7 $ cm
### Step-by-step:
$$
A = \pi (7)^2 = \pi \times 49 = 49\pi
$$
---
### Final Answer:
$$
\b... | 49\pi | augmented_math |
Let $P(x)$ be a polynomial with rational coefficients such that $P(\cos x)=0$ if and only if $\sin x$ is a root of the equation $x^3-x^2+4x+1=0$. Find the minimal possible degree of $P(x)$. | We are given that:
- $ P(x) $ is a polynomial with **rational coefficients**.
- $ P(\cos x) = 0 $ **if and only if** $ \sin x $ is a root of the equation:
$$
x^3 - x^2 + 4x + 1 = 0
$$
We are to find the **minimal possible degree** of such a polynomial $ P(x) $.
---
### Step 1: Understand the relationship betw... | 6 | augmented_math |
Triangle $DEF$ is an obtuse, isosceles triangle. Angle $D$ measures 25 degrees. What is the number of degrees in the measure of the largest interior angle of triangle $DEF$? | We are given that triangle $ DEF $ is **obtuse and isosceles**, and angle $ D $ measures $ 25^\circ $. We are to find the **measure of the largest interior angle** of the triangle.
---
### Step 1: Understand the properties
- **Isosceles triangle**: Two sides are equal, and the base angles (angles opposite the equal ... | 130 | augmented_math |
Consider a sequence of complex numbers defined by the recurrence relation:
\[b_n = \frac{b_{n-1} + b_{n+1}}{2}\]
for all $n \ge 2$, where $b_1 = 2 + 3i$ and $b_{2019} = 5 - 2i$. Determine $b_{2002}$, given that the sequence satisfies the additional condition that the imaginary part of $b_n$ is non-decreasing for all ... | We are given a recurrence relation for a sequence of complex numbers:
$$
b_n = \frac{b_{n-1} + b_{n+1}}{2}, \quad \text{for } n \ge 2
$$
and initial conditions:
$$
b_1 = 2 + 3i, \quad b_{2019} = 5 - 2i
$$
We are also told that the **imaginary part** of $ b_n $ is **non-decreasing** for all $ n $. That is:
$$
\text... | 5 - 2i | augmented_math |
If $4^6=8^n$, what is $n$? | We are given the equation:
$$
4^6 = 8^n
$$
### Step 1: Express both sides with the same base
Note that:
- $ 4 = 2^2 $
- $ 8 = 2^3 $
So we can rewrite both sides in terms of base 2:
$$
4^6 = (2^2)^6 = 2^{2 \cdot 6} = 2^{12}
$$
$$
8^n = (2^3)^n = 2^{3n}
$$
### Step 2: Set the exponents equal
Since both sides are p... | 4 | math |
What is the least positive integer value of $y$ such that $(3y)^2 + 2\cdot 43\cdot 3y + 43^2$ is a multiple of 53? | We are given the expression:
$$
(3y)^2 + 2 \cdot 43 \cdot 3y + 43^2
$$
and we are asked to find the **least positive integer value of $ y $** such that this expression is a **multiple of 53**.
---
### Step 1: Recognize the structure
The expression is:
$$
(3y)^2 + 2 \cdot 43 \cdot 3y + 43^2
$$
This is a **perfect... | 21 | augmented_math |
Lily invested a quarter of her annual savings in stocks and spent $50 from the remaining amount on a gift. If she now has $250 left, what was her total annual savings? | We are given the following information:
- Lily invested **a quarter of her annual savings** in stocks.
- She then **spent $50** from the **remaining amount** on a gift.
- After that, she had **$250 left**.
We are to find her **total annual savings**.
---
### Step 1: Let the total annual savings be $ x $.
- She inv... | 400 | augmented_gsm8k |
A bag contains 5 marbles numbered 1, 2, 3, 4, and 5. A die is rolled and a marble is drawn from the bag at random. What is the probability that the sum of the number on the die and the number on the marble is an odd number? Express your answer as a common fraction. | We are given:
- A **bag with 5 marbles** numbered **1 through 5**.
- A **die** is rolled.
- A **marble is drawn at random** from the bag.
- We are to find the **probability that the sum of the number on the die and the number on the marble is odd**.
---
### Step 1: Understand the possible outcomes
- A **die** has *... | \frac{1}{2} | augmented_math |
Aiden sells notebooks for $3 each and three for $8. If he started with 40 notebooks and made $72 from selling notebooks for $3 each, how much in total did he make from selling his notebooks? | We are told:
- Aiden sells **notebooks** at **$3 each** and **three for $8**.
