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7. 11. 3 227 " 3 18 " 8. 6 1.21 " 12. 1 2 28 " 16. 48b3, b 0 14. 15. 9 27x3y5, x. 0, y. 0 $ In 17–20, in each case, solve for the variable, using the set of real numbers as the replacement set. 17. y2 81 20. 2k2 144 0 18. m2 0.09 19. 3x2 600 0.25a8b10 " 498 Operations With Radicals 21. a. Use a calculator to evaluate 315.4176. " b. Is 315.4176 " a rational or an irrational number? Explain your answer. In 22–33, in each case, perform the indicated operation and simplify the result. 18 1 8 2 32 " 50 2 " 98 1 1 2 72 " 22. 24. 26. 28. 30. 32. " 2 8 2 " 2 " 7 " 16 " 2 " 3 " " 2 " 70 2 A A " 21 7 24 A " B B B 2 5 " 10 A " B 23. 3 20 2 2 45 25. 27. 29. 31. 33. " 75 2 3 " 12 " 3 A 5 " B 98 4 " 2 2 " 162 50 " 5 " 9 " 80 2 " " 1 6 " 60 A " B In 34 and 35, in each case, select the numeral preceding the correct choice. 34. The expression 108 2 (1) 105 " 35. The sum of (1) 9 34 " " (2) 35 and 9 2 " " (2) 13 is equivalent to (3) 5 3 " 3 " 3 " is 32 2 " (3) 10 34 " (4) 6 (4) 15 In 36–39, for each irrational number given, write a rational approximation: a. as shown on a calculator display b. rounded to four significant digits. 36. 194 3 16 " 40. The area of a square is 28.00 square meters. 38. 37. " " 2 0.7 39. 5 227 " a. Find, to the nearest thousandth of a meter, the length of one side of the square. b. Find, to the nearest thousandth of a meter, the perimeter of the square. c. Explain why the answer to part b is not equal to 4 times the answer to part a. 41. Write as a polynomial in simplest form: 42 |
. What is the product of 3.5x2 and 6.2x3? A 2x Exploration STEP 1. On a sheet of graph paper, draw the positive ray of the real number line. Draw a square, using the interval from 0 to 1 as one side of the square. Draw the diagonal of this square from 0 to its opposite vertex. Cumulative Review 499 2. Why? Place the point of a compass at The length of the diagonal is " 0 and the pencil of a compass at the opposite vertex of the square so 2. Keep that the measure of the opening of the pair of compasses is " the point of the compass at 0 and use the pencil to mark the endpoint of a segment of length on the number line. 2 " STEP 2. Using the same number line, draw a rectangle whose dimensions are 2 as one by 1, using the interval on the number line from 0 to " side. Draw the diagonal of this rectangle from 0 to the opposite vertex. 3. Place the point of a pair of compasses The length of the diagonal is at 0 and the pencil at the opposite vertex of the rectangle so that the 3. Keep the point measure of the opening of the pair of compasses is " of the compasses at 0 and use the pencil to mark the endpoint of a segment of length on the number line. " " 2 STEP 3. Repeat step 2, drawing a rectangle whose dimensions are by 1 to on the number line. This point should coincide with 2 on the " 4 3 3 " locate number line. " STEP 4. Explain how these steps can be used to locate integer n. n for any positive " CUMULATIVE REVIEW CHAPTERS 1–12 Part I Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. 1. A flagpole casts a shadow five feet long at the same time that a man who is six feet tall casts a shadow that is two feet long. How tall is the flagpole? (1) 12 feet (2) 15 feet (4) 24 feet (3) 18 feet 2. Which of the following is an irrational number? 6 " (1) 12 3 (2) 24 4 (3. Parallelogram ABCD is drawn on the coordinate plane with the vertices A(0, 0), B(8, 0), C(10, 5), and D(2, 5). The number of square units in the area of ABCD is |
(1) 40 (2) 20 4. The product (3a 2)(2a 3) can be written as (3) 16 (4) 10 " " 5 2 5 (4) 3 " (1) 6a2 6 (2) 6a2 a 6 (3) 6a2 5a 6 (4) 6a2 5a 6 500 Operations With Radicals 5. If 0.2x 8 x 4, then x equals (1) 120 (2) 12 (3) 15 (4) 15 6. If the height of a right circular cylinder is 12 centimeters and the measure of the diameter of a base is 8 centimeters, then the volume of the cylinder is (1) 768p (2) 192p 7. The identity 3(a 7) 3a 21 is an example of (4) 48p (3) 96p (1) the additive inverse property (2) the associative property for addition (3) the commutative property for addition (4) the distributive property of multiplication over addition 8. The value of a share of stock decreased from $24.50 to $22.05. The percent of decrease was (1) 1% (2) 10% (3) 11% (4) 90% 9. When written in scientific notation, 384.5 is equal to (1) 38.45 101 (2) 3.845 102 (3) 3.845 103 (4) 3.845 10–2 10. In right triangle ABC, C is the right angle. The cosine of B is AC BC AC AB BC AB BC AC (3) (2) (1) (4) Part II Answer all questions in this part. Each correct answer will receive 2 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 11. A car uses of a tank of gasoline to travel 600 kilometers. The tank holds 48 liters of gasoline. How far can the car go on one liter of gasoline? 3 4 12. A ramp that is 20.0 feet long makes an angle of 12.5° with the ground. What is the perpendicular distance from the top of the ramp to the ground? Part III Answer all questions in this part. Each correct answer will receive 3 credits. Clearly indicate the necessary steps, including appropriate formula substitutions |
, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 13. ABCD is a trapezoid with BC'AB and BC'CD and CD = 8. A line segment is drawn from A to E, the midpoint of, AB 13, BC 12,.CD Cumulative Review 501 a. Find the area of AED. b. Find the perimeter of AED. 14. Maria’s garden is in the shape of a rectangle that is twice as long as it is wide. Maria increases the width by 2 feet, making the garden 1.5 times as long as it is wide. What are the dimensions of the original garden? Part IV Answer all questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 15. The length of the base of an isosceles triangle is and the length of the. Express the perimeter as an exact value in simplest form. 2 " 10 altitude is 12 2 " 16. In the coordinate plane, O is the origin, A is a point on the y-axis, and B is g AB is 2 and its y-intercept is 8. a point on the x-axis. The slope of a. Write the equation of g AB. b. Draw g AB on graph paper. c. What are the coordinates of B? d. What is the x-intercept of g?AB CHAPTER 13 CHAPTER TABLE OF CONTENTS 13-1 Solving Quadratic Equations 13-2 The Graph of a Quadratic Function 13-3 Finding Roots from a Graph 13-4 Graphic Solution of a Quadratic-Linear System 13-5 Algebraic Solution of a Quadratic-Linear System Chapter Summary Vocabulary Review Exercises Cumulative Review 502 QUADRATIC RELATIONS AND FUNCTIONS When a baseball is hit, its path is not a straight line. The baseball rises to a maximum height and then falls, following a curved path throughout its flight.The maximum height to which it rises is determined by the force with which the ball was hit and the angle at which it was hit.The height of the ball at any time can be found by using an equation, as can the maximum |
height to which the ball rises and the distance between the batter and the point where the ball hits the ground. In this chapter we will study the quadratic equation that models the path of a baseball as well as functions and relations that are not linear. Solving Quadratic Equations 503 13-1 SOLVING QUADRATIC EQUATIONS The equation x2 3x 10 0 is a polynomial equation in one variable. This equation is of degree two, or second degree, because the greatest exponent of the variable x is 2. The equation is in standard form because all terms are collected in descending order of exponents in one side, and the other side is 0. A polynomial equation of degree two is also called a quadratic equation. The standard form of a quadratic equation in one variable is ax2 bx c 0 where a, b, and c are real numbers and a 0. To write an equation such as x(x 4) 5 in standard form, rewrite the left side without parentheses and add 5 to both sides to make the right side 0. x(x 4) 5 x2 4x 5 x2 4x 5 0 Solving a Quadratic Equation by Factoring When 0 is the product of two or more factors, at least one of the factors must be equal to 0. This is illustrated by the following examples: 5 0 0 0 7 0 (2) 0 0 0 (3 In general: When a and b are real numbers, ab 0 if and only if a 0 or b 0. This principle is used to solve quadratic equations. For example, to solve the quadratic equation x2 3x 2 0, we can write the left side as (x 2)(x 1). and (x 1) represent real numbers whose product is 0. The The factors equation will be true if the first factor is 0, that is, if (x 2) 0 or if the second factor is 0, that is, if (x 1) 0. (x 2 2) x2 3x 2 0 (x 2)(x 1 check will show that both 2 and 1 are values of x for which the equation is true. 504 Quadratic Relations and Functions Check for x 2: x2 3x 2 0 Check for x 1: x2 3x 2 0 (2)2 3(2) 2 4 6 2 0 5? 5? 0 0 0 ✔ (1)2 3(1) |
2 1 3 2 0 5? 5? 0 0 0 ✔ Since both 2 and 1 satisfy the equation x2 3x 2 0, the solution set of this equation is {2, 1}. The roots of the equation, that is, the values of the variable that make the equation true, are 2 and 1. Note that the factors of the trinomial x2 3x 2 are (x 2) and (x 1). If the trinomial x2 3x 2 is set equal to zero, then an equation is formed and this equation has a solution set, {2, 1}. Procedure To solve a quadratic equation by factoring: 1. If necessary, transform the equation into standard form. 2. Factor the quadratic expression. 3. Set each factor containing the variable equal to 0. 4. Solve each of the resulting equations. 5. Check by substituting each value of the variable in the original equation. The real number k is a root of ax2 bx c 0 if and only if (x k) is a factor of ax2 bx c. EXAMPLE 1 Solve and check: x2 7x 10 Solution How to Proceed (1) Write the equation in standard form: (2) Factor the quadratic expression: (3) Let each factor equal 0: (4) Solve each equation: x2 7x 10 x2 7x 10 0 (x 2)(x 55) Check both values in the original equation: Solving Quadratic Equations 505 Check for x 2: x2 7x 10 210 (2)2 7(2) 4 14 5? 5? 10 10 ✔ 210 Check for x 5: x2 7x 10 210 (5)2 7(5) 25 35 5? 5? 10 10 ✔ 210 Answer x 2 or x 5; the solution set is {2, 5}. EXAMPLE 2 Solution List the members of the solution set of 2x2 3x. How to Proceed (1) Write the equation in standard form: (2) Factor the quadratic expression: (3) Let each factor equal 0: (4) Solve each equation: (5) Check both values in the original equation. 2x2 3x 2x2 3x 0 x(2x 3) 0 x 0 2x 3 0 2x 3 x 5 3 2 Check for x 0: 2x2 3x 2(0)2 5? 3(0) |
0 0 ✔ Answer: The solution set is 0, 3 2. V U 3 Check for x : 2 2x2 3x 2 5? 3 3 2 B 9 5 Note: We never divide both sides of an equation by an expression containing a variable. If we had divided 2x2 = 3x by x, we would have obtained the equation 2x 3, whose solution is but would have lost the solution x 0. 3 2 506 Quadratic Relations and Functions EXAMPLE 3 Find the solution set of the equation x(x 8) 2x 25. Solution How to Proceed (1) Use the distributive property on the left side of the equation: (2) Write the equation in standard form: (3) Factor the quadratic expression: x(x 8) 2x 25 x2 8x 2x 25 x2 10x 25 0 (x 5)(x 5) 0 (4) Let each factor equal 0: (5) Solve each equation: (6) Check the value in the original equation(x 8) 2x 25 5? 2(5) 5(5 8) 5? 10 2 25 15 15 15 ✔ 25 Answer x 5; the solution set is {5}. Every quadratic equation has two roots, but as Example 3 shows, the two roots are sometimes the same number. Such a root, called a double root, is written only once in the solution set. EXAMPLE 4 The height h of a ball thrown into the air with an initial vertical velocity of 24 feet per second from a height of 6 feet above the ground is given by the equation h l6t2 24t 6 where t is the time, in seconds, that the ball has been in the air. After how many seconds is the ball at a height of 14 feet? Solution (1) In the equation, let h 14: (2) Write the equation in standard form: (3) Factor the quadratic expression: (4) Solve for t: h l6t2 24t 6 14 16t2 24t 6 0 16t2 24t 8 0 8(2t2 3t 1) 0 8(2t 1)(t 1) 2t 1 0 2t Answer The ball is at a height of 14 feet after second as it rises and after 1 second 1 2 as it falls. EXAMPLE 5 The area of a circle is equal to 3 times its circumference. What is the radius of the circle? |
Solving Quadratic Equations 507 Solution How to Proceed (1) Write an equation from the given information: (2) Set the equation in standard form: (3) Factor the quadratic expression: (4) Solve for r: (5) Reject the zero value. Use the positive value to write the answer. Answer The radius of the circle is 6 units. pr2 3(2pr) pr2 6pr 0 pr(r 6) 0 pr 0 r 0 r 6 0 r 6 EXERCISES Writing About Mathematics 1. Can the equation x2 9 be solved by factoring? Explain your answer. 2. In Example 4, the trinomial was written as the product of three factors. Only two of these factors were set equal to 0. Explain why the third factor was not used to find a solution of the equation. Developing Skills In 3–38, solve each equation and check. 3. x2 3x 2 0 6. r2 12r 35 0 9. x2 2x 1 0 12. x2 x 6 0 15. x2 x 12 0 18. m2 64 0 21. s2 s 0 24. x2 x 6 27. r2 4 30. s2 4s 33. 30 x x2 36. x(x 2) 35 4. z2 5z 4 0 7. c2 6c 5 0 10. y2 11y 24 0 13. x2 2x 15 0 16. x2 49 0 19. 3x2 12 0 22. x2 3x 0 25. y2 3y 28 28. x2 121 31. y2 8y 20 34. x2 3x 4 50 37. y(y 3) 4 5. x2 8x 16 0 8. m2 10m 9 0 11. x2 4x 5 0 14. r2 r 72 0 17. z2 4 0 20. d2 2d 0 23. z2 8z 0 26. c2 8c 15 29. y2 6y 32. x2 9x 20 35. 2x2 7 5 5x 38. x(x 3) 40 508 Quadratic Relations and Functions In 39–44, solve each equation and check. y 1 3 3 5 6 y 24x 12 x x 1 4 21 5 4 x 40. 42. 43. 39. 41. 44. x 3 5 12 x 2x Applying Skills 45. The height |
h of a ball thrown into the air with an initial vertical velocity of 48 feet per sec- ond from a height of 5 feet above the ground is given by the equation h 16t2 48t 5 where t is the time in seconds that the ball has been in the air. After how many seconds is the ball at a height of 37 feet? 46. A batter hit a baseball at a height of 4 feet with a force that gave the ball an initial vertical velocity of 64 feet per second. The equation h 16t2 64t 4 gives the height h of the baseball t seconds after the ball was hit. If the ball was caught at a height of 4 feet, how long after the batter hit the ball was the ball caught? 47. The length of a rectangle is 12 feet more than twice the width. The area of the rectangle is 320 square feet. a. Write an equation that can be used to find the length and width of the rectangle. b. What are the dimensions of the rectangle? 48. A small park is enclosed by four streets, two of which are parallel. The park is in the shape of a trapezoid. The perpendicular distance between the parallel streets is the height of the trapezoid. The portions of the parallel streets that border the park are the bases of the trapezoid. The height of the trapezoid is equal to the length of one of the bases and 20 feet longer than the other base. The area of the park is 9,000 square feet. a. Write an equation that can be used to find the height of the trapezoid. b. What is the perpendicular distance between the two parallel streets? 49. At a kennel, each dog run is a rectangle whose length is 4 feet more than twice the width. Each run encloses 240 square feet. What are the dimensions of the runs? 50. One leg of a right triangle is 14 centimeters longer than the other leg. The length of the hypotenuse is 26 centimeters. What are the lengths of the legs? 13-2 THE GRAPH OF A QUADRATIC FUNCTION A batter hits a baseball at a height of 3 feet off the ground, with an initial vertical velocity of 72 feet per second. The height, y, of the baseball can be found using the equation when x represents the number of seconds from the time the ball was hit. The graph of the equation––and, inci- y 5 216x2 1 72x 1 3 The |
Graph of a Quadratic Function 509 dentally, the actual path of the ball––is a curve called a parabola. The special properties of parabolas are discussed in this section. An equation of the form y ax2 bx c (a 0) is called a second-degree polynomial function or a quadratic function. It is a function because for every ordered pair in its solution set, each different value of x is paired with one and only one value of y. The graph of any quadratic function is a parabola. Because the graph of a quadratic function is nonlinear, a larger number of points are needed to draw the graph than are needed to draw the graph of a linear function. The graphs of two equations of the form y ax2 bx c, one that has a positive coefficient of x2 (a 0) and the other a negative coefficient of x2 (a 0), are shown below. CASE 1 The graph of ax2 bx c where a 0. Graph the quadratic function y x2 4x 1 for integral values of x from 1 to 5 inclusive: (1) Make a table using integral values of x from 1 to 5. (2) Plot the points associated with each ordered pair (x, y). (3) Draw a smooth curve through the points. x 1 0 1 2 3 4 5 x2 4x 12 1 16 16 1 25 20 –1 – The values of x that were chosen to draw this graph are not a random set of numbers. These numbers were chosen to produce the pattern of y-values shown in the table. Notice that as x increases from 1 to 2, y decreases from 6 to 3. Then the graph reverses and as x continues to increase from 2 to 5, y increases from 3 to 6. The smallest value of y occurs at the point (2, 3). The point is called the minimum because its y-value, 3, is the smallest value of y for the equation. The minimum point is also called the turning point or vertex of the parabola. 510 Quadratic Relations and Functions The graph is symmetric with respect to the vertical line, called the axis of symmetry of the parabola. The axis of symmetry of the parabola is determined by the formula x 5 2b 2a where a and b are the coefficients x2 and x, respectively, from the standard form of the quadratic equation. For the |
function y x2 4x 1, the equation of the vertical line of symor x 2. Every point on the parabola to the left of x 2 metry is matches a point on the parabola to the right of x 2, and vice versa. 2(24) 2(1) x 5 This example illustrates the following properties of the graph of the qua- dratic equation y x2 4x 1: 1. The graph of the equation is a parabola. 2. The parabola is symmetric with respect to the vertical line x 2. 3. The parabola opens upward and has a minimum point at (2, 3). 4. The equation defines a function. For every x-value there is one and only one y-value. 5. The constant term, 1, is the y-intercept. The y-intercept is the value of y when x is 0. CASE 2 The graph of ax2 bx c where a 0. Graph the quadratic function y x2 2x 5 using integral values of x from 4 to 2 inclusive: (1) Make a table using integral values of x from 4 to 2. (2) Plot the points associated with each ordered pair (x, y). (3) Draw a smooth curve through the points. x 4 3 2 1 0 1 2 x2 2x 5 16 1 – Again, the values of x that were chosen to produce the pattern of y-values shown in the chart. Notice that as x increases from 4 to 1, y increases from The Graph of a Quadratic Function 511 3 to 6. Then the graph reverses, and as x continues to increase from 1 to 2, y decreases from 6 to 3. The largest value of y occurs at the point (1, 6). This point is called the maximum because its y-value, 6, is the largest value of y for the equation. In this case, the maximum point is the turning point or vertex of the parabola. The graph is symmetric with respect to the vertical line whose equation is x 1. As shown in Case 1, this value of x is again given by the formula x 5 2b 2a where a and b are the coefficients of x2 and x, respectively. For the function y x2 2x 5, the equation of the axis of symmetry is x or x 1. This example illustrates the following properties of the graph of the qua- 2( |
22) 2(21) dratic equation y 5 2x2 2 2x 1 5 : 1. The graph of the equation is a parabola. 2. The parabola is symmetric with respect to the vertical line x 1. 3. The parabola opens downward and has a maximum point at (1, 6) 4. The equation defines a function. 5. The constant term, 5, is the y-intercept. When the equation of a parabola is written in the form y ax2 bx c 0, the equation of the axis of symmetry is x and the x-coordinate of the turn. This can be used to find a convenient set of values of x to be used ing point is when drawing a parabola. Use as a middle value of x with three values that are smaller and three that are larger. 2b 2a 2b 2a 2b 2a KEEP IN MIND 1. The graph of y ax2 bx c, with a 0, is a parabola. 2. The axis of symmetry of the parabola is a vertical line whose equation is 2b x. 2a 3. A parabola has a turning point on the axis of symmetry. The x-coordinate of the turning point is by substituting the vertex of the parabola. 2b 2a 2b 2a. The y-coordinate of the turning point is found into the equation of the parabola. The turning point is 4. If a is positive, the parabola opens upward and the turning point is a minimum. The minimum value of y for the parabola is the y-coordinate of the turning point. 5. If a is negative, the parabola opens downward and the turning point is a maximum. The maximum value of y for the parabola is the y-coordinate of the turning point. 512 Quadratic Relations and Functions EXAMPLE 1 a. Write the equation of the axis of symmetry of y x2 3. b. Graph the function. c. Does the function have a maximum or a minimum? d. What is the maximum or minimum value of the function? e. Write the coordinates of the vertex. Solution a. In this equation, a 1. Since there is no x term in the equation, the equation can be written as y x2 0x 3 with b 0. The equation of the axis of symmetry is x 2b 2a |
5 2(0) 2(1) 0 or x 0. b. (1) Since the vertex of the parabola is on the axis of symmetry, the x-coordinate of the vertex is 0. Use three values of x that are less than 0 and three values of x that are greater than 0. Make a table using integral values of x from 3 to 3. (2) Plot the points associated with each ordered pair (x, y). (3) Draw a smooth curve through the points to draw a parabola. x –3 –2 –1 0 1 2 3 x2 3 (3)2 3 (2)2 3 (1)2 3 (0)2 3 (1)2 3 (2)2 3 (3)1 –1 O 1 x c. Since a 1 0, the function has a minimum. d. The minimum value of the function is the y-coordinate of the vertex, 3, which can be read from the table of values. e. Since the vertex is the turning point of this parabola, the coordinates of the vertex are (0, 3). The Graph of a Quadratic Function 513 Calculator Solution a. Determine the equation of the axis of symmetry as before. b. Enter the equation in the Y list of functions and graph the function. Clear any equations already in the list. ENTER: Y X,T,,n x2 3 ZOOM 6 DISPLAY: 1 2 Plot1 Plot2 Plot3 = = X 2 –3 \Y = \Y = \Y = \Y = \Y = \Y = \Y 3 4 5 6 7 c. The graph shows that the function has a minimum. d. Since the minimum of the function occurs at the vertex, use value 2nd CALC 1 from the CALC menu to evaluate the Q function at x 0, the x-coordinate of the vertex: R ENTER: 2nd CALC 1 0 ENTER DISPLAY: Calculate 1: value 2: zero 3: minimum 4: minimum 5: intersect 6: dy/dx 7: f [x] dx Y1 = X2 –3 * X = 0 Y = –3 The calculator displays the minimum value, 3. e. The coordinates of the vertex are (0, 3). Answers a. The axis of symmetry is the y-axis. The equation is x 0. b. Graph c. The function has a minimum. d. The minimum value is |
