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) ↓ 0.5r ↓ (4) Solve the equation. (a) Write the equation: (b) Use the distributive property: 0.5r 2(r 15) 150 0.5r 2r 30 150 Using Formulas to Solve Problems 137 (c) Combine like terms: (d) Add 30, the opposite of 30 to each side of the equation: (e) Divide each side by 2.5: 2.5r 2.5r 30 150 30 30 120 2.5 5 120 2.5r 2.5 r 48 (5) Find the average speed for each part of the trip. Sabrina’s speed on local roads r 48 mph. Sabrina’s speed on the highway r 15 48 15 63 mph. Check On local roads: On the interstate highway: The total distance traveled: 0.5(48) 24 miles 2(63) 126 miles 150 miles ✔ Answer Sabrina traveled at an average speed of 48 miles per hour on local roads and 63 miles per hour on the interstate highway. EXERCISES Writing About Mathematics 1. In Example 4, step 4 uses the equivalent equation 2.5r 30 150. Explain what each term in the left side of this equation represents in the problem. 2. Antonio solved the problem in Example 4 by letting r represent Sabrina’s rate of speed on the interstate highway. To complete the problem correctly, how should Antonio represent her rate of speed on the local roads? Explain your answer. Developing Skills In 3–19, state the meaning of each formula, tell what each variable represents, and find the required value. In 3–12, express each answer to the correct number of significant digits. 3. If P a b c, find c when P 85 in., a 25 in., and b 12 in. 4. If P 4s, find s when P 32 m. 5. If P 4s, find s when P 6.8 ft. 6. If P 2l 2w, find w when P 26 yd and l 8 yd. 7. If P 2a b, find b when P 80 cm and a 30 cm. 8. If P 2a b, find a when P 18.6 m and b 5.8 m. 9. If A bh, find b when A 240 cm2 and h 15 cm. 138 First Degree Equations and Inequalities in One Variable 10. If A b |
h, find h when A 3.6 m2 and b 0.90 m. 1 2bh 11. If A, find h when A 24 sq ft and b 8.0 ft. 12. If V lwh, find w when V 72 yd3, l 0.75 yd, and h 12 yd. 13. If d rt, find r when d 120 mi and t 3 hr. 14. If I prt, find the principal, p, when the interest, I, is $135, the yearly rate of interest, r, is 2.5%, and the time, t, is 3 years. 15. If I prt, find the rate of interest, r, when I $225, p $2,500, and t 2 years. 16. If T nc, find the number of items purchased, n, if the total cost, T, is $19.80 and the cost of one item, c, is $4.95. 17. If T nc, find the cost of one item purchased, c, if T $5.88 and n 12. 18. If S nw, find the hourly wage, w, if the salary earned, S, is $243.20 and the number of hours worked, n, is 38. 19. If S nw, find the number of hours worked, n, if S $315.00 and w $8.40. In 20–32, a. write a formula that can be used to solve each problem, b. use the formula to solve each problem and check the solution. All numbers may be considered to be exact values. 20. Find the length of a rectangle whose perimeter is 34.6 centimeters and whose width is 5.7 centimeters. 21. The length of the second side of a triangle is 2 inches less than the length of the first side. The length of the third side is 12 inches more than the length of the first side. The perimeter of the triangle is 73 inches. Find the length of each side of the triangle. 22. Two sides of a triangle are equal in length. The length of the third side exceeds the length of one of the other sides by 3 centimeters. The perimeter of the triangle is 93 centimeters. Find the length of each side of the triangle. 23. The length of a rectangle is 5 meters more than its width. The perimeter is 66 meters. Find the dimensions of the rectangle. 24. The width of |
a rectangle is 3 yards less than its length. The perimeter is 130 yards. Find the length and the width of the rectangle. 25. The length of each side of an equilateral triangle is 5 centimeters more than the length of each side of a square. The perimeters of the two figures are equal. Find the lengths of the sides of the square and of the triangle. 26. The length of each side of a square is 1 centimeter more than the width of a rectangle. The length of the rectangle is 1 centimeter less than twice its width. The perimeters of the two figures are equal. Find the dimensions of the rectangle. 27. The area of a triangle is 36 square centimeters. Find the measure of the altitude drawn to the base when the base is 8 centimeters. Using Formulas to Solve Problems 139 28. The altitude of a triangle is 4.8 meters. Find the length of the base of the triangle if the area is 8.4 square meters. 29. The length of a rectangle is twice the width. If the length is increased by 4 inches and the width is decreased by 1 inch, a new rectangle is formed whose perimeter is 198 inches. Find the dimensions of the original rectangle. 30. The length of a rectangle exceeds its width by 4 feet. If the width is doubled and the length is decreased by 2 feet, a new rectangle is formed whose perimeter is 8 feet more than the perimeter of the original rectangle. Find the dimensions of the original rectangle. 31. A side of a square is 10 meters longer than the side of an equilateral triangle. The perimeter of the square is 3 times the perimeter of the triangle. Find the length of each side of the triangle. 32. The length of each side of a hexagon is 4 inches less than the length of a side of a square. The perimeter of the hexagon is equal to the perimeter of the square. Find the length of a side of the hexagon and the length of a side of the square. Applying Skills 33. The perimeter of a rectangular parking lot is 146 meters. Find the dimensions of the lot, using the correct number of significant digits, if the length is 7.0 meters less than 4 times the width. 34. The perimeter of a rectangular tennis court is 228 feet. If the length of the court exceeds twice its width by 6.0 feet, find the dimensions of the court using the correct number of significant digits. In 35–48, make a table to organize the information according to the formula |
to be used. All numbers may be considered to be exact values. 35. Rahul has 25 coins, all quarters and dimes. Copy the table given below and organize the facts in the table using the answers to a through c. Number of coins in one denomination b a Value of one coin a b Total value of the coins of that denomination Number of Coins Value of One Coin Total Value Dimes Quarters 140 First Degree Equations and Inequalities in One Variable a. If x is the number of dimes Rahul has, express, in terms of x, the number of quarters he has. b. Express the value of the dimes in terms of x. c. Express the value of the quarters in terms of x. d. If the total value of the dimes and quarters is $4.90, write and solve an equation to find how many dimes and how many quarters Rahul has. e. Check your answer in the words of the problem. 36. If the problem had said that the total value of Rahul’s 25 dimes and quarters was $5.00, what conclusion could you draw? 37. When Ruth emptied her bank, she found that she had 84 coins, all nickels and dimes. The value of the coins was $7.15. How many dimes did she have? (Make a table similar to that given in exercise 35.) 38. Adele went to the post office to buy stamps and postcards. She bought a total of 25 stamps, some 39-cent stamps and the rest 23-cent postcards. If she paid $8.47 altogether, how many 39-cent stamps did she buy? 39. Carlos works Monday through Friday and sometimes on Saturday. Last week Carlos worked 38 hours. Copy the table given below and organize the facts in the table using the answers to a through c. Hours Worked Wage Per Hour Earnings Monday–Friday Saturday a. If x is the total number of hours Carlos worked Monday through Friday, express, in terms of x, the number of hours he worked on Saturday. b. Carlos earns $8.50 an hour when he works Monday through Friday. Express, in terms of x, his earnings Monday through Friday. c. Carlos earns $12.75 an hour when he works on Saturday. Express, in terms of x, his earnings on Saturday. d. Last week Carlos earned $340. How many hours did he work on Saturday? 40. Janice earns $6.00 an hour |
when she works Monday through Friday and $9.00 an hour when she works on Saturday. Last week, her salary was $273 for 42 hours of work. How many hours did she work on Saturday? (Make a table similar to that given in exercise 39.) 41. Candice earns $8.25 an hour and is paid every two weeks. Last week she worked 4 hours longer than the week before. Her pay for these two weeks, before deductions, was $594. How many hours did she work each week? Using Formulas to Solve Problems 141 42. Akram drove from Rochester to Albany, a distance of 219 miles. After the first 1.5 hours of travel, it began to snow and he reduced his speed by 26 miles per hour. It took him another 3 hours to complete the trip. Copy the table given below and fill in the entries using the answers to a through c. Rate Time Distance First part of the trip Last part of the trip a. If r is the average speed at which Akram traveled for the first part of the trip, express, in terms of r, his average speed for the second part of the trip. b. Express, in terms of r, the distance that Akram traveled in the first part of the trip. c. Express, in terms of r, the distance that Akram traveled in the second part of the trip. d. Find the speed at which Akram traveled during each part of the trip. 43. Vera walked from her home to a friend’s home at a rate of 3 miles per hour. She rode to 5 6 of an hour work with her friend at an average rate of 30 miles per hour. It took Vera a total of 50 minutes tance of 16 miles. How long did she walk and how long did she ride with her friend to get to work? (Make a table similar to that given in exercise 42.) to walk to her friend’s home and to get to work, traveling a total dis- A B 44. Peter drove a distance of 189 miles. Part of the time he averaged 65 miles per hour and for the remaining time, 55 miles per hour. The entire trip took 3 hours. How long did he travel at each rate? 45. Shelly and Jack left from the same place at the same time and drove in opposite directions along a straight road. Jack traveled 15 miles per hour faster than Shelly. After 3 hours, they were 315 miles apart. Find the rate at which each |
traveled. 46. Carla and Candice left from the same place at the same time and rode their bicycles in the same direction along a straight road. Candice bicycled at an average speed that was threequarters of Carla’s average speed. After 2 hours they were 28 miles apart. What was the average speed of Carla and Candice? 47. Nolan walked to the store from his home at the rate of 5 miles per hour. After spending one-half hour in the store, his friend gave him a ride home at the rate of 30 miles per hour. 1 1 12 hours He arrived home 1 hour and 5 minutes after he left. How far is the store from Nolan’s home? A B 48. Mrs. Dang drove her daughter to school at an average rate of 45 miles per hour. She returned home by the same route at an average rate of 30 miles per hour. If the trip took one-half hour, How long did it take to get to school? How far is the school from their home? 142 First Degree Equations and Inequalities in One Variable 4-5 SOLVING FOR A VARIABLE IN TERMS OF ANOTHER VARIABLE An equation may contain more than one variable. For example, the equation ax b 3b contains the variables a, b, and x. To solve this equation for x means to express x in terms of the other variables. To plan the steps in the solution, it is helpful to use the strategy of using a simpler related problem, that is, to compare the solution of this equation with the solution of a simpler equation that has only one variable. In Example 1, the solution of ax b 3b is compared with the solution of 2x 5 15. The same operations are used in the solution of both equations. EXAMPLE 1 Solve for x in ax b = 3b. Solution Compare with 2x 5 15. Check 2x 5 15 5 5 10 2x 2x 2 5 10 2 x 5 ax b 3b b 2b b ax a 5 2b ax a x 5 2b a Answer x 5 2b a EXAMPLE 2 Solve for x in x a b. Solution Compare with x 5 9. x 5 9 5 5 14 x Answer EXAMPLE 3 Solve for x in 2ax 10a2 3ax (a 0). ax b 3b 2b? 3b 1 b 5 a B? 2b 1 b 5 3b a |
A 3b 3b ✔ Check x a b b 1 a 2 a 5? b b b ✔ Solution Compare with 2x 10 3x. 2x 10 3x 3x 3ax 3x 2ax 10a2 3ax 3ax 5x 10 5x 5 5 10 5 x 2 5ax 10a2 5a 5 10a2 5ax 5a x 2a Transforming Formulas 143 Check 2ax 10a2 3ax 2a(2a) 5? 10a2 2 3a(2a) 5? 10a2 2 6a2 4a2 4a2 4a2 ✔ Answer x 2a EXERCISES Writing About Mathematics 1. Write a simpler related equation in one variable that can be used to suggest the steps needed to solve the equation a(x b) 4ab for x. 2. Write a simpler related equation in one variable that can be used to suggest the steps needed to solve the equation 5cy d 2cy for y. Developing Skills In 3–24, solve each equation for x or y and check. 3. 5x b 7. x 5r 7r 11. d y 9 15. ax b 3b 19. bx 9b2 22. m2x 3m2 12m2 4. sx 8 8. x a 4a 12. 3x q 5q 16. dx 5c 3c 5. ry s 9. y c 9c 13. 3x 8r r 17. r sy t 6. hy m 10. 4 x k 14. cy d 4d 18. m 2(x n) 20. cx c2 5c2 7cx 23. 9x 24a 6a 4x 21. rsx rs2 0 24. 8ax 7a2 19a2 5ax 4-6 TRANSFORMING FORMULAS A formula is an equation that contains more than one variable. Sometimes you want to solve for a variable in the formula that is different from the subject of the formula. For example, the formula for distance, d, in terms of rate, r, and time, t, is d = rt. Distance is the subject of the formula, but you might want to rewrite the formula so that it expresses time in terms of distance and rate. You do this by solving the equation d = rt for t in terms of d and r. 144 First Degree Equations and Inequalities in One Variable EXAMPLE 1 a. Sol |