- He **started with 40 notebooks**.
- He **made $72 from selling notebooks for $3 each**.
- We are to find the **total amount he made** from selling all his notebooks.
---
### Step 1: Determine how many notebooks were sold at $3 each
He m... | 112 | augmented_gsm8k |
There are 20 boys and 11 girls in the second grade and twice that number in the third grade. How many students are in grades 2 and 3? | We are given:
- In the **second grade**, there are **20 boys** and **11 girls**.
- In the **third grade**, there are **twice that number** of students as in the second grade.
---
### Step 1: Total students in second grade
$$
20 \text{ boys} + 11 \text{ girls} = 31 \text{ students}
$$
---
### Step 2: Total student... | 93 | gsm8k |
For what values of $x$ does the equation $\frac{x^2-2x-35}{x+5} > 0$ hold true? Express your answer in interval notation. | We are given the inequality:
$$
\frac{x^2 - 2x - 35}{x + 5} > 0
$$
---
### Step 1: Factor the numerator
We factor the quadratic in the numerator:
$$
x^2 - 2x - 35 = (x - 7)(x + 5)
$$
So the expression becomes:
$$
\frac{(x - 7)(x + 5)}{x + 5}
$$
---
### Step 2: Simplify the expression
We can cancel the common ... | (7, \infty) | augmented_math |
Samantha wants to purchase olive oil in bulk for her bakery. She can buy a 2-liter bottle of olive oil at a wholesale store for $15. At her local grocery store, a 250-milliliter bottle costs $4. How much money does she save by buying the larger bottle? | To determine how much Samantha saves by buying the larger 2-liter bottle instead of multiple smaller 250-milliliter bottles, we need to compare the total cost of the same amount of olive oil from both stores.
---
### Step 1: Convert units to be consistent
- 1 liter = 1000 milliliters
- So, 2 liters = 2000 millilit... | 17 | augmented_gsm8k |
At Academic Academy, to pass an algebra test you must score at least $80\%$. If there are 35 problems on the test, what is the greatest number you can miss and still pass? | We are told that to **pass** an algebra test at Academic Academy, you must score **at least 80%**. The test has **35 problems**.
---
### Step 1: Find 80% of 35
To find the minimum number of problems you must get correct to pass:
$$
0.80 \times 35 = 28
$$
So, you must get **at least 28 problems correct** to pass.
... | 7 | math |
Find $\tan \left( -\frac{3 \pi}{4} \right).$ | We are asked to find:
$$
\tan \left( -\frac{3\pi}{4} \right)
$$
---
### Step 1: Use the identity for tangent of a negative angle
$$
\tan(-\theta) = -\tan(\theta)
$$
So,
$$
\tan \left( -\frac{3\pi}{4} \right) = -\tan\left( \frac{3\pi}{4} \right)
$$
---
### Step 2: Evaluate $\tan\left( \frac{3\pi}{4} \right)$
Th... | 1 | math |
A 20 person classroom is filled with 40% girls. If 5 new boys join the classroom, what is the new percentage of girls in the class? | We are given:
- Total number of students initially: **20**
- Percentage of girls: **40%**
- **5 new boys** join the class.
---
### Step 1: Find the number of girls initially
$$
\text{Number of girls} = 40\% \text{ of } 20 = 0.40 \times 20 = 8 \text{ girls}
$$
---
### Step 2: Find the new total number of students
... | 32 | gsm8k |
A cylinder with radius $R$ and height $H$ has a tangent cone with the same radius $R$ and height $h$. If the cylinder and the cone have the same volume, find the ratio $\frac{H}{h}$. | We are given:
- A **cylinder** with **radius** $ R $ and **height** $ H $
- A **cone** with **radius** $ R $ and **height** $ h $
- The **volumes** of the cylinder and the cone are **equal**
We are to find the ratio $ \frac{H}{h} $.