3. e. The vertex is (0, 3). 514 Quadratic Relations and Functions When the coordinates of the turning point are rational numbers, we can use from the or maximum minimum CALC menu to find the vertex. In Example 1, since the turning point is a minimum, use minimum: CALC CALC 2nd 2nd 3 4 R Q Q R ENTER: 2nd CALC 3 When the calculator asks “LeftBound?” move the cursor to any point to the left of the vertex using the left or right arrow keys, and then press enter. When the calculator asks “RightBound?” move the cursor to the right of the vertex, and then press enter. When the calculator asks “Guess?” move the cursor near the vertex and then press enter. The calculator displays the coordinates of the vertex at the bottom of the screen. Y1=X^2–3 Y1=X^2–3 * * LeftBound? X = -.8510638 Y = –2.27569 RightBound? X = 1.0638298 Y = –1.868266 Y1=X^2–3 * Y = -3 Guess? X = 0 * Y = -3 Minimum X = 0 EXAMPLE 2 Sketch the graph of the function y x2 7x 10. Solution (1) Find the equation of the axis of symmetry and the x-coordinate of the vertex: 2b x 2a 2(27) 2(1) 3.5 (2) Make a table of values using three integral values smaller than and three larger than 3.5. (3) Plot the points whose coordinates are given in the table and draw a smooth curve through them. x x2 7x 10 y 1 2 3 3.5 4 5 6 4 1 7 10 4 14 10 0 9 21 10 2 12.25 24.5 10 2.25 16 28 10 2 25 35 10 0 36 42 10 4 The Graph of a Quadratic Function 515 y 1 –1 –1 O 1 x y = x2 – 7x + 10 Note: The table can also be displayed on the calculator. First enter the equation into Y1. ENTER: Y X,T,,n x2 7 X,T,,n 10 Then enter the starting value and the interval between the x values. We will use 1 as the starting value and 0.5 as the interval in order to include 3.5, |
the x-value of the vertex. ENTER: 2nd TBLSET 1 ENTER.5 ENTER Before creating the table, make sure that “Indpnt:” and “Depend:” are set to “auto.” If. they are not, press ENTER ENTER 2nd Finally, press to create the table. Scroll up and down to view the values of x and y. TABLE X Y1 4 1.75 0 -1.25 -2 -2.25 -2 1 1.5 2 2.5 3 3.5 4 X=1 Calculator Solution Enter the equation in the Y= menu and sketch the graph of the function in the standard window. ENTER: Y X,T,,n x2 7 X,T,,n 10 ZOOM 6 DISPLAY: 516 Quadratic Relations and Functions EXAMPLE 3 The perimeter of a rectangle is 12. Let x represent the measure of one side of the rectangle and y represent the area. a. Write an equation for the area of the rectangle in terms of x. b. Draw the graph of the equation written in a. c. What is the maximum area of the rectangle? Solution a. Let x be the measure of the length of the rectangle. Use the formula for perimeter to express the measure of the width in terms of x: Write the formula for area in terms of x and y: P 2l 2w 12 2x 2w 6 x w w 6 x A lw y x(6 x) y 6x x2 y x2 6x b. The equation of the axis of symmetry is x values using values of x on each side of 3. 26 2(21) or x 3. Make a table of y x 0 1 2 3 4 5 6 x2 6x 0 0 1 6 4 12 9 18 16 24 25 30 36 36 1 –1 1 x c. The maximum value of the area, y, is 9. Note: The graph shows all possible values of x and y. Since both the measure of a side of the rectangle, x, and the area of the rectangle, y, must be positive, 0 x 6 and 0 y 9. Since (2, 8) is a point on the graph, one possible rectangle has dimensions 2 by (6 2) or 2 by 4 and an area of 8. The rectangle with maximum area, 9, has dimensions 3 by (6 3) or 3 by 3, a square. The Graph |
of a Quadratic Function 517 Translating, Reflecting, and Scaling Graphs of Quadratic Functions Just as linear and absolute value functions can be translated, reflected, or scaled, graphs of quadratic functions can also be manipulated by working with the graph of the quadratic function y x2. y 5 x2 1 2.5 For instance, the graph of shifted 2.5 units up. The graph of y x2 is the graph of y x2 reflected in the x-axis. The y 5 3x2 stretched vertically by a factor of 3, graph of y 5 x2 while the graph of compressed vertically by a fac1 tor of. 3 is the graph of y 5 1 3x2 y 5 x2 is the graph of is the graph of y 5 x2 2 + 2.5 y = x y 1 y = x2 –1 1 x y 1 –2 –1 y = x2 x 2 y = – x2 1 1 x Translation Rules for Quadratic Functions If c is positive: The graph of y 5 x2 1 c is the graph of y 5 x2 shifted c units up. The graph of y 5 x2 2 c is the graph of y 5 x2 shifted c units down. The graph of y 5 (x 1 c)2 is the graph of y 5 x2 shifted c units to the left. The graph of y 5 (x 2 c)2 is the graph of y 5 x2 shifted c units to the right. Reflection Rule for Quadratic Functions The graph of y 5 2x2 is the graph of y 5 x2 reflected in the x-axis. Scaling Rules for Quadratic Functions When c 1, the graph of y 5 cx2 is the graph of y 5 x2 stretched vertically by a factor of c. When 0 c 1, the graph of y 5 cx2 is the graph of y 5 x2 compressed vertically by a factor of c. 518 Quadratic Relations and Functions EXAMPLE 4 In a–e, write an equation for the resulting function if the graph of y 5 x2 is: a. shifted 5 units down and 1.5 units to the left b. stretched vertically by a factor of 4 and shifted 2 units down c. compressed vertically by a factor of and reflected in the x-axis 1 6 d. reflected in the x-axis, shifted 2 units up, and shifted 2 units to the right Solution a. y |
5 (x 1 1.5)2 2 5 Answer b. First, stretch vertically by a factor of 4: y 5 4x2 Then, translate the resulting function 2 units down: 1 c. First, compress vertically by a factor of : 6 Then, reflect in the x-axis: d. First, reflect in the x-axis: Then, translate the resulting function 2 units up: Finally, translate the resulting function 2 units to the right: y 5 4x2 2 2 Answer 1 y 6x2 21 y 6x2 y 5 2x2 Answer y 5 2x2 1 2 y 5 2(x 2 2)2 1 2 Answer EXERCISES Writing About Mathematics h(x) 5 1 1. Penny drew the graph of 4x2 2 2x x 4 to x 4. Her graph is shown to the right. Explain why Penny’s graph does not look like a parabola. from y 1 1O x The Graph of a Quadratic Function 519 2. What values of x would you choose to draw the graph of h(x) 5 1 4x2 2 2x so that points on both sides of the turning point would be shown on the graph? Explain your answer. Developing Skills In 3–14: a. Graph each quadratic function on graph paper using the integral values for x indicated in parentheses to prepare the necessary table of values. b. Write the equation of the axis of symmetry of the graph. c. Write the coordinates of the turning point of the graph. 3. y x2 (3 x 3) 5. y x2 1 (3 x 3) 7. y x2 4 (3 x 3) 9. y x2 2x (2 x 4) 11. y x2 4x 3 (1 x 5) 13. y x2 2x 3 (4 x 2) 4. y x2 (3 x 3) 6. y x2 1 (3 x 3) 8. y x2 2x (2 x 4) 10. y x2 6x 8 (0 x 6) 12. y x2 2x 1 (2 x 4) 14. y x2 4x 3 (1 x 5) In 15–20: a. Write the equation of the axis of symmetry of the graph of the function. coordinates of the vertex. three points with integral coefficients on each side of the vertex. b. Find the c. Draw the graph on graph |
paper or on a calculator, showing at least 15. y x2 6x 1 18. y x2 4x 3 21. Write an equation for the resulting function if the graph of y x2 is: 16. y x2 2x 8 19. y x2 3x 7 17. y x2 8x 12 20. y x2 x 5 a. reflected in the x-axis and shifted 3 units left. b. compressed vertically by a factor of and shifted 9 units up. 2 7 c. reflected in the x-axis, stretched vertically by a factor of 6, shifted 1 unit down, and shifted 4 units to the right. In 22–25, each graph is a translation and/or a reflection of the graph of y x2. For each graph, a. determine the vertex and the axis of symmetry, and b. write the equation of each graph. 22. y 23. y x x 520 Quadratic Relations and Functions 24. y 25. y x x 26. Of the graphs below, which is the graph of a quadratic function and the graph of an absolute value function? (1) y (3) y (2) y x x (4) y x x Applying Skills 27. The length of a rectangle is 4 more than its width. a. If x represents the width of the rectangle, represent the length of the rectangle in terms of x. b. If y represents the area of the rectangle, write an equation for y in terms of x. c. Draw the graph of the equation that you wrote in part b. d. Do all of the points on the graph that you drew represent pairs of values for the width and area of the rectangle? Explain your answer. The Graph of a Quadratic Function 521 28. The height of a triangle is 6 less than twice the length of the base. a. If x represents the length of the base of the triangle, represent the height in terms of x. b. If y represents the area of the triangle, write an equation for y in terms of x. c. Draw the graph of the equation that you wrote in part b. d. Do all of the points on the graph that you drew represent pairs of values for the length of the base and area of the triangle? Explain your answer. 29. The perimeter of a rectangle is 20 centimeters. Let x represent the measure of one side of the rectangle and y represent the area of the rectangle. a. Use the formula |
for perimeter to express the measure of a second side of the rectangle. b. Write an equation for the area of the rectangle in terms of x. c. Draw the graph of the equation written in b. d. What are the dimensions of the rectangle with maximum area? e. What is the maximum area of the rectangle? f. List four other possible dimensions and areas for the rectangle. 30. A batter hit a baseball at a height 3 feet off the ground, with an initial vertical velocity of 64 feet per second. Let x represent the time in seconds, and y represent the height of the baseball. The height of the ball can be determined over a limited period of time by using the equation y 16x2 64x 3. a. Make a table using integral values of x from 0 to 4 to find values of y. b. Graph the equation. Let one horizontal unit second, and one vertical unit 10 1 4 feet. (See suggested coordinate grid below.) h 100 80 60 40 20. If the ball was caught after 4 seconds, what was its height when it was caught? d. From the table and the graph, determine: (1) the maximum height reached by the baseball; (2) the time required for the ball to reach this height. 522 Quadratic Relations and Functions 13-3 FINDING ROOTS FROM A GRAPH In Section 1, you learned to find the solution of an equation of the form ax2 bx c 0 by factoring. In Section 2, you learned to draw the graph of a function of the form. How are these similar expressions related? To answer this question, we will consider three possible cases. y 5 ax2 1 bx 1 c CASE 1 A quadratic equation can have two distinct real roots. The equation x2 7x 10 0 has exactly two roots or solutions, 5 and 2, that make the equation true. The function, y x2 7x 10 has infinitely many pairs of numbers that make the equation true. The graph of this function shown at the right intersects the x-axis in two points, (5, 0) and (2, 0). Since the y-coordinates of these points are 0, the x-coordinates of these points are the roots of the equation x2 7x 10 0. y 1 O –1 – CASE 2 A quadratic equation can have only one distinct real root. The two roots of the equation x2 6x 9 0 are equal.There is only |
one number, 3, that makes the equation true. The function y x2 6x 9 has infinitely many pairs of numbers that make the equation true. The graph of this function shown at the right intersects the x-axis in only one point, (3, 0). Since the y-coordinate of this point is 0, the x-coordinate of this point is the root of the equation x2 6x 9 0. y 1 –1 – Recall from Section 1 that when a quadratic equation has only one distinct root, the root is said to be a double root. In other words, when the root of a quadratic equation is a double root, the graph of the corresponding quadratic function intersects the x-axis exactly once. CASE 3 A quadratic equation can have no real roots. Finding Roots from a Graph 523 The equation x2 2x 3 0 has no real roots. There is no real number that makes the equation true. The function, y x2 2x 3 has infinitely many pairs of numbers that make the equation true. The graph of this function shown at the right does not intersect the x-axis. There is no point on the graph whose y-coordinate is 0. Since there is no point whose y-coordinate is 0, there are no real numbers that are roots of the equation x2 2x 3 01 –1 O 1 x The equation of the x-axis is y 0. The x-coordinates of the points at which the graph of y ax2 bx c intersects the x-axis are the roots of the equation ax2 bx c 0. The graph of the function y ax2 bx c can intersect the x-axis in 0, 1, or 2 points, and the equation ax2 bx c 0 can have 0, 1, or 2 real roots. A real number k is a root of the quadratic equation ax2 bx c 0 if and only if the graph of y ax2 bx c intersects the x-axis at (k, 0). EXAMPLE 1 Use the graph of y x2 5x 6 to find the roots of x2 5x 6 0. y Solution The graph intersects the x-axis at (2, 0) and (3, 0). The x-coordinates of these points are the roots of x2 5x 6 0. Answer The roots are 2 and 3. The solution set is {2, 3 |
}. 1 –1 1 Note that in Example 1 the quadratic expression x2 5x 6 can be factored into (x 2)(x 3), from which we can obtain the solution set 2, 3. 6 5 524 Quadratic Relations and Functions EXAMPLE 2 Use the graph of y x2 3x 4 to find the linear factors of x2 1 3x 2 4. y Solution The graph intersects the x-axis at (4, 0) and (1, 0). Therefore, the roots are 4 and 1. If x 4, then x (4) (x 4) is a factor. If x 1, then (x 1) is a factor. O 1 –1 –1 x Answer The linear factors of x2 3x 4 are (x 4) and (x 1). y = x2 + 3x – 4 EXERCISES Writing About Mathematics 1. The coordinates of the vertex of a parabola y x2 2x 5 are (1, 4). Does the equation x2 2x 5 0 have real roots? Explain your answer. 2. The coordinates of the vertex of a parabola y x2 2x 3 are (1, 4). Does the equa- tion x2 2x 3 0 have real roots? Explain your answer. Developing Skills In 3–10: a. Draw the graph of the parabola. b. Using the graph, find the real numbers that are elements of the solution set of the equation. c. Using the graph, factor the corresponding quadratic expression if possible. 3. y x2 6x 5; 0 x2 6x 5 5. y x2 2x 3; 0 x2 2x 3 7. y x2 2x 1; 0 x2 2x 1 9. y x2 4x 5; 0 x2 4x 5 4. y x2 2x 1; 0 x2 2x 1 6. y x2 x 2; 0 x2 x 2 8. y x2 3x 2; 0 x2 3x 2 10. y x2 5x 6; 0 x2 5x 6 11. If the graph of a quadratic function, f(x), crosses the x-axis at x 6 and x 8, what are two factors of f(x)? 12. If the factors of a quadratic function, h(x), are (x 6) and (x 3), what is the |
solution set for the equation h(x) 0? Graphic Solution of a Quadratic-Linear System 525 In 13–15, refer to the graph of the parabola shown below. 13. Which of the following is the equation of the parabola? (1) y (x 2)(x 3) (2) y (x 2)(x 3) (3) y (x 2)(x 3) (4) y (x 2)(x 3) y 1 14. Set the equation of the parabola equal to 0. What are the 1O x roots of this quadratic equation? 15. If the graph of the parabola is reflected in the x-axis, how many roots will its corresponding quadratic equation have? 13-4 GRAPHIC SOLUTION OF A QUADRATIC-LINEAR SYSTEM In Chapter 10 you learned how to solve a system of linear equations by graphing. For example, the graphic solution of the given system of linear equations is shown below. y 5 21 2x 1 4 y 2x 1 Since the point of intersection, (2, 3), is a solution of both equations, the common solution of the system is x 2 and y 3. A quadratic-linear system consists of a quadratic equation and a linear equation. The solution of a quadratic-linear system is the set of ordered pairs of numbers that make both equations true. As shown below, the line may intersect the curve in two, one, or no points. Thus the solution set may contain two ordered pairs, one ordered pair, or no ordered pairs2, 3) y = ––x + 4 2 1 O 1 x Two points of intersection One point of intersection No point of intersection 526 Quadratic Relations and Functions EXAMPLE 1 Solve the quadratic-linear system graphically: y x2 6x 6 y x 4 Solution (1) Draw the graph of y x2 6x 6. The axis of symmetry of a parabola is x. Therefore, the axis of symme- 2b 2a try for the graph of y x2 6x 6 is x or x 3. Make a table of values using integral values of x less than 3 and greater than 3. Plot the points associated with each pair (x, y) and join them with a smooth curve. 2(26) 2(1) x 0 1 2 3 4 5 6 x2 6x 6 0 |
0 6 1 6 6 4 12 6 9 18 6 16 24 6 25 30 6 36 36 2) On the same set of axes, draw the graph of y x 4. Make a table of val- ues and plot the points2, –2) y = x – 4 (5, 1) x Graphic Solution of a Quadratic-Linear System 527 The line could also have been graphed by using the y-intercept, 4. Starting at the point (0, 4) move 1 unit up and 1 unit to the right to locate a second point. Then again, move 1 unit up and 1 unit to the right to locate a third point. Draw a line through these points., and the slope 5 1 or 1 1 (3) Find the coordinates of the points at which the graphs intersect. The graphs intersect at (2, 2) and at (5, 1). Check each solution in each equation. Four checks are required in all. The checks are left for you. Calculator Solution (1) Enter the equations into the Y= menu. ENTER: Y X,T,,n x2 6 DISPLAY: X,T,,n X,T,,2) Calculate the first intersection by choosing intersect from the CALC menu. Accept Y1 X2 6X 6 as the first curve and Y2 X 4 as the second curve. Then press enter when the screen prompts “Guess?”. 2nd CALC 5 ENTER ENTER ENTER ENTER: DISPLAY: * Y = –2 intersection X = 2 The calculator displays the coordinates of the intersection point at the bottom of the screen. (3) To calculate the second intersection point, repeat the process from (2), but when the screen prompts “Guess?” move the cursor with the left and right arrow keys to the approximate position of the second intersection point. * Answer The solution set is {(2, 2), (5, 1)}. intersection X = 5 Y = 1 528 Quadratic Relations and Functions EXERCISES Writing About Mathematics 1. What is the solution set of a system of equations when the graphs of the equations do not intersect? Explain your answer. 2. Melody said that the equations y x2 and y 2 do not have a common solution even before she drew their graphs. Explain how Melody was able to justify her conclusion. Developing Skills In 3–6, use the graph on the right to find the common solution of the system. 3. |
y x2 2x 3 y 0 4. y x2 2x 3 y 3 5. y x2 2x 3 y 4 6. y x2 2x 1 –1 x 7. For what values of c do the equations y x2 2x 3 and y c have no points in common? 8. a. Draw the graph of y x2 4x 2, in the interval 1 x 5. b. On the same set of axes, draw the graph of y x 2. c. Write the coordinates of the points of intersection of the graphs made in parts a and b. d. Check the common solutions found in part c in both equations. In 9–16, find graphically and check the solution set of each system of equations. 9. y x2 y x 2 11. y x2 2x 1 y 2x 5 13. y x2 8x 15 x y 5 15. y x2 4x 1 y 2x 1 10. y x2 2x 4 y x 12. y 4x x2 y x 4 14. y x2 6x 5 y 3 16. y x2 x 4 2x y 2 Algebraic Solution of a Quadratic-Linear System 529 Applying Skills 17. When a stone is thrown upward from a height of 5 feet with an initial velocity of 48 feet per second, the height of the stone y after x seconds is given by the function y 5 216x2 1 48x 1 5. a. Draw a graph of the given function. Let each horizontal unit equal second and each vertical unit equal 5 feet. Plot points for 0,, 2, b. On the same set of axes, draw the graph of y 25. c. From the graphs drawn in parts a and b, determine when the stone is at a height of, 1, 1 2 3 2 5 2 1 2, and 3 seconds. 25 feet. 18. If you drop a baseball on Mars, the gravity accelerates the baseball at 12 feet per second squared. Let’s suppose you drop a baseball from a height of 100 feet. A formula for the height, y, of the baseball after x seconds is given by y 6x2 100. a. On a calculator, graph the given function. Set Xmin to –10, Xmax to 10, Xscl to 1, Ymin to 5, Ymax to 110, and Yscl to 10. b. Graph the functions y 46 and y 0 as |
Y2 and Y3. c. Using a calculator method, determine, to the nearest tenth of a second, when the baseball has a height of 46 feet and when the baseball hits the ground (reaches a height of 0 feet). 13-5 ALGEBRAIC SOLUTION OF A QUADRATIC-LINEAR SYSTEM = x2 – 4 x In the last section, we learned to solve a quadratic-linear system by finding the points of intersection of the graphs. The solutions of most of the systems that we solved were integers that were easy to read from the graphs. However, not all solutions are integers. For example, the graphs of y x2 4x 3 and 2x 1 1 are shown at the right. They intersect in two points. One of those points, (4, 3), has integral coordinates and can be read easily from the graph. However, the coordinates of the other point are not integers, and we are not able to identify the exact values of x and y from the graph. y 5 1 530 Quadratic Relations and Functions In Chapter 10 we learned that a system of linear equations can be solved by an algebraic method of substitution. This method can also be used for a quadratic-linear system. The algebraic solution of the system graphed on the previous page is shown in Example 1. EXAMPLE 1 Solve algebraically and check: y x2 4x 3 y 5 1 2x 1 1 Solution (1) Since y is expressed in terms of x in the linear equation, substitute 1 2x 1 1 the expression for y in the quadratic equation to form an equation in one variable: y x2 4x 3 1 2x 1 1 x2 4x 3 (2) To eliminate fractions as coefficients, multiply both sides of the equation by 2: 2 A 1 2x 1 1 B (3) Write the quadratic equation in standard form: (4) Solve the quadratic equation by factoring: (5) Set each factor equal to 0 and solve for x. (6) Substitute each value of x in the linear equation to find the corresponding value of y: (7) Write each solution as coordinates: 2(x2 4x 3) x 2 2x2 8x 6 0 2x2 9x 4 0 (2x 1)(x 4) 2x 1 0 2x 1 x 5 1 2 1 2x x, y) y y y y 1 2 |