ve the formula d = rt for t. b. Use the answer obtained in part a to find the value of t when d 200 miles and r 40 miles per hour. Solution a. d rt d r 5 rt r d r 5 t b. t 5 d r 5 200 40 5 Answers a. t d r b. t 5 hours Note that the rate is 40 miles per hour, that is, 40 miles 1 hour. Therefore, 200 miles 40 miles 1 hour 200 miles 3 1 hour 40 miles 5 hours We can think of canceling miles in the numerator and the denominator of the fractions being multiplied. EXAMPLE 2 a. The formula for the volume of a cone is V. Solve this formula for h. b. Find the height of a cone that has a volume of 92.0 cubic centimeters and a circular base with a radius of 2.80 centimeters. Express the answer using the correct number of significant digits. 1 3Bh Solution a. V 3V 1 3Bh 1 3 3Bh B A 3V Bh 3V B 5 Bh B 3V h B b. Find B, the area of the base of the cone. Since the base is a circle, its area is p times the square of the radius, r. B pr2 p(2.80)2 ENTER: 2nd p 2.80 x2 ENTER Transforming Formulas 145 DISPLAY Now use the answer to part a to find h: h 5 3V B < 3(92.0) 24.63 ENTER: 3 92.0 24.63 ENTER DISPLAY Since each measure is given to three significant digits, round the answer to three significant digits. Answers a. h 5 3V B b. The height of the cone is 11.2 centimeters. EXERCISES Developing Skills In 1–14, transform each given formula by solving for the indicated variable. 3. d rt for r 6. I prt for t 2. A bh for h 5. P br for r 1. P 4s for s 4. V lwh for l 1 7. A 2bh 10. P 2a b for b for h 13. F 9 5C 1 32 for C 1 8. V 3Bh 11. P 2a b for a for B 14. 2S n(a l) for a 1 9. s 2gt 12. P 2l + 2w for w for g Applying Skills 15. The concession |
stand at a movie theater wants to sell popcorn in containers that are in the shape of a cylinder. The volume of the cylinder is given by the formula V = pr2h, where V is the volume, r is the radius of the base, and h is the height of the container. a. Solve the formula for h. b. If the container is to hold 1,400 cubic centimeters of popcorn, find, to the nearest tenth, the height of the container if the radius of the base is: (1) 4.0 centimeters (2) 5.0 centimeters (3) 8.0 centimeters c. The concession stand wants to put an ad with a height of 20 centimeters on the side of the container. Which height from part b do you think would be the best for the container? Why? 146 First Degree Equations and Inequalities in One Variable 16. A bus travels from Buffalo to Albany, stopping at Rochester and Syracuse. At each city there is a 30-minute stopover to unload and load passengers and baggage. The driving distance from Buffalo to Rochester is 75 miles, from Rochester to Syracuse is 85 miles, and from Syracuse to Albany is 145 miles. The bus travels at an average speed of 50 miles per hour. a. Solve the formula d rt for t to find the time needed for each part of the trip. b. Make a schedule for the times of arrival and departure for each city if the bus leaves Buffalo at 9:00 A.M. 4-7 PROPERTIES OF INEQUALITIES The Order Property of Real Numbers If two real numbers are graphed on the number line, only one of the following three situations can be true: x is to the left of y x and y are at the same point x is to the right of These three cases illustrate the order property of real numbers: If x and y are two real numbers, then one and only one of the following can be true: x y or x y or x y Let y be a fixed point, for example, y 3. Then y separates the real numbers into three sets. For any real number, one of the following must be true: x 3, x 3, x 3. The real numbers, x, that make the inequality x 3 true are to the left of 3 on the number line. The circle at 3 indicates that 3 is the boundary value of the set. The circle is not filled in, indicating that 3 does not belong to this set. 0 |
3 x < 3 Properties of Inequalities 147 The real number that makes the corresponding equality, x 3, true is a single point on the number line. This point, x 3, is also the boundary between the values of x that make x 3 true and the values of x that make x 3 true. The circle is filled in, indicating that 3 belongs to this set. Here, 3 is the only element of the set. The real numbers, x, that make x 3 true are to the right of 3 on the number line. Again, the circle at 3 indicates that 3 is the boundary value of the set. The circle is not filled in, indicating that 3 does not belong to this set The Transitive Property of Inequality From the graph at the right, you can see that, if x lies to the left of y, and y lies to the left of z, then x lies to the left of z. The graph illustrates the transitive property of inequality: x y z For the real numbers x, y, and z: If x y and y z, then x z; and if z y and y x, then z x. The Addition Property of Inequality The following table shows the result of adding a number to both sides of an inequality. True Sentence 9 2 Order is “greater than.” 9 2 Order is “greater than.” 2 9 Order is “less than.” 2 9 Order is “less than.” Number to Add to Both Sides 9 3? 2 3 Add a positive number. 9 (–3)? 2 (–3) Add a negative number. 2 3? 9 3 Add a positive number. 2 (–3)? 9 (–3) Add a negative number. Result 12 5 Order is unchanged. 6 1 Order is unchanged. 5 12 Order is unchanged 1 6 Order is unchanged 148 First Degree Equations and Inequalities in One Variable The table illustrates the addition property of inequality: For the real numbers x, y, and z: If x y, then x z y z; and if x y, then x z y z. Since subtracting the same number from both sides of an inequality is equivalent to adding the additive inverse to both sides of the inequality, the following is true: When the same number is added to or subtracted from both sides of an inequality, the order of the new inequality is the same as the order of the original one. EXAM |
PLE 1 Use the inequality 5 9 to write a new inequality: a. by adding 6 to both sides b. by adding 9 to both sides Solution a. 5 6 9 6 11 15 b. 5 (–9) 9 (–9) –4 0 Answers a. 11 15 b. 4 0 The Multiplication Property of Inequality The following table shows the result of multiplying both sides of an inequality by the same number. True Sentence 9 2 Order is “greater than.” 5 9 Order is “less than.” 9 2 Order is “greater than.” 5 9 Order is “less than.” Number to Multiply Both Sides 9(3)? 2(3) Multiply by a positive number 5(3)? 9(3) Multiply by a positive number. 9(–3)? 2(–3) Multiply by a negative number 5(–3)? 9(–3) Multiply by a negative number. Result 27 6 Order is unchanged. 15 27 Order is unchanged. 27 6 Order is changed. 15 27 Order is changed. Properties of Inequalities 149 The table illustrates that the order does not change when both sides are multiplied by the same positive number, but does change when both sides are multiplied by the same negative number. In general terms, the multiplication property of inequality states: For the real numbers x, y, and z: If z is positive (z 0) and x y, then xz yz. If z is positive (z 0) and x y, then xz yz. If z is negative (z 0) and x y, then xz yz. If z is negative (z 0) and x y, then xz yz. Dividing both sides of an inequality by a number is equivalent to multiplying both sides by the multiplicative inverse of the number. A number and its multiplicative inverse always have the same sign. Therefore, the following is true: When both sides of an inequality are multiplied or divided by the same positive number, the order of the new inequality is the same as the order of the original one. When both sides of an inequality are multiplied or divided by the same negative number, the order of the new inequality is the opposite of the order of the original one. EXAMPLE 2 Use the inequality 6 9 to write a new inequality: a. by multiplying both sides by 2 |
. 21 3 b. by multiplying both sides by. Solution a. 6 9 b. 6 9 6(2)? 9(2) 12 18 6 A Answers a. 12 18 b. 2 3 21 3 21? 9 3 A 2 3 B B 150 First Degree Equations and Inequalities in One Variable EXERCISES Writing About Mathematics 1. Sadie said that if 5 4, then it must be true that 5x 4x. Do you agree with Sadie? Explain why or why not. 2. Lucius said that if x y and a b then x a y b. Do you agree with Lucius? Explain why or why not. 3. Jason said that if x y and a b then x a y b. Do you agree with Jason? Explain why or why not. Developing Skills In 4–31, replace each question mark with the symbol or so that the resulting sentence will be true. 4. Since 8 2, 8 1? 2 1. 8. Since 7 3, 6. Since 9 5, 9 2? 5 2. 2 2 3(3) 3(7)? 21 3 10. Since 9 6, 9 4.? 6 12. If y 6, then y 2? 6 2. A B 5. Since 6 2, 6 (–4)? 2 (–4). 1 4 B 9. Since 8 4, (8) (4)? (4) (4). 7. Since 2 8,? 28 2 22 2 1 4 A A B. 21 3. B A 11. If 5 x, then 5 7? x 7. 13. If 20 r, then 4(20)? 4(r). 14. If t 64, then t 8? 64 8. 15. If x 8, then 2x? 2(8). 16. If y 8, then y (–4)? 8 (–4). 17. If x 2 7, the x 2 (–2)? 7 (–2) or x? 5. 18. If y 3 12, then y 3 3? 12 3 or y? 15. 19. If a 5 14, then a 5 5? 14 5 or a? 9. 8 2 20. If 2x 8, then or x? 4. 1 3y 21. If 4, then A 22. If 3x 36, then 2x 2? 1 3 3y B 23x 36 23 23 2(22x)? 21 21 |
24. If x 5 and 5 y, then x? y. 23. If 2x 6, then?? 3(4) or y? 12. or x? 12. 2(6) 26. If 3 7, then 7? 3. 28. 1f 9 x, then x? 9. or x? 3. 25. If m 7 and 7 a, then m? a. 27. If 4 12, then 12? 4. 29. If 7 a, then a? 7. 30. If x 10 and 10 z, then x? z. 31. If a b and c b, then a? c. 4-8 FINDING AND GRAPHING THE SOLUTION OF AN INEQUALITY Finding and Graphing the Solution of an Inequality 151 When an inequality contains a variable, the domain or replacement set of the inequality is the set of all possible numbers that can be used to replace the variable. When an element from the domain is used in place of the variable, the inequality may be true or it may be false. The solution set of an inequality is the set of numbers from the domain that make the inequality true. Inequalities that have the same solution set are equivalent inequalities. To find the solution set of an inequality, solve the inequality by methods similar to those used in solving an equation. Use the properties of inequalities to transform the given inequality into a simpler equivalent inequality whose solution set is evident. In Examples 1–5, the domain is the set of real numbers. EXAMPLE 1 Find and graph the solution of the inequality x 4 1. Solution How to Proceed (1) Write the inequality: (2) Use the addition property of inequality. Add 4 to each side: x 4 1 4 4 5 x –2 – The graph above shows the solution set. The circle at 5 indicates that 5 is the boundary between the numbers to the right, which belong to the solution set, and the numbers to the left, which do not belong. Since 5 is not included in the solution set, the circle is not filled in. Check (1) Check one value from the solution set, for example, 7. This value will make the inequality true. 7 4 1 is true. (2) Check the boundary value, 5. This value, which separates the values that make the inequality true from the values that make it false, will make the corresponding equality, x 4 1, true. 5 4 1 is true. Answer x 5 An |
alternative method of expressing the solution set is interval notation. When this notation is used, the solution set is written as (5, ). The first number, 5 names the lower boundary. The symbol, often called infinity, indicates that there is no upper boundary, that is, that the set of real numbers continues without end. The parentheses indicate that the boundary values are not elements of the set. 152 First Degree Equations and Inequalities in One Variable EXAMPLE 2 Find and graph the solution of 5x 4 11 2x. Solution The solution set of 5x 4 11 2x includes all values of the domain for which either 5x 4 11 2x is true or 5x 4 11 2x is true. How to Proceed (1) Write the inequality: (2) Add 2x to each side: (3) Add 4 to each side: (4) Divide each side by 7: 5x 4 11 2x 2x 2x 7x 4 11 4 4 7 7x 7 # 7 7x 7 x 1 The solution set includes 1 and all of the real numbers less than 1. This is shown on the graph below by filling in the circle at 1 and drawing a heavy line to the left of 1. Answer x 1 –4 –3 –2 –1 0 1 2 3 The solution set can also be written in interval notation as (, 1]. The symbol, often called negative infinity, indicates that there is no lower boundary, that is, all negative real numbers less than the upper boundary are included. The number, 1, names the upper boundary. The right bracket indicates that the upper boundary value is an element of the set. EXAMPLE 3 Find and graph the solution set: 2(2x 8) 8x 0. Solution How to Proceed (1) Write the inequality: (2) Use the distributive property: (3) Combine like terms in the left side: (4) Add 16 to each side: 2(2x 8) 8x 0 4x 16 8x 0 4x 16 0 16 16 16 4x Finding and Graphing the Solution of an Inequality 153 (5) Divide both sides by 4. Dividing by a negative number reverses the inequality: (6) The graph of the solution set includes 4 and all of the real numbers to the right of 4 on the number line: Answer x 4 or [4, ) 24x 24 $ 16 24 x 4 –5 –4 –3 –2 –1 0 1 |
2 3 Graphing the Intersection of Two Sets The inequality 3 x 6 is equivalent to (3 x) and (x 6). This statement is true when both simple statements are true and false when one or both statements are false. The solution set of this inequality consists of all of the numbers that are in the solution set of both simple inequalities. The graph of 3 x 6 can be drawn as shown below. How to Proceed (1) Draw the graph of the solution set of the first inequality, 3 x, a few spaces above the number line: (2) Draw the graph of the solution set of the second inequality, x 6, above the number line, but below the graph of the first inequality: Solution 3 < x –2 –2 –3) Draw the graph of the intersection of these two sets by shading, on the number line, the points that belong to the solution set of both simple inequalities: Since 3 is in the solution set of x 6 but not in the solution set of 3 x, 3 is not in the intersection of the two sets. Also, since 6 is in the solution set of 3 x but not in the solution set of x 6, 6 is not in the intersection of the two sets. Therefore, the circles at 3 and 6 are not filled in, indicating that these boundary values are not elements of the solution set of 3 x 6. –2 – This set can also be written as (3, 6), a pair of numbers that list the left and right boundaries of the set. The parentheses indicate that the boundary values do not belong to the set. Similarly, the set of numbers 3 x 6 can be written as [3, 6]. The brackets indicate that the boundary values do belong to the set. 154 First Degree Equations and Inequalities in One Variable Although this notation is similar to that used for an ordered pair that names a point in the coordinate plane, the context in which the interval or ordered pair is used will determine the meaning. EXAMPLE 4 Solve the inequality and graph the solution set: 7 x 5 0. Solution How to Proceed (1) First solve the inequalities for x2) Draw the graphs of 2 x and x 5 above the number line: –2 < x x < 5 –4 –3 –2 –3) Draw the graph of all points that are common to the graphs of 2 x and x 5: –2 < x < 5 –4 –3 –2 – Answer 2 x 5 or (–2, 5 |