---
### Step 1: Write the volume formulas
- Volume of a **cylinder** is:
$$
V_... | \frac{1}{3} | augmented_math |
Given that 2.2 pounds is approximately 1 kilogram, how many pounds are in 5 kilograms? Express your answer as a decimal to the nearest tenth. | We are given that:
$$
1 \text{ kilogram} \approx 2.2 \text{ pounds}
$$
To find how many pounds are in **5 kilograms**, we multiply:
$$
5 \text{ kg} \times 2.2 \text{ pounds/kg} = 11.0 \text{ pounds}
$$
So, the answer is:
$$
\boxed{11.0}
$$
**Answer: 11.0** (to the nearest tenth) | 11.0 | augmented_math |
Tom is going to buy a new video game that costs $g$ dollars. He also wants to purchase additional items in the game. Each item costs $i$ dollars, and he wants to buy 5 items. However, there's a promotion that gives him a 10% discount on the total cost of the items if he buys them together with the game. Tom has $50$ do... | We are given the following:
- The cost of the video game is $ g $ dollars.
- Each additional item costs $ i $ dollars, and he wants to buy **5 items**.
- There is a **10% discount** on the **total cost of the items** (the 5 items) **if** he buys them **together with the game**.
- Tom has **$50** to spend.
---
### St... | 50 - g - 4.5i | augmented_math |
Emily has some stamps, and a few are from foreign countries. One quarter of Emily's stamps are from foreign countries, and Tom has 3 times as many foreign stamps as Emily. If Tom has 48 foreign stamps, how many stamps does Emily have? | We are given the following information:
- One **quarter** of Emily's stamps are from foreign countries.
- Tom has **3 times as many foreign stamps** as Emily.
- Tom has **48 foreign stamps**.
---
### Step 1: Find how many foreign stamps Emily has
Since Tom has 3 times as many foreign stamps as Emily, and Tom has 48... | 64 | augmented_gsm8k |
Ava has 3 different containers of cookies. Each of her containers has 8 cookies left. Her sister, Emma, also has 3 containers of cookies. One of Emma's containers has 12 cookies left, another has 5 cookies left, and the third has 9 cookies left. How many more cookies does Emma have? | We are given the number of cookies in each container for both Ava and Emma, and we are asked how many **more** cookies Emma has than Ava.
---
### Step 1: Calculate the total number of cookies Ava has
Ava has **3 containers**, and **each** has **8 cookies**.
$$
\text{Ava's total} = 3 \times 8 = 24 \text{ cookies}
$$... | 2 | augmented_gsm8k |
For how many positive integers $n\geq 2$ is $1001_n$ a prime number? | We are given the number $ 1001_n $, which is written in base $ n $, and we are to find how many **positive integers $ n \geq 2 $** make $ 1001_n $ a **prime number**.
---
### Step 1: Convert $ 1001_n $ to base 10
The number $ 1001_n $ in base $ n $ means:
$$
1001_n = 1 \cdot n^3 + 0 \cdot n^2 + 0 \cdot n + 1 = n^3 ... | 0 | math |
A solution containing 20% alcohol is mixed with a solution containing 30% alcohol. The resulting mixture contains 25% alcohol. Find the ratio of the amount of the first solution to the amount of the second solution. | We are given:
- A **20% alcohol** solution is mixed with a **30% alcohol** solution.
- The **resulting mixture** is **25% alcohol**.
- We are to find the **ratio** of the **amount of the first solution** to the **amount of the second solution**.
---
### Step 1: Let’s define variables
Let:
- $ x $ = amount of the *... | 1 | augmented_math |
Ashley wants to decorate her room with a combination of flowers and vases. She has 5 different types of flowers (roses, lilies, daisies, sunflowers, and tulips) and 3 different types of vases (glass, ceramic, and wooden). However, she has a constraint that she cannot pair roses with ceramic vases, and she cannot pair s... | We are given:
- **5 types of flowers**: roses, lilies, daisies, sunflowers, and tulips.
- **3 types of vases**: glass, ceramic, and wooden.
- **Restrictions**:
- Roses **cannot** be paired with **ceramic** vases.
- Sunflowers **cannot** be paired with **wooden** vases.
We are to find the **total number of valid f... | 13 | augmented_math |
A fair coin is flipped seven times. What is the probability that the sum of all seven flips is even? Express your answer as a common fraction. | We are given that a **fair coin is flipped 7 times**, and we are to find the **probability that the sum of all seven flips is even**.
---
### Step 1: Understand the Problem
Each coin flip results in either **0** (for tails) or **1** (for heads). So the **sum of the flips** is the **number of heads** in the 7 flips.