2x 1 1 1 2(4x, y) (4, 3) (8) Check each ordered pair in each of the given equations: Check for x2 4x 3 5 1 4 5? 2 A A 5 4 5 2x 1 1 5 1 4 5? 1 2 2 B A 5 4 5 Algebraic Solution of a Quadratic-Linear System 531 Check for x 4, y 3 y x2 4x 3 3 5? (4)2 2 4(4) 1 3 3 5? 16 2 16 1 3 ✔ 3 5 3 y 3 5? 1 1 2x 1 1 2(4) 1 1 3 5? 2 1 1 ✔ 3 5 3 Answer EXAMPLE 2 2, 5 1 4 B U A, (4, 3) V The length of the longer leg of a right triangle is 2 units more than twice the length of the shorter leg. The length of the hypotenuse is 13 units. Find the lengths of the legs of the triangle. Solution Let a the length of the longer leg and b the length of the shorter leg. (1) Use the Pythagorean Theorem to write an equation: (2) Use the information in the first sentence of the problem to write another equation: (3) Substitute the expression for a from step 2 in the equation in step 1: (4) Square the binomial and write the equation in standard form: (5) Factor the left member of the equation: a2 b2 (13)2 a2 b2 169 a 2b 2 a2 b2 169 (2b 2)2 b2 169 4b2 8b 4 b2 169 5b2 8b 165 0 (5b 33)(b 5) 0 (6) Set each factor equal to 0 and solve 5b 33 0 for b: (7) For each value of b, find the value of a. Use the linear equation in step 2: (8) Reject the negative values. Use the pair of positive values to write the answer. a Answer The lengths of the legs are 12 units and 5 units. 5b 33 233 b 5 a 2b 2 b 5 0 b 5 b 5 a 2b 2 a 2 233 5 A 256 5 1 2 B a 2(5) 2 a 12 532 Quadratic Relations and Functions EXERCISES Writing About Mathematics 1. Explain why the equations y x2 and y 4 have no common solution in the set of |
real numbers. 2. Explain why the equations x2 y2 49 and x 8 have no common solution in the set of real numbers. 3. y x2 2x Developing Skills In 3–20, solve each system of equations algebraically and check all solutions. 4. y x2 5 y x 5 7. x2 y 9 y x 9 y x 1 8. x2 y 2 y x 6. y x2 2x 1 y x 5. y x2 4x 3 y x 3 9. x2 2y 5 y x 1 12. y 3x2 8x 5 x y 3 15. x2 y2 25 x 2y 5 18. x2 y2 40 y 2x 2 Applying Skills 10. y 2x2 6x 5 11. y 2x2 2x 3 y x 2 13. y x2 3x 1 y 5 1 3x 1 2 16. x2 y2 100 y x 2 19. x2 y2 20 y x 2 y x 3 14. y x2 6x 8 y 5 21 2x 1 2 17. x2 y2 50 x y 20. x2 y2 2 y x 2 21. A rectangular tile pattern consists of a large square, two small squares, and a rectangle arranged as shown in the diagram. The height of the small rectangle is 1 unit and the area of the tile is 70 square units. x a. Write a linear equation that expresses the relationship between the length of the sides of the quadrilaterals that make up the tile. b. Write a second-degree equation using the sum of the areas of the quadrilaterals that make up the tile. c. Solve algebraically the system of equations written in parts a and b. d. What are the dimensions of each quadrilateral in the tile? y 1 y " " Chapter Summary 533 22. A doorway to a store is in the shape of an arch whose equation is h 5 23, where x represents the horizontal distance, in feet, from the left end of the base of the doorway and h is the height, in feet, of the doorway x feet from the left end of the base. 4x2 1 6x h a. How wide is the doorway at its base? b. What is the maximum height of the doorway? (0, 0) x c. Can a box that is 6 feet wide, 6 feet long, and 5 feet high be moved through the door- way |
? Explain your answer. 23. The length of the diagonal of a rectangle is 85 meters. The length of the rectangle is 1 meter longer than the width. Find the dimensions of the rectangle. 24. The length of one leg of an isosceles triangle is 29 feet. The length of the altitude to the base of the triangle is 1 foot more than the length of the base a. Let a the length of the altitude to the base and b the distance from the vertex of a base angle to the vertex of the right angle that the altitude makes with the base. Use the Pythagorean Theorem to write an equation in terms of a and b. b. Represent the length of the base in terms of b. c. Represent the length of the altitude, a, in terms of b. d. Solve the system of equations from parts a and c. e. Find the length of the base and the length of the altitude to the base. f. Find the perimeter of the triangle. g. Find the area of the triangle. CHAPTER SUMMARY The equation y ax2 bx c, where a 0, is a quadratic function whose domain is the set of real numbers and whose graph is a parabola. The axis of symmetry of the parabola is the vertical line x. The vertex or turning point of the parabola is on the axis of symmetry. If a 0, the parabola opens upward and the y-value of the vertex is a minimum value for the range of the function. If a 0, the parabola opens downward and the y-value of the vertex is a maximum value for the range of the function. 2b 2a A quadratic-linear system consists of two equations one of which is an equation of degree two and the other a linear equation (an equation of degree one). The common solution of the system may be found by graphing the equations on the same set of axes or by using the algebraic method of substitution. A quadratic-linear system of two equations may have two, one, or no common solutions. 534 Quadratic Relations and Functions The roots of the equation ax2 bx c 0 are the x-coordinates of the points at which the function y ax2 bx c intersects the x-axis. The real number k is a root of ax2 bx c 0 if and only if (x k) is a factor of ax2 bx c. When the |
graph of y x2 is translated by k units in the vertical direction, the equation of the image is y x2 k. When the graph of y x2 is translated k units in the horizontal direction, the equation of the image is y (x k)2. When the graph of y x2 is reflected over the x-axis, the equation of the image is y x2. The graph of y kx2 is the result of stretching the graph of y x2 in the vertical direction when k 1 or of compressing the graph of y x2 when 0 k 1. VOCABULARY 13-1 Standard form • Polynomial equation of degree two • Quadratic equation • Roots of an equation • Double root 13-2 Parabola • Second-degree polynomial function • Quadratic function • Minimum • Turning point • Vertex • Axis of symmetry of a parabola • Maximum 13-4 Quadratic-linear system REVIEW EXERCISES 1. Explain why x y2 is not a function when the domain and range are the set of real numbers. 2. Explain why x y2 is a function when the domain and range are the set of positive real numbers. In 3–6, the set of ordered pairs of a relation is given. For each relation, a. list the c. determine if the elements of the domain, b. list the elements of the range, relation is a function. 3. {(1, 3), (2, 2), (3, 1), (4, 0), (5, 1)} 4. {(1, 3), (1, 2), (1, 1), (1, 0), (1, 1)} 5. {(3, 9), (2, 4), (1, 1), (0, 0), (1, 1), (2, 4), (3, 9)} 6. {(0, 1), (1, 1), (2, 1), (3, 1), (4, 1)} In 7–10, for each of the given functions: a. Write the equation of the axis of symmetry. b. Draw the graph. c. Write the coordinates of the turning point. Review Exercises 535 d. Does the function have a maximum or a minimum? e. What is the range of the function? 7. y x2 6x 6 9. f(x) x2 2x 6 8. y x2 |
4x 1 10. f(x) x2 6x 1 In 11–16, solve each system of equations graphically and check the solution(s) if they exist. 11. y x2 6 x y 6 14. y x2 4x x y 4 y x 16. y 2x x2 x y 2 y x 5 15. y 5 x2 y 4 12. y x2 2x 1 13. y x2 x 3 In 17–22, solve each system of equations algebraically and check the solutions. 18. y x2 4x 9 17. x2 y 5 y 1 2x y 3x 1 21. x2 y2 40 20. y x2 6x 5 y 3x y x 1 19. x2 2y 11 y x 4 22. x2 y2 5 y 5 1 2x y 5 x2 23. Write an equation for the resulting function if the graph of is shifted 3 units up, 2.5 units to the left, and is reflected over the x-axis. 24. The sum of the areas of two squares is 85. The length of a side of the larger square minus the length of a side of the smaller square is 1. Find the length of a side of each square. 25. Two square pieces are cut from a rectangular piece of carpet as shown in the diagram. The area of the original piece is 144 square feet, and the width of the small rectangle that is left is 2 feet. Find the dimensions of the original piece of carpet. Exploration Write the square of each integer from 2 to 20. Write the prime factorization of each of these squares. What do you observe about the prime factorization of each of these squares? Let n be a positive integer and a, b, and c be prime numbers. If the prime rational or irra-, is n a perfect square? Is n 5 a3 3 b2 3 c4 n factorization of tional? Express in terms of a, b, and c. n " " 536 Quadratic Relations and Functions CUMULATIVE REVIEW CHAPTERS 1–13 Part I Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. 1. When a 7, 5 2a2 equals (1) 147 (2) 93 (3) 103 (4) 147 2. Which expression is rational? p (1) 1 3 $ 3. When factored completely, 3 |
x2 75 can be expressed as 2 8 $ (2) (3) (4) 0.4 " (1) (3x 15)(x 5) (2) (x 5)(3x 15) (3) 3x(x 5)(x 5) (4) 3(x 5)(x 5) 4. When b2 4b is subtracted from 3b2 3b the difference is (1) 3 7b (2) 2b2 7b 5. The solution set of the equation 0.5x 4 2x 0.5 is (3) 2b2 7b (4) 2b2 b (1) {3} (2) {1.4} (3) {30} (4) {14} 6. Which of these represents the quadratic function y x2 5 shifted 2 units down and 4 units to the right? (1) y (x 4)2 2 (2) y (x 4)2 3 (3) y (x 4)2 7 (4) y (x 2)2 9 7. The slope of the line whose equation is x 2y 4 is (1) 2 (2) 2 (3) 1 2 (4) 21 2 8. The solution set of the equation x2 7x 10 0 is (1) {–2, 5} (2) {2, 5} (3) (x 2)(x 5) (4) (x 2)(x 5) 9. Which of these shows the graph of a linear function intersecting the graph of an absolute value function? (1) y (2) y x x (3) y (4) y Cumulative Review 537 x x 10. In ABC, mA 72 and mB 83. What is the measure of C? (1) 155° (2) 108° (3) 97° (4) 25° Part II Answer all questions in this part. Each correct answer will receive 2 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 11. In a class of 300 students, 242 take math, 208 take science, and 183 take both math and science. How many students take neither math nor science? 12. Each of the equal sides of an isosceles triangle is 3 centimeters longer than the |
base. The perimeter of the triangle is 54 centimeters, what is the measure of each side of the triangle? Part III Answer all questions in this part. Each correct answer will receive 3 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 13. The base of a right circular cylinder has a diameter of 5.00 inches. Sally measured the circumference of the base of the cylinder and recorded it to be 15.5 inches. What is the percent of error in her measurement? Express your answer to the nearest tenth of a percent. 14. Solve the following system of equations and check your solutions. y x2 3x 1 y x 1 538 Quadratic Relations and Functions Part IV Answer all questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 15. Find to the nearest degree the measure of the acute angle that the graph of y 2x 4 makes with the x-axis. 16. Jean Forester has a small business making pies and cakes. Today, she must make at least 4 cakes to fill her orders and at least 3 pies. She has time to make a total of no more than 10 pies and cakes. a. Let x represent the number of cakes that Jean makes and y represent the number of pies. Write three inequalities that can be used to represent the number of pies and cakes that she can make. b. In the coordinate plane, graph the inequalities that you wrote and indi- cate the region that represents their common solution. c. Write at least three ordered pairs that represent the number of pies and cakes that Jean can make. ALGEBRAIC FRACTIONS,AND EQUATIONS AND INEQUALITIES INVOLVING FRACTIONS Although people today are making greater use of decimal fractions as they work with calculators, computers, and the metric system, common fractions still surround us. 1 4 1 2 11 3 -yard gain in football, pint of cream, We use common fractions in everyday measures: 1 21 2 2 1 -inch 4 cups of flour. nail, We buy dozen eggs, not 0.5 dozen eggs. We describe 15 minutes as hour, not 0.25 hour |
. Items are sold at a third off, or at a fraction of the original price. B Fractions are also used when sharing. For example, Andrea designed some beautiful Ukrainian eggs this year. She gave onefifth of the eggs to her grandparents.Then she gave one-fourth of the eggs she had left to her parents. Next, she presented her aunt with one-third of the eggs that remained. Finally, she gave one-half of the eggs she had left to her brother, and she kept six eggs. Can you use some problem-solving skills to discover how many Ukrainian eggs Andrea designed? 1 3 A CHAPTER 14 CHAPTER TABLE OF CONTENTS 14-1 The Meaning of an Algebraic Fraction 14-2 Reducing Fractions to Lowest Terms 14-3 Multiplying Fractions 14-4 Dividing Fractions 14-5 Adding or Subtracting Algebraic Fractions 14-6 Solving Equations with Fractional Coefficients 14-7 Solving Inequalities with Fractional Coefficients 14-8 Solving Fractional Equations Chapter Summary Vocabulary Review Exercises Cumulative Review In this chapter, you will learn operations with algebraic fractions and methods to solve equations and inequalities that involve fractions. 539 540 Algebraic Fractions, and Equations and Inequalities Involving Fractions 14-1 THE MEANING OF AN ALGEBRAIC FRACTION A fraction is a quotient of any number divided by any nonzero number. For example, the arithmetic fraction indicates the quotient of 3 divided by 4. An algebraic fraction is a quotient of two algebraic expressions. An algebraic fraction that is the quotient of two polynomials is called a fractional expression or a rational expression. Here are some examples of algebraic fractions that are rational expressions: 3 4 x 2 2 x 4c 3d x 1 5 x 2 2 x2 1 4x 1 3 x 1 1 a b The fraction means that the number represented by a, the numerator, is to be divided by the number represented by b, the denominator. Since division by 0 is not possible, the value of the denominator, b, cannot be 0. An algebraic fraction is defined or has meaning only for values of the variables for which the denominator is not 0. EXAMPLE 1 Find the value of x for which 12 x 2 9 is not defined. Solution The fraction 12 x 2 9 is not defined |
when the denominator, x 9, is equal to 0. x 9 0 x 9 Answer EXERCISES Writing About Mathematics 1. Since any number divided by itself equals 1, the solution set for 1 is the set of all real x x numbers. Do you agree with this statement? Explain why or why not. 2. Aaron multiplied by (equal to 1) to obtain the fraction b2 2 b b 1 1. Is the fraction b 1 1 1 b equal to the fraction for all values of b? Explain your answer. b b b 1 1 1 b b2 2 b b 1 1 Developing Skills In 3–12, find, in each case, the value of the variable for which the fraction is not defined. 3. 2 x 8. y 1 5 y 1 2 4. 9. 25 6x 10 2x 2 1 5. 10. 12 y2 2y 1 3 4y 1 2 6. 1 x 2 5 11. 1 x2 2 4 7. 7 2 2 x 12. 3 x2 2 5x 2 14 Reducing Fractions to Lowest Terms 541 Applying Skills In 13–17, represent the answer to each problem as a fraction. 13. What is the cost of one piece of candy if five pieces cost c cents? 14. What is the cost of 1 meter of lumber if p meters cost 980 cents? 15. If a piece of lumber 10x 20 centimeters in length is cut into y pieces of equal length, what is the length of each of the pieces? 16. What fractional part of an hour is m minutes? 17. If the perimeter of a square is 3x 2y inches, what is the length of each side of the square? 14-2 REDUCING FRACTIONS TO LOWEST TERMS A fraction is said to be reduced to lowest terms or is a lowest terms fraction when its numerator and denominator have no common factor other than 1 or 1. Each of the fractions 5 10 The arithmetic fraction and 5 10 a 2a 1 can be expressed in lowest terms as. 2 is reduced to lowest terms when both its numer- ator and denominator are divided by 5: The algebraic fraction a 2a tor and denominator are divided by a, where a 0: 5 10 5 5 4 5 10 4 5 5 1 2 is reduced to lowest terms when both its numera- a 2a 5 a 4 a Fractions that are equal in value are called equivalent fractions. Thus, 2a |
4 a 5 1 2 5 10 and 1 2 are equivalent fractions, and both are equivalent to a 2a, when a 0. The examples shown above illustrate the division property of a fraction: if the numerator and the denominator of a fraction are divided by the same nonzero number, the resulting fraction is equal to the original fraction. In general, for any numbers a, b, and x, where b 0 and x 0: bx 5 ax 4 x ax bx 4 x 5 a b When a fraction is reduced to lowest terms, we list the values of the variables that must be excluded so that the original fraction is equivalent to the reduced form and also has meaning. For example: 5x 4 x 5 4 5 5x 5 4x 4 x 4x cy 4 y dy 4 y 5 c d cy dy 5 (where x 0) (where y 0, d 0) 542 Algebraic Fractions, and Equations and Inequalities Involving Fractions When reducing a fraction, the division of the numerator and the denomina- tor by a common factor may be indicated by a cancellation. Here, we use cancellation to divide the numerator and the denominator by 3: 3(x 1 5) 18 5 1 3(x 1 5) 18 5 x 1 5 6 6 Here, we use cancellation to divide the numerator and the denominator by (a 2 3) : 1 a2 2 9 3a 2 9 5 (a 2 3)(a 1 3) 3(a 2 3) 5 a 1 3 3 1 (where a 3) By re-examining one of the examples just seen, we can show that the multi- plication property of one is used whenever a fraction is reduced: 3(x 1 5) 18 5 3? (x 1 5) 3? 6 5 3 3? (x 1 5) 6 5 1? (x 1 5) 6 5 x 1 5 6 However, when the multiplication property of one is applied to fractions, it is referred to as the multiplication property of a fraction. In general, for any numbers a, b, and x, where b 0 and x 0 Procedure To reduce a fraction to lowest terms: METHOD 1 1. Factor completely both the numerator and the denominator. 2. Determine the greatest common factor of the numerator and the denominator. 3. Express the given fraction as the product of two fractions, one of which has as its numerator and its |
denominator the greatest common factor determined in step 2. 4. Use the multiplication property of a fraction. METHOD 2 1. Factor both the numerator and the denominator. 2. Divide both the numerator and the denominator by their greatest common factor. Reducing Fractions to Lowest Terms 543 EXAMPLE 1 Reduce 15x2 35x4 to lowest terms. Solution METHOD 1 15x2 5x2 35x4 5 3 7x2? 5x2 5 3 7x2? 1 5 3 7x2 Answer 3 7x2 (x 0) EXAMPLE 2 METHOD 2 15x2 35x4 5 3? 5x2? 5x2 7x2 1 5 3? 5x2 7x2 1? 5x2 5 3 7x2 Express 2x2 2 6x 10x as a lowest terms fraction. Solution METHOD 1 2x2 2 6x 10x 5 2x(x 2 3) 2x? 5 METHOD 2 2x2 2 6x 10x 5 2x(x 2 3) 10x 5 2x 2x? 5 1? (x 2 3) 5 (x 2 3) 5 1 2x(x 2 3) 10x Answer x 2 3 5 (x 0) EXAMPLE 3 Reduce each fraction to lowest terms. a. x2 2 16 x2 2 5x 1 4 Solution a. Use Method 1: x2 2 16 x2 2 5x 1 4 5 (x 1 4)(x 2 4) (x 2 1)(x 2 4. 2 2 x 4x 2 8 b. Use Method 2: 2 2 x 4x 2 8 5 21(x 2 2) 4(x 2 2) 1 5 21(x 2 2) 4(x 2 2) 1 5 21 4 Answers a. x 1 4 x 2 1 (x 1, x 4) b. 21 4 (x 2) 544 Algebraic Fractions, and Equations and Inequalities Involving Fractions EXERCISES Writing About Mathematics 1. Kevin used cancellation to reduce a 1 4 a 1 8 Kevin’s work? to lowest terms as shown below. What is the error in. Kevin let a 4 to prove that when reduced to lowest terms. Explain to Kevin why his reasoning is incorrect. 1 2 Developing Skills In 3–54, reduce each fraction to lowest terms. In each case, list the values of the variables for |
which the fractions are not defined. 3. 4x 12x 7. ab cb 11. 15x2 5x 15. 112a2b 28ac 19. 3x 1 6 4 23. 2ax 1 2bx 6x2 27. 18b2 1 30b 9b3 31. 2a2 6a2 2 2ab 35. x2 2 1 5x 2 5 39. 16 2 a2 2a 2 8 43. x2 1 7x 1 12 x2 2 16 47. x2 2 25 x2 2 2x 2 15 51. r2 2 4r 2 5 r2 2 2r 2 15 4. 8. 12. 16. 20. 24. 27y2 36y 3ay2 6by2 5x2 25x4 220x2y2 290xy2 8y 2 12 6 5a2 2 10a 5a2 28. 4x 4x 1 8 32. 14 7r 2 21s 36. 1 2 x x 2 1 40. 44. 48. 52. x2 2 y2 3y 2 3x x2 1 x 2 2 x2 1 4x 1 4 a2 2 a 2 6 a2 2 9 48 1 8x 2 x2 x2 1 x 2 12 5. 9. 13. 17. 24c 36d 5xy 9xy 27a 36a2 232a3b3 148a3b3 21. 5x 2 35 5x 25. 12ab 2 3b2 3ab 29. 7d 7d 1 14 33. 12a 1 12b 3a 1 3b 37. 3 2 b b2 2 9 41. 45. 49. 53. 2b(3 2 b) b2 2 9 3y 2 3 y2 2 2y 1 1 a2 2 6a a2 2 7a 1 6 2x2 2 7x 1 3 (x 2 3)2 6. 9r 10r 10. 2abc 4abc 14. 18. 8xy2 24x2y 5xy 45x2y2 22. 8m2 1 40m 8m 26. 30. 6x2y 1 9xy2 12xy 5y 5y 1 5x 34. x2 2 9 3x 1 9 38. 2s 2 2r s2 2 r2 42. r2 2 r 2 6 3r 2 9 46. x2 2 3x x2 2 4x 1 3 50. 2x |
2 2 50 x2 1 8x 1 15 54. x2 2 7xy 1 12y2 x2 1 xy 2 20y2 Multiplying Fractions 545 55. a. Use substitution to find the numerical value of x2 2 5x x 2 5, then reduce each numerical frac- tion to lowest terms when: (1) x 7 (4) x 2 (2) x 10 (5) x 4 (3) x 20 (6) x 10 b. What pattern, if any, do you observe for the answers to part a? c. Can substitution be used to evaluate x2 2 5x x 2 5 d. Reduce the algebraic fraction x2 2 5x x 2 5 to lowest terms. when x 5? Explain your answer. e. Using the answer to part d, find the value of x 38,756. x2 2 5x x 2 5, reduced to lowest terms, when f. If the fraction is multiplied by x2 2 5x x 2 5? Explain your answer. x2 2 5x x 2 5 x x to become x(x2 2 5x) x(x 2 5), will it be equivalent to 14-3 MULTIPLYING FRACTIONS The product of two fractions is a fraction with the following properties: 1. The numerator is the product of the numerators of the given fractions. 2. The denominator is the product of the denominators of the given fractions. In general, for any numbers a, b, x, and y, when b 0 and y 0: a b? y 5 ax x by We can find the product of 7 27 and 9 4 in lowest terms by using either of two methods. METHOD 1. 7 27 3 4 5 7 3 9 9 27 3 4 5 63 108 5 7 3 9 12 3 9 5 7 12 3 9 9 5 7 12 3 1 5 7 12 METHOD 2. 27 3 9 7 4 5 7 27 3 1 3 9 4 5 7 12 Notice that Method 2 requires less computation than Method 1 since the reduced form of the product was obtained by dividing the numerator and the denominator by a common factor before the product was found. This method may be called the cancellation method. The properties that apply to the multiplication of arithmetic fractions also apply to the multiplication of algebraic fractions. 546 Algebraic Fractions, and Equations and Inequalities Involving Fractions To multiply and express the product in lowest |