) Graphing the Union of Two Sets The inequality (x 3) or (x 6) is true when one or both of the simple statements are true. It is false when both simple statements are false. The solution set of the inequality consists of the union of the solution sets of the two simple statements. The graph of the solution set can be drawn as shown below. How to Proceed (1) Draw the graph of the solution set of the first inequality a few spaces above the number line: (2) Draw the graph of the solution set of the second inequality above the number line, but below the graph of the first inequality: Solution x > 3 12 2 – EXAMPLE 5 Solution Finding and Graphing the Solution of an Inequality 155 (3) Draw the graph of the union by shading, on the number line, the points that belong to the solution of one or both of the simple inequalities: –2 – Since 3 is not in the solution set of either inequality, 3 is not in the union of the two sets. Therefore, the circle at 3 is not filled in. Answer: x 3 or (3, ) Solve the inequality and graph the solution set: (x 2 0) or (x 3 0). How to Proceed (1) Solve each inequality for x: (2) Draw the graphs of x 2 and x 3 above the number line: (3) Draw the graph of all points of the graphs of x 2 or x 3: x 2 0 2 2 2 x or 2 x > 3 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 Answer (x 2) or (x 3) or (, 2) or (3, ) Note: Since the solution is the union of two sets, the answer can also be (2`,22) < (3,`) expressed using set notation: (x, 22) < (x. 3) or. EXERCISES Writing About Mathematics 1. Give an example of a situation that can be modeled by the inequality x 5 in which a. the solution set has a smallest value, and b. the solution set does not have a smallest value. 2. Abram said that the solution set of (x 4) or (x 4) is the set of all real numbers. Do you agree with Abram? Explain why or why not. 156 First Degree Equations and |
Inequalities in One Variable Developing Skills In 3–37, find and graph the solution set of each inequality. The domain is the set of real numbers. 3. x 2 4 7. x 3 6 11. y 4 4 15. 15 3y 19. –10x 20 4. z 6 4 8. 19 y 17 12. 25 d 22 16. –10 4h 20. 12 1.2r 24. –10 2.5z 28. 5x 4 4 3x 5. 31 9. 4 13. 3t 6 17. –6y 24 1 3x. 2 21. 25. 2x 1 5 29. 8y 1 3y 29 6. x 1.5 3.5 10. –3.5 c 0.5 14. 2x 12 18. 27 9y 22 3z $ 6 22. 26. 3y 6 12 30. 6x 2 8x 14 x 2. 1 23. 27. 5y 3 13 31. 8m 2(2m 3) 34. 0 2x 4 6 37. (x 5 2x) or ( x 8 3x) 38. Which of the following is equivalent to y 4 9? (3) y 13 32. 0 x 3 6 35. (x 1 3) or (x 1 9) (2) y 5 39. Which of the following is equivalent to 4x 5x 6? (2) x 6 40. The smallest member of the solution set of 3x 7 8 is (1) x 6 (1) y 5 (3) x 6 (1) 3 (2) 4 (3) 5 41. The largest member of the solution set of 4x 3x 2 is (1) 1 (2) 2 (3) 3 33. –5 x 2 7 36. (2x 2) or (x 5 10) (4) y 13 (4) x 6 (4) 6 (4) 4 In 42–47, write an inequality for each graph using interval notation. 42. 43. 44. 45. 46. –4 –3 –2 –1 –4 –3 –2 –1 –4 –3 –2 –1 –4 –3 –2 –1 –4 –3 –2 – 47. –4 –3 4 –1 48. a. Graph the inequality (x 2) or (x 2). –2 3 0 1 2 b. Write an inequality equivalent to (x 2) |
or (x 2). Using Inequalities to Solve Problems 157 4-9 USING INEQUALITIES TO SOLVE PROBLEMS Many problems can be solved by writing an inequality that describes how the numbers in the problem are related and then solving the inequality. An inequality can be expressed in words in different ways. For example: x 12 A number is more than 12. A number exceeds 12. A number is greater than 12. A number is over 12. x 12 A number is less than 12. A number is under 12. x 12 A number is at least 12. A number has a minimum value of 12. A number is not less than 12. A number is not under 12. x 12 A number is at most 12. A number has a maximum value of 12. A number is not greater than 12. A number does not exceed 12. A number is not more than 12. Procedure To solve a problem that involves an inequality: 1. Choose a variable to represent one of the unknown quantities in the problem. 2. Express other unknown quantities in terms of the same variable. 3. Choose an appropriate domain for the problem. 4. Write an inequality using a relationship given in the problem, a previously known relationship, or a formula. 5. Solve the inequality. 6. Check the solution using the words of the problem. 7. Use the solution of the inequality to answer the question in the problem. EXAMPLE 1 Serafina has $53.50 in her pocket and wants to purchase shirts at a sale price of $14.95 each. How many shirts can she buy? Solution (1) Choose a variable to represent the number of shirts Serafina can buy and the cost of the shirts. Let x the number of shirts that she can buy. Then, 14.95x the cost of the x shirts. The domain is the set of whole numbers, since she can only buy a whole number of shirts. 158 First Degree Equations and Inequalities in One Variable (2) Write an inequality using a relationship given in the problem. The cost of the shirts is less than or equal to $53.50. |___________________| |_____________________| ↓ 14.95x ↓ ↓ $53.50 (3) Solve the inequality. 14.95x 53.50 14.95x 14.95 # 53.50 14.95 Use a calculator to complete the computation. ENTER: 53.50 14.95 ENTER DISPLAY |
Therefore, x 3.578595318. Since the domain is the set of whole numbers, the solution set is {x : x is a counting number less than or equal to 3} or {0, 1, 2, 3}. (4) Check the solution in the words of the problem. 0 shirt costs $14.95(0) $0 1 shirt costs $14.95(1) $14.95 2 shirts cost $14.95(2) $29.90 3 shirts cost $14.95(3) $44.85 4 or more shirts cost more than $14.95 (4) or more than $59.80 Answer Serafina can buy 0, 1, 2, or 3 shirts. EXAMPLE 2 The length of a rectangle is 5 centimeters more than its width. The perimeter of the rectangle is at least 66 centimeters. Find the minimum measures of the length and width. Solution If the perimeter is at least 66 centimeters, then the sum of the measures of the four sides is either equal to 66 centimeters or is greater than 66 centimeters. Let x the width of the rectangle. Then, x + 5 the length of the rectangle. The domain is the set of positive real numbers. Using Inequalities to Solve Problems 159 The perimeter of the rectangle is at least 66 centimeters. |____________________________| |_________| |_____________| ↓ x (x 5) x (x 5) ↓ ↓ 66 4x 10 66 4x 10 10 66 10 4x 56 x 14 The width can be any real number that is greater than or equal to 14 and the length is any real number that is 5 more than the width. Since we are looking for the minimum measures, the smallest possible width is 14 and the smallest possible length is 14 5 or 19. Answer The minimum width is 14 centimeters, and the minimum length is 19 centimeters. EXERCISES Writing About Mathematics 1. If there is a number x such that x 3, is it true that x 3? Explain why or why not. 2. Is the solution set of x 3 the same as the solution set of x 3? Explain why or why not. Developing Skills In 3–12, represent each sentence as an algebraic inequality. 3. x is less than or equal to 15. 4. y is greater than or equal to 4. 5. x is at most 50. 6. x is more than 50. 7. The greatest possible value of 3y |
is 30. 9. The maximum value of 4x 6 is 54. 11. The product of 3x and x 1 is less than 35. 8. The sum of 5x and 2x is at least 70. 10. The minimum value of 2x 1 is 13. 12. When x is divided by 3 the quotient is greater than 7. In 13–19, in each case write and solve the inequality that represents the given conditions. Use n as the variable. 13. Six less than a number is less than 4. 14. Six less than a number is greater than 4. 15. Six times a number is less than 72. 16. A number increased by 10 is greater than 50. 160 First Degree Equations and Inequalities in One Variable 17. A number decreased by 15 is less 18. Twice a number, increased by 6, is less than 35. than 48. 19. Five times a number, decreased by 24, is greater than 3 times the number. Applying Skills In 20–29, in each case write an inequality and solve the problem algebraically. 20. Mr. Burke had a sum of money in a bank. After he deposited an additional sum of $100, he had at least $550 in the bank. At least how much money did Mr. Burke have in the bank originally? 21. The members of a club agree to buy at least 250 tickets for a theater party. If they expect to buy 80 fewer orchestra tickets than balcony tickets, what is the least number of balcony tickets they will buy? 22. Mrs. Scott decided that she would spend no more than $120 to buy a jacket and a skirt. If the price of the jacket was $20 more than 3 times the price of the skirt, find the highest possible price of the skirt. 23. Three times a number increased by 8 is at most 40 more than the number. Find the greatest value of the number. 24. The length of a rectangle is 8 meters less than 5 times its width. If the perimeter of the rec- tangle is at most 104 meters, find the greatest possible width of the rectangle. 25. The length of a rectangle is 12 centimeters less than 3 times its width. If the perimeter of the rectangle is at most 176 centimeters, find the greatest possible length of the rectangle. 26. Mrs. Diaz wishes to save at least $1,500 in 12 months. If she saves $300 during the first 4 months, what is the least possible average amount that she |
must save in each of the remaining 8 months? 27. Two consecutive even numbers are such that their sum is greater than 98 decreased by twice the larger. Find the smallest possible values for the integers. 28. Minou wants $29 to buy music online. Her father agrees to pay her $6 an hour for gardening in addition to her $5 weekly allowance for helping around the house. What is the minimum number of hours Minou must work at gardening to receive at least $29 this week? 29. Allison has more than 2 but less than 3 hours to spend on her homework. She has work in math, English, and social studies. She plans to spend equal amounts of time studying English and studying social studies, and to spend twice as much time studying math as in studying English. a. What is the minimum number of minutes she can spend on English homework? b. What is the maximum number of minutes she can spend on social studies? c. What is the maximum number of minutes she can devote to math? CHAPTER SUMMARY Chapter Summary 161 The properties of equality allow us to write equivalent equations to solve an equation. 1. The addition property of equality: If equals are added to equals the sums are equal. 2. The subtraction property of equality: If equals are subtracted from equals the differences are equal. 3. The multiplication property of equality: If equals are multiplied by equals the products are equal. 4. The division property of equality: If equals are divided by nonzero equals the quotients are equal. 5. The substitution property: In any statement of equality, a quantity may be substituted for its equal. Before solving an equation, simplify each side if necessary. To solve an equation that has the variable in both sides, transform it into an equivalent equation in which the variable appears in only one side. Do this by adding the opposite of the variable term on one side to both sides of the equation. Use the properties of equality. Any equation or formula containing two or more variables can be transformed so that one variable is expressed in terms of all other variables. To do this, think of solving a simpler but related equation that contains only one variable. Order property of numbers: For real numbers x and y, one and only one of the following can be true: x y, x y, or x y. Transitive property of inequality: For real numbers x, y, and z, if x y and y z, then x z, and if x y and y z, then x z. |
Addition property of inequality: When the same number is added to or subtracted from both sides of an inequality, the order of the new inequality is the same as the order of the original one. Multiplication property of inequality: When both sides of an inequality are multiplied or divided by the same positive number, the order of the new inequality is the same as the order of the original one. When both sides of an inequality are multiplied or divided by the same negative number, the order of the new inequality is the opposite of the order of the original one. The domain or replacement set of an inequality is the set of all possible numbers that can be used to replace the variable. The solution set of an inequality is the set of numbers from the domain that make the inequality true. Inequalities that have the same solution set are equivalent inequalities. 162 Algebraic Expressions and Open Sentences VOCABULARY 4-1 Equation • Left side • Left member • Right side • Right member • Root • Solution • Solution set • Identity • Equivalent equations • Solve an equation • Conditional equation • Check 4-2 First degree equation in one variable • Like terms • Similar terms • Unlike terms 4-4 Perimeter • Distance formula 4-7 Order property of the real numbers • Transitive property of inequality • Addition property of inequality • Multiplication property of inequality 4-8 Domain of an inequality • Replacement set of an inequality • Solution set of an inequality • Equivalent inequalities • Interval notation REVIEW EXERCISES 1. Compare the properties of equality with the properties of inequality. Explain how they are alike and how they are different. In 2–9, solve for the variable and check. 2. 8w 60 4w 4. 4h 3 23 h 6. 8a (6a 5) 1 8. 3(4x 1) 2 17x 10 3. 8w 4w 60 5. 5y 3 2y 7. 2(b 4) 4(2b 1) 9. (x 2) (3x 2) x 3 In 10–15, solve each equation for x in terms of a, b, and c. 10. a x bc 11. cx a b c 14. ax b 12. bx a 5a c 15. ax 2b c 13. a1c 2 x b 16. a. Solve A 1 2bh for h in terms of A and b. b. Find h when A 5.4 and |