... | \frac{1}{2} | augmented_math |
In an isosceles triangle, the two congruent sides measure 8 inches each, and the perimeter is 24 inches. What is the length of the base in inches? | We are given:
- An **isosceles triangle** has **two congruent sides** of **8 inches** each.
- The **perimeter** of the triangle is **24 inches**.
We are to find the **length of the base**.
---
### Step 1: Let the base be $ b $ inches.
Since it's an isosceles triangle, two sides are equal (each 8 inches), and the t... | 8 | augmented_math |
Maria has 36 yards of fabric. She gave one-third of it to her sister. Then, she used three-fifths of the remaining fabric to make a quilt. How many yards of fabric does Maria have left? | Let's break the problem down step by step.
### Step 1: Total fabric Maria has
Maria starts with **36 yards** of fabric.
---
### Step 2: She gives one-third of the fabric to her sister
One-third of 36 is:
$$
\frac{1}{3} \times 36 = 12 \text{ yards}
$$
So, Maria gives **12 yards** to her sister.
---
### Step 3: Fa... | 9.6 | augmented_gsm8k |
A student has nine test scores, with six of them being 85, 90, 78, 92, 88, and 76. If the remaining three scores are all distinct, what is the smallest possible value of the range of the nine test scores? | We are given:
- A student has **9 test scores**.
- **6 of them** are known: **85, 90, 78, 92, 88, and 76**.
- The **remaining 3 scores are all distinct**.
- We are to find the **smallest possible value of the range** of the **nine test scores**.
---
### Step 1: Understand the range
The **range** of a set of numbers... | 16 | augmented_math |
In a hotel lobby, there are three identical-looking elevators, labeled A, B, and C. Each elevator goes to a different set of floors: A goes to floors 1-10, B goes to floors 11-20, and C goes to floors 21-30. If you randomly choose an elevator and press a button for a random floor, what is the probability that you will ... | We are given three identical-looking elevators:
- **Elevator A** goes to floors **1–10**
- **Elevator B** goes to floors **11–20**
- **Elevator C** goes to floors **21–30**
You **randomly choose an elevator** and then **randomly press a button for a floor**. We are to find the **probability** that you arrive on a **f... | \frac{1}{3} | augmented_math |
Compute: $\frac{1}{4} \cdot \frac{2}{3} \cdot \frac{6}{5} \cdot 10.$ | We are given the expression:
$$
\frac{1}{4} \cdot \frac{2}{3} \cdot \frac{6}{5} \cdot 10
$$
### Step 1: Multiply the fractions and the whole number
We can rewrite the expression as:
$$
\left( \frac{1}{4} \cdot \frac{2}{3} \cdot \frac{6}{5} \right) \cdot 10
$$
### Step 2: Multiply the fractions
Multiply the numerat... | 2 | augmented_math |
In a regular hexagon of side length 2, a set of parallel lines is drawn, spaced 1 unit apart, such that they are perpendicular to one of the diagonals of the hexagon. Another set of parallel lines is drawn, spaced 1 unit apart, such that they are perpendicular to another diagonal of the hexagon. These lines cut the hex... | We are given a **regular hexagon** of side length **2**, and two sets of **parallel lines** are drawn:
- One set is spaced **1 unit apart**, and is **perpendicular to one of the diagonals**.
- Another set is spaced **1 unit apart**, and is **perpendicular to another diagonal**.