terms, we may use 5x2 7y by 14y2 15x3 either of the two methods. In this example, x 0 and y 0. METHOD 1. 5x2 7y? 14y2 15x3 5 5x2? 14y2 7y? 15x3 5 70x2y2 105x3y 5 2y 3x? 35x2y 35x2y 5 2y 3x? 1 5 2y 3x METHOD 2. 5x2 7x? 14y2 15x3 5 5x2 1 7y 1? 2y 14y2 15x3 3x 5 2y 3x (the cancellation method) While Method 1 is longer, it has the advantage of displaying each step as a property of fractions. This can be helpful for checking work. Procedure To multiply fractions: METHOD 1 1. Multiply the numerators of the given fractions. 2. Multiply the denominators of the given fractions. 3. Reduce the resulting fraction, if possible, to lowest terms. METHOD 2 1. Factor any polynomial that is not a monomial. 2. Use cancellation to divide a numerator and a denominator by each common factor. 3. Multiply the resulting numerators and the resulting denominators to write the product in lowest terms. EXAMPLE 1 Multiply and express the product in reduced form: 5a3 9bx? 6bx a2 Solution METHOD 1 (1) Multiply the numerators and denominators of the given fractions: (2) Reduce the resulting fraction to lowest terms: 5a3 9bx? 6bx a2 5 5a3? 6bx 9bx? a2 5 30a3bx 9a2bx 5 10a 3? 3a2bx 3a2bx 5 10a 3? 1 5 10a 3 Multiplying Fractions 547 METHOD 2 (1) Divide the numerators and denominators by the common factors 3bx and a2: 5a3 9bx? a 6bx a2 5 5a3 9bx 3? 2 6bx a2 1 (2) Multiply the resulting numerators and the resulting denominators: 5 10a 3 Answer 10a 3 (a 0, b 0, x 0) EXAMPLE 2 Multiply and express the product in simplest form: 12a? 3 |
8a Solution Think of 12a as 12a 1. 12a? 8a 5 36a 8a 5 12a 3 3 1? 8a 9 9 2 5 9 5 4a 4a? 2 5 1? 2 Answer 9 2 (a 0) EXAMPLE 3 Multiply and simplify the product: x2 2 5x 1 6 3x? 2 4x 2 12 Solution x2 2 5x 1 6 3x? 2 4x 2 12 5 1 (x23) (x 2 2) 3x 1 2 (x 2 3) 1? 4 2 5 x 2 2 6x Answer x 2 2 6x (x 0, 3) EXERCISES Writing About Mathematics 1. When reduced to lowest terms, a fraction whose numerator is x2 3x 2 equals 1. What is the denominator of the fraction? Explain your answer. 2. Does x2 xz 1 z2? x2 2 z2 x2 2 xz 5 x z for all values of x and z? Explain your answer. 548 Algebraic Fractions, and Equations and Inequalities Involving Fractions Developing Skills In 3–41, find each product in lowest terms. In each case, list any values of the variable for which the fractions are not defined. 3. 8. 13. 18. 30a 36 8 12? mn? 30m2 18n? 12a 2 4 b 8 m2n2 6n 5m? b3 12 21. 2r r 2 1? r 2 1 10 24. 1 x2 2 1? 2x 1 2 6 27. 30. 33. 36. 39. a(a 2 b)2 4b? 4b a(a2 2 b2)? 3y x 1 1? x2 1 6x 1 5 9y2 x2 2 25 4x2 2 9 8x 2x2 2 8 b2 1 81 b2 2 81?? 2x 1 3 x 2 5 8x 1 16 32x2 81 2 b2 81 1 b2 4. 9. 14. 36? 5y 9y 14y 24x 35y? 8x 24a3b2 7c3? 21c2 12ab 5. 10. 15. 1 2? 20x 12x 5y? 7 8? 15y2 36x2 2x 1 4 21 b3 2 b2 a? ab 2 a b2 7s s 1 |
2? a2 2 9 3 2s 1 4 21? 12 2a 2 6 (a 2 2)2 4b? 16b3 4 2 a2 y2 2 2y 2 3 2c3? 4c2 2y 1 2? 5x 1 15 x2 2 4 3x 3x 2 6 4x 1 8 6x 1 18 2 2 x 2x? d2 2 25 4 2 d2? 5d2 2 20 d 1 5 when x 65,908? 19. 22. 25. 28. 31. 34. 37. 40. 6. 11. 16. 5 d? d2 m2 32 8? 3m 3a 1 9 15a? 20. a3 18 7. 12. 17. x2 36? 20 6r2 10rs 5s2? 6r3 5x 2 5y x2y? xy2 25 x2 2 1 x2? 3x2 2 3x 15 8x 2x 1 6? x 1 3 x2 x2 2 x 2 2 3? 21 x2 2 4 23. 26. 29. a2 2 7a 2 8 2a 1 2? 5 a 2 8 32. 35. 38. 41. 10y 1 90 5y 2 45 6a 1 12 5a 2 15 4a 2 6 4a 1 8? y2 2 81 (y 1 9)2? x2 2 3x 1 2 2x2 2 2 a2 1 12a 1 36 a2 2 36? 2x x 2 2? 36 2 a2 36 1 a2 42. What is the value of x2 2 4 6x 1 12? 4x 2 12 x2 2 5x 1 6 14-4 DIVIDING FRACTIONS We know that the operation of division may be defined in terms of the multiplicative inverse, the reciprocal. A quotient can be expressed as the dividend times the reciprocal of the divisor. Thus 24 35 and We use the same rule to divide algebraic fractions. In general, for any num- bers a, b, c, and d, when b 0, c 0, and d 0 ad d bc Procedure To divide by a fraction, multiply the dividend by the reciprocal of the divisor. EXAMPLE 1 Divide: 16c3 21d2 4 24c4 14d3 Solution How to Proceed (1) Multiply the dividend by the reciprocal of the divisor: |
(2) Divide the numerators and denominators by the common factors: (3) Multiply the resulting numerators and the resulting denominators: Dividing Fractions 549 16c3 21d2 4 24c4 14d3 5 16c3 21d2? 14d3 24c4 2 5 16c3 21d2 3 2d? 14d3 24c4 3c 5 4d 9c Answer 4d 9c (c 0, d 0) EXAMPLE 2 Divide: 8x 1 24 x2 2 25 4 4x x2 1 8x 1 15 Solution How to Proceed (1) Multiply the dividend by the reciprocal of the divisor: (2) Factor the numerators and denominators, and divide by the common factors: (3) Multiply the resulting numerators and the resulting denominators: Answer 2(x 1 3)2 x(x 2 5) (x 0, 5, 5, 3) 8x 1 24 x2 2 25 4 4x x2 1 8x 1 15 5 8x 1 24 x2 2 25? x2 1 8x 1 15 4x 5 2 8 (x 1 5) 1 (x 1 3) (x 2 5)? 1 (x 1 5) 4 1 (x 1 3) x 5 2(x 1 3)2 x(x 2 5) Note: If x 5, 5, or 3, the dividend and the divisor will not be defined. If x 0, the reciprocal of the divisor will not be defined. 550 Algebraic Fractions, and Equations and Inequalities Involving Fractions EXERCISES Writing About Mathematics 1. Explain why the quotient 2. To find the quotient 3 2 4 1 nator and wrote 3 2(x 2 4 15, Ruth canceled (x 4) in the numerator and denomi- is undefined for x 2 and for x 3.. Is Ruth’s answer correct? Explain why or why not. Developing Skills In 3–27, find each quotient in lowest terms. In each case, list any values of the variables for which the quotient is not defined. 3. 7. 11. 15. 18. 21. 24. 27. 4. 8. 12. 7a 10 4 21 5 3x 5y 4 21x 2y 4x 1 4 9 4 3 8x x2 2 5x 1 4 |
2x 4 2x 2 2 8x2 4a 4 (a2 2 b2) a2 2 ab 35 4 4b 12 7 7ab2 10cd 3y2 1 9y 4 14b3 5c2d2 18 4 16. 19. 5. 9. 8 4 x 2y xy2 x2y a3 2 a b 4 x y3 4 a3 4b3 6. 10. 14. 9 4 x x 3 6a2b2 8c 4 3ab x2 2 1 5 4 x 2 1 10 4 b2 2 4 b2 4 21x 3x 1 6 17. 20. b2 2 b 2 6 2b (x 2 2)2 4x2 2 16 13. 5y2 27 10 4 10a 1 15 25 4a2 2 9 12y 2 6 8 4 (2y2 2 3y 1 1) 5x 1 10y 3x 1 12y x2 2 2xy 2 8y2 x2 2 16y2 4 2x 1 2 x 1 2 4 4x a 1 b)2 a2 2 b2 4 a 1 b b2 2 a2? a 2 b (a 2 b)2 22. x2 2 4x 1 4 3x 2 6 4 (2 2 x) 25. x 1 y x2 1 y2? x x 2 y 4 (x 1 y)2 x4 2 y4 23. (9 2 y2) 4 y2 1 8y 1 15 2y 1 10 26. 2a 1 6 a2 28. For what value(s) of a is a2 2 2a 1 1 a2 29. Find the value of 30. If x y a and y2 2 6y 1 9 y2 2 9 y 4 z 5 1 a undefined? 4 a2 2 1 a when y 70. 10y 2 30 y2 1 3y 4, what is the value of x z? 14-5 ADDING OR SUBTRACTING ALGEBRAIC FRACTIONS We know that the sum (or difference) of two arithmetic fractions that have the same denominator is another fraction whose numerator is the sum (or difference) of the numerators and whose denominator is the common denominator of the given fractions. We use the same rule to add algebraic fractions that have the same nonzero denominator. Thus: Adding or Subtracting Algebraic Fractions 551 Ar |
ithmetic fractions Algebraic fractions Procedure To add (or subtract) fractions that have the same denominator: 1. Write a fraction whose numerator is the sum (or difference) of the numerators and whose denominator is the common denominator of the given fractions. 2. Reduce the resulting fraction to lowest terms. Add and reduce the answer to lowest terms: 4x 1 9 5 4x 4x 1 9 5 4x 5 5 1 9 4x 5 14 4x 5 7 2x EXAMPLE 1 Solution Answer 7 2x (x 0) EXAMPLE 2 Subtract: 4x 1 7 6x 2 2x 2 4 6x Solution 4x 1 7 6x 2 2x 2 4 6x 5 (4x 1 7) 2 (2x 2 4) 6x 5 4x 1 7 2 2x 1 4 6x 5 2x 1 11 6x Answer 2x 1 11 6x (x 0) Note: In Example 2, since the fraction bar is a symbol of grouping, we enclose numerators in parentheses when the difference is written as a single fraction. In this way, we can see all the signs that need to be changed for the subtraction. In arithmetic, in order to add (or subtract) fractions that have different denominators, we change these fractions to equivalent fractions that have the same denominator, called the common denominator. Then we add (or subtract) the equivalent fractions. 3 4 For example, to add and, we use any common denominator that has 4 and 1 6 6 as factors. METHOD 1. Use the product of the denominators as the common denominator. Here, a common denominator is 4 6, or 24. 3 4 1 1 22 24 5 4 24 5 18 24 Answer 6 3 4 3 11 12 552 Algebraic Fractions, and Equations and Inequalities Involving Fractions METHOD 2. To simplify our work, we use the least common denominator (LCD), that is, the least common multiple of the given denominators. The LCD of is 12. 3 4 and 12 1 2 12 5 11 12 Answer To find the least common denominator of two fractions, we factor the denominators of the fractions completely. The LCD is the product of all of the factors of the first denominator times the factors of the second denominator that are not factors of the first. 4 2 2 3 6 2 LCD 2 2 3 Then, to change each fraction to an equivalent |
form that has the LCD as the, where x is the number by which the original denominator, we multiply by denominator must be multiplied to obtain the LCD 12 5 9 12 1 6 1 6 5 12 5 2 12 x x 2 2 B B A A Note that the LCD is the smallest possible common denominator. Procedure To add (or subtract) fractions that have different denominators: 1. Choose a common denominator for the fractions. 2. Change each fraction to an equivalent fraction with the chosen common denominator. 3. Write a fraction whose numerator is the sum (or difference) of the numerators of the new fractions and whose denominator is the common denominator. 4. Reduce the resulting fraction to lowest terms. Algebraic fractions are added in the same manner as arithmetic fractions, as shown in the examples that follow. Adding or Subtracting Algebraic Fractions 553 EXAMPLE 3 Add: 5 a2b 1 2 ab2 Solution How to Proceed (1) Find the LCD of the fractions: (2) Change each fraction to an equivalent fraction with the least common denominator, a2b2: (3) Write a fraction whose numerator is the sum of the numerators of the new fractions and whose denominator is the common denominator: a2b a a b ab2 a b b LCD a a b b a2b2 5 a2b ab2? 1 2 1 2 b b a a ab2 5 5 5 5b a2b? a2b2 1 2a a2b2 5 5b 1 2a a2b2 Answer 5b 1 2a a2b2 (a 0, b 0) EXAMPLE 4 Solution Subtract: 2x 1 5 3 2 x 2 2 4 2x 1 5 3 2 x 2 2 LCD 3 4 12 2x 1 5 4? 3 5 8x 1 20 12 4 5 4 2 3 3? 2 3x 2 6 12 x 2 2 4 (8x 1 20) 2 (3x 2 6) 12 5 5 8x 1 20 2 3x 1 6 12 5 5x 1 26 Answer 12 EXAMPLE 5 Express as a fraction in simplest form: y 1 1 2 1 y 2 1 Solution LCD? Answer y2 2 2 y 2 1 (y 1) 2 1 y 2 1 5 y2 y2 2 1 2 1 y 2 1 5 y2 554 Algebraic Fractions, and Equations and |
Inequalities Involving Fractions EXAMPLE 6 Solution Subtract: 6x x2 2 4 2 3 x 2 2 x2 4 (x 2)(x 2) x 2 (x 2) LCD (x 2)(x 2) 6x x2 2 4 2 3 x 2 2 3(x 1 2) (x 2 2)(x 1 2) 6x 5 (x 2 2)(x 1 2) 2 6x 2 3(x 1 2) 5 (x 2 2)(x 1 2) 5 6x 2 3x 2 6 (x 2 2)(x 1 2) 3x 2 6 (x 2 2)(x 1 2) 3(x 2 2) (x 2 2)(x 1 2) 5 5 5 3 x 1 2 Answer 3 x 1 2 (x 2, 2) EXERCISES Writing About Mathematics 1. In Example 2, the answer is 2x 1 11 6x. Can we divide 2x and 6x by 2 to write the answer in lowest terms as? Explain why or why not. x 1 11 3x. Joey said that if a x. Do you agree with Joey? Explain why or why not. Developing Skills In 3–43, add or subtract the fractions as indicated. Reduce each answer to lowest terms. In each case, list the values of the variables for which the fractions are not defined. 3. 6. 9. 12. 15. 18. 4c 2 6 4c 1 5 11 4c 5x 2 6 6x 2 5 x2 2 1 x2 2 1 5x 6 2 2x 3 8x 4 1 7x 5 2 3x 10 9 4x 1 3 2x 4. 7. 10. 13. 16. 19. t 2 2s 5r t 7 2 2y 6y 2 4 4y 1 3 1 4y 1 3 1 8 2 r2 r2 1 4r r2 3a 5a 4 1 x 1 1 2x 2 1 8x r2 2 r 2 6 5. 8. 11. 14. 17. 10c 1 9 10c 2 3 6 10c 9d 1 6 2d 1 1 2 7d 1 5 2d 1 1 x 3 1 x 2 5 1 ab ab 4 a 7 1 b 14 8b 2 3a 4b 20. 9a Adding or Subtracting Algebraic Fractions 555 21. 24. 27. 30. 33. 36. 39. d 1 |
7 5d b 2 3 5b 2 b 1 2 10b 4y 1 4 1x 8x 2 8 2 3x 4x 2 4 4 3x 1 18 x x2 2 36 x 2 5 2 x 2 x 1 3 22. 25. 28. 31. 34. 37. 40 3y 2 5 4y2 1 3y 5 2 2x x 1 y 2 3a 2 1 1 7 15a y2 2 9 y 2 3 1 2 1 2x 2 1 x 1 2 1 2x 2 3 23. 26. 29. 32. 35. 38. 41. 2 y 2 2 4 2 3c 2 3 6c2 7 2x 2 6 3y 2 4 5 3c 2 7 2c 5 x 2 3 1 10 3 2x 2 4 3x 2 6 1 6 y2 2 16 2y 3x 1 12y 2 6x 2 y x2 1 3xy 2 4y2 42. 7a (a 2 1)(a 1 3) 1 2a 2 5 (a 1 3)(a 1 2) 43. 2a 1 7 a2 2 2a 2 15 2 3a 2 4 a2 2 7a 1 10 Applying Skills In 44–46, represent the perimeter of each polygon in simplest form. 44. The lengths of the sides of a triangle are represented by x 2, 3x 5, and 7x 10. 45. The length of a rectangle is represented by x 1 3 4, and its width is represented by x 2 4 3. 46. Each leg of an isosceles triangle is represented by 6x 2 18 21. 2x 2 3 7, and its base is represented by In 47 and 48, find, in each case, the simplest form of the indicated length. 47. The perimeter of a triangle is the length of the third side. 17x 24, and the lengths of two of the sides are 3x 8 and 2x 2 5 12. Find 48. The perimeter of a rectangle is 14x 15, and the measure of each length is x 1 2 3. Find the mea- sure of each width. 49. The time t needed to travel a distance d at a rate of speed r can be found by using the for- mula t 5 d r. a. For the first part of a trip, a car travels x miles at 45 miles per hour. Represent the time that the car traveled at that speed in terms of x. b |
. For the remainder of the trip, the car travels 2x 20 miles at 60 miles per hour. Represent the time that the car traveled at that speed in terms of x. c. Express, in terms of x, the total time for the two parts of the trip. 50. Ernesto walked 2 miles at a miles per hour and then walked 3 miles at (a 1) miles per hour. Represent, in terms of a, the total time that he walked. 51. Fran rode her bicycle for x miles at 10 miles per hour and then rode (x 3) miles farther at 8 miles per hour. Represent, in terms of x, the total time that she rode. 556 Algebraic Fractions, and Equations and Inequalities Involving Fractions 14-6 SOLVING EQUATIONS WITH FRACTIONAL COEFFICIENTS The following equations contain fractional coefficients: 1 2x 5 10 x 2 5 10 3x 1 60 5 5 1 6x 3 1 60 5 5x x 6 Each of these equations can be solved by finding an equivalent equation that does not contain fractional coefficients. This can be done by multiplying both sides of the equation by a common denominator for all the fractions present in the equation. We usually multiply by the least common denominator, the LCD., since a dec- 1 Note that the equation 0.5x 10 can also be written as 2x 5 10 imal fraction can be replaced by a common fraction. Procedure To solve an equation that contains fractional coefficients: 1. Find the LCD of all coefficients. 2. Multiply both sides of the equation by the LCD. 3. Solve the resulting equation using the usual methods. 4. Check in the original equation. EXAMPLE 1 Solve and check: 3 1 x x 5 5 8 Solution How to Proceed (1) Write the equation: (2) Find the LCD: 3 1 x x 5 5 8 LCD 3 5 15 (3) Multiply both sides of the equation by the LCD: 15 (4) Use the distributive property: (5) Simplify: (6) Solve for x: 15 x 3 A B Answer x 15 5 15(8) 5 15(8 15 A B B 5x 3x 120 8x 120 x 15 Check 3 1 x x 15 3 1 15 5 3 5 5 8 5 5? 8 5? 8 8 8 ✔ EXAMPLE 2 Solution Solve: a. 4 5 20 1 x 3 |
x 4 a. 4 5 20 1 x 3x 4 LCD 4 4 4 A A 5 4 20 1 x 4 B 5 4(20) 1 4 3x 4 3x 4 3x 5 80 1 x B B A x 4 A B 2x 5 80 x 5 40 Answer Solving Equations with Fractional Coefficients 557 b. 2x 1 7 6 2 2x 2 9 10 5 3 b. 30 2x 1 7 6 2 2x 2 9 10 5 3 LCD 30 B B 30 A 2x 1 7 6 2x 1 7 6 2 30 2 2x 2 9 10 2x 2 9 10 5 30(3) 5 30(3) B A A 5(2x 7) 3(2x 9) 90 10x 35 6x + 27 90 4x 62 90 4x 28 x 7 Answer In Example 2, the check is left to you. EXAMPLE 3 A woman purchased stock in the PAX Company over 3 months. In the first month, she purchased one-half of her present number of shares. In the second month, she bought two-fifths of her present number of shares. In the third month, she purchased 14 shares. How many shares of PAX stock did the woman purchase? Solution Let x total number of shares of stock purchased. 1 Then number of shares purchased in month 1, 2x 2 number of shares purchased in month 2, 5x 14 number of shares purchased in month 3. The sum of the shares purchased over 3 months is the total number of shares. month 1 1 month 2 1 month 3 5 total 10 A 2 5x 2 5x 14 x 1 2x 1 2x 5x 4x 140 10x 9x 140 10x 14 B 10(x) 140 x 558 Algebraic Fractions, and Equations and Inequalities Involving Fractions 70, month 2 2 5(140) 70 56 14 140 ✔ 56, month 3 14 Check month 1 1 2(140) Answer 140 shares EXAMPLE 4 In a child’s coin bank, there is a collection of nickels, dimes, and quarters that amounts to $3.20. There are 3 times as many quarters as nickels, and 5 more dimes than nickels. How many coins of each kind are there? Solution Let x the number of nickels. Then 3x the number of quarters, and x 5 the number of dimes. Also, 0.05x |
the value of the nickels, 0.25(3x) the value of the quarters, and 0.10(x 5) the value of the dimes. Write the equation for the value of the coins. To simplify the equation, which contains coefficients that are decimal fractions with denominators of 100, multiply each side of the equation by 100. The total value of the coins is $3.20. 0.05x 0.25(3x) 0.10(x 5) 3.20 100[0.05x 0.25(3x) 0.10(x 5)] 100(3.20) 5x 25(3x) 10(x 5) 320 5x 75x 10x 50 320 90x 50 320 90x 270 x 3 Check There are 3 nickels, 3(3) 9 quarters, and 3 5 8 dimes. The value of 3 nickels is $0.05(3) $0.15 The value of 9 quarters is $0.25(9) $2.25 The value of 8 dimes is $0.10(8) $0.80 $3.20 ✔ Answer There are 3 nickels, 9 quarters, and 8 dimes. Solving Equations with Fractional Coefficients 559 Note: In a problem such as this, a chart such as the one shown below can be used to organize the information: Coins Number of Coins Value of One Coin Total Value Nickels Quarters Dimes x 3x x 5 0.05 0.25 0.10 0.05x 0.25(3x) 0.10(x 5) EXERCISES Writing About Mathematics 1. Abby solved the equation 0.2x 0.84 3x as follows: 3x 0.2x 0.84 0.1x 0.2x 0.84 0.2x 8.4 x Is Abby’s solution correct? Explain why or why not. 2. In order to write the equation 0.2x 0.84 3x as an equivalent equation with integral coefficients, Heidi multiplied both sides of the equation by 10. Will Heidi’s method lead to a correct solution? Explain why or why not. Compare Heidi’s method with multiplying by 100 or multiplying by 1,000. Developing Skills In 3–37, solve each equation and check. 3. 6. 15. 12. 2x 1 1 5y 2 |
30 4 5 6 7 5 0 3 5 6x 2 9 3 1 x 10 26 6 2 t 2 25 t 2 3 5 5 4 21. 24. 0.4x 0.08 4.24 27. 1.7x 30 0.2x 18. 1 6t 5 18 m 2 2 9 5 3 5x 2 5 15 4 3y 1 1 4 5 44 2 y 5 4. 7. 10. 13. 16. 19. 3 2 r r 7y 12 2 1 3m 1 1 6 5 2 4 5 2y 2 5 3 4 5 2 2 3 2 2m 6 22. 25. 2c 0.5c 50 28. 0.02(x 5) 8 5. 3x 5 5 15 2r 1 6 8. 17. 11. 14. 5 5 24 35 15 4 2 6 5 t 3t 12 20. 2 23. 0.03y 1.2 8.7 26. 0.08y 0.9 0.02y 29. 0.05(x 8) 0.07x 560 Algebraic Fractions, and Equations and Inequalities Involving Fractions 30. 0.4(x 9) 0.3(x 4) 32. 0.04x 0.03(2,000 x) 75 34. 0.05x 10 0.06(x 50) 36. 3 1 0.2a 0.4a 4 5 2 31. 0.06(x 5) 0.04(x 8) 33. 0.02x 0.04(1,500 x) 48 35. 0.08x 0.03(x 200) 4 37. 6 2 0.3a 0.1a 4 5 3 38. The sum of one-half of a number and one-third of that number is 25. Find the number. 39. The difference between one-fifth of a positive number and one-tenth of that number is 10. Find the number. 40. If one-half of a number is increased by 20, the result is 35. Find the number. 41. If two-thirds of a number is decreased by 30, the result is 10. Find the number. 42. If the sum of two consecutive integers is divided by 3, the quotient is 9. Find the integers. 43. If the sum of two consecutive odd integers is divided by 4, the quotient is 10. Find the integers. 44. |
In an isosceles triangle, each of the congruent sides is two-thirds of the base. The perimeter of the triangle is 42. Find the length of each side of the triangle. 45. The larger of two numbers is 12 less than 5 times the smaller. If the smaller number is equal to one-third of the larger number, find the numbers. 46. The larger of two numbers exceeds the smaller by 14. If the smaller number is equal to three-fifths of the larger, find the numbers. 47. Separate 90 into two parts such that one part is one-half of the other part. 48. Separate 150 into two parts such that one part is two-thirds of the other part. Applying Skills 49. Four vegetable plots of unequal lengths and of equal widths are arranged as shown. The length of the third plot is one-fourth the length of the second plot. 1 2 3 4 The length of the fourth plot is one-half the length of the second plot. The length of the first plot is 10 feet more than the length of the fourth plot. If the total length of the four plots is 100 feet, find the length of each plot. 50. Sam is now one-sixth as old as his father. In 4 years, Sam will be one-fourth as old as his father will be then. Find the ages of Sam and his father now. 51. Robert is one-half as old as his father. Twelve years ago, he was one-third as old as his father was then. Find their present ages. Solving Equations with Fractional Coefficients 561 52. A coach finds that, of the students who try out for track, 65% qualify for the team and 90% of those who qualify remain on the team throughout the season. What is the smallest number of students who must try out for track in order to have 30 on the team at the end of the season? 53. A bus that runs once daily between the villages of Alpaca and Down makes only two stops in between, at Billow and at Comfort. Today, the bus left Alpaca with some passengers. At Billow, one-half of the passengers got off, and six new ones got on. At Comfort, again onehalf of the passengers got off, and, this time, five new ones got on. At Down, the last 13 passengers on the bus got off. How many passengers were aboard when the bus left Al |
paca? 54. Sally spent half of her money on a present for her mother. Then she spent one-quarter of the cost of the present for her mother on a treat for herself. If Sally had $6.00 left after she bought her treat, how much money did she have originally? 55. Bob planted some lettuce seedlings in his garden. After a few days, one-tenth of these seedlings had been eaten by rabbits. A week later, one-fifth of the remaining seedlings had been eaten, leaving 36 seedlings unharmed. How many lettuce seedlings had Bob planted originally? 56. May has 3 times as many dimes as nickels. In all, she has $1.40. How many coins of each type does she have? 57. Mr. Jantzen bought some cans of soup at $0.39 per can, and some packages of frozen vegetables at $0.59 per package. He bought twice as many packages of vegetables as cans of soup. If the total bill was $9.42, how many cans of soup did he buy? 58. Roger has $2.30 in dimes and nickels. There are 5 more dimes than nickels. Find the number of each kind of coin that he has. 59. Bess has $2.80 in quarters and dimes. The number of dimes is 7 less than the number of quarters. Find the number of each kind of coin that she has. 60. A movie theater sold student tickets for $5.00 and full-price tickets for $7.00. On Saturday, the theater sold 16 more full-price tickets than student tickets. If the total sales on Saturday were $1,072, how many of each kind of ticket were sold? 61. Is it possible to have $4.50 in dimes and quarters, and have twice as many quarters as dimes? Explain. 62. Is it possible to have $6.00 in nickels, dimes, and quarters, and have the same number of each kind of coin? Explain. 63. Mr. Symms invested a sum of money in 7% bonds. He invested $400 more than this sum in 8% bonds. If the total annual interest from these two investments is $257, how much did he invest at each rate? 64. Mr. Charles borrowed a sum of money at 10% interest. He borrowed a second sum, which was $1, |