b 0.9. 17. If P 2l + 2w, find w when P 17 and l 5. 18. If F 9 5C 32, find C when F 68. In 19–26, find and graph the solution set of each inequality. 20. 2x 3 5 19. 6 x 3 23. 3 x 1 2 22. x 4 1 1 3x 21. 24. (x 2 5) and (2x 14) Review Exercises 163 25. (x 2) or (x 0) 26. (x 4 1) and (2x 18) In 27–30, tell whether each statement is sometimes, always, or never true. Justify your answer by stating a property of inequality or by giving a counterexample. 27. If x y, then a x a y. 29. If x y and y z, then x z. 28. If x y, then ax ay. 30. If x y, then x y. In 31–33, select the answer choice that correctly completes the statement or answers the question. 31. An inequality that is equivalent to 4x 3 5 is 1 2 (1) x 2 (2) x 2 (3) x (4) x 1 2 –1 0 1 2 3 4 The solution set of which inequality is shown in the graph above? (1) x 2 0 (3) x 2 0 (2) x 2 0 (4) x 2 0 32. 33. –5 –4 –3 –2 –1 0 1 2 The above graph shows the solution set of which inequality? (1) 4 x 1 (2) 4 x 1 (3) 4 x l (4) 4 x 1 34. The figure on the right consists of two squares and two isosceles right triangles. Express the area of the figure in terms of s, the length of one side of a square. 35. Express in terms of w the number of days in w weeks and 4 days. 36. The length of a rectangular room is 5 feet more than 3 times the width. The perimeter of the room is 62 feet. Find the dimensions of the room. 37. A truck must cross a bridge that can support a maximum weight of 24,000 pounds. The weight of the empty truck is 1,500 pounds, and the driver weighs 190 pounds. What is the weight of a load that the truck can carry? 38. In an apartment building there is one elevator, and |
the maximum load that it can carry is 2,000 pounds. The maintenance supervisor wants to move a replacement part for the air-conditioning unit to the roof. The part weighs 1,600 pounds, and the mechanized cart on which it is being moved weighs 250 pounds. When the maintenance supervisor drives the cart onto the elevator, the alarm sounds to signify that the elevator is overloaded. 164 Algebraic Expressions and Open Sentences a. How much does the maintenance supervisor weigh? b. How can the replacement part be delivered to the roof if the part can- not be disassembled? 39. A mail-order digital photo developer charges 8 cents for each print plus a $2.98 shipping fee. A local developer charges 15 cents for each print. How many digital prints must be ordered in order that: a. the local developer offers the lower price? b. the mail-order developer offers the lower price? 40. a. What is an appropriate replacement set for the problem in the chapter opener on page 116 of this chapter? b. Write and solve the equations suggested by this problem. c. Write the solution set for this problem. Exploration The figure at the right shows a circle inscribed in a square. Explain how pr2, (2r)2. this figure shows that CUMULATIVE REVIEW CHAPTERS 1–4 Part I Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. 1. The rational numbers are a subset of (1) the integers (2) the counting numbers (3) the whole numbers (4) the real numbers 2. If x 12.6 8.4 0.7x, then x equals (1) 0.07 (2) 0.7 (3) 7 (4) 70 3. The solution set of 2x 4 5x 14 is (1) 210 3 U V (2) 10 3 U V (3) {6} (4) {–6} 4. Which of the following inequalities is false? 2 (3) 0.6 3 2 (1) 0.6 3 2 3 # 0.6 (2) 2 (4) 0.6 3 Cumulative Review 165 5. Which of the following identities is an illustration of the commutative property of addition? (1) (x 3) 2 x (3 2) (2) x 3 3 x (3) 5(x 3) 5x 15 (4) x 0 |
x 6. Which of the following sets is closed under division? (1) nonzero whole numbers (2) nonzero integers (3) nonzero even integers (4) nonzero rational numbers 7. The measure of one side of a rectangle is 20.50 feet. This measure is given to how many significant digits? (1) 1 (2) 2 (3) 3 (4) 4 8. In the coordinate plane, the vertices of quadrilateral ABCD have the coor- dinates A(–2, 0), B(7, 0), C(7, 5), and D(0, 5). The quadrilateral is (1) a rhombus (2) a rectangle (3) a parallelogram (4) a trapezoid 9. One element of the solution set of (x 3) or (x 5) is (1) 4 (4) None of the above. The solution set is the empty set. (2) 2 (3) 5 10. When x 3, x2 is (1) 6 (2) 6 (3) 9 (4) 9 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 quadrilateral has four sides. Quadrilateral ABCD has three sides that have equal measures. The measure of the fourth side is 8.0 cm longer than each of the other sides. If the perimeter of the quadrilateral is 28.0 m, find the measure of each side using the correct number of significant digits. 12. To change degrees Fahrenheit, F, to degrees Celsius, C, subtract 32 from the Fahrenheit temperature and multiply the difference by five-ninths. a. Write an equation for C in terms of F. b. Normal body temperature is 98.6° Fahrenheit. What is normal body temperature in degrees Celsius? 166 Algebraic Expressions and Open Sentences 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. Is it possible for the remainder to be 2 when a prime number that |
is greater than 2 is divided by 4? Explain why or why not. 14. A plum and a pineapple cost the same as three peaches. Two plums cost the same as a peach. How many plums cost the same as a pineapple? 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. h 2(b11b2) 15. A trapezoid is a quadrilateral with only one pair of parallel sides called the bases of the trapezoid. The formula for the area of a trapezoid is A where h represents the measure of the altitude to the bases, and b1 and b2 represent the measures of the bases. Find the area of a trapezoid if h 5.25 cm, b1 9.50 cm. Express your answer 12.75 cm, and b2 to the number of significant digits determined by the given data. 16. Fred bought three shirts, each at the same price, and received less than $12.00 in change from a $50.00 bill. a. What is the minimum cost of one shirt? b. What is the maximum cost of one shirt? OPERATIONS WITH ALGEBRAIC EXPRESSIONS Marvin is planning two rectangular gardens that will have the same width. He wants one to be 5 feet longer than it is wide and the other to be 8 feet longer than it is wide. How can he express the area of each of the gardens and the total area of the two gardens in terms of w, the width of each? Problems like this often occur in many areas of business, science and technology as well as every day life. When we use variables and the rules for adding and multiplying expressions involving variables, we can often write general expressions that help us investigate many possibilities in the solution of a problem. In this chapter, you will learn to add, subtract, multiply, and divide algebraic expressions. CHAPTER 5 CHAPTER TABLE OF CONTENTS 5-1 Adding and Subtracting Algebraic Expressions 5-2 Multiplying Powers That Have the Same Base 5-3 Multiplying by a Monomial 5-4 Multiplying Polynomials 5-5 Dividing Powers That Have the Same Base 5-6 Powers With Zero and Negative Exponents 5-7 |
Scientific Notation 5-8 Dividing by a Monomial 5-9 Dividing by a Binomial Chapter Summary Vocabulary Review Exercises Cumulative Review 167 168 Operations with Algebraic Expressions 5-1 ADDING AND SUBTRACTING ALGEBRAIC EXPRESSIONS Recall that an algebraic expression that is a number, a variable, or a product or quotient of numbers and variables is called a term. Examples of terms are: 7 a 2b 24 7y2 0.7ab5 2 5 w Two or more terms that contain the same variable or variables with corresponding variables having the same exponents, are called like terms or similar terms. For example, the following pairs are like terms. 6k and k 5x2 and 7x2 9ab and 0.4ab 9 2x2y3 and 211 3 x2y3 Two terms are unlike terms when they contain different variables, or the same variable or variables with different exponents. For example, the following pairs are unlike terms. 3x and 4y 5x2 and 5x3 9ab and 0.4a 8 3x3y2 and 4 7x2y3 To add or subtract like terms, we use the distributive property of multipli- cation over addition or subtraction. 9x 2x (9 2)x 11x 16cd 3cd (16 3)cd 13cd 18y2 5y2 (18 5)y2 13y2 7ab ab 7ab 1ab (7 1)ab 6ab Since the distributive property is true for any number of terms, we can express the sum or difference of any number of like terms as a single term. 3ab2 4ab2 2ab2 (3 4 2)ab2 l ab2 ab2 x3 11x3 8x3 4x3 (1 11 8 4)x3 0x3 0 Recall that when like terms are added: 1. The sum or difference has the same variable or variables as the original terms. 2. The numerical coefficient of the sum or difference is the sum or difference of the numerical coefficients of the terms that were added. The sum of unlike terms cannot be expressed as a single term. For example, the sum of 2x and 3y cannot be written as a single term but is written 2x 3y. EXAMPLE 1 Add: a. 3a (8a) b. 12b2 (5b2) c |
. 15abc 6abc d. 8x2 y x2y e. 9y 9y f. 2(a b) 6(a b) EXAMPLE 2 Adding and Subtracting Algebraic Expressions 169 Answers [3 (8)]a 5a [12 (5)]b2 [12 5]b2 7b2 (15 6)abc 9abc (8 1)x2y 7x2y (9 9)y 0y 0 (2 6)(a b) 8(a b) An isosceles triangle has two sides that are equal in length. The length of each of the two equal sides of an isosceles triangle is twice the length of the third side of the triangle. If the length of the third side is represented by n, represent in simplest form the perimeter of the triangle. Solution n represents the length of the base. 2n represents the length of one of the equal sides. 2n represents the length of the other equal side. Perimeter n 2n 2n (1 2 2)n 5n. Note that the length of a side of a geometric figure is a positive number. Therefore, the variable n must represent a positive real number, that is, the replacement set for n must be the set of positive real numbers. Answer 5n Monomials and Polynomials A term that has no variable in the denominator is called a monomial. For example, 5, 5w, and 5 w A monomial or the sum of monomials is called a polynomial. A polynomial may have one or more terms. Some polynomials are given special names to indicate the number of terms. are monomials, but is not a monomial. 3w2 5 • A monomial such as 4x2 may be considered to be a polynomial of one term. (Mono- means “one”; poly- means “many.”) • A polynomial of two unlike terms, such as 10a 12b, is called a binomial. (Bi- means “two.”) 170 Operations with Algebraic Expressions • A polynomial of three unlike terms, such as x2 3x 2, is called a trinomial. (Tri- means “three.”) • A polynomial such as 5x2 (2x) (4) is usually written as 5x2 |
2x 4. A polynomial has been simplified or is in simplest form when it contains no like terms. For example, 5x3 8x2 5x3 7, when expressed in simplest form, becomes 8x2 7. A polynomial is said to be in descending order when the exponents of a particular variable decrease as we move from left to right. The polynomial x3 5x2 4x 9 is in a descending order of powers of x. A polynomial is said to be in ascending order when the exponents of a particular variable increase as we move from left to right. The polynomial 4 5y y2 is in an ascending order of powers of y. To add two polynomials, we use the commutative, associative, and distribu- tive properties to combine like terms. EXAMPLE 3 Solution Simplify: 3ab + 5b ab 4ab 2b How to Proceed (1) Write the expression: (2) Group like terms together by using the commutative and associative properties: (3) Use the distributive property: (4) Simplify the numerical expressions that are in parentheses: 3ab 5b ab 4ab 2b 3ab ab 4ab 5b 2b (3ab ab 4ab) (5b 2b) (3 1 4)ab (5 2)b 6ab 3b Answer 6ab 3b EXAMPLE 4 Solution Find the sum: (3x2 5) (6x2 8) How to Proceed (1) Write the expression: (2) Use the associative property: (3) Use the commutative property: (4) Use the associative property: (5) Add like terms: Answer 9x2 13 (3x2 5) (6x2 8) 3x2 (5 6x2) 8 3x2 (6x2 5) 8 (3x2 6x2) (5 8) 9x2 13 Adding and Subtracting Algebraic Expressions 171 The sum of polynomials can also be arranged vertically, placing like terms under one another. The sum of 3x2 5 and 6x2 8 can be arranged as shown at the right. Addition can be checked by substituting any convenient value for the variable and evaluating each polynomial and the sum. 3x2 5 6x2 8 9x2 13 Check Let x 4 3x2 |
5 3(4)2 5 53 6x2 8 6(4)2 8 104 9x2 13 9(4)2 13 157 ✔ EXAMPLE 5 Simplify: 6a [5a (6 3a)] Solution When one grouping symbol appears within another, first simplify the expres- sion within the innermost grouping symbol. How to Proceed (1) Write the expression (2) Use the commutative property: (3) Use the associative property: (4) Combine like terms: (5) Use the associative property: (6) Combine like terms: 6a [5a (6 3a)] 6a [5a (3a 6)] 6a [(5a 3a) 6] 6a [2a 6] (6a 2a) 6 8a 6 Answer 8a 6 EXAMPLE 6 Solution Express the difference (4x2 2x 3) (2x2 5x 3) in simplest form. How to Proceed (1) Write the subtraction problem: (2) To subtract, add the opposite (4x2 2x 3) (2x2 5x 3) (4x2 2x 3) (2x2 5x 3) of the polynomial to be subtracted: (3) Use the commutative and associative properties to group like terms: (4) Add like terms: Answer 2x2 7x (4x2 2x2) (2x 5x) (3 3) 2x2 7x 0 2x2 7x 172 Operations with Algebraic Expressions EXERCISES Writing About Mathematics 1. Christopher said that 3x x 3. a. Use the distributive property to show Christopher that his answer is incorrect. b. Substitute a numerical value of x to show Christopher that his answer is incorrect. 2. Explain how the procedure for adding like terms is similar to the procedure for adding frac- tions. Developing Skills In 327, write each algebraic expression in simplest form. 3. (8c) (7c) 5. (20r) (5r) 7. (5ab) (9ab) 9. 5y 6y 9y 14y 11. (8x2) (x2) (12x2) (2x2) 13. 7b (4b 6) 15. (6x 4) 5x 17. 8d2 (6d2 4d |