These lines divide the hexagon into **s... | 24 | augmented_math |
Consider a function $g : \mathbb{Z}^+ \to \mathbb{Z}^+$ such that for any positive integers $m$, $n$, and $p$,
\[g(m \times n + p) = g(m) + g(p)\times g(n).\]
Given that there exists a unique positive integer $a$ such that $g(a) = a$, determine the possible values of $a$. | We are given a function $ g : \mathbb{Z}^+ \to \mathbb{Z}^+ $ satisfying the identity:
$$
g(m \times n + p) = g(m) + g(p) \times g(n)
$$
for all positive integers $ m, n, p $, and we are told that **there exists a unique positive integer $ a $** such that $ g(a) = a $. Our goal is to determine the **possible values o... | 1 | augmented_math |
Calculate the sum of the geometric series $1+\left(\frac{1}{5}\right)+\left(\frac{1}{5}\right)^2 + \left(\frac{1}{5}\right)^3 + \dots$. Express your answer as a common fraction. | We are given the infinite geometric series:
$$
1 + \left(\frac{1}{5}\right) + \left(\frac{1}{5}\right)^2 + \left(\frac{1}{5}\right)^3 + \dots
$$
---
### Step 1: Identify the first term and the common ratio
This is a geometric series of the form:
$$
a + ar + ar^2 + ar^3 + \dots
$$
- First term $ a = 1 $
- Common r... | \frac{5}{4} | math |
Emily's science project requires her to collect 120 different types of leaves. Oak trees average 8 leaves per branch, maple trees average 5 leaves per branch, and pine trees average 4 needles per branch. Emily has gathered 5 oak branches, 7 maple branches, and 10 pine branches to collect their leaves. How many more lea... | We are given:
- **Oak trees** average **8 leaves per branch**
- **Maple trees** average **5 leaves per branch**
- **Pine trees** average **4 needles per branch** (though the question asks about **leaves**, we will assume this is a typo and that "needles" should be "leaves" for the purpose of this problem)
Emily has c... | 5 | augmented_gsm8k |
In $\triangle XYZ$, we have $\angle X = 90^\circ$ and $\tan Z = 3$. What is $\cos Z$? | We are given:
- Triangle $ \triangle XYZ $ is a **right triangle** with $ \angle X = 90^\circ $, so $ \angle X $ is the right angle.
- $ \tan Z = 3 $
- We are to find $ \cos Z $
---
### Step 1: Understand the triangle
Since $ \angle X = 90^\circ $, triangle $ XYZ $ is a **right triangle** with the right angle at $ ... | \frac{\sqrt{10}}{10} | math |
A $4\times4$ grid consists of 16 squares, each of side length one unit. Two edges of a square can only be joined by a diagonal of a single unit square if they share a vertex. A Hamiltonian path visits every square exactly once using only these diagonals.
Find the maximum possible number of different Hamiltonian paths... | We are given a **$4 \times 4$ grid** of squares (16 squares total), and we are to find the **maximum possible number of different Hamiltonian paths** that can pass through **one single square** in the grid.
---
### 🔍 Understanding the Problem
- A **Hamiltonian path** visits **every square exactly once**.
- The path... | 8 | augmented_math |
A hexagon is obtained by joining, in order, the points $(0,1)$, $(1,2)$, $(2,2)$, $(2,1)$, $(3,1)$, $(2,0)$, and $(0,1)$. The perimeter of the hexagon can be written in the form $a+b\sqrt{2}+c\sqrt{5}$, where $a$, $b$ and $c$ are whole numbers. Find $a+b+c$. | We are given a hexagon formed by joining the points in order:
$$
(0,1), (1,2), (2,2), (2,1), (3,1), (2,0), (0,1)
$$
We are to find the **perimeter** of this hexagon and express it in the form:
$$
a + b\sqrt{2} + c\sqrt{5}
$$
where $a$, $b$, and $c$ are whole numbers. Then we are to compute $a + b + c$.
---
### St... | 6 | math |
A popular food festival is held every year for four days. The total attendance over the four days this year was 4800 people. The third day was extremely hot, resulting in only two-thirds of the number of people that attended the first day attending the third day. The fourth day was the grand finale, so attendance was d... | We are given the following information about the attendance at a 4-day food festival:
- Total attendance over 4 days: **4800 people**
- **Day 3** had **two-thirds** of the attendance of **Day 1**
- **Day 4** had **double** the attendance of **Day 1**
- **Day 2** had **150 more** people than **Day 1**
Let’s define:
-... | 665 | augmented_gsm8k |
Two sequences $X$ and $Y$ start with the same number, which is an integer. The first sequence increases by 2 and the second decreases by 3. What is the smallest positive integer with which both sequences can start such that the 15th term of both sequences is a multiple of 7? | We are given two sequences:
- Sequence $ X $: starts with some integer $ a $, and increases by 2 each time.
- Sequence $ Y $: starts with the same integer $ a $, and decreases by 3 each time.
We are to find the **smallest positive integer** $ a $ such that the **15th term** of both sequences is a **multiple of 7**.
... | 7 | augmented_math |
In the diagram below, the shaded region represents a water tank with a circular base of radius 12 feet and height 36 feet. The water in the tank is currently at 75% capacity. What is the length of the line segment $AB$, where $A$ is the point on the top of the tank's water level that is directly above the center of the... | We are given a **cylindrical tank** with:
- **Radius of the base**: $ r = 12 $ feet
- **Height of the tank**: $ h = 36 $ feet
- **Water is at 75% capacity**, so the **height of the water** is:
$$
\text{Water height} = 0.75 \times 36 = 27 \text{ feet}
$$
We are to find the **length of the line segment $ AB $** wh... | 29.55 | augmented_math |
Find the smallest integer n such that the sum of the digits of n^2 is equal to the sum of the digits of n.