500 less than the first sum, at 11% interest. If the annual interest on these two loans is $202.50, how much did he borrow at each rate? 562 Algebraic Fractions, and Equations and Inequalities Involving Fractions 14-7 SOLVING INEQUALITIES WITH FRACTIONAL COEFFICIENTS In our modern world, many problems involve inequalities. A potential buyer may offer at most one amount for a house, while the seller will accept no less than another amount. Inequalities that contain fractional coefficients are handled in much the same way as equations that contain fractional coefficients. The chart on the right helps us to translate words into algebraic symbols. Procedure Words Symbols a is greater than b a is less than b a is at least b a is no less than b a is at most b a is no greater than b a b a b a b a b To solve an inequality that contains fractional coefficients: 1. Find the LCD, a positive number. 2. Multiply both sides of the inequality by the LCD. 3. Solve the resulting inequality using the usual methods. EXAMPLE 1 Solve the inequality, and graph the solution set on a number line: a. 3 2 x x 6. 2 Solution a. 6. 2. 6(2 2x 2 x. 12. 12 A b. b. 8 2 4y 7 # 3 8 2 4y 3y 2 1 7 # 3 3y 2 1 3y 2 1 A 1 14 # 14(3) 14 8 2 4y 7 8 2 4y 7 B 21y 1 16 2 8y # 42 # 42 B A B 14 3y 2 A x. 12 Since no domain was given, use the domain of real numbers. Answer: x 12 0 2 4 6 8 10 12 14 16 13y # 26 y # 2 Since no domain was given, use the domain of real numbers. Answer: y 2 –1 0 1 2 3 Solving Inequalities with Fractional Coefficients 563 EXAMPLE 2 Two boys want to pool their money to buy a comic book. The younger of the boys has one-third as much money as the older. Together they have more than $2.00. Find the smallest possible amount of money each can have. Solution Let x the number of cents that the older boy has. Then the number of cents that the younger boy has. 1 3x The sum of their money in |
cents is greater than 200. 1 x 200 3x x 1 1 3x. 3(200) 3 A B 3x + x 600 4x 600 x 150 50 1 3x The number of cents that the younger boy has must be an integer greater than 50. The number of cents that the older boy has must be a multiple of 3 that is greater than 150. The younger boy has at least 51 cents. The older boy has at least 153 cents. The sum of 51 and 153 is greater than 200. Answer The younger boy has at least $0.51 and the older boy has at least $1.53. EXERCISES Writing About Mathematics 1. Explain the error in the following solution of an inequality. x 23. 2 2 x 23. 23(22) B x. 6 23 A 2. In Example 2, what is the domain for the variable? Developing Skills In 3–23, solve each inequality, and graph the solution set on a number line. 3. 6. 9. 5x. 9 4x 2 1 1 20 8 # 5 x 4 2 x 8 t 10 # 4 1 t 5 4. 7. 10. y 2 2 3y, 5 y y 6 $ 12 1 1 1 1 2x 3 $ x 2 5. 5 6c. 1 3c. 36 11. 2.5x 1.7x 4 564 Algebraic Fractions, and Equations and Inequalities Involving Fractions 12. 2y 3 0.2y 15. 18. 21. 2d 1 1 3x 2 30 4, 7d 6, x 2 r 2 3 12 1 5 3 3 2 2 3 # 2 2r 2 3 5 13. 16. 19. 22. 3x 4c 2. 37 6x 2 3 3 $ 2t 1 4 2 1 7 6 10 1 x 1 2 5 1 5t 2 1 3t 2 4 9 6 14. 17. 20. 23. 1 1 5y 2 30 7 # 0 3 $ 7 2 m 2m 2y, 10 5 2 2a 1 3 6 24. If one-third of an integer is increased by 7, the result is at most 13. Find the largest possible integer. 25. If two-fifths of an integer is decreased by 11, the result is at least 4. Find the smallest possi- ble integer. 26. The sum of one-fifth of an integer and one-tenth of that integer is less |
than 40. Find the greatest possible integer. 27. The difference between three-fourths of a positive integer and one-half of that integer is greater than 28. Find the smallest possible integer. 28. The smaller of two integers is two-fifths of the larger, and their sum is less than 40. Find the largest possible integers. 29. The smaller of two positive integers is five-sixths of the larger, and their difference is greater than 3. Find the smallest possible integers. Applying Skills 30. Talk and Tell Answering Service offers customers two monthly options. OPTION 1 Measured Service base rate is $15 each call costs $0.10 OPTION 2 Unmeasured Service base rate is $20 no additional charge per call Find the least number of calls for which unmeasured service is cheaper than measured service. 31. Paul earned some money mowing lawns. He spent one-half of this money for a book, and then one-third for a CD. If he had less than $3 left, how much money did he earn? 32. Mary bought some cans of vegetables at $0.89 per can, and some cans of soup at $0.99 per can. If she bought twice as many cans of vegetables as cans of soup, and paid at least $10, what is the least number of cans of vegetables she could have bought? 33. A coin bank contains nickels, dimes, and quarters. The number of dimes is 7 more than the number of nickels, and the number of quarters is twice the number of dimes. If the total value of the coins is no greater than $7.20, what is the greatest possible number of nickels in the bank? 34. Rhoda is two-thirds as old as her sister Alice. Five years from now, the sum of their ages will be less than 60. What is the largest possible integral value for each sister’s present age? Solving Fractional Equations 565 35. Four years ago, Bill was 11 4 times as old as his cousin Mary. The difference between their present ages is at least 3. What is the smallest possible integral value for each cousin’s present age? 36. Mr. Drew invested a sum of money at 71 2% interest. He invested a second sum, which was $200 less than the first, at 7% interest. If the total annual interest from these two investments is at least $160 |
, what is the smallest amount he could have invested at 71 2%? 37. Mr. Lehtimaki wanted to sell his house. He advertised an asking price, but knew that he would accept, as a minimum, nine-tenths of the asking price. Mrs. Patel offered to buy the house, but her maximum offer was seven-eighths of the asking price. If the difference between the seller’s lowest acceptance price and the buyer’s maximum offer was at least $3,000, find: a. the minimum asking price for the house; b. the minimum amount Mr. Lehtimaki, the seller, would accept; c. the maximum amount offered by Mrs. Patel, the buyer. 38. When packing his books to move, Philip put the same number of books in each of 12 boxes. Once packed, the boxes were too heavy to lift so Philip removed one-fifth of the books from each box. If at least 100 books in total remain in the boxes, what is the minimum number of books that Philip originally packed in each box? 14-8 SOLVING FRACTIONAL EQUATIONS An equation is called an algebraic equation when a variable appears in at least one of its sides. An algebraic equation is a fractional equation when a variable appears in the denominator of one, or more than one, of its terms. For example, a2 11 3d 1 1 2 6d y2 1 2y 2 3 y 2 2 are all fractional equations. To simplify such an equation, clear it of fractions by multiplying both sides by the least common denominator of all fractions in the equation. Then, solve the simpler equation. As is true of all algebraic fractions, a fractional equation has meaning only when values of the variable do not lead to a denominator of 0. KEEP IN MIND When both sides of an equation are multiplied by a variable expression that may represent 0, the resulting equation may not be equivalent to the given equation. Such equations will yield extraneous solutions, which are solutions that satisfy the derived equation but not the given equation. Each solution, therefore, must be checked in the original equation. 566 Algebraic Fractions, and Equations and Inequalities Involving Fractions EXAMPLE 1 Solve and check: 3 1 1 1 x 5 1 2 Solution Multiply both sides of the equation by the least common denominator, 6x. 6x x 5 1 2 5 6x |
6x B B 2x 1 6 5 3x 5 6x 6x 1 3 A B Answer x 6 EXAMPLE 2 Solve and check: 5x 1 10 x 1 2 5 7 Solution Multiply both sides of the equation by the least common denominator, x 2. 5x 1 10 x 1 2 5 7 (x 1 2) A 5x 1 10 x 1 2 5x 1 10 5 7x 1 14 5 (x 1 2)(7) B 22x 5 4 x 5 22 Check? 1 3 1 1 1 2 6 5? 1 2 1 2 5 1 2 ✔ Check 5x 1 10 x 1 2 5 7 5(22) 1 10 22 1 2 210 1 10 0 5? 5? 7 7 0 0 5 7 ✘ The only possible value of x is a value for which the equation has no meaning because it leads to a denominator of 0. Therefore, there is no solution for this equation. Answer The solution set is the empty set, or { }. EXAMPLE 3 Solve and check: 2 x 5 6 2 x 4 Solution METHOD 1 METHOD 2 Solving Fractional Equations 567 Multiply both sides of the equation by the LCD, 4x: 2 x 5 4x 4x (6 2 x) 8 5 6x 2 x2 x2 2 6x 1 8 5 0 (x 2 2)(x 2 4 Use the rule for proportion: the product of the means equals the product of the extremes. x 5 6 2 x 2 4 x(6 2 x) 5 8 6x 2 x2 5 8 2x2 1 6x 2 8 5 0 x2 2 6x 1 8 5 0 (x 2 2)(x 2 4 Check Answer x 2 or x 4 EXAMPLE 4 Solve and check(x 2 1) Solution Multiply both sides of the equation by the LCD, 2(x 1 2)(x 2 1) : 2(x 1 2)(x 2 1) 2(x 1 2)(x 2 1) x 1 2 2(x 1 2)(x 2 1) x 1 2 5 1 2(x 2 1(x 1 2)(x 2 1) x 2 1 2(x 1 2)(x 2 1) x 2 1 2(x 2 1) 1 2(x 1 2) 5 x 1 2 B A 2(x 1 2)(x 2 1) 2(x 2 1) 2(x 1 |
2)(x 2 1) 2(x 2 1) 1 5 5 2(x 1 2)(x 2 1) 4x 1 2 5 x 1 2 3x 5 0 x 5 0 Answer x 0 Check? 2 1 1 1 21 5? 21 2 5 21 2 1 2(21) ✔ 1 2(0 2 1) 568 Algebraic Fractions, and Equations and Inequalities Involving Fractions EXERCISES Writing About Mathematics 1. Nathan said that the solution set of r 2 5 5 10 2 5r 2 25 agree with Nathan? Explain why or why not. is the set of all real numbers. Do you 2. Pam multiplied each side of the equation y 1 5 y2 2 25 5 3 y 1 5 by (y + 5)(y 5) to obtain the 5 3 equation y 5 3y 15, which has as its solution y 10. Pru said that the equation y 1 5 is a proportion and can be solved by writing the product of the means equal y2 2 25 to the product of the extremes. She obtained the equation 3(y2 25) (y 5)2, which has as its solution 10 and 5. Both girls used a correct method of solution. Explain the difference in their answers. y 1 5 Developing Skills In 3–6, explain why each fractional equation has no solution. 3. 6x x 5 3 4. 4a 1 4 a 1 1 5 5 5. x 5 4 1 2 2 x 6 In 7–45, solve each equation, and check. 7. 10. 13. 16. 19. 22. 25. 28. 31. 34. 37. 40. 43. 10 x 5 5 15 4x 5 1 8 9 2x 5 7 2x 6x 5 3 2 1 x 5x 1 1 30 6 3x 2 1 5 3 4 5 2 3a 12 y 5 1 3 8. 11. 14. 17. 20. 23. 26. 29. 32. 35. 38. 41. 44. 15 y 5 3 10 18 30 2x y 1 9 2y 1 3 5 15 3x 2 4 5 1 4 4z 7 1 5z 15 12 x2 18 9. 12. 15. 18. 21. 24. 27. 30. 33. 36. 39. 42. 45. 3 2x 5 1 2 15 y 2 3 y 5 4 y 2 2 2y 5 3 8 3a 1 5 12 5 2 1 a |
5 1 x 2x 2x 1 2x 1 1 3x 5 1 5x 3m 2 1 12y 8y 2b 1 1 5 b 2 2x 1 4 5 23 1 x 1 2 1 2x 1 2 3 In 46–49, solve each equation for x in terms of the other variables. Chapter Summary 569 46. t x 2 k 5 0 by c, y 5 c2 a x 5 47. t x 2 k 5 5k 48. a 1 b x 5 c 49, and c 0, is it possible to know the numerical value of x 50. If without knowing numerical values of a, b, c, and y? Explain your answer Applying Skills 51. If 24 is divided by a number, the result is 6. Find the number. 52. If 10 is divided by a number, the result is 30. Find the number. 53. The sum of 20 divided by a number, and 7 divided by the same number, is 9. Find the num- ber. 54. When the reciprocal of a number is decreased by 2, the result is 5. Find the number. 55. The numerator of a fraction is 8 less than the denominator of the fraction. The value of the fraction is. Find the fraction. 3 5 56. The numerator and denominator of a fraction are in the ratio 3 : 4. When the numerator is decreased by 4 and the denominator is increased by 2, the value of the new fraction, in simplest form, is. Find the original fraction. 1 2 57. The ratio of boys to girls in the chess club is 4 to 5. After 2 boys leave the club and 2 girls join, the ratio is 1 to 2. How many members are in the club? 58. The length of Emily’s rectangular garden is 4 feet greater than its width. The width of Sarah’s rectangular garden is equal to the length of Emily’s and its length is 18 feet. The two gardens are similar rectangles, that is, the ratio of the length to the width of Emily’s garden equals the ratio of the length to the width of Sarah’s garden. Find the possible dimensions of each garden. (Two answers are possible.) CHAPTER SUMMARY An algebraic fraction is the quotient of two algebraic expressions. If the algebraic expressions are polynomials, the fraction is called a rational expression or a fractional expression. An algebraic fraction is |
defined only if values of the variables do not result in a denominator of 0. Fractions that are equal in value are called equivalent fractions. A fraction is reduced to lowest terms when an equivalent fraction is found such that its numerator and denominator have no common factor other than 1 or 1. This fraction is considered a lowest terms fraction. 570 Algebraic Fractions, and Equations and Inequalities Involving Fractions Operations with algebraic fractions follow the same rules as operations with arithmetic fractions: Multiplication Division a x? y 5 ab b xy (x 0, y 0 ay bx (x 0, y 0, b 0) Addition/subtraction with the same denominator c 0), c 0) Addition/subtraction with different denominators (first, obtain the common denominator): bd 1 bc bd 5 ad 1 bc b 5 ad b d 1 c d d? b? bd (b 0, d 0) b 1 c a d 5 a A fractional equation is an equation in which a variable appears in the denominator of one or more than one of its terms. To simplify a fractional equation, or any equation or inequality containing fractional coefficients, multiply both sides by the least common denominator (LCD) to eliminate the fractions. Then solve the simpler equation or inequality and check for extraneous solutions. VOCABULARY 14-1 Algebraic fraction • Fractional expression • Rational expression 14-2 Reduced to lowest terms • Lowest terms fraction • Equivalent fractions • Division property of a fraction • Cancellation • Multiplication property of a fraction 14-3 Cancellation method 14-5 Common denominator • Least common denominator 14-8 Algebraic equation • Fractional equation • Extraneous solution REVIEW EXERCISES 1. Explain the difference between an algebraic fraction and a fractional expression. 2. What fractional part of 1 centimeter is x millimeters? 3. For what value of y is the fraction 4. Factor completely: 12x3 27x y 2 1 y 2 4 undefined? Review Exercises 571 In 5–8, reduce each fraction to lowest terms. 5. 8bg 12bg 6. 14d 7d2 7. 5x2 2 60 5 8. 8y2 2 12y 8y In 9–23, in each case, perform the indicated operation and express the answer in lowest terms. 3xy 2 |
x 9. 10. 12. 13. 15. 2x 2 2 3x2 8 4? 9x 7b 4 18a 3a 35 3 1 ax ax 4 x2 2 5x x? x2 2x 2 10 x2 2 25 12 4 x2 2 10x 1 25 3y? 6 2 m 5m 6 5 xy 2 2 yz a 1 b 1 2b 2a a 1 b c 2 3 12 1 c 1 3 8 b 24. If the sides of a triangle are represented by, 2 perimeter of the triangle in simplest form. 16. 22. 21. 19. 18. 4 11. 14. 17. 20. 23. 6c2 3x 1 2x 1 1 4x 1 5 2x y 1 3 y 1 7 5 2 4 3a 2 9a2 a 4 (1 2 9a2) 5b 6, and 2b 3, express the 25. If a 2, b 3, and c 4, what is the sum of b a 1 a c? In 26–31, solve each equation and check. 26. 29. 20 5 3 k 4 6 m 5 20 m 2 2 27. 30. x 2 3 10 5 4 5 2t 5 2 t 2 2 10 5 2 28. 31 20 In 32–34, solve each equation for r in terms of the other variables. 32. S h 5 2pr 33. c 2r 5 p 34. a r 2 n 5 0 35. Mr. Vroman deposited a sum of money in the bank. After a few years, he found that the interest equaled one-fourth of his original deposit and he had a total sum, deposit plus interest, of $2,400 in the bank. What was the original deposit? 36. One-third of the result obtained by adding 5 to a certain number is equal to one-half of the result obtained when 5 is subtracted from the number. Find the number. 37. Of the total number of points scored by the winning team in a basketball game, one-fifth was scored in the first quarter, one-sixth was scored in the second quarter, one-third was scored in the third quarter, and 27 was scored in the fourth quarter. How many points did the winning team score? 38. Ross drove 300 miles at r miles per hour and 360 miles at r 10 miles per hour. If the time needed to drive 300 miles was equal to the time needed to drive |
360 miles, find the rates at which Ross drove. (Express the time t 5 d needed for each part of the trip as r.) 572 Algebraic Fractions, and Equations and Inequalities Involving Fractions 39. The total cost, T, of n items that cost a dollars each is given by the equa- tion T na. a. Solve the equation T na for n in terms of T and a. b. Use your answer to a to express n1, the number of cans of soda that cost $12.00 if each can of soda costs a dollars. c. Use your answer to a to express n2, the number of cans of soda that cost $15.00 if each can of soda costs a dollars. d. If the number of cans of soda purchased for $12.00 is 4 less than the number purchased for $15.00, find the cost of a can of soda and the number of cans of soda purchased. 40. The cost of two cups of coffee and a bagel is $1.75. The cost of four cups of coffee and three bagels is $4.25. What is the cost of a cup of coffee and the cost of a bagel? 41. A piggybank contains nickels, dimes, and quarters. The number of nickels is 4 more than the number of dimes, and the number of quarters is 3 times the number of nickels. If the total value of the coins is no greater than $8.60, what is the greatest possible number of dimes in the bank? Exploration Some rational numbers can be written as terminating decimals and others as infinitely repeating decimals. (1) Write each of the following fractions as a decimal: 1 2, 1 4, 1 5, 1 8, 1 10, 1 16, 1 20, 1 25, 1 50, 1 100 (2) What do you observe about the decimals written in (1)? (3) Write each denominator in factored form. (4) What do you observe about the factors of the denominators? (5) Write each of the following fractions as a decimal: 1 3, 1 6, 1 9, 1 11, 1 12, 1 15, 1 18, 1 22, 1 24, 1 30 What do you observe about the decimals written in (5)? (7) Write each denominator in factored form. (8) What |
do you observe about the factors of the denominators? (9) Write a statement about terminating and infinitely repeating decimals based on your observations. CUMULATIVE REVIEW Part I Cumulative Review 573 CHAPTERS 1–14 Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. 1. The product of 3a2 and 5a5 is (1) 15a10 (2) 15a7 (3) 8a10 (4) 8a7 2. In the coordinate plane, the point whose coordinates are (2, 1) is in quadrant (1) I (2) II (3) III (4) IV 3. In decimal notation, 3.75 10–2 is (1) 0.0375 (2) 0.00375 4. The slope of the line whose equation is 3x y 5 is (3) 37.5 (1) 5 (2) 5 (3) 3 5. Which of the following is an irrational number? 2 3 (1) 1.3 (3) (2) 9 " 6. The factors of x2 7x 18 are (1) 9 and 2 (2) 9 and 2 (3) (x 9) and (x 2) (4) (x 9) and (x 2) (4) 375 (4) 3 (4) 5 " 7. The dimensions of a rectangular box are 8 by 5 by 9. The surface area is (1) 360 cubic units (2) 360 square units (3) 157 square units (4) 314 square units 8. The length of one leg of a right triangle is 8 and the length of the hypotenuse is 12. The length of the other leg is (1) 4 9. The solution set of (1) {1, 2} 4 5 (2) " a2 2) {2} is (3) 4 13 " (3) {0, 1} (4) 80 (4) {1, 2} 10. In the last n times a baseball player was up to bat, he got 3 hits and struck out the rest of the times. The ratio of hits to strike-outs is (1) 3 n Part II (2) n 2 3 n (3) 3 n 2 3 (4) n 2 3 3 Answer all questions in this part. Each correct answer will receive 2 credits. Clearly indicate the necessary steps, |
including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 574 Algebraic Fractions, and Equations and Inequalities Involving Fractions 11. Mrs. Kniger bought some stock on May 1 for $3,500. By June 1, the value of the stock was $3,640. What was the percent of increase of the cost of the stock? 12. A furlong is one-eighth of a mile. A horse ran 10 furlongs in 2.5 minutes. What was the speed of the horse in feet per second? Part III Answer all questions in this part. Each correct answer will receive 3 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 13. Two cans of soda and an order of fries cost $2.60. One can of soda and two orders of fries cost $2.80. What is the cost of a can of soda and of an order of fries? 14. a. Draw the graph of y x2 2x. b. From the graph, determine the solution set of the equation x2 2x 3. Part IV Answer all questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 15. If the measure of the smallest angle of a right triangle is 32° and the length of the shortest side is 36.5 centimeters, find the length of the hypotenuse of the triangle to the nearest tenth of a centimeter. 16. The area of a garden is 120 square feet. The length of the garden is 1 foot less than twice the width. What are the dimensions of the garden? PROBABILITY Mathematicians first studied probability by looking at situations involving games of chance.Today, probability is used in a wide variety of fields. In medicine, it helps us to determine the chances of catching an infection or of controlling an epidemic, and the likelihood that a drug will be effective in curing a disease. In industry, probability tells us how long a manufactured product should last or how many defective items may be expected in a production run. In |
biology, the study of genes inherited from one’s parents and grandparents is a direct application of probability. Probability helps us to predict when more tellers are needed at bank windows, when and where traffic jams are likely to occur, and what kind of weather we may expect for the next few days. While the list of applications is almost endless, all of them demand a strong knowledge of higher mathematics. As you study this chapter, you will learn to solve problems such as the following: A doctor finds that, as winter approaches, 45% of her patients need flu shots, 20% need pneumonia shots, and 5% need both. What is the probability that the next patient that the doctor sees will need either a flu shot or a pneumonia shot? Like the early mathematicians, we will begin a formal study of probability by looking at games and other rather simple applications. CHAPTER 15 CHAPTER TABLE OF CONTENTS 15-1 Empirical Probability 15-2 Theoretical Probability 15-3 Evaluating Simple Probabilities 15-4 The Probability of (A and B) 15-5 The Probability of (A or B) 15-6 The Probability of (Not A) 15-7 The Counting Principle, Sample Spaces, and Probability 15-8 Probabilities With Two or More Activities 15-9 Permutations 15-10 Permutations With Repetition 15-11 Combinations 15-12 Permutations, Combinations, and Probability Chapter Summary Vocabulary Review Exercises Cumulative Review 575 576 Probability 15-1 EMPIRICAL PROBABILITY Probability is a branch of mathematics in which the chance of an event happening is assigned a numerical value that predicts how likely that event is to occur. Although this prediction tells us little about what may happen in an individual case, it can provide valuable information about what to expect in a large number of cases. A decision is sometimes reached by the toss of a coin: “Heads, we’ll go to the movies; tails, we’ll go bowling.” When we toss a coin, we don’t know whether the coin will land with the head side or the tail side facing upward. However, we believe that heads and tails have equal chances of happening whenever we toss a fair coin. We can describe this situation by saying that the probability of heads is and the probability of tails is, symbolized as: 1 2 1 2 P(heads) 5 1 2 or P( |