) 19. 9y [7 (6y 7)] 21. 5a [3b (2a 4b)] 23. 3y2 [6y2 (3y 4)] 25. d2 [9d (2 4d2)] 27. (x3 9x 5) (4x2 12x 5) 4. (4a) (6a) 6. (7w) (7w) 8. (6x) (4x) (5x) (10x) 10. 4m 9m 12m m 12. 4a (9a 3) 14. 8c (7 9c) 16. r (s 2r) 18. (5x 3) (6x 5) 20. (5 6y) (9y 2) 22. (5x2 4) (3x2 9) 24. (x3 3x2) (2x2 9) 26. (x2 5x 24) (x2 4x 9) In 28–31, state whether each expression is a monomial, a binomial, a trinomial, or none of these. 30. 2a2 3a 6 31. x3 2x2 x 7 28. 8x 3 29. 7y 32. a. Give an example of the sum of two binomials that is a binomial. b. Give an example of the sum of two binomials that is a monomial. c. Give an example of the sum of two binomials that is a trinomial. d. Give an example of the sum of two binomials that has four terms. e. Can the sum of two binomials have more than four terms? Multiplying Powers That Have the Same Base 173 Applying Skills In 33–41, write each answer as a polynomial in simplest form. 33. A cheeseburger costs 3 times as much as a soft drink, and an order of fries costs twice as much as a soft drink. If a soft drink costs s cents, express the total cost of a cheeseburger, an order of fries, and a soft drink in terms of s. 34. Jack deposited some money in his savings account in September. In October he deposited twice as much as in September, and in November he deposited one-half as much as in September. If x represents the amount of money deposited in September, represent, in terms of x, the total amount Jack |
deposited in the 3 months. 35. On Tuesday, Melita read 3 times as many pages as she read on Monday. On Wednesday she read 1.5 times as many pages as on Monday, and on Thursday she read half as many pages as on Monday. If Melita read p pages on Monday, represent in terms of p, the total number of pages she read in the 4 days. 36. The cost of 12 gallons of gas is represented by 12x, and the cost of a quart of oil is repre- sented by 2x 30. Represent the cost of 12 gallons of gas and a quart of oil. 37. In the last basketball game of the season, Tom scored 2x points, Tony scored x 5 points, Walt scored 3x 1 points, Dick scored 4x 7 points, and Dan scored 2x 2 points. Represent the total points scored by these five players. 38. Last week, Greg spent twice as much on bus fare as he did on lunch, and 3 dollars less on entertainment than he did on bus fare. If x represents the amount, in dollars, spent on lunch, express in terms of x the total amount Greg spent on lunch, bus fare, and entertainment. 39. The cost of a chocolate shake is 40 cents less than the cost of a hamburger. If h represents the cost, in cents, of a hamburger, represent in terms of h the cost of a hamburger and a chocolate shake in dollars. 40. Rosie spent 12 dollars more for fabric for a new dress than she did for buttons, and 1 dollar less for thread than she did for buttons. If b represents the cost, in dollars, of the buttons, represent in terms of b the total cost of the materials needed for the dress. 41. The length of a rectangle is 7z2 3 inches and the width is 9z2 2 inches. Represent the perimeter of the rectangle. 5-2 MULTIPLYING POWERS THAT HAVE THE SAME BASE Finding the Product of Powers We know that y2 means y y and y3 means y y y. Therefore, 2 3 5 y2 y3 (y y)(y y y) (y y y y y) y5 174 Operations with Algebraic Expressions Similarly, and 2 4 6 c2 c4 (c c)(c c c c) (c c c c c c) c6 1 4 3 x x3 (x)(x x x) (x x x x) x4 |
The exponent in each product is the sum of the exponents in the factors, as shown in these examples. In general, when x is a real number and a and b are positive integers: xa xb xa b EXAMPLE 1 Simplify each of the following products: a. x5 x2 b. a7 a c. 32 34 Answers a. x5 x2 x52 x7 b. a7 a a71 a8 c. 32 34 324 36 Note: When we multiply powers with like bases, we do not actually perform the operation of multiplication but rather count up the number of times that the base is to be used as a factor to find the product. In Example 1c above, the answer does not give the value of the product but indicates only the number of times that 3 must be used as a factor to obtain the product. We can use to evaluate the products 32 34 and 36 to show that they the power key are equal. ^ ENTER: 3 ^ 2 3 ^ 4 ENTER ENTER: 3 ^ 6 ENTER DISPLAY: 3 ^ 2 * 3 ^ 4 DISPLAY Finding a Power of a Power Since (x3)4 x3 x3 x3 x3, then (x3)4 x12. The exponent 12 can be obtained by addition: 3 3 3 3 12 or by multiplication: 4 3 12. In general, when x is a real number and a and c are positive integers: (xa)c xac Multiplying Powers That Have the Same Base 175 An expression such as (x5y2)3 can be simplified by using the commutative and associative properties: (x5y2)3 (x5y2)(x5y2)(x5y2) (x5 x5 x5)(y2 y2 y2) x15y6 When the base is the product of two or more factors, we apply the rule for the power of a power to each factor. (x5y2)3 (x5)3 (y2)3 x5(3)y2(3) x15y6 Thus, (xayb)c (xa)c(yb)c xacybc The expression (5 4)3 can be evaluated in two ways. (5 4)3 53 43 125 64 8,000 (5 4)3 203 8,000 EXAMPLE 2 Simplify each expression in two ways. a. (a2)3 b |
. (ab2)4 c. (32 42)3 Solution a. (a2)3 a2 a2 a2 a222 a6 or (a2)3 a2(3) a6 Answer a6 b. (ab2)4 ab2 ab2 ab2 ab2 (a a a a)(b2 b2 b2 b2) a4b8 or (ab2)4 a1(4)b2(4) a4b8 Answer a4b8 c. (32 42)3 (32)3 (42)3 36 46 (3 4)6 126 or (32 42)3 ((3 4)2)3 (122)3 126 Answer 126 176 Operations with Algebraic Expressions To evaluate the expression in Example 2c, use a calculator. Evaluate (32 42)3: Evaluate (12)6: ENTER: ( ^ 3 3 x2 4 x2 ) ENTER: 12 ^ 6 ENTER ENTER DISPLAY DISPLAY: 1 2 ^ 6 EXERCISES Writing About Mathematics 1. Does 53 53 253? Use the commutative and associative properties of multiplication to explain why or why not. 2. Does 24 4 26? Use the commutative and associative properties of multiplication to explain why or why not. Developing Skills In 3–26, multiply in each case. 3. a2 a3 7. z3 z3 z5 11. e4 e5 e 15. 43 4 19. (z3)2 (z4)2 23. (22 32)3 4. b3 b4 8. t8 t 4 t2 12. 23 22 16. 24 25 2 20. (x2y3)2 24. (5 23)4 5. r2 r4 r5 9. x x 13. 34 33 17. (x3)2 21. (ab2)4 25. (1002 103)5 6. r3 r3 10. a2 a 14. 52 54 18. (a4)2 22. (rs)3 26. (a2)5 a In 27–31, multiply in each case. (All exponents are positive integers.) 27. xa x2a 28. yc y2 29. cr c2 30. xm x 31. (3y)a (3y)b In 32–39, state whether each sentence is true or false. 32. 104 103 |
107 36. 33 22 66 33. 24 22 28 37. 54 5 55 34. 33 22 65 38. (22)3 25 35. 1480 1410 1490 39. (63)4 (64)3 Multiplying by a Monomial 177 Applying Skills 40. Two students attended the first meeting of the Chess Club. At that meeting, they decided that each person would bring one additional person to the next meeting, doubling the membership. At the second meeting, they again decided that each person would bring one additional person to the next meeting, again doubling the membership. If this plan was carried out for n meetings, the membership would equal 2n persons. a. How many persons attended the fifth meeting? b. At which meeting would the membership be twice as large as at the fifth meeting? 41. In the metric system, 1 meter 102 centimeters and 1 kilometer 103 meters. How many centimeters equal one kilometer? 5-3 MULTIPLYING BY A MONOMIAL Multiplying a Monomial by a Monomial We know that the commutative property of multiplication makes it possible to arrange the factors of a product in any order and that the associative property of multiplication makes it possible to group the factors in any combination. For example: (5x)(6y) (5)(6)(x)(y) (5 6)(x y) 30xy (3x)(7x) (3)(7)(x)(x) (3 7) (x x) 21x2 (2x2)(+5x4) (2)(x2)(+5)(x4) [(2)(+5)] [(x2)(x4)] 10x6 (3a2b3)(4a4b) (3)(a2) (b3)(4)(a4) (b) [(3)(4)][(a2)(a4)][(b3)(b)] 12a6b4 In the preceding examples, the factors may be rearranged and grouped mentally. Procedure To multiply a monomial by a monomial: 1. Use the commutative and associative properties to rearrange and group the fac- tors.This may be done mentally. 2. Multiply the numerical coefficients. 3. Multiply powers with the same base by adding exponents. 4. Multiply the products obtained in Steps 2 and 3 and any other variable factors by writing them with no sign between them. 178 Operations |
with Algebraic Expressions EXAMPLE 1 Multiply: Answers 24xyz a. (8xy)(3z) c. (6y3)(y) 6y4 e. (5x2y3)(2xy2) 10x3y5 g. (3x2)3 b. (4a3)(5a5) Answers 20a8 12a5b7 d. (3a2b3)(4a3b4) f. (6c2d4)(0.5d) 3c2d5 (3x2)(3x2)(3x2) 27x6 or (3)3(x2)3 27x6 EXAMPLE 2 Solution Represent the area of a rectangle whose length is 3x and whose width is 2x. How to Proceed (1) Write the area formula: (2) Substitute the values of l and w: (3) Perform the multiplication: A lw (3x)(2x) (3 2)(x x) 6x2 Answer 6x2 Multiplying a Polynomial by a Monomial The distributive property of multiplication over addition is used to multiply a polynomial by a monomial. Therefore, a(b c) ab ac x(4x 3) x(4x) x(3) 4x2 3x This result can be illustrated geometrically. Let us separate a rectangle, whose length is 4x 3 and whose width is x, into two smaller rectangles such that the length of one rectangle is 4x and the length of the other is 3. x 3 4x x(4x + 3) 4x + 3 = x (x)(4x) + x (x)(3) 4x 3 Multiplying by a Monomial 179 Since the area of the largest rectangle is equal to the sum of the areas of the two smaller rectangles: x(4x 3) x(4x) x(3) 4x2 3x Procedure To multiply a polynomial by monomial, use the distributive property: Multiply each term of the polynomial by the monomial and write the result as the sum of these products. Multiplication and Grouping Symbols When an algebraic expression involves grouping symbols such as parentheses, we follow the general order of operations and perform operations with algebraic terms. In the example at the right, first simplify the expression within parentheses |
: Next, multiply: Finally, combine like terms by addition 8y 2(7y 4y) 5 8y 2(3y) 5 8y 6y 5 2y 5 or subtraction: In many expressions, however, the terms within parentheses cannot be combined because they are unlike terms. When this happens, we use the distributive property to clear parentheses and then follow the order of multiplying before adding. Here, clear the parentheses by using the distributive property: Next, multiply: Finally, combine like terms by addition: 3 7(2x 3) 3 7(2x) 7(3) 3 14x 21 24 14x The multiplicative identity property states that a l a. By using this property, we can say that 5 (2x 3) 5 1(2x 3) and then follow the procedures shown above. 5 (2x 3) 5 1(2x 3) 5 1(2x) 1(3) 5 2x 3 2 2x Also, since a l a, we can use this property to simplify expressions in which a parentheses is preceded by a negative sign: 6y (9 7y) 6y 1(9 7y) 6y 1(9) 1(7y) 6y 9 7y 13y 9 180 Operations with Algebraic Expressions EXAMPLE 3 Multiply: a. 5(r 7) b. 8(3x 2y 4z) c. 5x(x2 2x 4) d. 3a2b2(4ab2 3b2) Answers 5r 35 24x 16y 32z 5x3 10x2 20x 12a3b4 9a2b4 EXAMPLE 4 Simplify: a. 5x(x2 2) 7x b. 3a (5 7a) a. How to Proceed (1) Write the expression: (2) Use the distributive property: 5x(x2) 5x(2) 7x (3) Multiply: (4) Add like terms: 5x3 10x 7x 5x3 3x 5x(x2 2) 7x Answer 5x3 3x b. How to Proceed (1) Write the expression: (2) Use the distributive property: (3) Add like terms: 3a (5 7a) 3a 1(5 7a) 3a 5 7a 10a 5 Answer 10a 5 EXERCIS |
ES Writing About Mathematics 1. In an algebraic term, how do you show the product of a constant times a variable or the product of different variables? 2. In the expression 2 3(7y), which operation is performed first? Explain your answer. 3. In the expression (2 3)(7y), which operation is performed first? Explain your answer. 4. In the expression 5y(y 3), which operation is performed first? Explain your answer. 5. Can the sum x2 x3 be written in simpler form? Explain your answer. 6. Can the product x2(x3) be written in simpler form? Explain your answer. Multiplying by a Monomial 181 Developing Skills In 7–29, find each product. A B A 18b 7. (4b)(6b) 10. (8r)(2r) 23 4a 13. 16. (7r)(5st) 19. (3s)(4s)(5t) 22. (20y3))(7y2) 25. (8y5)(5y) 28. (7a3b)(5a2b2) B 8. (5)(2y)(3y) 11. (7x)(2y)(3z) A B A 1 2y 26x 21 3z 14. B A 17. (2)(6cd)(e) 20. (5a2)(4a2) 23. (18r5)(5r2) 26. (9z)(8z4)(z3) 29. (4ab2)(2a2b3) B 9. (4a)(5b) 12. (6x)(0.5y) 15. (5ab)(3c) 18. (9xy)(2x) 21. (6x4)(3x3) 24. (3z2)(4z) 27. (6x2y3)(4x4y2) 30. 3(6c 3d) 33. 10(2x 0.2y) 4c 2 5 3 8d In 30–47, write each product as a polynomial. 31. 5(4m 6n) 2 3m 2 4n 34. A 37. 4x(5x 6) 40. mn(m n) 43. r3s3(2r4s 3s4) 46. 3xy(x2 x |
y y2) 36. B A 39. 5c2(15c 4c) 42. 3ab(5a2 7b2) 45. 8(2x2 3x 5) 216 12 B 32. 2(8a 6b) B 28 4r 2 1 4s 35. A 38. 5d(d2 3d) 41. ab(a b) 44. 10d(2a 3c 4b) 47. 5r2s2(2r2 3rs 4s2) In 48–50, represent the area of each rectangle whose length l and width w are given. 48. l 5y, w 3y 51. The dimensions of the outer rectangle pictured at the right are 4x by 3x 6. The dimen- 50. l 3c, w 8c 2 49. l 3x, w 5y sions of the inner rectangle are 2x by x 2. a. Express the area of the outer rectangle in terms of x. 3x – 6 b. Express the area of the inner rectangle in terms of x. c. Express as a polynomial in simplest form the area of the 4x x + 2 2x shaded region. In 52–73, simplify each expression. 52. 5(d 3) 10 55. 2(x 1) 6 53. 3(2 3c) 5c 56. 4(3 6a 8s) 54. 7 2(7x 5) 57. 5 4(3e 5) 182 Operations with Algebraic Expressions 58. 8 (4e 2) 61. 9 (5t 6) 59. a (b a) 62. 4 (2 8s) 60. (6b 4) 2b 63. (6x 7) 14 64. 5x(2x 3) 9x 65. 12y 3y(2y 4) 66. 7x 3(2x 1) 8 67. 7c 4d 2(4c 3d) 68. 3a 2a(5a a) a2 69. (a 3b) (a 3b) 70. 4(2x 5) 3(2 7x) 71. 3(x y) 2(x 3y) 72. 5x(2 3x) x(3x 1) 73. y(y 4) y(y 3) 9y Applying Skills In 74–86, write each answer |