In other words, we are looking for the smallest integer n such that the sum of the digits of n squared is equal to the sum of the digits of n itself.
For example, if n = 12, then n^2 = 144, and the sum of the di... | We are looking for the **smallest integer** $ n $ such that:
$$
\text{sum of digits of } n = \text{sum of digits of } n^2
$$
---
### Step-by-step Plan:
1. **Define a helper function** to compute the sum of digits of a number.
2. **Iterate through integers starting from 1**, compute $ n^2 $, and check if the sum of ... | 1 | augmented_math |
In trapezoid $ABCD$, $AB$ is parallel to $CD$, and $AB=6$ inches. Points $E$ and $F$ lie on sides $BC$ and $AD$, respectively, such that $BE=DF=2$ inches. The area of quadrilateral $AEFD$ is equal to half the area of triangle $CDE$. What is the length of segment $CD$? | We are given a trapezoid $ABCD$ with $AB \parallel CD$, and $AB = 6$ inches. Points $E$ and $F$ lie on sides $BC$ and $AD$ respectively such that $BE = DF = 2$ inches. Also, the area of quadrilateral $AEFD$ is equal to half the area of triangle $CDE$. We are to find the length of segment $CD$.
---
### Step 1: Underst... | 12 | augmented_math |
Mrs. Thompson has a vitamin supplement that lasts for 45 days. After three-fourths of the days, she has 18 pills remaining. How many pills is Mrs. Thompson supposed to take daily if her dosage is consistent? | We are told:
- Mrs. Thompson's vitamin supplement lasts for **45 days**.
- After **three-fourths of the days**, she has **18 pills remaining**.
- We are to find out how many pills she takes **daily**, assuming her **dosage is consistent**.
---
### Step 1: Determine how many days have passed
Three-fourths of 45 days... | 2 | augmented_gsm8k |
A bakery sells a total of $250$ loaves of bread per day. They offer two types of bread: whole wheat and white bread. If the ratio of whole wheat bread to white bread is $3:5$, and each loaf of whole wheat bread weighs $1.2$ pounds and each loaf of white bread weighs $1.0$ pounds, how many pounds of bread does the baker... | We are given:
- Total loaves sold per day: **250**
- Ratio of whole wheat to white bread: **3:5**
- Weight of each whole wheat loaf: **1.2 pounds**
- Weight of each white loaf: **1.0 pound**
---
### Step 1: Find the number of whole wheat and white loaves
The ratio of whole wheat to white bread is **3:5**, so the to... | 268.75 | augmented_math |
Emily and Michael have a snow shoveling business. They charge $5 for each driveway they clear. On Thursday they cleared 4 driveways. On Friday they cleared 2 driveways. On Saturday, they counted their money and found they had $85 for all three days. How many driveways did they clear on Saturday? | We are told that:
- Emily and Michael charge **$5 per driveway**.
- On **Thursday**, they cleared **4 driveways**.
- On **Friday**, they cleared **2 driveways**.
- On **Saturday**, they had **$85 total** for all three days.
We are asked: **How many driveways did they clear on Saturday?**
---
### Step 1: Calculate e... | 11 | augmented_gsm8k |
The graph of $y = f(x)$ is shown below.
[asy]
unitsize(0.3 cm);
real func(real x) {
real y;
if (x >= -3 && x <= 0) {y = -2 - x;}
if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}
if (x >= 2 && x <= 3) {y = 2*(x - 2);}
return(y);
}
int i, n;
for (i = -8; i <= 8; ++i) {
draw((i,-8)--(i,8),gray(0.7));
... | We are given the graph of $ y = f(x) $ and the graph of $ y = g(x) $, and we are asked to express $ g(x) $ in terms of $ f(x) $.