H) 5 1 2 P(tails) 5 1 2 or P(T) 5 1 2 Before we define probability, let us consider two more situations. 1. Suppose we toss a coin and it lands heads up. If we were to toss the coin a second time, would the coin land tails up? Is your answer “I don’t know”? Good! We cannot say that the coin must now be tails because we cannot predict the next result with certainty. 2. Suppose we take an index card Card and fold it down the center. If we then toss the card and let it fall, there are only three possible results. The card may land on its side, it may land on its edge, or it may form a tent when it lands. Can we say P(edge), P(side) 5 1 3, and P(tent) 5 1 3? 5 1 3 Side Edge Tent Again, your answer should be “I don’t know.” We cannot assign a number as a probability until we have some evidence to support our claim. In fact, if we were to gather evidence by tossing this card, we would find that the probabili3, 1 1 ties are not because, unlike the result of tossing the coin, each result is not equally likely to occur. 3, and 1 3 Variables that might affect the experiment include the dimensions of the index card, the weight of the cardboard, and the angle measure of the fold. (An index card with a 10° opening would be much less likely to form a tent than an index card with a 110° opening.) Empirical Probability 577 An Empirical Study Let us go back to the problem of tossing a coin. While we cannot predict the 1 result of one toss of a coin, we can still say that the probability of heads is 2 based on observations made in an empirical study. In an empirical study, we perform an experiment many times, keep records of the results, and then analyze these results. 20 20 20 20 20 20 20 20 20 20 Number of Heads Number of Tosses For example, ten students decided to take turns tossing a coin. Each student completed 20 tosses and the number of heads was recorded as shown at the right. If we look at the results and think of the probability of heads as a fraction comparing the number of heads to the total number of tosses, only Maria, with 10 heads out of 20 tosses, had |
results 1 where the probability was, or. This 2 fraction is called the relative frequency. Elizabeth had the lowest rel- 10 20 Albert Peter Thomas Maria Elizabeth Joanna Kathy Jeanne 8 13 12 10 6 12 11 7 6 20 ative frequency of heads,. Peter and Debbie tied for the highest relative. Maria’s relative frefrequency with 1 quency was 2 other students were incorrect; the coins simply fell that way. 13 20 or. This does not mean that Maria had correct results while the James 10 20 9 Debbie 13 The students decided to combine their results, by expanding the chart, to see what happened when all 200 tosses of the coin were considered. As shown in columns 3 and 4 of the table on the next page, each cumulative result is found by adding all the results up to that point. For example, in the second row, by adding the 8 heads that Albert tossed and the 13 heads that Peter tossed, we find the cumulative number of heads up to this point to be 21, the total number of heads tossed by Albert and Peter together. Similarly, when the 20 tosses that Albert made and the 20 tosses that Peter made are added, the cumulative number of tosses up to this point is 40. Since each student completed 20 coin tosses, the cumulative number of tosses should increase by 20 for each row. The cumulative number of heads should increase by varying amounts for each row since each student experienced different results. 578 Probability (COL. 1) (COL. 2) (COL. 3) (COL. 4) (COL. 5) Number Number of Tosses of Heads Albert Peter Thomas Maria Elizabeth Joanna Kathy Jeanne Debbie James 8 13 12 10 6 12 11 7 13 9 20 20 20 20 20 20 20 20 20 20 Cumulative Cumulative Cumulative Number of Tosses Relative Frequency Number of Heads 8 21 33 43 49 61 72 79 92 101 20 40 60 80 100 120 140 160 180 200 8 20 5 21 40 5 33 60 5 43 80 5 49 100 5 61 120 5 72 140 5 79 160 5 92 180 5 101 200 5.400.525.550.538.490.508.514.494.511.505 In column 5, the cumulative relative frequency is found by dividing the total number of heads at or above a row by the total number of tosses at or above that row. The cumulative relative frequency is shown as a fraction and then, for easy comparison, as a decimal. The decimal is given to the nearest thousandth |
. 6 20 for Peter and Debbie, the cumulative relative frequency, after While the relative frequency for individual students varied greatly, from for Elizabeth to 1 all 200 tosses were combined, was a number very close to. 2 A graph of the results of columns 4 and 5 will tell us even more. In the graph, the horizontal axis is labeled “Number of tosses” to show the cumulative results of column 4; the vertical axis is labeled “Cumulative relative frequency of heads” to show the results of column 5. 13 20 CUMULATIVE RELATIVE FREQUENCY OF HEADS FOR 0 TO 200 TOSSES OF A COIN.6 0.5 0.4 0.3 0.2 0.1 0 0 20 40 60 80 140 100 Number of tosses 120 160 180 200 Empirical Probability 579 On the graph, we have plotted the points that represent the data in columns 4 and 5 of the preceding table, and we have connected these points to form a line graph. Notice how the line moves up and down around the relative frequency of 0.5, or. The graph shows that the more times the coin is tossed, the 1 closer the relative frequency comes to. In other words, the line seems to level 2 out at a relative frequency of. We say that the cumulative relative frequency converges to the number and the coin will land heads up about one-half of the time. 1 2 1 2 1 2 Even though the cumulative relative frequency of, we sense that the line will approach the number. When we use carefully collected evidence about tossing a fair coin to guess that the probability of heads is, we have arrived at this conclusion empirically, that is, by experimentation and observation. is not exactly 1 2 1 2 101 200 1 2 Empirical probability may be defined as the most accurate scientific estimate, based on a large number of trials, of the cumulative relative frequency of an event happening. Experiments in Probability A single attempt at doing something, such as tossing a coin only once, is called a trial. We perform experiments in probability by repeating the same trial many times. Experiments are aimed at finding the probabilities to be assigned to the occurrences of an event, such as heads coming up on a coin. The objects used in an experiment may be classified into one of two categories: 1. Fair and unbiased objects have not been weighted or made unbalanced. An object is fair when the different possible results have equal chances of happening. Objects such as |
coins, cards, and spinners will always be treated in this book as fair objects, unless otherwise noted. 2. Biased objects have been tampered with or are weighted to give one result a better chance of happening than another. The folded index card described earlier in this section is a biased object because the probability of each of three results is not. The card is weighted so that it will fall on its side more often than it will fall on its edge. 1 3 You have seen how to determine empirical probability by the tables and graph previously shown in this section. Sometimes, however, it is possible to guess the probability that should be assigned to the result described before you start an experiment. In Examples 1–5, use common sense to guess the probability that might be assigned to each result. (The answers are given without comment here. You will learn how to determine these probabilities in the next section.) 580 Probability EXAMPLE 1 A die is a six-sided solid object (a cube). Each side (or face) is a square. The sides are marked with 1, 2, 3, 4, 5, and 6 pips, respectively. (The plural of die is dice.) In rolling a fair die, getting a 4 means that the side showing four pips is facing up. Find the probability of getting a 4, or P(4). Die Faces of a die Answer P(4) 5 1 6 EXAMPLE 2 A standard deck of cards contains 52 cards. There are four suits: hearts, diamonds, spades, and clubs. Each suit contains 13 cards: 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king, and ace. The diamonds and hearts are red; the spades and clubs are black. In selecting a card from the deck without looking, find the probability of drawing: a. the 7 of diamonds b. a 7 c. a diamond Answers a. There is only one 7 of diamonds. P(7 of diamonds) 5 1 52 b. There are four 7s. P(7) 5 4 52 or 1 13 c. There are 13 diamonds. P(diamond) 5 13 52 or 1 4 EXAMPLE 3 There are 10 digits in our numeral system: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. In selecting a digit without looking, what is the probability it will be: a. the 8? b. an odd digit |
? Answers a. P(8) 5 1 10 b. P(odd) 5 10 or 1 2 EXAMPLE 4 A jar contains eight marbles: three are white and the remaining five are blue. In selecting a marble without looking, what is the probability it will be blue? (All marbles are the same size.) Answer P(blue) 5 5 8 Empirical Probability 581 EXAMPLE 5 The English alphabet contains 26 letters. There are 5 vowels (A, E, I, O, U). The other 21 letters are consonants. If a person turns 26 tiles from a word game facedown and each tile represents a different letter of the alphabet, what is the probability of turning over: a. the A? b. a vowel? c. a consonant? Answers a. P(A) 5 1 26 b. P(vowel) 5 5 26 c. P(consonant) 5 21 26 EXERCISES Writing About Mathematics 1. Alicia read that, in a given year, one out of four people will be involved in an automobile accident. There are four people in Alicia’s family. Alicia concluded that this year, one of the people in her family will be involved in an automobile accident. Do you agree with Alicia’s conclusion? Explain why or why not. 2. A library has a collection of 25,000 books. Is the probability that a particular book will be checked out 1 25,000? Explain why or why not. Developing Skills In 3–8, in each case, a fair, unbiased object is involved. These questions should be answered without conducting an experiment; take a guess. 3. The six sides of a number cube are labeled 1, 2, 3, 4, 5, and 6. Find P(5), the probability of getting a 5 when the die is rolled. 4. In drawing a card from a standard deck without looking, find P(any heart). 5. Each of 10 pieces of paper contains a different number from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. The pieces of paper are folded and placed in a bag. In selecting a piece of paper without looking, find P(7). 6. A jar contains five marbles, all the same size. Two marbles are black, and the other three are white. In selecting a marble without looking, find P(black). 7. Using the let |
tered tiles from a word game, a boy places 26 tiles, one tile for each letter of the alphabet, facedown on a table. After mixing up the tiles, he picks one. What is the probability that the tile contains one of the letters in the word MATH? 582 Probability 8. A tetrahedron is a four-sided object. Each side, or face, is an equilateral triangle. The numerals 1, 2, 3, and 4 are used to number the different faces, as shown in the figure. A trial consists of rolling the tetrahedron and reading the number that is facedown. Find P(4). 1 3 2 4 Tetrahedron Faces of a tetrahedron 9. By yourself, or with some classmates, conduct any of the experiments described in Exercises 3–8 to verify that you have assigned the correct probability to the event or result described. A good experiment should contain at least 100 trials. 10. The figure at the right shows a spinner that has an equal chance of landing on one of four sectors. The regions are equal in size and are numbered 1, 2, 3, and 4. An experiment was conducted by five people to find the probability that the arrow will land on the 2. Each person spun the arrow 100 times. When the arrow landed on a line, the result did not count and the arrow was spun again. a. Before doing the experiment, what probability would you assign to the arrow landing on the 2? (In symbols, P(2)?) 1 3 2 4 b. Copy and complete the table below to find the cumulative results of this experiment. In the last column, record the cumulative relative frequencies as fractions and as decimals to the nearest thousandth. Number of Times Arrow Number of Landed on 2 Spins Cumulative Number of Times Arrow Landed on 2 Cumulative Number of Spins Barbara Tom Ann Eddie Cathy 29 31 19 23 24 100 100 100 100 100 29 60 100 200 Cumulative Relative Frequency 29 100 5.290 c. Did the experiment provide evidence that the probability you assigned in part a was correct? d. Form a group of five people and repeat the experiment, making a table similar to the one shown above. Do your results provide evidence that the probability you assigned in a was correct? Empirical Probability 583 In 11–15, a biased object is described. A probability can be assigned to a result only by conducting an experiment to determine the cumulative relative frequency |
of the event. While you may wish to guess at the probability of the event before starting the experiment, conduct at least 100 trials to determine the best probability to be assigned. 11. An index card is folded in half and tossed. As described earlier in this section, the card may land in one of three positions: on its side, on its edge, or in the form of a tent. In tossing the folded card, find P(tent), the probability that the card will form a tent when it lands. 12. A paper cup is tossed. It can land in one of three positions: on its top, on its bottom, or on its side, as shown in the figure. In tossing the cup, find P(top), the probability that the cup will land on its top. Top Bottom Side 13. A nickel and a quarter are glued or taped together so that the two faces seen are the head of the quarter and the tail of the nickel. (This is a very crude model of a weighted coin.) In tossing the coin, find P(head). 14. A paper cup in the shape of a cone is tossed. It can land in one of two positions: on its base or on its side, as shown in the figure. In tossing this cup, find P(side), the probability that the cup will land on its side. Base Side 15. A thumbtack is tossed. It may land either with the pin up or with the pin down, as shown in the figure. In tossing the thumbtack, find P(pin up). Pin up Pin down 16. The first word is selected from a page of a book written in English. a. What is the probability that the word contains at least one of the letters a, e, i, o, u, or y? b. In general, what is the largest possible probability? c. What is the probability that the word does not contain at least one of the letters a, e, i, o, u, or y? d. In general, what is the smallest possible probability? Applying Skills 17. An insurance company’s records show that last year, of the 1,000 cars insured by the com- pany, 210 were involved in accidents. What is the probability that an insurance policy, chosen at random from their files, is that of a car that was not involved in an accident? 18. A school’s attendance records for last year show that of the 885 students enrolled |
, 15 had no absences for the year. What is the probability that a student, chosen at random, had no absences? 19. A chess club consists of 45 members of whom 24 are boys and 21 are girls. If a member of the club is chosen at random to represent the club at a tournament, what is the probability that the person chosen is a boy? 584 Probability 15-2 THEORETICAL PROBABILITY An empirical approach to probability is necessary whenever we deal with biased objects. However, common sense tells us that there is a simple way to define the probability of an event when we deal with fair, unbiased objects. For example, let us suppose that Alma is playing a game in which each player must roll a die. To win, Alma must roll a number greater than 4. What is the probability that Alma will win on her next turn? Common sense tells us that: 1. The die has an equal chance of falling in any one of six ways: 1, 2, 3, 4, 5, and 6. 2. There are two ways for Alma to win: rolling a 5 or a 6. 3. Therefore: P(Alma wins) number of winning results number of possible results 5 2 6 5 1 3. Terms and Definitions Using the details of the preceding example, let us examine the correct terminology to be used. An outcome is a result of some activity or experiment. In rolling a die, 1 is an outcome, 2 is an outcome, 3 is an outcome, and so on. There are six outcomes when rolling a six-sided die. A sample space is a set of all possible outcomes for the activity. When rolling a die, there are six possible outcomes in the sample space: 1, 2, 3, 4, 5, and 6. We say that the sample space is {1, 2, 3, 4, 5, 6}. An event is a subset of the sample space. We use the term event in two ways. In ordinary conversation, it means a situation or happening. In the technical language of probability, it is the subset of the sample space that lists all of the outcomes for a given situation. When we focus on a particular event, such as heads facing up when we toss a coin, we refer to it as the favorable event. The other event or events in the sample space, in this case tails, are called unfavorable. When we roll a die, we may define many different situations. Each of these is called an event. |
1. For Alma, the event of rolling a number greater than 4 contains only two outcomes: 5 and 6. 2. For Lee, a different event might be rolling a number less than 5. This event contains four outcomes: 1, 2, 3, and 4. Theoretical Probability 585 3. For Sandi, the event of rolling a 2 contains only one outcome: 2. When there is only one outcome, we call this a singleton event. We can now define theoretical probability for fair, unbiased objects: The theoretical probability of an event is the number of ways that the event can occur, divided by the total number of possibilities in the sample space. In symbolic form, we write: where P(E) n(E) n(S) P(E) n(E) n(S) represents the probability of event E; represents the number of ways event E can occur or the number of outcomes in event E; represents the total number of possibilities, or the number of outcomes in sample space S. Since theoretical probability relies on calculation as opposed to experimen- tation, it is sometimes referred to as calculated probability. Let us now use this formula to write the probabilities of the three events described above. 1. For Alma, there are two ways to roll a number greater than 4, and there are six possible ways that the die may fall. We say: E the set of numbers on a die that are greater than 4: {5, 6}. n(E) 2 since there are 2 outcomes in this event. S the set of all possible outcomes: {1, 2, 3, 4, 5, 6}. n(S) 6 since there are six outcomes in the sample space. Therefore: 2. For Lee, the probability of rolling a number less than 5 is P(E) n(E) n(S) 5 2 6 5 1 3 n(E) n(S) 5 4 3. For Sandi, the probability of rolling a 2 is P(E) 6 5 2 3 P(E) n(E) n(S) 5 1 6 586 Probability Uniform Probability A sample space is said to have uniform probability, or to contain equally likely outcomes, when each of the possible outcomes has an equal chance of occurring. In rolling a die, there are six possible outcomes in the sample space; each is equally likely to occur. Therefore(6) ; P(5) ; P(4) ; P(3 |
) ; P(2) P(1) We say that the die has uniform probability. If, however, a die is weighted to make it biased, then one or more sides will have a probability greater than while one or more sides will have a probabil- 1 6, ity less than theoretical probability does not apply to weighted objects.. A weighted die does not have uniform probability; the rule for 1 6 Random Selection When we select an object from a collection of objects without knowing any of the special characteristics of the object, we are making a random selection. Random selections are made when drawing a marble from a bag, when taking a card from a deck, or when picking a name out of a hat. In the same way, we may use the word random to describe the outcomes when tossing a coin or rolling a die; the outcomes happen without any special selection on our part. Procedure To find the simple probability of an event: 1. Count the total number of outcomes in the sample space S: n(S). 2. Count all the possible outcomes of the event E: n(E). 3. Substitute these values in the formula for the probability of event E: P(E) n(E) n(S) Note: The probability of an event is usually written as a fraction. A standard calculator display, however, is in decimal form. Therefore, if you use a calculator when working with probability, it is helpful to know fraction-decimal equivalents. Recall that some common fractions have equivalent decimals that are ter- minating decimals: 1 2 5 0.5 1 4 5 0.25 1 5 5 0.2 1 8 5 0.125 Theoretical Probability 587 Others have equivalent decimals that are repeating decimals: 1 3 5 0.3 2 3 5 0.6 1 6 5 0.16 1 9 5 0.1 The graphing calculator has a function that will change a decimal fraction to a common fraction in lowest terms. For example, the probability of a die showing a number greater than 4 can be displayed on a calculator as follows: ENTER: 2 6 ENTER ENTER: MATH 1 ENTER DISPLAY: 2 / 6 DISPLAY EXAMPLE 1 A standard deck of 52 cards is shuffled. Daniella draws a single card from the deck at random. What is the probability that the card is a jack? Solution S sample space of all possible outcomes, or 52 cards. Thus, n(S) 52. |