as a polynomial in simplest form. 74. If 1 pound of grass seed costs 25x cents, represent in terms of x the cost of 7 pounds of seed. 75. If a bus travels at the rate of 10z miles per hour for 4 hours, represent in terms of z the dis- tance traveled. 76. If Lois has 2n nickels, represent in terms of n the number of cents she has. 77. If the cost of a notebook is 2x 3, express the cost of five notebooks. 78. If the length of a rectangle is 5y 7 and the width is 3y, represent the area of the rectangle. 79. If the measure of the base of a triangle is 3b 2 and the height is 4b, represent the area of the triangle. 80. Represent the distance traveled in 3 hours by a car traveling at 3x 7 miles per hour. 81. Represent in terms of x and y the amount saved in 3y weeks if x 2 dollars are saved each week. 82. The length of a rectangular skating rink is 2 less than 3 times the width. If w represents the width of the rink, represent the area in terms of w. 83. An internet bookshop lists used books for 3x 5 dollars each. The cost for shipping and handling is 2 dollars for five books or fewer. Represent the total cost of an order for four used books. 84. A store advertises skirts for x 5 dollars and allows an additional 10-dollar reduction on the total purchase if three or more skirts are bought. Represent the cost of five skirts. 85. A store advertises skirts for x 5 dollars and allows an additional two-dollar reduction on each skirt if three or more skirts are purchased. Represent the cost of five skirts. 86. A store advertises skirts for x 5 dollars and tops for 2x 3 dollars. Represent the cost of two skirts and three tops. Multiplying Polynomials 183 5-4 MULTIPLYING POLYNOMIALS As discussed in Section 5-3, to find the product (x 4)(a), we use the distributive property of multiplication over addition: (x 4)(a) x(a) 4(a) Now, let us use this property to find the product of two binomials, for example, (x 4)(x 3). (a) x (a) 4 (a) (x 4) (x 4)(x 3) x(x 3) 4 |
(x 3) x2 3x 4x 12 x2 7x 12 This result can also be illustrated geometricallyx + 4)(x + 3) = 3 + x x(x + 3) 4(x + 3) = 3 + x x 3x x2 4 12 4x 3 x (x + 4)(x + 3) x(x + 3) + 4(x + 3) x2 + 3x + 4x + 12 In general, for all a, b, c, and d: (a b)(c d) a(c d) b(c d) ac ad bc bd Notice that each term of the first polynomial multiplies each term of the second. At the right is a convenient vertical arrangement of the preceding multiplication, similar to the arrangement used in arithmetic multiplication. Note that multiplication is done from left to right. x 3 x 4 x(x 3) → x2 3x 4(x 3) → 4x 12 x2 7x 12 The word FOIL serves as a convenient way to remember the steps neces- sary to multiply two binomials. Last First Outside (2x 5)(x 4) 2x(x) 2x(4) 5(x) 5(4) Inside ➤ ➤ ➤ ➤ 2x2 8x 5x 20 2x2 3x 20 184 Operations with Algebraic Expressions Procedure To multiply a polynomial by a polynomial, first arrange each polynomial in descending or ascending powers of the same variable.Then use the distributive property: multiply each term of the first polynomial by each term of the other. EXAMPLE 1 Solution Simplify: (3x 4)(4x 5) METHOD 1 ➤ ➤ (3x 4)(4x 5) 3x(4x 5) 4(x 5) 12x2 15x 16x 20 12x2 x 20 ➤ ➤ Answer 12x2 x 20 EXAMPLE 2 Solution Simplify: (x2 3xy 9y2)(x 3y) METHOD 2 3x 4 4x 5 12x2 16x 15x 20 12x2 x 20 ➤ ➤ ➤ ➤ (x2 3xy 9y2)(x 3y) x2(x 3y) 3xy(x 3y) 9y2(x |
3y) x3 3x2y 3x2y 9xy2 9xy2 27y3 x3 0x2y 0xy2 27y3 ➤ ➤ Answer x3 27y3 EXAMPLE 3 Solution Simplify: (2x 5)2 (x 3) (2x 5)2 (x 3) (2x 5)(2x 5) (x 3) 2x(2x) 2x(5) 5(2x) 5(5) (x) (+3) 4x2 10x 10x 25 x 3 4x2 21x 28 Answer 4x2 21x 28 Multiplying Polynomials 185 EXERCISES Writing About Mathematics 1. The product of two binomials in simplest form can have four terms, three terms, or two terms. a. When does the product of two binomials have four terms? b. When is the product of two binomials a trinomial? c. When is the product of two binomials a binomial? 2. Burt wrote (a 3)2 as a2 9. Prove to Burt that he is incorrect. Developing Skills In 3–35, write each product as a polynomial. 3. (a 2)(a 3) 6. (x 7)(x 2) 9. (b 8)(b 10) 12. (12 r)(6 r) 15. (5a 9)(5a 9) 18. (2x 3)(2x 3) 21. (a b)(a b) 24. (x 4y)(x 4y) 27. (r2 5)(r2 2) 30. (2c 1)(2c2 3c 1) 33. (x 4)(x 4)(x 4) 4. (x 5)(x 3) 7. (m 3)(m 7) 10. (6 y)(5 y) 13. (x 5)(x 5) 16. (2x 1)(x 6) 19. (3d 8)(3d 8) 22. (a + b)(a b) 25. (x 4y)2 28. (x2 y2)(x2 y2) 31. (3 2a a2)(5 2a) 34. (a 5)3 5. (d 9)(d 3) 8. (t 15)(t 6) 11. (8 e)(6 e |
) 14. (2y 7)(2y 7) 17. (5y 2)(3y 1) 20. (x y)(x y) 23. (a b)2 26. (9x 5y)(2x 3y) 29. (x 2)(x2 3x 5) 32. (2x 1)(3x 4)(x 3) 35. (x y)3 In 36–43, simplify each expression. 36. (x 7)(x 2) x2 38. r(r 2) (r 5) 40. (x 4)(x 3) (x 2)(x 5) 42. (y 4)2 (y 3)2 37. 2(3x 1)(2x 3) 14x 39. 8x2 (4x 3)(2x 1) 41. (3y 5)(2y 3) (y 7)(5y 1) 43. a[(a 2)(a 2) 4] Applying Skills In 44–46, use grouping symbols to write an algebraic expression that represents the answer. Then, express each answer as a polynomial in simplest form. 44. The length of a rectangle is 2x 5 and its width is x 7. Express the area of the rectangle as a trinomial. 186 Operations with Algebraic Expressions 45. The dimensions of a rectangle are represented by 11x 8 and 3x 5. Represent the area of the rectangle as a trinomial. 46. A train travels at a rate of (15x 100) kilometers per hours. a. Represent the distance it can travel in (x 3) hours as a trinomial. b. If x 2, how fast does the train travel? c. If x 2, how far does it travel in (x 3) hours? 5-5 DIVIDING POWERS THAT HAVE THE SAME BASE We know that any nonzero number divided by itself is 1. Therefore, x x 1 and y3 y3 1. In general, when x 0 and a is a positive integer: Therefore, xa xa 1 x5 x3 5 x2? x3 x3 y5? y4 y9 y4 5 y4 c5 c 5 c4? c c x2 1 x2 y5 1 y5 c4 1 c4 These same results can be obtained by using the relationship between divi- sion and multiplication: If a b c, then c |
b a. • Since x2 x3 x5, then x5 x3 x2. • Since y5 y4 y9, then y9 y4 y5. • Since c4 c c5, then c5 c1 c4. Observe that the exponent in each quotient is the difference between the exponent of the dividend and the exponent of the divisor. In general, when x 0 and a and b are positive integers with a b: xa xb xab Procedure To divide powers of the same base, find the exponent of the quotient by subtracting the exponent of the divisor from the exponent of the dividend.The base of the quotient is the same as the base of the dividend and of the divisor. Dividing Powers That Have the Same Base 187 EXAMPLE 1 Simplify by performing each indicated division. b. y5 y a. x9 x5 c. c5 c5 d. 105 103 Answers a. x95 x4 b. y51 y4 c. 1 d. 1053 102 EXAMPLE 2 Write 57? 54 58 in simplest form: a. by using the rules for multiplying and dividing powers with like bases. b. by using a calculator. Solution a. First simplify the numerator. Then, apply the rule for division of powers with the same base. 57 b. On a calculator:? 54 58 5 5714 58 5 511 58 5 511–8 5 53 5 125 ENTER ENTER DISPLAY Answer 125 EXERCISES Writing About Mathematics 1. Coretta said that 54 5 14. Do you agree with Coretta? Explain why or why not. 2. To evaluate the expression, 38 35? 32 a. in what order should the operations be performed? Explain your answer. b. does 38 35? 32 38 35 32? Explain why or why not. Developing Skills In 3–18, divide in each case. 3. x8 x2 7. e9 e3 4. a10 a5 5. c5 c4 8. m12 m4 9. n10 n9 6. x7 x7 10. r6 r6 188 Operations with Algebraic Expressions 11. x8 x 15. 106 104 12. z10 z 16. 34 32 13. t5 t 17. 53 5 14. 25 22 18. 104 10 In 19–24, divide in each case. (All exponents are positive integers.) 19. x |
5a x2a 22. sx s2 (x 2) 20. y10b y2b 23. ab ab 21. rc rd (c d) 24. 2a 2b (a b) In 25–32: a. Simplify each expression by using the rules for multiplying and dividing powers with like bases. b. Evaluate the expression using a calculator. Compare your answers to parts a and b. 25. 29. 23? 24 22 106 102? 104 26. 30. 58 54? 5 108? 102 (105)2 27. 31. 102? 103 104 64? 69 62? 63 28. 32. 33? 32 32 45? 45 (42)4 In 33–35, tell whether each sentence is true or false. 33. 10099 1098 1002 34. a6 a2 a4 (a 0) 35. 45045 45040 15 5-6 POWERS WITH ZERO AND NEGATIVE EXPONENTS Integers that are negative or zero, such as 0, 1, and 2 can also be used as exponents. We will define powers having zero and negative integral exponents in such a way that the properties that were valid for positive integral exponents will also be valid for zero and negative integral exponents. In other words, the following properties will be true when the exponents a and b are positive integers, negative integers, or 0: xa xb xa b xa xb xa b (xa)b xab The Zero Exponent x3 We know that, for x 0, 1. If x3 ment, we must let x0 1 since x0 and 1 are each equal to lowing definition: x3 5 x3 – 3 5 x0 x3 x3 x3 is to be a meaningful state-. This leads to the fol- DEFINITION x0 1 if x is a number such that x 0. It can be shown that all the laws of exponents remain valid when x0 is defined as 1. For example: • Using the definition 100 1, we have 103 100 103 1 103 Powers with Zero and Negative Exponents 189 • Using the law of exponents, we have 103 100 1030 103. The two procedures result in the same product. • Using the definition 100 1, we have 103 100 103 1 103 • Using the law of exponents, we have 103 100 1030 103. The two procedures result in the same quotient. The definition x0 |
1 (x 0) permits us to say that the zero power of any number except 0 equals 1. 40 1 (4)0 1 (4x)0 1 (4x)0 1 A calculator will return this value. For example, to evaluate 40: ENTER: 4 ^ 0 ENTER DISPLAY: 4 ^ 0 1 Note that 4x0 41 x0 4 1 4 but (4x)0 40 x0 1 1 1. The Negative Integral Exponent We know that, for x 0, x3 x5 5 1? x3? x3 5 1 x2 x2? is to be a meaningful statement, we must let x2? 1 5 1 x2 x3 5 1 x3. x22 5 1 x2 since x2 x3 If x5 5 x3–5 5 x22 1 x2 x3 x5 and are each equal to. This leads to the following definition: DEFINITION xn if x 0. 1 xn A graphing calculator will return equal values for x2 and. For example, 1 x2 let x 5. Evaluate 52. ENTER: 5 ^ (-) 2 ENTER DISPLAY: 5 ^ - 2 Evaluate ENTER: 1. 1 52 5 ^ 2 ENTER DISPLAY 190 Operations with Algebraic Expressions as It can be shown that all the laws of exponents remain valid if xn is defined 1. For example: xn 24 5 22 • Using the definition 24, we have 22 24 • Using the law of exponents, we have 22 24 22(4) 22. 22 5 222. 24 5 1 1 24? 22 1 The two procedures give the same result. Now we can say that, for all integral values of a and b, xa xb 5 xa2b (x 2 0) EXAMPLE 1 Transform each given expression into an equivalent one with a positive exponent. Answers a. 43 b. 101 c. d. 1 225 5 3 A B 22 10 1 43 1 101 5 1 1 4 225 5 1 4 1 32 5 1 52 4 1 322 5 1 522 25 5 1 3 25 52 3 32 1 5 32 1 5 25 52 5 2 3 5 B A EXAMPLE 2 Compute the value of each expression. a. 30 b. 102 c. (5)0 24 d. 6(33) EXAMPLE 3 Answers 1 1 100 102 5 1 11 1 24 5 1 1 1 1 5 6 33 1 27 6 A |
B A B 16 5 1 1 16 5 6 27 5 2 9 Use the laws of exponents to perform the indicated operations. Answers Answers a. 27 23 27(3) 24 b. 36 32 36(2) 362 34 c. (x4)3 x4(3) x12 d. (y2)4 y2(4) y8 Scientific Notation 191 EXERCISES Writing About Mathematics 1. Sasha said that for all x 0, x2 is a positive number less than 1. Do you agree with Sasha? Explain why or why not. 2. Brandon said that, when n is a whole number, the number 10n when written in ordinary dec- imal notation uses n 1 digits. Do you agree with Brandon? Explain why or why not. Developing Skills In 3–7, transform each given expression into an equivalent expression involving a positive exponent. 3. 104 4. 21 5. 22 6. m6, m 0 7. r3, r 0 2 3 A B In 8–19, compute each value using the definitions of zero and negative exponents. Compare your answers with the results obtained using a calculator. 8. 100 12. 101 9. (4)0 13. 102 10. 40 14. 103 16. 1.5(10)3 17. 70 62 18. 0 1323 1 2 A B 11. 32 15. 4(10)2 19. 2 41 In 20–27, use the laws of exponents to perform each indicated operation. 21. 34 32 25. (41)2 20. 102 105 24. 34 30 28. Find the value of 7x0 (6x)0, (x 0). 29. Find the value of 5x0 2x1 when x 4. 22. 103 105 26. (33)2 23. (42)2 44 27. 20 25 5-7 SCIENTIFIC NOTATION Scientists and mathematicians often work with numbers that are very large or very small. In order to write and compute with such numbers more easily, these workers use scientific notation. A number is expressed in scientific notation when it is written as the product of two quantities: the first is a number greater than or equal to 1 but less than 10, and the second is a power of 10. In other words, a number is in scientific notation when it is written as where 1 a 10 and n is an integer. a 10n 192 Operations with Algebraic Expressions Writing Numbers in |
Scientific Notation To write a number in scientific notation, first write it as the product of a number between 1 and 10 times a power of 10. Then express the power of 10 in exponential form. The table at the right shows some integral powers of 10. When the exponent is a positive integer, the power can be written as 1 followed by the number of 0’s equal to the exponent of 10. When the exponent is a negative integer, the power can be written as a decimal value with the number of decimal places equal to the absolute value of the exponent of 10. 3,000,000 3 1,000,000 3 106 780 7.8 100 7.8 102 3 3 1 3 100 0.025 2.5 0.01 2.5 102 0.0003 3 0.0001 3 104 Powers of 10 105 100,000 104 10,000 103 1,000 102 100 101 10 100 1 101 102 103 104 101 5 1 102 5 1 103 5 1 104 5 1 10 100 1 1 1 1 0.1 0.01 0.001 0.0001 1,000 10,000 When writing a number in scientific notation, keep in mind the following: • A number equal to or greater than 10 has a positive exponent of 10. • A number equal to or greater than 1 but less than 10 has a zero exponent of 10. • A number between 0 and 1 has a negative exponent of 10. EXAMPLE 1 The distance from the earth to the sun is approximately 93,000,000 miles. Write this number in scientific notation. Solution How to Proceed (1) Write the number, placing a decimal point after 93,000,000. the last digit. (2) Place a caret (^) after the first nonzero digit so that replacing the caret with a decimal point will give a number between 1 and 10. (3) Count the number of digits between the caret and the decimal point. This is the exponent of 10 in scientific notation. The exponent is positive because the given number is greater than 10. 9 3,000,000. ^ 9 3,000,000. ^ 7 (4) Write the number in the form a 10n, where where a is found by replacing the caret with a decimal point and n is the exponent found in Step 3. Answer 9.3 107 Scientific Notation 193 9.3 107 EXAMPLE 2 Express 0.0000029 in scientific notation. Solution Since the number is between 0 and 1, |