---
### Step 1: Understand the graph of $ f(x) $
From the first Asymptote code, we can deduce the definition of $ f(x) $:
$$
f(x) =
\begin{cases}
-2 - x, & \text{if } -3 \le x \le 0 \\
\... | f(4-x) | math |
The function $f(x)$ has a domain of $\{2,3,4,5\}$ and a range of $\{0,3,6,9\}$. The function $g(x)$ has a domain of $\{0,3,6,9\}$ and is defined by $g(x) = x - 2$. What is the product of all possible values of $g(f(x))$? | We are given:
- The **domain** of $ f(x) $ is $ \{2, 3, 4, 5\} $
- The **range** of $ f(x) $ is $ \{0, 3, 6, 9\} $
- The function $ g(x) = x - 2 $
- We are to find the **product of all possible values of $ g(f(x)) $**
---
### Step 1: Understand what $ g(f(x)) $ means
We are composing the functions:
$$
g(f(x)) = g... | -56 | augmented_math |
The volumes of two spheres are in the ratio $8:27$. What is the ratio of their surface areas? Express your answer in the form $a:b$. | We are given that the **volumes** of two spheres are in the ratio $ 8:27 $, and we are to find the **ratio of their surface areas**.
---
### Step 1: Use the volume formula for a sphere
The volume $ V $ of a sphere is given by:
$$
V = \frac{4}{3} \pi r^3
$$
Let the radii of the two spheres be $ r_1 $ and $ r_2 $, a... | 4:9 | augmented_math |
Find the smallest positive integer $n$ such that the function $g(x) = \lfloor x^n \rfloor$ satisfies $g(1.1) = g(g(1.1))$. | We are given the function:
$$
g(x) = \lfloor x^n \rfloor
$$
and we are to find the **smallest positive integer** $ n $ such that:
$$
g(1.1) = g(g(1.1))
$$
---
### Step 1: Compute $ g(1.1) $
We compute:
$$
g(1.1) = \lfloor 1.1^n \rfloor
$$
We'll compute this for small values of $ n $, and check whether the condi... | 1 | augmented_math |
What is the smallest positive integer that is divisible by all the prime numbers from 2 to 7, inclusive? | We are asked to find the **smallest positive integer** that is divisible by **all the prime numbers from 2 to 7, inclusive**.
### Step 1: Identify the prime numbers from 2 to 7
The prime numbers in this range are:
$$
2, 3, 5, 7
$$
### Step 2: Understand the problem
We are looking for the **least common multiple (LCM... | 210 | augmented_math |
Find all values of $x$ that satisfy \[\frac{x^2}{x+1} \ge \frac{2}{x-1} + \frac{5}{4}.\]
Determine the values of $x$ that satisfy the inequality \[\frac{x^2+2x-3}{x^2+3x-4} \le \frac{1}{x+1}.\] | We are given two inequalities to solve:
---
### **Problem 1:**
$$
\frac{x^2}{x+1} \ge \frac{2}{x-1} + \frac{5}{4}
$$
---
### **Step 1: Combine the right-hand side**
We combine the two terms on the right-hand side:
$$
\frac{2}{x-1} + \frac{5}{4} = \frac{8 + 5(x-1)}{4(x-1)} = \frac{8 + 5x - 5}{4(x-1)} = \frac{5x + ... | (-1, 1) \cup (3, \infty) | augmented_math |
A sequence $(b_n)$ is defined as follows: $b_1 = 2,$ $b_2 = \frac{1}{3},$ and
$b_n = \frac{2 - b_{n - 1}}{3b_{n - 2}}$ for all $n \ge 3.$ Find $b_{100}.$ | We are given a recursive sequence:
- $ b_1 = 2 $
- $ b_2 = \frac{1}{3} $
- For $ n \geq 3 $, $ b_n = \frac{2 - b_{n-1}}{3b_{n-2}} $
We are to find $ b_{100} $.
---
### Step 1: Compute the first few terms
Let’s compute the first few terms of the sequence to see if there's a **pattern** or **cycle**.
#### Compute $... | \frac{31}{18} | augmented_math |
What is the smallest positive integer $n$ such that $3^n$ has a remainder of 1 when divided by 17? | We are asked to find the **smallest positive integer** $ n $ such that:
$$
3^n \equiv 1 \pmod{17}
$$
This means that $ 3^n - 1 $ is divisible by 17, or equivalently, $ 3^n \equiv 1 \pmod{17} $.
This is a classic problem in **modular arithmetic**, and the smallest such $ n $ is called the **order** of 3 modulo 17.