J event of selecting a jack. There are four jacks in the deck: jack of hearts, of diamonds, of spades, and of clubs. Thus, n(J) 4. P(J) number of possible jacks number of possible cards 5 n(J) n(S) 5 4 52 5 1 13 Answer Calculator Solution ENTER: 4 52 MATH 1 ENTER DISPLAY EXAMPLE 2 An aquarium at a pet store contains 8 goldfish, 6 angelfish, and 7 zebrafish. David randomly chooses a fish to take home for his fishbowl. a. How many possible outcomes are in the sample space? b. What is the probability that David takes home a zebrafish? Solution a. Each fish represents a distinct outcome. Therefore, if S is the sample space of all possible outcomes, then n(S) 8 6 7 21. b. Since there are 7 zebrafish: P(zebrafish) 5 7 21 5 1 3. Answers a. 21 b. 1 3 588 Probability EXAMPLE 3 A spinner contains eight regions, numbered 1 through 8, as shown in the figure. The arrow has an equally likely chance of landing on any of the eight regions. If the arrow lands on a line, the result is not counted and the arrow is spun again. a. How many possible outcomes are in the sample space S. What is the probability that the arrow lands on the 4? That is, what is P(4)? c. List the set of possible outcomes for event O, in which the arrow lands on an odd number. d. Find the probability that the arrow lands on an odd number. Answers a. n(S) 8 b. Since there is only one region numbered 4 out of eight regions: P(4) c. Event O {1, 3, 5, 7} d. Since O {1, 3, 5, 7}, n(O) 4. P(O) n(O) n(S EXERCISES Writing About Mathematics 1. A spinner is divided into three sections numbered 1, 2, and 3, as shown in the figure. Explain why the probability of the arrow landing on the region num1 bered 1 is not. 3 2 3 1 2. Mark said that since there are 50 states, the probability that the next baby born in the United States will be born in New Jersey is why not. 1 50. Do you agree with Mark? Explain why or |
Developing Skills 3. A fair coin is tossed. a. List the sample space. b. What is P(head), the probability that a head will appear? c. What is P(tail)? Theoretical Probability 589 In 4–9, a fair die is tossed. For each question: a. List the possible outcomes for the event. b. State the probability of the event. 4. The number 3 appears. 5. An even number appears. 6. A number less than 3 appears. 7. An odd number appears. 8. A number greater than 3 appears. 9. A number greater than or equal to 3 appears. In 10–15, a spinner is divided into five equal regions, numbered 1 through 5, as shown below. The arrow is spun and lands in one of the regions. For each question: a. List the outcomes for the event. b. State the probability of the event. 10. The arrow lands on number 3. 11. The arrow lands on an even number. 12. The arrow lands on a number less than 3. 13. The arrow lands on an odd number. 14. The arrow lands on a number greater than 3. 15. The arrow lands on a number greater than or equal to 3. 1 2 5 3 4 16. A standard deck of 52 cards is shuffled, and one card is drawn. What is the probability that the card is: a. the queen of hearts? b. a queen? d. a red card? g. an ace? e. the 7 of clubs? h. a red 7? c. a heart? f. a club? i. a black 10? j. a picture card (king, queen, jack)? 17. A person does not know the answer to a test question and takes a guess. Find the probabil- ity that the answer is correct if the question is: a. a multiple-choice question with four choices b. a true-false question c. a question where the choices given are “sometimes, always, or never” 18. A marble is drawn at random from a bag. Find the probability that the marble is green if the bag contains marbles whose colors are: a. 3 blue, 2 green d. 6 blue, 4 green b. 4 blue, 1 green e. 3 green, 9 blue c. 5 red, 2 green, 3 blue f. 5 red, 2 green, 9 blue 19. The digits of the |
number 1,776 are written on disks and placed in a jar. What is the probabil- ity that the digit 7 will be chosen on a single random draw? 20. A letter is chosen at random from a given word. Find the probability that the letter is a vowel if the word is: a. APPLE b. BANANA c. GEOMETRY d. MATHEMATICS 590 Probability Applying Skills 21. There are 16 boys and 14 girls in a class. The teacher calls students at random to the chalk- board. What is the probability that the first person called is: a. a boy? b. a girl? 22. There are 840 tickets sold in a raffle. Jay bought five tickets, and Lynn bought four tickets. What is the probability that: a. Jay has the winning ticket? b. Lynn has the winning ticket? 23. The figures that follow are eight polygons: a square; a rectangle; a parallelogram that is not a rectangle; a right triangle; an isosceles triangle that does not contain a right angle; a trapezoid that does not contain a right angle; an equilateral triangle; a regular hexagon. One of the figures is selected at random. What is the probability that this polygon: a. contains a right angle? b. is a quadrilateral? c. is a triangle? d. has at least one acute angle? e. has all sides congruent? f. has at least two sides congruent? g. has fewer than five sides? h. has an odd number of sides? i. has four or more sides? j. has at least two obtuse angles? 15-3 EVALUATING SIMPLE PROBABILITIES We have called an event for which there is only one outcome a singleton. For example, when rolling a die only once, getting a 3 is a singleton. However, when rolling a die only once, some events are not singletons. For example: 1. The event of rolling an even number on a die is {2, 4, 6}. 2. The event of rolling a number less than 6 on a die is {1, 2, 3, 4, 5}. Evaluating Simple Probabilities 591 The Impossible Case On a single roll of a die, what is the probability that the number 7 will appear? We call this case an impossibility because there is no way in which this event can occur |
. In this example, event E rolling a 7; so, E { } or, and n(E) 0. The sample space S for rolling a die contains six possible outcomes, and n(S) 6. Therefore: P(E) number of ways to roll a 7 number of outcomes for the die 5 n(E) n(S) 5 0 6 5 0 In general, for any sample space S containing k possible outcomes, we say n(S) k. For any impossible event E, which cannot occur in any way, we say n(E) 0. Thus, the probability of an impossible event is: P(E) n(E) n(S) 5 0 k 5 0 and we say: The probability of an impossible event is 0. There are many other impossibilities where the probability must equal zero. For example, the probability of selecting the letter E from the word PROBABILITY is 0. Also, selecting a coin worth 9 cents from a bank containing a nickel, a dime, and a quarter is an impossible event. The Certain Case On a single roll of a die, what is the probability that a whole number less than 7 will appear? We call this case a certainty because every one of the possible outcomes in the sample space is also an outcome for this event. In this example, the event E rolling a whole number less than 7, so n(E) = 6. The sample space S for rolling a die contains six possible outcomes, so n(S) 6. Therefore: P(E) number of ways to roll a number less than 7 number of outcomes for the die 5 n(E) n(S) 5 6 6 5 1 When an event E is certain, the event E is the same as the sample space S, that is, E 5 S and n(E) n(S). In general, for any sample space S containing k possible outcomes, n(S) k. When the event E is certain, every possible outcome for the sample space is also an outcome for event E, or n(E) = k. Thus, the probability of a certainty is given as: P(E) n(E) n(S) 5 k k 5 1 and we say: The probability of an event that is certain to occur is 1. 592 Probability There are many other certainties where the probability must equal 1. Examples include the probability of selecting a consonant from the letters JFK or selecting a red sweater |
from a drawer containing only red sweaters. The Probability of Any Event The smallest possible probability is 0, for an impossible case; no probability can be less than 0. The largest possible probability is 1, for a certain event; no probability can be greater than 1. Many other events, as seen earlier, however, have probabilities that fall between 0 and 1. Therefore, we conclude: The probability of any event E must be equal to or greater than 0, and less than or equal to 1: 0 P(E) 1 Subscripts in Sample Spaces A sample space may sometimes contain two or more objects that are exactly alike. To distinguish one object from another, we make use of subscripts. A subscript is a number, usually written in smaller size to the lower right of a term. For example, a box contains six jellybeans: two red, three green, and one yellow. Using R, G, and Y to represent the colors red, green, and yellow, respectively, we can list this sample space in full, using subscripts: {R1, R2, G1, G2, G3, Y1} Since there is only one yellow jellybean, we could have listed the last out- come as Y instead of Y1. EXAMPLE 1 An arrow is spun once and lands on one of three equally likely regions, numbered 1, 2, and 3, as shown in the figure. a. List the sample space for this experiment. b. List all eight possible events for one spin of the arrow. 1 2 3 Solution a. The sample space S {1, 2, 3}. b. Since events are subsets of the sample space S, the eight possible events are the eight subsets of S: { }, the empty set for impossible events. The arrow lands on a number other than 1, 2, or 3. {1}, a singleton. The arrow lands on 1 or the arrow lands on a number less than 2. Evaluating Simple Probabilities 593 {2}, a singleton. The arrow lands on 2 or the arrow lands on an even number. {3}, a singleton. The arrow lands on 3 or the arrow lands on a number greater than 2. {1, 2}, an event with two possible outcomes. The arrow does not land on 3 or does land on a number less than 3. {1, 3}, an event with two possible outcomes. The arrow lands on an odd number or does not land on 2. { |
t least 10 cents? c. exactly 3 cents? d. more than 3 cents? Solution The sample space for this example is {N, D1, D2, Q}. Therefore, n(S) 4. a. There are two coins worth exactly 10 cents, D1 and D2. Therefore, n(E) 2 and P(coin worth 10 cents) n(E) n(S) 5 2 4 5 1 2. b. There are three coins worth at least 10 cents, D1, D2, and Q. Therefore, n(E) 3 and P(coin worth at least 10 cents) n(E) n(S) 5 3 4. c. There is no coin worth exactly 3 cents. This is an impossible event. Therefore, P(coin worth 3 cents) 0. d. Each of the four coins is worth more than 3 cents. This is a certain event. Therefore, P(coin worth more than 3 cents) 1. Answers a. 1 2 b. 3 4 c. 0 d. 1 594 Probability EXAMPLE 3 In the Sullivan family, there are two more girls than boys. At random, Mrs. Sullivan asks one of her children to go to the store. If she is equally likely to ask any one of her children, and the probability that she asks a girl is, how many boys and how many girls are there in the Sullivan family? 2 3 Check number of girls number of children P(girl) 5 2 3 5? 4 6 2 3 5 2 3 ✔ Solution Let x the number of boys x 2 the number of girls 2x 2 the number of children. Then: P(girl) 5 number of girls number of children 3 5 x 1 2 2 2x 1 2 2(2x 1 2) 5 3(x 1 2) 4x 1 4 5 3x 1 6 x 5 2 Then x 2 4 and 2x 2 6 Answer There are two boys and four girls. EXERCISES Writing About Mathematics 1. Describe three events for which the probability is 0. 2. Describe three events for which the probability is 1. Developing Skills 3. A fair coin is tossed, and its sample space is S {H, T}. a. List all four possible events for the toss of a fair coin. b. Find the probability of each event named in part a. In 4–11, a spinner is divided into seven equal sectors, numbered 1 through 7. An |
arrow is spun to fall into one of the regions. For each question, find the probability that the arrow lands on the number described. 4. the number 5 6. a number less than 5 5. an even number 7. an odd number Evaluating Simple Probabilities 595 8. a number greater than 5 9. a number greater than 1 10. a number greater than 7 11. a number less than 8 12. A marble is drawn at random from a bag. Find the probability that the marble is black if the bag contains marbles whose colors are: a. 5 black, 2 green b. 2 black, 1 green c. 3 black, 4 green, 1 red d. 9 black e. 3 green, 4 red f. 3 green 13. Ted has two quarters, three dimes, and one nickel in his pocket. He pulls out a coin at ran- dom. Find the probability that the coin is worth: a. exactly 5 cents b. exactly 10 cents c. exactly 25 cents d. exactly 50 cents e. less than 25 cents f. less than 50 cents g. more than 25 cents h. more than l cent i. less than l cent 14. A single fair die is rolled. Find the probability for each event. a. The number 8 appears. b. A whole number appears. c. The number is less than 5. d. The number is less than 1. e. The number is less than 10. f. The number is negative. 15. A standard deck of 52 cards is shuffled, and a card is picked at random. Find the probability that the card is: a. a jack d. a red club b. a club c. a star e. a card from the deck f. a black club g. the jack of stars h. a 17 i. a red 17 In 16–20, a letter is chosen at random from a given word. For each question: a. Write the sample space, using subscripts to designate events if needed. b. Find the probability of the event. 16. Selecting the letter E from the word EVENT 17. Selecting the letter S from the word MISSISSIPPI 18. Selecting a vowel from the word TRIANGLE 19. Selecting a vowel from the word RECEIVE 20. Selecting a consonant from the word SPRY Applying Skills 21. There are 15 girls and 10 boys in a class. The teacher calls on a student in the class at random to answer |
a question. Express, in decimal form, the probability that the student called upon is: a. a girl b. a boy c. a pupil in the class d. a person who is not a student in the class 596 Probability 22. The last digit of a telephone number can be any of the following: 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. Express, as a percent, the probability that the last digit is: a. 7 b. odd c. greater than 5 d. a whole number e. the letter R 23. A girl is holding five cards in her hand: the 3 of hearts, the 3 of diamonds, the 3 of clubs, the 4 of diamonds, the 7 of clubs. A player to her left takes one of these cards at random. Find the probability that the card selected from the five cards in the girl’s hand is: a. a 3 d. a black 4 g. a 5 b. a diamond c. a 4 e. a club f. the 4 of hearts h. the 7 of clubs i. a red card j. a number card k. a spade l. a number greater than 1 and less than 8. 24. A sack contains 20 marbles. The probability of drawing a green marble is. How many 2 5 green marbles are in the sack? 25. There are three more boys than girls in the chess club. A member of the club is to be chosen at random to play in a tournament. Each member is equally likely to be chosen. If the probability that a girl is chosen is, how many boys and how many girls are members of the club? 3 7 26. A box of candy contains caramels and nut clusters. There are six more caramels than nut clusters. If a piece of candy is to be chosen at random, the probability that it will be a caramel is. How many caramels and how many nut clusters are in the box? 3 5 27. At a fair, each ride costs one, two, or four tickets. The number of rides that cost two tickets is three times the number of rides that cost one ticket. Also, seven more rides cost four tickets than cost two tickets. Tycella, who has a book of tickets, goes on a ride at random. If the probability that the ride cost her four tickets is, how many rides are there at the fair? 4 7 15-4 THE PROBA |
BILITY OF (A AND B) If a fair die is rolled, we can find simple probabilities, since we know that S {1, 2, 3, 4, 5, 6}. For example, let event A be rolling an even number. Then A {2, 4, 6} and P(A) Let event B be rolling a number less than 3. Then B {1, 2} and n(S) 5 3 6 n(A) P(B) n(B) n(S) 5 2 6 Now, what is the probability of obtaining a number on the die that is even and less than 3? We may think of this as event (A and B), which consists of those elements of the sample space that are in A and also in B. In set notation we say, The Probability of (A and B) 597 (A and B) A B {2} Only 2 is both even and less than 3. Notice that we use the symbol for inter- section () to denote and. Since n(A and B) 1, P(A and B) n(A and B) n(S) 5 1 6 Let us consider another example in which a fair die is rolled. What is the probability of rolling a number that is both odd and a 4? Event C {1, 3, 5}. Three numbers on a die are odd. P(C) n(C) n(S) 5 3 6 Event D {4}. One number on the die is 4. n(D) n(S) 5 1 6 (C and D) {numbers on a die that are odd and 4}= C D, the empty P(D) set. Since there is no outcome common to both C and D, n(C and D) 0. Therefore, P(C D) Conclusions n(C d D) n(S) 5 0 6 5 0 There is no simple rule or formula whereby the n(A) and n(B) can be used to find n(A and B). We must simply count the number of outcomes that are common in both events or write the intersection of the two sets and count the number of elements in that intersection. KEEP IN MIND Event (A and B) consists of the outcomes that are in event A and in event B. Event (A and B) may be regarded as the intersection of sets, namely, A B. EXAMPLE 1 A fair die |
is rolled once. Find the probability of obtaining a number that is greater than 3 and less than 6. Solution Event A {numbers greater than 3} {4, 5, 6}. Event B {numbers less than 6} {1, 2, 3, 4, 5}. Event (A and B) {numbers greater than 3 and less than 6} {4, 5}. n(A d B) n(S) 5 2 6 Therefore: P(A and B) or P(A B) n(A and B) n(S) 5 2 6. Answer P(number greater than 3 and less than 6) or 1 3 2 6 598 Probability EXERCISES Writing About Mathematics 1. Give an example of two events A and B, such that P(A and B) P(A). 2. If P(A and B) P(A), what must be the relationship between set A and set B? Developing Skills 3. A fair die is rolled once. The sides are numbered 1, 2, 3, 4, 5, and 6. Find the probability that the number rolled is: a. greater than 2 and odd b. less than 4 and even c. greater than 2 and less than 4 d. less than 2 and even e. less than 6 and odd f. less than 4 and greater than 3 4. From a standard deck of cards, one card is drawn. Find the probability that the card is: a. the king of hearts b. a red king c. a king of clubs d. a black jack g. a 2 of spades e. a 10 of diamonds f. a red club h. a black 2 i. a red picture card Applying Skills 5. A set of polygons consists of an equilateral triangle, a square, a rhombus that is not a square, and a rectangle. One of the polygons is selected at random. Find the probability that the polygon contains: a. all sides congruent and all angles congruent b. all sides congruent and all right angles c. all sides congruent and two angles not congruent d. at least two congruent sides and at least two congruent angles e. at least three congruent sides and at least two congruent angles 6. In a class of 30 students, 23 take science, 28 take math, and all take either science or math. a. How many students take |
both science and math? b. A student from the class is selected at random. Find: (1) P(takes science) (2) P(takes math) (3) P(takes science and math) The Probability of (A or B) 599 7. At a karaoke party, some of the boys and girls take turns singing songs. Of the five boys, Patrick and Terence are teenagers while Brendan, Drew, and Kevin are younger. Of the seven girls, Heather and Claudia are teenagers while Maureen, Elizabeth, Gwen, Caitlin, and Kelly are younger. Find the probability that the first song is sung by: a. a girl c. a teenager e. a boy under 13 g. a teenage girl b. a boy d. someone under 13 years old f. a girl whose initial is C h. a girl under 13 i. a boy whose initial is C j. a teenage boy 15-5 THE PROBABILITY OF (A OR B) If a fair die is rolled, we can find simple probabilities. For example, let event A be rolling an even number. Then: P(A) n(A) n(S) 5 3 6 Let event C be rolling a number less than 2. Then: P(C) n(C) n(S) 5 1 6 Now, what is the probability of obtaining a number on the die that is even or less than 2? We may think of this as event (A or C). For the example above, there are four outcomes in the event (A or C): 1, 2, 4, and 6. Each of these numbers is either even or less than 2 or both. Since n(A or C) 4 and there are six elements in the sample space: Observe that P(A) 5 3 6, P(C) 5 1 6 that P(A or C) n(A or C) n(S) 5 4 6 and P(A or C) 5 4 6. In this case, it appears P(A)1P(C) 5 P(A or C). Will this simple addition rule hold true for all problems? Before you say “yes,” consider the next example, in which a fair die is rolled. Event A {even numbers on a die} {2, 4, 6}. P(A) n(A) n(S) 5 3 6 Event B {numbers |
less than 3 on a die} {1, 2}. P(B) Then, event (A or B) {numbers that are even or less than 3}. n(B) n(S) 5 2 6 P(A or B) n(A or B) n(S) 5 4 6 600 Probability 5 3 6 5 2 6 Here P(A). In this case, the simple rule of addition does not work: P(A) P(B) P(A or B). What makes this example different from P(A or C), shown previously?, and P(A or B), P(B) 5 4 6 A Rule for the Probability of (A or B) Probability is based on the number of outcomes in a given event. For the event (A or B) in our example, we observe that the outcome 2 is found in event A and in event B. Therefore, we may describe rolling a 2 as the event (A and B). The simple addition rule does not work for the event (A or B) because we have counted the event (A and B) twice: first in event A, then again in event B. In order to count the event (A and B) only once, we must subtract the number of shared elements,, from the overall number of elements in (A or B). n(A and B) Thus, the rule becomes: n(A or B) n(A) n(B) n(A and B) For this example: n(A or B) 3 2 1 4 Dividing each term by n(S), we get an equivalent equation: For this example: n(A or B) n(S) n(A or B) n(S) n(A) n(S) 3 6 n(B) n(S) 2 6 n(A and B) n(S) 1 6 4 6 Since P(A or B) n(A or B) n(S), we can write a general rule: P(A or B) P(A) P(B) P(A and B) In set terminology, the rule for probability becomes: P(A or B) P(A B) P(A) P(B) P(A B) " Note the use of the union symbol () to indicate or. Mutually Exclusive Events We have been examining three different events that occur when a die |