the exponent will be negative. Place a caret after the first nonzero digit to indicate the position of the dec- imal point in scientific notation. Answer 0.0000029 0.000002 9 2.9 106 ^ 6 Graphing calculators can be placed in scientific notation mode and will return the results shown in Examples 1 and 2 when the given numbers are entered. ENTER: MODE ENTER CLEAR.0000029 ENTER DISPLAY This display is read as 2.9 106, where the integer following “E” is the exponent to the base 10 used to write the number in scientific notation. Changing to Ordinary Decimal Notation We can change a number that is written in scientific notation to ordinary decimal notation by expanding the power of 10 and then multiplying the result by the number between 1 and 10. 194 Operations with Algebraic Expressions EXAMPLE 3 The approximate population of the United States is 2.81 108. Find the approximate number of people in the United States. Solution How to Proceed (1) Evaluate the second factor, which 2.81 108 2.81 100,000,000 is a power of 10: (2) Multiply the factors: 2.81 108 281,000,000 Answer 281,000,000 people Note: We could have multiplied 2.81 by 108 quickly by moving the decimal point in 2.81 eight places to the right. EXAMPLE 4 The diameter of a red blood corpuscle is expressed in scientific notation as 7.5 104 centimeters. Write the number of centimeters in the diameter as a decimal fraction. Solution How to Proceed (1) Evaluate the second factor, which 7.5 104 7.5 0.0001 is a power of 10: (2) Multiply the factors: 7.5 104 0.00075 Answer 0.00075 cm Note: We could have multiplied 7.5 by 104 quickly by moving the decimal point in 7.5 four places to the left. EXAMPLE 5 Calculator Solution Use a calculator to find the product: 45,000 570,000. ENTER: 45000 570000 ENTER DISPLAY calculator will shift to scientific notation when the number is too large or too small for the display. The number in this display can be changed to decimal notation by using the procedure shown in Examples 3 and 4. Answer 2.565 1010 2.565 10,000,000,000 25,650,000,000 Scientific Notation 195 EXAMPLE 6 Use a calculator to find |
the mass of 2.70 1015 hydrogen atoms if the mass of one hydrogen atom is 1.67 1024 grams. Round the answer to three significant digits. Solution Multiply the mass of one hydrogen atom by the number of hydrogen atoms. (1.67 1024) (2.70 1015) (1.67 2.70) (1024 1015) (1.67 2.70) (1024 15) 4.509 109 Round 4.509 to 4.51, which has three significant digits. Answer 4.51 109 4.51 0.000000001 0.00000000451 grams Calculator Solution Use a calculator to multiply the mass of one hydrogen atom by the number of hydrogen atoms. Enter the numbers in scientific notation. ENTER: 1.67 2nd EE (-) 24 2.7 2nd EE 15 ENTER DISPLAY Round 4.509 to three significant digits. Answer 4.51 109 4.51 0.000000001 0.00000000451 grams EXERCISES Writing About Mathematics 1. Jared said that when a number is in scientific notation, a 10n, the number of digits in a is the number of significant digits. Do you agree with Jared? Explain why or why not. 2. When Corey wanted to enter 2.54 105 into his calculator, he used this sequence of keys: 2.54 or why not. 2nd EE 5 ENTER. Is this a correct way to enter the number? Explain why 196 Operations with Algebraic Expressions Developing Skills In 3–8, write each number as a power of 10. 3. 100 6. 0.0001 4. 10,000 7. 1,000,000,000 5. 0.01 8. 0.0000001 In 9–20, find the number that is expressed by each numeral. 11. 103 15. 6 101 19. 1.27 103 10. 1010 14. 4 108 18. 8.3 1010 9. 107 13. 3 105 17. 1.3 104 12. 105 16. 9 107 20. 6.14 102 In 21–32, find the value of n that will make each resulting statement true. 21. 120 1.2 10n 24. 0.00161 1.61 10n 27. 0.00000000375 3.75 10n 30. 2.54 2.54 10n 22. 9,300 9.3 10n 25. 0.0000760 7.60 10n 28. 872,000,000 8.72 |
10n 31. 0.00456 4.56 10n 23. 5,280 5.28 10n 26. 52,000 5.2 10n 29. 0.800 8.00 10n 32. 7,123,000 7.123 10n In 33–44, express each number in scientific notation. 33. 8,400 37. 0.00061 41. 453,000 34. 27,000 38. 0.0000039 42. 0.00381 35. 54,000,000 36. 320,000,000 39. 0.0000000140 40. 0.156 43. 375,000,000 44. 0.0000763 In 45–48, compute the result of each operation. Using the correct number of significant digits: a. write the result in scientific notation, b. write the result in ordinary decimal notation. 45. (2.9 103)(3.0 103) 47. (7.50 104) (2.5 103) 46. (2.55 102)(3.00 103) 48. (6.80 105) (3.40 108) Applying Skills In 49–52, express each number in scientific notation. 49. A light-year, which is the distance light travels in 1 year, is approximately 9,500,000,000,000 kilometers. 50. A star that is about 12,000,000,000,000,000,000,000 miles away can be seen by the Palomar telescope. 51. The radius of an electron is about 0.0000000000005 centimeters. 52. The diameter of some white blood corpuscles is approximately 0.0008 inches. Dividing by a Monomial 197 In 53–57, express each number in ordinary decimal notation. 53. The diameter of the universe is 2 109 light-years. 54. The distance from the earth to the moon is 2.4 105 miles. 55. In a motion-picture film, the image of each picture remains on the screen approximately 6 102 seconds. 56. Light takes about 2 108 seconds to cross a room. 57. The mass of the earth is approximately 5.9 1024 kilograms. 5-8 DIVIDING BY A MONOMIAL Dividing a Monomial by a Monomial We know that a b? c We can rewrite this equality interchanging the left and right members. bd 5 a ac d 5 ac bd b? c d Using this relationship, we can |
write: 2x4 5 230 2 a5 5 221 a4?? 23 y2 3y0z1 3 1 z 3z y2? 7a1b3 7ab3 221a5b4 23a4b 15x2 12y2z2 4y2z x6 x4 b4 b 230x6?? 5 12 z2 4 z Procedure To divide a monomial by a monomial: 1. Divide the numerical coefficients. 2. When variable factors are powers of the same base, divide by subtracting exponents. 3. Multiply the quotients from steps 1 and 2. If the area of a rectangle is 42 and its length is 6, we can find its width by dividing the area, 42, by the length, 6. Thus, 42 6 7, which is the width. 198 Operations with Algebraic Expressions Similarly, if the area of a rectangle is represented by 42x2 and its length by 6x, we can find its width by dividing the area, 42x2, by the length, 6x: Therefore, the width can be represented by 7x. 42x2 6x 7x EXAMPLE 1 Divide: Answers a. b. c. 24a5 23a2 218x3y2 26x2y 20a3c4d2 25a3c3 EXAMPLE 2 24 23 218 26 20 25??? a5 a2 5 28a3 x3 y2 x2? y c4 a3 a3? c3? d2 3xy 4(1)cd2 4cd2 The area of a rectangle is 24x4y3. Express, in terms of x and y, the length of the rectangle if the width is 3xy2. Solution The length of a rectangle can be found by dividing the area by the width. 24x4y3 3xy2 5 8x3y Answer Dividing a Polynomial by a Monomial We know that to divide by a number is the same as to multiply by its reciprocal. Therefore, a 1 c b 5 1 b(a 1 c) 5 a b 1 c b Similarly, and 2x 1 2y 2 5 1 2(2x 1 2y) 5 2x 2 1 2y 2 5 x 1 y 21a2b 2 3ab 3ab 5 1 3ab(21a2b 2 3ab) 5 21a2b 3ab 2 3ab 3 |
ab 5 7a 2 1 Usually, the two middle steps are done mentally. Procedure To divide a polynomial by a monomial, divide each term of the polynomial by the monomial. Dividing by a Monomial 199 EXAMPLE 3 Divide: a. (8a5 6a4) 2a2 b. 24x3y4 2 18x2y2 2 6xy 26xy Answers 4a3 3a2 4x2y3 3xy 1 EXERCISES Writing About Mathematics 1. Mikhail divided (12ab2 6ab) by (6ab) and got 2b for his answer. Explain to Mikhail why his answer is incorrect. 2. Angelique divided (15cd 11c) by 5c and got (3d 2.2) as her answer. Do you agree with Angelique? Explain why or why not. Developing Skills In 3–26, divide in each case. 3. 14x2y2 7 4. 36y10 6y2 7. 249c4b3 7c2b2 11. (14x 7) 7 15. p 1 prt p 19. 9y9 2 6y6 23y3 23. 22a2 2 3a 1 1 21 8. 12. 224x2y 23xy cm 1 cn c 16. y2 2 5y 2y 20. 24. 8a3 2 4a2 24a2 2.4y5 1 1.2y4 2 0.6y3 20.6y3 Applying Skills 5. 9. 13. 18x6 2x2 256abc 8abc tr 2 r r 17. 18d3 1 12d2 6d 21. 25. 3ab2 2 4a2b ab a3 2 2a2 0.5a2 6. 10. 14. 18. 22. 26. 5x2y3 25y3 227xyz 9xz 8c2 2 12d2 24 18r5 1 12r3 6r2 4c2d 2 12cd2 4cd 1.6cd 2 4.0c2d 0.8cd 27. If five oranges cost 15y cents, represent the average cost of one orange. 28. If the area of a triangle is 32ab and the base is 8a, represent the height of the triangle. 29. If a train traveled 54r miles in 9 hours, represent the average distance traveled in 1 hour |
. 30. If 40ab chairs are arranged in 5a rows with equal numbers of chairs in each row, represent the number of chairs in one row. 200 Operations with Algebraic Expressions 5-9 DIVIDING BY A BINOMIAL When we divide 736 by 32, we use repeated subtraction of multiples of 32 to determine how many times 32 is contained in 736. To divide a polynomial by a binomial, we will use a similar procedure to divide x2 6x 8 by x 2. How to Proceed (1) Write the usual division form: (2) Divide the first term of the dividend by the first term of the divisor to obtain the first term of the quotient: (3) Multiply the whole divisor by the first term of the quotient. Write each term of the product under the like term of the dividend: (4) Subtract and bring down the next term of the dividend to obtain a new dividend: (5) Divide the first term of the new dividend by the first term of the divisor to obtain the next term of the quotient: (6) Repeat steps (3) and (4), multiplying the whole divisor by the new term of the quotient. Subtract this product from the new dividend. Here the remainder is zero and the division is complete: x 1 2qx2 1 6x 1 8 x x 1 2qx2 1 6x 1 8 x x 1 2qx2 1 6x 1 8 x2 2x x x 1 2qx2 1 6x 1 8 x2 2x 4x 8 x 4 x 1 2qx2 1 6x 1 8 x2 2x 4x 8 x 4 x 1 2qx2 1 6x 1 8 x2 2x 4x 8 4x 8 0 The division can be checked by multiplying the quotient by the divisor to obtain the dividend: (x 4)(x 2) x(x 2) 4(x 2) x2 2x 4x 8 x2 6x 8 EXAMPLE 1 Divide 5s 6s2 6 by 2s 3 and check. Solution First arrange the terms of the dividend in descending order: 6s2 5s 6 Dividing by a Binomial 201 3s 2 2s 1 3q6s2 1 5s 2 6 6s2 9s (-) 4s 6 4s 6 0 Check ( |
3s 2)(2s 3) 3s(2s 3) 2(2s 3) 6s2 9s 4s 6 6s2 5s 6 ✔ Note that we subtracted 9s from 5s by adding 9s to 5s. Answer 3s 2 EXERCISES Writing about Mathematics 1. Nate said that 2. Mason wrote x3 1 as x3 0x2 0x 1 before dividing by x 1. 1 5 x2 2 1 x 1 1 5 x3 x3 2 1 x 1 21. Is Nate correct? Explain why or why not. a. Does x3 1 x3 0x2 0x 1? b. Divide x3 1 by x 1 by writing x3 0x2 0x 1 as the dividend. Check your answer to show that your computation is correct. Developing Skills In 3–14, divide and check. 3. (b2 5b 6) (b 3) 4. (y2 3y 2) (y 2) 5. (m2 8m 7) (m 1) 6. w2 1 2w 2 15 w 1 5 7. y2 1 21y 1 68 y 1 17 8. 9. (3a2 8a 4) (3a 2) 10. (15t2 19t 56) (5t 7) 11. x2 1 7x 1 10 x 1 5 10y2 2 y 2 24 2y 1 3 8 – 22c 1 12c2 4c 2 2 13. (17x 66 x2) (x 6) 12. 15. One factor of x2 4x 21 is x 7. Find the other factor. 14. x2 2 64 x 2 8 Applying Skills 16. The area of a rectangle is represented by x2 8x 9. If its length is represented by x 1, how can the width be represented? 17. The area of a rectangle is represented by 3y2 8y 4. If its length is represented by 3y 2, how can the width be represented? 202 Operations with Algebraic Expressions CHAPTER SUMMARY Two or more terms that contain the same variable, with corresponding variables having the same exponents, are called like terms. The sum of like terms is the sum of the coefficients of the terms times the common variable factor of the terms. A term that has no variable in the denominator is called a monomial. A polynomial is the sum of monom |
ials. To subtract one polynomial from another, add the opposite of the polynomial to be subtracted (the subtrahend) to the polynomial from which it is to be subtracted (the minuend). When x is a nonzero real number and a and b are integers: xa xb xa b (xa)b xab xa xb xa b x0 1 x2a 5 1 xa A number is in scientific notation when it is written as a 10n, where 1 a 10 and n is an integer. If x a 10n. Then: • When x 10, n is positive. • When 1 x 10, n is zero. • When 0 x 1, n is negative. To multiply a polynomial by a polynomial, multiply each term of one polynomial by each term of the other polynomial and write the product as the sum of these results in simplest form. To divide a polynomial by a monomial, divide each term of the polynomial by the monomial and write the quotient as the sum of these results. To divide a polynomial by a binomial, subtract multiples of the divisor from the dividend until the remainder is 0 or of degree less than the degree of the divisor. VOCABULARY 5-1 Term • Like terms (similar terms) • Unlike terms • Monomial • Polynomial • Binomial • Trinomial • Simplest form • Descending order • Ascending order 5-4 FOIL 1 5-5 Zero exponent (x0 1) • Negative integral exponent (xn ) xn 5-7 Scientific notation Review Exercises 203 REVIEW EXERCISES 1. Explain why scientific notation is useful. 2. Is it possible to write a general rule for simplifying an expression such as an + bn? In 3–17, simplify each expression. 3. 5bc bc 6. 8mg(3g) 9. (6ab3)2 12. (2a 5)2 6y4 22y3 5y 1 15. 4. 3y2 2y y2 8y 2 7. 3x2(4x2 2x 1) 10. (6a b)2 13. 2x x(2x 5) 16. 6w3 2 8w2 1 2w 2w 5. 5t (4 8t) 8. (4x 3 |