-... | 16 | augmented_math |
A restaurant plans to hire 30 new servers to help with the increased customer traffic during the summer season. Each server will work 6 hours a day and will be paid $12.00 an hour. After 7 days, how much will the restaurant pay all 30 new servers? | To calculate how much the restaurant will pay all 30 new servers after 7 days, we can follow these steps:
---
### **Step 1: Calculate daily pay for one server**
- Each server works **6 hours per day**.
- Each hour, a server is paid **$12.00**.
$$
\text{Daily pay per server} = 6 \, \text{hours/day} \times \$12.00/\t... | 15120 | augmented_gsm8k |
Determine the digit $B$ such that the number $23B6$ is divisible by $4$. | We are given the number $ 23B6 $, and we are to determine the digit $ B $ such that the number is divisible by 4.
---
### Step 1: Understand the Rule for Divisibility by 4
A number is divisible by 4 **if and only if** the number formed by its **last two digits** is divisible by 4.
So, for the number $ 23B6 $, we on... | 1 | augmented_math |
A lottery game offers three possible prizes: a $1000 cash prize with a 10% chance of winning, a $500 cash prize with a 20% chance of winning, and a $200 cash prize with a 70% chance of winning. What is the expected value of the total amount of money that will be won by a player who plays the game 8 times? Give your ans... | We are given the probabilities and values of three possible prizes in a lottery game:
- **$1000 prize** with a **10% (0.10)** chance
- **$500 prize** with a **20% (0.20)** chance
- **$200 prize** with a **70% (0.70)** chance
### Step 1: Find the **expected value of a single play**
The expected value $ E $ for one pl... | 2720.0 | augmented_math |
In a class, 30% of the students are given a special badge every year. However, due to a policy change, the percentage of students receiving the badge increased by 20% each year for the next three years. If 60 students received the badge in the first year, how many students will receive the badge in the fourth year? | We are told that:
- In the **first year**, **30%** of the students received a badge, and **60 students** received it.
- The **percentage of students receiving the badge increases by 20% each year** for the next **three years**.
- We are to find how many students will receive the badge in the **fourth year**.
---
###... | 104 | augmented_math |
Stan weighs 5 more pounds than Steve. Steve is eight pounds lighter than Jim. If Jim weighs 110 pounds and the three of them crowd onto a scale at the same time, what is their total weight? | We are given the following information:
- **Jim** weighs **110 pounds**.
- **Steve** is **8 pounds lighter** than Jim.
- **Stan** weighs **5 pounds more** than Steve.
Let's calculate each person's weight step by step.
---
### Step 1: Steve's weight
Steve is 8 pounds lighter than Jim:
$$
\text{Steve} = 110 - 8 = 10... | 319 | gsm8k |
Tom needs 24 oz of cream cheese for a cheesecake recipe. He can either buy a single 24 oz package of the store-brand cream cheese for $10, or he can buy a 12 oz package of store cream cheese for $6 and use a coupon to get an additional two 6 oz packages that normally sell for $3.0 each at a 30% discount (which is appli... | We are given two options for Tom to buy 24 oz of cream cheese, and we need to determine which is the **cheapest**.
---
### **Option 1: Buy one 24 oz package**
- **Price:** $10
---
### **Option 2: Buy 12 oz package + 2 x 6 oz packages with coupon**
Let’s break this down:
#### Step 1: Buy a 12 oz package
- **Price... | 10 | augmented_gsm8k |
How many positive five-digit integers are divisible by 12? | We are asked to find how many **positive five-digit integers** are divisible by **12**.
---
### Step 1: Understand the range of five-digit numbers
A five-digit number ranges from:
$$
10000 \quad \text{to} \quad 99999
$$
---
### Step 2: Find the smallest five-digit number divisible by 12
We want the **smallest** ... | 7500 | augmented_math |
End of preview. Expand in Data Studio
Generated dataset
This dataset was pushed from existing parquet chunks with the following command:
python push_only.py --work_dir ./regen_melhoushi__OpenMathInstruct-2-Qwen3-8B-nothink/ --output_repo melhoushi/OpenMathInstruct-Q2-wen3-8B-nothink --split train --delete_existing_repo
Parameters
- work_dir:
./regen_melhoushi__OpenMathInstruct-2-Qwen3-8B-nothink/ - output_repo:
melhoushi/OpenMathInstruct-Q2-wen3-8B-nothink - dataset_config:
None - split:
train - private:
False
Notes
This README was generated automatically by the standalone push-only script.
- Downloads last month
- 105