is tossed. Event A {an even number} {2, 4, 6} Event B {a number less than 3} {1, 2} Event C {a number less than 2} {1} The Probability of (A or B) 601 We found that P(A or C) P(A) P(C) P(A or B) P(A) P(B) P(A and B) Why are these results different? Of these sets, A and C are disjoint, that is they have no element in common. Events A and C are said to be mutually exclusive events because only one of the events can occur at any one throw of the dice. Events that are disjoint sets are mutually exclusive. If two events A and C are mutually exclusive: P(A or C) P(A) P(C) Sets A and B are not disjoint sets. They have element 2 in common. Events A and B are not mutually exclusive events because both can occur at any one throw of the dice. Events that are not disjoint sets are not mutually exclusive. If two events A and B are not mutually exclusive: P(A or B) P(A) P(B) P(A and B) Some examples of mutually exclusive events include the following. • Drawing a spade or a red card when one card is drawn from a standard deck. • Choosing a ninth-grade boy or a tenth-grade girl from a student body that consists of boys and girls in each grade 9 through 12. • Drawing a quarter or a dime when one coin is drawn from a purse that contains three quarters, two dimes, and a nickel. • Drawing a consonant or a vowel when a letter is drawn from the word PROBABILITY. Some examples of events that are not mutually exclusive include the fol- lowing. • Drawing an ace or a red card when one card is drawn from a standard deck. • Choosing a girl or a ninth-grade student from a student body that consists of boys and girls in each grade 9 through 12. • Drawing a dime or a coin worth more than five cents when one coin is drawn from a purse that contains a quarter, two dimes, and a nickel. • Drawing a Y or a letter that follows L in the alphabet when a letter is drawn from the word PROBABILITY. 602 Probability EXAMPLE 1 A standard deck of 52 cards is shuffled, and one card |
is drawn at random. Find the probability that the card is: a. a king or an ace b. red or an ace Solution a. There are four kings in the deck, so P(king) 5 4 52. There are four aces in the deck, so P(ace) 5 4 52. These are mutually exclusive events. The set of kings and the set of aces are disjoint sets, having no elements in common. P(king or ace) P(king) P(ace) 4 52 1 4 52 5 8 52 or 2 13 Answer b. There are 26 red cards in the deck, so P(red) 5 26 52. There are four aces in the deck, so P(ace) 5 4 52. There are two red aces in the deck, so P(red and ace) 5 2 52. These are not mutually exclusive events. Two cards are both red and ace. P(red or ace) P(red) P(ace) P(red and ace) 52 2 2 52 5 28 52 or 7 13 52 1 4 Answer 5 26 Alternative Solution There are 26 red cards and two more aces not already counted (the ace of spades, the ace of clubs). Therefore, there are 26 2, or 28, cards in this event. Then: P(red or ace) 52 or 7 28 13 Answer EXAMPLE 2 There are two events, A and B. Given that P(A).3, P(B).5, and P(A d B) 5.1, find P(A B). Solution Since the probability of A B is not 0, A and B are not mutually exclusive. P(A B) P(A) P(B) P(A B).3.5.1.7 Answer P(A B).7 EXAMPLE 3 A town has two newspapers, the Times and the Chronicle. One out of every two persons in the town subscribes to the Times, three out of every five persons in the town subscribe to the Chronicle, and three out of every ten persons in the The Probability of (A or B) 603 town subscribe to both papers. What is the probability that a person in the town chosen at random subscribes to the Times or the Chronicle? Solution Subscribing to the Times and subscribing to the Chronicle are not mutually exclusive events. 5 1 2 P(Times) 5 3 5 P(Times or Chronicle) P |
(Times) P(Chronicle) P(Times and Chronicle) P(Times and Chronicle) P(Chronicle) 5 3 10 5 1 5 5 2 1 3 10 1 6 5 2 3 10 10 2 3 10 5 8 10 or 4 5 Answer EXERCISES Writing About Mathematics 1. Let A and B be two events. Is it possible for P(A or B) to be less that P(A)? Explain why or why not. 2. If B is a subset of A, which of the following is true: P(A or B) P(A), P(A or B) P(A)? Explain your answer. P(A or B) 5 P(A), Developing Skills 3. A spinner consists of five equal sectors of a circle. The sectors are numbered 1 through 5, and when an arrow is spun, it is equally likely to stop on any sector. For a single spin of the arrow, determine whether or not the events are mutually exclusive and find the probability that the number on the sector is: a. 3 or 4 b. odd or 2 c. 4 or less d. 2 or 3 or 4 e. odd or 3 4. A fair die is rolled once. Determine whether or not the events are mutually exclusive and find the probability that the number rolled is: a. 3 or 4 e. odd or 3 b. odd or 2 f. less than 2 or more than 5 c. 4 or less than 4 g. less than 5 or more than 2 d. 2, 3, or 4 h. even or more than 3 5. From a standard deck of cards, one card is drawn. Determine whether or not the events are mutually exclusive and find the probability that the card will be: a. a queen or an ace b. a queen or a 7 c. a heart or a spade d. a queen or a spade e. a queen or a red card f. a jack or a queen or a king g. a 7 or a diamond h. a club or a red card i. an ace or a picture card 604 Probability 6. A bank contains two quarters, six dimes, three nickels, and five pennies. A coin is drawn at random. Determine whether or not the events are mutually exclusive and find the probability that the coin is: a. a quarter or a dime b. a dime or a nickel c. worth 10 cents or more than 10 |
cents d. worth 10 cents or less e. worth 1 cent or more f. a quarter, a nickel, or a penny In 7–12, in each case choose the numeral preceding the expression that best completes the statement or answers the question. 7. If a single card is drawn from a standard deck, what is the probability that it is a 4 or a 9? (1) 2 52 (2) 8 52 (3) 13 52 (4) 26 52 8. If a single card is drawn from a standard deck, what is the probability that it is a 4 or a diamond? 8 52 (1) (2) 16 52 (3) 17 52 9. If P(A).2, P(B).5, and P(A B).1, then P(A B) = (1).6 (2).7 (3).8 10. If P(A) 5 1 3, P(B) 5 1 2, and P(A and B) 5 1 6, then P(A or B) = (1) 2 5 (2) 2 3 (3) 5 6 11. If P(A) 5 1 4, P(B) 5 1 2, and P(A B) 5 1 8, then P(A B) = (1) 1 8 (2) 5 8 (3) 3 4 12. If P(A).30, P(B).35, and (A B), then P(A B) = (1).05 (2).38 (3).65 (4) 26 52 (4).9 (4) 1 (4) 7 8 (4) 0 Applying Skills 13. Linda and Aaron recorded a CD together. Each sang some solos, and both of them sang some duets. Aaron recorded twice as many duets as solos, and Linda recorded six more solos than duets. If a CD player selects one of these songs at random, the probability that it 1 will select a duet is. Find the number of: 4 a. solos by Aaron b. solos by Linda c. duets 14. In a sophomore class of 340 students, some students study Spanish, some study French, some study both languages, and some study neither language. If P(Spanish).70, P(French).40, and P(Spanish and French).25, find: |
a. the probability that a sophomore studies Spanish or French b. the number of sophomores who study one or more of these languages The Probability of (Not A) 605 15. The Greenspace Company offers lawn care services and snow plowing in the appropriate seasons. Of the 600 property owners in town, 120 have contracts for lawn care, 90 for snow plowing, and 60 for both with the Greenspace Company. A new landscape company, the Earthpro Company offers the same services and begins a telephone campaign to attract customers, choosing telephone numbers of property owners at random. What is the probability that the Earthpro Company reaches someone who has a contract for lawn care or snow plowing with the Greenspace Company? 15-6 THE PROBABILITY OF (NOT A) In rolling a fair die, we know that P(4) since there is only one outcome for the event (rolling a 4). We can also say that P(not 4) since there are five outcomes for the event (not rolling a 4): 1, 2, 3, 5, and 6. 5 6 5 1 6 We can think of these probabilities in another way. The event (4) and the event (not 4) are mutually exclusive events. Also, the event (4 or not 4) is a certainty whose probability is 1. P(4 or not 4) P(4) P(not 4) 1 P(4) P(not 4) 1 P(4) P(not 4) 1 1 6 5 5 6 5 P(not 4) P(not 4) The event (not A) is the complement of event A, when the universal set is the sample space, S. In general, if P(A) is the probability that some given result will occur, and P(not A) is the probability that the given result will not occur, then: 1. P(A) P(not A) 1 2. P(A) 1 P(not A) 3. P(not A) 1 P(A) Probability as a Sum When sets are disjoint, we have seen that the probability of the union can be found by the rule P(A B) = P(A) P(B). Since the possible outcomes that are singletons represent disjoint sets, we can say: The probability of any event is equal to the sum of the probabilities of the singleton outcomes in the event. 606 Probability For example, when |
we draw a card from a standard deck, there are 52 sin. Since all singleton events are dis- gleton outcomes, each with a probability of joint, we can say: 1 52 P(king) P(king of hearts) P(king of diamonds) P(king of spades) P(king of clubs) 1 1 52 52 52 or 1 4 13 1 52 1 52 P(king) We also say: The sum of the probabilities of all possible singleton outcomes for any sam- ple space must always equal 1. For example, in tossing a coin, P(S) P(heads) P (tails) 2 1 1 1 2 1. Also, in rolling a die, P(S) P(1) P(2) P(3) P(4) P(5) P(6 EXAMPLE 1 Dr. Van Brunt estimates that 4 out every 10 patients that he will see this week will need a flu shot. What is the probability that the next patient he sees will not need a flu shot? Solution The probability that a patient will need a flu shot is 4 10 or 2 5 The probability that a patient will not need a flu shot is 1. 5 5 3 2 5. Answer EXAMPLE 2 A letter is drawn at random from the word ERROR. a. Find the probability of drawing each of the different letters in the word. b. Demonstrate that the sum of these probabilities is 1. Solution a. P(E) 5 1 5 ; P(R) 5 3 5 ; P(O Answer 5 5 5 5 b. P(E) P(R) P(O) 1 Answer The Probability of (Not A) 607 EXERCISES Writing About Mathematics 1. If event A is a certainty, P(A) 1. What must be true about P(not A)? Explain your answer. 2. If A and B are disjoint sets, what is P(not A or not B)? Explain your answer. Developing Skills 3. A fair die is rolled once. Find each probability: a. P(3) b. P(not 3) c. P(even) d. P(not even) e. P(less than 3) f. P(not less than 3) g. P(odd or even) h. P[not (odd or even)] i. P[not (2 or 3)] 4. From a standard deck of cards, one |
card is drawn. Find the probability that the card is: a. a heart d. not a picture card b. not a heart e. not an 8 c. a picture card f. not a red 6 g. not the queen of spades h. not an 8 or a 6 5. One letter is selected at random from the word PICNICKING. a. Find the probability of drawing each of the different letters in the word. b. Demonstrate that the sum of these probabilities is 1. 6. If the probability of an event happening is, what is the probability of that event not 1 7 happening? 7. If the probability of an event happening is.093, what is the probability of that event not happening? 8. A jar contains seven marbles, all the same size. Three are red and four are green. If a marble is chosen at random, find each probability: a. P(red) b. P(green) c. P(not red) d. P(red or green) e. P(red and green) f. P[not (red or green)] 9. A box contains 3 times as many black marbles as green marbles, all the same size. If a mar- ble is drawn at random, find the probability that it is: a. black b. green c. not black d. black or green e. not green 10. A letter is chosen at random from the word PROBABILITY. Find each probability: a. P(A) d. P(A or B) b. P(B) e. P(A or I) c. P(C) f. P(a vowel) g. P(not a vowel) h. P(A or B or L) i. P(A or not A) 608 Probability 11. A single card is drawn at random from a well-shuffled deck of 52 cards. Find the probability that the card is: a. a 6 d. a 6 or a club g. a 6 or a 7 j. a black 6 Applying Skills b. a club e. not a club h. not the 6 of clubs c. the 6 of clubs f. not a 6 i. a 6 and a 7 k. a 6 or a black card l. a black card or not a 6 12. A bank contains three quarters, four dimes, and five nickels. A coin is drawn at random. a |
. Find the probability of drawing: (1) a quarter (2) a dime (3) a nickel b. Demonstrate that the sum of the three probabilities given as answers in part a is 1. c. Find the probability of not drawing: (1) a quarter (2) a dime (3) a nickel 13. The weather bureau predicted a 30% chance of rain. Express in fractional form: a. the probability that it will rain b. the probability that it will not rain 14. Mr. Jacobsen’s mail contains two letters, three bills, and five ads. He picks up the first piece of mail without looking at it. Express, in decimal form, the probability that this piece of mail is: a. a letter b. a bill d. a letter or an ad e. a bill or an ad g. not an ad h. a bill and an ad c. an ad f. not a bill 15. The square dartboard shown at the right, whose side measures 30 inches, has at its center a shaded square region whose side measures 10 inches. If darts directed at the board are equally likely to land anywhere on the board, what is the probability that a dart does not land in the shaded region? 30 in. 10 in. 10 in. 30 in. 16. A telephone keypad contains the ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Mabel is calling a friend. Find the probability that the last digit in her friend’s telephone number is: a. 6 b. 6 or a larger number c. a number smaller than 6 d. 6 or an odd number e. 6 or a number smaller than 6 f. not 6 The Counting Principle, Sample Spaces, and Probability 609 g. 6 and an odd number h. a number not larger than 6 i. a number smaller than 2 and larger than 6 j. a number smaller than 2 or larger than 6 k. a number smaller than 6 and larger than 2 l. a number smaller than 6 or larger than 2 15-7 THE COUNTING PRINCIPLE, SAMPLE SPACES, AND PROBABILITY So far, we have looked at simple problems involving a single activity, such as rolling one die or choosing one card. More realistic problems occur when there are two or more activities, such as rolling two dice or dealing a hand of five cards. An event consisting of two or |
more activities is called a compound event. Before studying the probability of events based on two or more activities, let us study ways to count the number of elements or outcomes in a sample space for two or more activities. For example: A store offers five flavors of ice cream: vanilla, chocolate, strawberry, peach, and raspberry. A sundae can be made with either a hot fudge topping or a marshmallow topping. If a sundae consists of one flavor of ice cream and one topping, how many different sundaes are possible? We will use letters to represent the five flavors of ice cream (V, C, S, P, R) and the two toppings (F, M). We can show the number of elements in the sample space in three ways: 1. Tree Diagram. The tree diagram at the right first branches out to show five flavors of ice cream. For each of these flavors the tree again branches out to show the two toppings. In all, there are 10 paths or branches to follow, each with one flavor of ice cream and one topping. These 10 branches show that the sample space consists of 10 possible outcomes, in this case, sundaes. V C S P R Tree diagram. List of Ordered Pairs. It is usual to order a sundae by telling the clerk the flavor of ice cream first and then the type of topping. This suggests a listing of ordered pairs. The first component of the ordered pair is the icecream flavor, and the second component is the type of topping. The set of pairs (ice cream, topping) is shown below. {(V, F), (C, F), (S, F), (P, F), (R, F), (V, M), (C, M), (S, M), (P, M), (R, M)} These 10 ordered pairs show that the sample space consists of 10 possible sundaes. 610 Probability 3. Graph of Ordered Pairs. Instead of listing pairs, we may construct a graph of the ordered pairs. At the right, the five flavors of ice cream appear on the horizontal scale or line, and the two toppings are on the vertical line. Each point in the graph represents an ordered pair. For example, the point circled shows the ordered pair (P, F), that is, (peach ice cream, fudge topping). M R V C F P S This graph of 10 points, or 10 ordered pairs, shows that the sample |
space consists of 10 possible sundaes. Graph of ordered pairs Whether we use a tree diagram, a list of ordered pairs, or a graph of ordered pairs, we recognize that the sample space consists of 10 sundaes. The number of elements in the sample space can be found by multiplication: number of flavors number of toppings number of of ice cream possible sundaes e 5 f f 2 10 Suppose the store offered 30 flavors of ice cream and seven possible top- pings. To find the number of elements in the sample space, we multiply: 30 7 210 possible sundaes This simple multiplication procedure is known as the counting principle, because it enables us to count the number of elements in a sample space. The Counting Principle: If one activity can occur in any of m ways and, following this, a second activity can occur in any of n ways, then both activities can occur in the order given in m n ways. We can extend this rule to include three or more activities by extending the multiplication process. We can also display three or more activities by extending the branches on a tree diagram, or by listing ordered elements such as ordered triples and ordered quadruples. For example, a coin is tossed three times in succession. • On the first toss, the coin may fall in either of two ways: a head or a tail. • On the second toss, the coin may also fall in either of two ways. • On the third toss, the coin may still fall in either of two ways. By the counting principle, the sample space contains 2 2 2 or 8 possible outcomes. By letting H represent a head and T represent a tail, we can illustrate the sample space by a tree diagram or by a list of ordered triples: We did not attempt to draw a graph of this sample space because we would need a horizontal scale, a vertical scale, and a third scale, making the graph The Counting Principle, Sample Spaces, and Probability 611 Three Tosses of a Coin H, H, H) (H, H, T) (H, T, H) (H, T, T) (T, H, H) (T, H, T) (T, T, H) (T, T, T) Tree diagram List of ordered triples three-dimensional. Although such a graph can be drawn, it is too difficult at this time. We can conclude that: 1. Tree diagrams, or lists of ordered elements, are effective ways to |
indicate any compound event of two or more activities. 2. Graphs should be limited to ordered pairs, or to events consisting of exactly two activities. EXAMPLE 1 The school cafeteria offers four types of salads, three types of beverages, and five types of desserts. If a lunch consists of one salad, one beverage, and one dessert, how many possible lunches can be chosen? Solution By the counting principle, we multiply the number of possibilities for each choice: 4 3 5 12 5 60 possible lunches Answer Independent Events The probability of rolling 5 on one toss of a die is. What is the probability of rolling a pair of 5’s when two dice are tossed? 1 6 When we roll two dice, the number obtained on the second die is in no way determined by of the result obtained on the first die. When we toss two coins, the face that shows on the second coin is in no way determined by the face that shows on the first coin. When the result of one activity in no way influences the result of a second activity, the results of these activities are called independent events. In cases where two events are independent, we may extend the counting principle to find the probability that both independent events occur at the same time. 612 Probability For instance, what is the probability that, when two dice are thrown, a 5 will appear on each of the dice? Let S represent the sample space and F represent the event (5 on both dice). (1) Use the counting principle to find the number of elements in the sample space. There are 6 ways in which the first die can land and 6 ways in which the second die can land. Therefore, there are 6 6 or 36 pairs of numbers in the sample space, that is, n(S) 36. (2) There is only one face on each die that has a 5. Therefore there is 1 1 or 1 pair in the event F, that is n(F) 1. (3) P(5 on both dice) n(F) n(S) 5 1 36 The probability of 5 on both dice can also be determined by using the prob- ability of each of the independent events. P(5 on first die) P(5 on second die) 1 6 1 6 P(5 on both dice) P(5 on first) P(5 on second) 6 3 1 1 6 5 1 36 We can extend the counting principle to help us find the probability of any two or more independent events. The |
Counting Principle for Probability: E and F are independent events. The probability of event E is m (0 m 1) and the probability of event F is n (0 n 1). The probability of the event in which E and F occur jointly is the product m n. Note 1: The product m n is within the range of values for a probability, namely, 0 # m 3 n # 1. Note 2: Not all events are independent, and this simple product rule cannot be used to find the probability when events are not independent. EXAMPLE 2 Mr. Gillen may take any of three buses, A or B or C, to get to the train station. He may then take the 6th Avenue train or the 8th Avenue train to get to work. The buses and trains arrive at random and are equally likely to arrive. What is the probability that Mr. Gillen takes the B bus and the 6th Avenue train to get to work? Solution P(B bus) 1 3 1 and P(6th Ave. train) 2 Since the train taken is independent of the bus taken: P(B bus and 6th Ave. train) P(B bus) P(6th Ave. train) Answer 231 1 3 1 6 The Counting Principle, Sample Spaces, and Probability 613 EXERCISES Writing About Mathematics 1. Judy said that if a quarter and a nickel are tossed, there are three equally likely outcomes; two heads, two tails, or one head and one tail. Do you agree with Judy? Explain why or why not. 2. a. When a green die and a red die are rolled, is the probability of getting a 2 on the green die and a 3 on the red die the same as the probability of getting 3 on both dice? Explain why or why not. b. When rolling two fair dice, is the probability of getting a 2 and a 3 the same as the probability of getting two 3’s? Explain why or why not. Developing Skills 3. A quarter and a penny are tossed simultaneously. Each coin may fall heads or tails. The tree diagram at the right shows the sample space involved. a. List the sample space as a set of ordered pairs. b. Use the counting principle to demonstrate that there are four outcomes in the sample space. c. In how many outcomes do the coins both fall heads up? H T H T H T Quarter Penny d. In how many outcomes do the coins land showing one head |
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