)(2x 1) 11. (2a 5)(2a 5) 14. 17. 40b3c6 28b2c x2 1 x 2 30 x 2 5 In 18–21, use the laws of exponents to perform the operations, and simplify. 18. 35 34 21. 120 122 12 20. [2(102)]3 19. (73)2 In 22–25, express each number in scientific notation. 22. 5,800 23. 14,200,000 24. 0.00006 25. 0.00000277 In 26–29, find the decimal number that is expressed by each given numeral. 26. 4 104 27. 3.06 103 29. 1.03 104 28. 9.7 108 30. Express the area of each of the gardens and the total area of the two gar- dens described in the chapter opener on page 167. 31. If the length of one side of a square is 2h 3, express in terms of h: a. the perimeter of the square. b. the area of the square. 32. The perimeter of a triangle is 41px. If the lengths of two sides are 18px and 7px, represent the length of the third side. 33. If the length of a rectangle can be represented by x 5, and the area of the rectangle by x2 7x 10, find the polynomial that represents: a. the width of the rectangle. b. the perimeter of the rectangle. 204 Operations with Algebraic Expressions 34. The cost of a pizza is 20 cents less than 9 times the cost of a soft drink. If x represents the cost, in cents, of a soft drink, express in simplest form the cost of two pizzas and six soft drinks. Exploration Study the squares of two-digit numbers that end in 5. From what you observe, can you devise a method for finding the square of such a number mentally? Can this method be applied to the square of a three-digit number that ends in 5? Study the squares of the integers from 1 to 12. From what you observe, can you devise a method that uses the square of an integer to find the square of the next larger integer? CUMULATIVE REVIEW CHAPTERS 1–5 Part I Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. 1. Which of the numbers listed below has the largest value? (1) 12 3 |
(2) 1.67 (3) 1.67 (4) 12 7 1 2. For which of the following values of x is x2 x? x (1) 1 (2) 0 (3) 3 (4) 3. Which of the numbers given below is not a rational number? (2) 11 2 (3) 1.3 (4) 2 3 7 3 (1) 2 " 4. Which of the following inequalities is false? (1) 1.5 11 2 (2) 1.5 11 2 (3) 1.5 1.5 (4) 1.5 1 5. Which of the following identities is an illustration of the associative prop- erty? (1) x 7 7 x (2) 3(x 7) 3x 3(7) (3) (x 7) 3 3 (7 x) (4) (x 7) 3 x ( 7 3) 6. The formula C 5 9(F 2 32) can be used to find the Celsius temperature, C, for a given Fahrenheit temperature, F. What Celsius temperature is equal to a Fahrenheit temperature of 68°? (1) 3° (2) 20° (3) 35° (4) 180° 7. If the universe is the set of whole numbers, the solution set of x 3 is (4) {1, 2, 3} (2) {0, 1, 2, 3} (1) {0, 1, 2} (3) {1, 2} 8. The perimeter of a square whose area is 81 square centimeters is Cumulative Review 205 (1) 9 cm (2) 18 cm 9. In simplest form, (2x 4)2 3(x 1) is equal to (3) 20.25 cm (4) 36 cm (1) 4x2 13x 13 (2) 4x2 13x 19 (3) 4x2 3x 19 (4) 4x2 3x 13 10. To the nearest tenth of a meter, the circumference of a circle whose radius is 12.0 meters is (1) 37.6 m (2) 37.7 m (3) 75.3 m (4) 75.4 m 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. The formula for the volume V of a cone is V 5 1 3Bh where B is the area of the base and h is the height. Solve the formula for h in terms of V and B. 12. Each of the numbers given below is different from the others. Explain in what way each is different. 2 7 77 84 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. Solve the given equation for x. Show each step of the solution and name the property that is used in each step. 3(x 4) 5x 8 14. Simplify the following expression. Show each step of the simplification and name the property that you used in each step. 4a 7 (7 3a) 206 Operations with Algebraic Expressions 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. A small park is in the shape of a rectangle that measures 525 feet by 468 feet. a. Find the number of feet of fencing that would be needed to enclose the park. Express your answer to the nearest foot. b. If the entire park is to be planted with grass seed, find the number of square feet to be seeded. Express your answer to the correct number of significant digits based on the given dimensions. c. The grass seed to be purchased is packaged in sacks, each of which holds enough seed to cover 25,000 square feet of ground. How many sacks of seed are needed to seed the park? 16. An ice cream stand sells single-dip cones for $1.75 and double-dip cones for $2.25. Yesterday, 500 cones were sold for $930. How many single-dip and how many double-dip cones were sold? RATIO AND PROPORTION Everyone likes to save money by purchasing something at a reduced price. Because merchants realize that a reduced price may entice a prospective buyer to buy on impulse or to buy at one store rather than another, they |
offer discounts and other price reductions.These discounts are often expressed as a percent off of the regular price. When the Acme Grocery offers a 25% discount on frozen vegetables and the Shop Rite Grocery advertises “Buy four, get one free,” the price-conscious shopper must decide which is the better offer if she intends to buy five packages of frozen vegetables. In this chapter, you will learn how ratios, and percents which are a special type of ratio, are used in many everyday problems. CHAPTER 6 CHAPTER TABLE OF CONTENTS 6-1 Ratio 6-2 Using a Ratio to Express a Rate 6-3 Verbal Problems Involving Ratio 6-4 Proportion 6-5 Direct Variation 6-6 Percent and Percentage Problems 6-7 Changing Units of Measure Chapter Summary Vocabulary Review Exercises Cumulative Review 207 208 Ratio and Proportion 6-1 RATIO A ratio, which is a comparison of two numbers by division, is the quotient obtained when the first number is divided by the second, nonzero number. Since a ratio is the quotient of two numbers divided in a definite order, care must be taken to write each ratio in its intended order. For example, the ratio of 3 to 1 is written 3 1 (as a fraction) while the ratio of 1 to 3 is written 1 3 (as a fraction) or or 3 : 1 (using a colon) 1 : 3 (using a colon) In general, the ratio of a to b can be expressed as a b or a b or a : b To find the ratio of two quantities, both quantities must be expressed in the same unit of measure before their quotient is determined. For example, to compare the value of a nickel and a penny, we first convert the nickel to 5 pennies and then find the ratio, which is or 5 : 1. Therefore, a nickel is worth 5 times as much as a penny. The ratio has no unit of measure. 5 1 Equivalent Ratios is a fraction, we can use the multiplication property of 1 to find 5 Since the ratio 1 many equivalent ratios. For example: 5 2 5 10 15 3 1 5 5 5 x 5 5x 1x 1 3 x (x 0) From the last example, we see that 5x and lx represent two numbers whose ratio is 5 : 1. In general, if a, b, and x are numbers (b 0, x 0), ax and bx represent two numbers whose |
ratio is a : b because b 3 x 5 ax x bx a b 5 a 24 16 b 3 1 5 a is a fraction, we can divide the numerator and the Also, since a ratio such as denominator of the fraction by the same nonzero number to find equivalent ratios. For example: 24 16 5 24 4 2 16 4 4 5 6 4 A ratio is expressed in simplest form when both terms of the ratio are whole numbers and when there is no whole number other than 1 that is a factor of 16 5 24 4 8 24 16 4 2 5 12 8 24 16 5 24 4 4 16 4 8 5 3 2 both of these terms. Therefore, to express the ratio both terms by 8, the largest integer that will divide both 24 and 16. Therefore, 3 in simplest form is. 2 in simplest form, we divide 24 16 24 16 Ratio 209 Continued Ratio Comparisons can also be made for three or more quantities. For example, the length of a rectangular solid is 75 centimeters, the width is 60 centimeters, and the height is 45 centimeters. The ratio of the length to the width is 75 : 60, and the ratio of the width to the height is 60 : 45. We can write these two ratios in an abbreviated form as the continued ratio 75 : 60 : 45. 45 cm 7 5 c m 60 cm A continued ratio is a comparison of three or more quantities in a definite order. Here, the ratio of the measures of the length, width, and height (in that order) of the rectangular solid is 75 : 60 : 45 or, in simplest form, 5 : 4 : 3. In general, the ratio of the numbers a, b, and c (b 0, c 0) is a : b : c. An oil tank with a capacity of 200 gallons contains 50 gallons of oil. a. Find the ratio of the number of gallons of oil in the tank to the capacity of the tank. b. What part of the tank is full? EXAMPLE 1 Solution a. Ratio number of gallons of oil in tank capacity of tank 5 50 200 5 1 4. b. The tank is 1 4 full. Answers a. 1 4 b. 1 4 full EXAMPLE 2 Compute the ratio of 6.4 ounces to 1 pound. Solution First, express both quantities in the same unit of measure. Use the fact that 1 pound 16 ounces. 1 pound 5 6.4 ounces 6.4 ounces 16 ounces 5 6.4 16 5 6. |
4 16 3 10 10 5 64 160 5 64 4 32 160 4 32 5 2 5 210 Ratio and Proportion Calculator Solution On a calculator, divide 6.4 ounces by 16 ounces. ENTER: 6.4 16 ENTER DISPLAY: 6. 4 / 1 6. 4 Change the decimal in the display to a fraction. ENTER: DISPLAY: 2nd ANS MATH ENTER ENTER Answer The ratio is 2 : 5. EXAMPLE 3 Express the ratio 13 4 to 11 2 in simplest form. Solution Since a ratio is the quotient obtained when the first number is divided by the second, divide 13 4 by 11. 2 13 4 4 11 14 2 12 5 7 6 Answer The ratio in simplest form is or 7 : 6. 7 6 EXERCISES Writing About Mathematics 1. Last week, Melanie answered 24 out of 30 questions correctly on a test. This week she answered 20 out of 24 questions correctly. On which test did Melanie have better results? Explain your answer. 2. Explain why the ratio 1.5 : 4.5 is not in simplest form. Developing Skills In 3–12, express each ratio in simplest form: a. as a fraction b. using a colon 3. 36 to 12 4. 48 to 24 5. 40 to 25 6. 12 to 3 7. 5 to 4 8. 8 to 32 9. 40 to 5 10. 0.2 to 8 11. 72 to 1.2 12. 3c to 5c Ratio 211 13. If the ratio of two numbers is 10 : 1, the larger number is how many times the smaller num- ber? 14. If the ratio of two numbers is 8 : 1, the smaller number is what fractional part of the larger number? In 15–19, express each ratio in simplest form. 15. 3 4 to 1 4 16. 11 8 to 3 8 17. 1.2 to 2.4 18. 0.75 to 0.25 19. 6 to 0.25 In 20–31, express each ratio in simplest form. 20. 80 m to 16 m 21. 75 g to 100 g 22. 36 cm to 72 cm 23. 54 g to 90 g 26. 11 2 hr to hr 1 2 29. 1 yd to 1 ft Applying Skills 24. 75 cm to 350 cm 25. 8 ounces to 1 pound 27. 3 in. to 1 2 in. 30. 1 hr to 15 min 28. 1 ft to 1 in. 31. 6 dollars to 50 |
cents 32. A baseball team played 162 games and won 90. a. What is the ratio of the number of games won to the number of games played? b. For every nine games played, how many games were won? 33. A student did six of ten problems correctly. a. What is the ratio of the number right to the number wrong? b. For every two answers that were wrong, how many answers were right? 34. A cake recipe calls for 11 4 cups of milk to 13 4 cups of flour. Write, in simplest form, the ratio of the number of cups of milk to the number of cups of flour in this recipe. 35. The perimeter of a rectangular garden is 30 feet, and the width is 5 feet. Find the ratio of the length of the rectangle to its width in simplest form. 36. In a freshman class, there are b boys and g girls. Express the ratio of the number of boys to the total number of pupils. 37. The length of a rectangular classroom is represented by 3x and its width by 2x. Find the ratio of the width of the classroom to its perimeter. 38. The ages of three teachers are 48, 28, and 24 years. Find, in simplest form, the continued ratio of these ages from oldest to youngest. 39. A woodworker is fashioning a base for a trophy. He starts with a block of wood whose length is twice its width and whose height is one-half its width. Write, in simplest form, the continued ratio of length to width to height. 40. Taya and Jed collect coins. The ratio of the number of coins in their collections, in some order, is 4 to 3. If Taya has 60 coins in her collection, how many coins could Jed have? 212 Ratio and Proportion 6-2 USING A RATIO TO EXPRESS A RATE When two quantities have the same unit of measure, their ratio has no unit of measure. A rate, like a ratio, is a comparison of two quantities, but the quantities may have different units of measures and their ratio has a unit of measure. For example, if a plane flies 1,920 kilometers in 3 hours, its rate of speed is a ratio that compares the distance traveled to the time that the plane was in flight. Rate 5 640 kilometers 1 hour The abbreviation km/h is read “kilometers per hour.” 1,920 kilometers 3 hours time 5 distance 640 km/h A rate may be expressed |
in lowest terms when the numbers in its ratio are whole numbers with no common factor other than 1. However, a rate is most frequently written as a ratio with 1 as its second term. As shown in the example above, the second term may be omitted when it is 1. A rate that has a denominator of 1 is called a unit rate. A rate that identifies the cost of an item per unit is called the unit price. For example, $0.15 per ounce or $3.79 per pound are unit prices. Kareem scored 175 points in seven basketball games. Express, in lowest terms, the average rate of the number of points Kareem scored per game. EXAMPLE 1 Solution Rate 175 points 7 games 5 25 points 1 game 25 points per game Answer Kareem scored points at an average rate of 25 points per game. EXAMPLE 2 Solution There are 5 grams of salt in 100 cubic centimeters of a solution of salt and water. Express, in lowest terms, the ratio of the number of grams of salt per cubic centimeters in the solution. 5 g 100 cm3 5 1 g 20 cm3 5 1 20 g/cm3 Answer The solution contains 1 20 solution. grams or 0.05 grams of salt per cubic centimeter of Using a Ratio to Express a Rate 213 EXERCISES Writing About Mathematics 1. How is a rate a special kind of ratio? 2. How does the way in which a rate is usually expressed differ from a ratio in simplest form? Developing Skills In 3–8, express each rate in lowest terms. 3. The ratio of 36 apples to 18 people. 4. The ratio of 48 patients to 6 nurses. 5. The ratio of $1.50 to 3 liters. 6. The ratio of 96 cents to 16 grams. 7. The ratio of $2.25 to 6.75 ounces. 8. The ratio of 62 miles to 100 kilometers. Applying Skills In 9–12, in each case, find the average rate of speed, expressed in miles per hour. 9. A vacationer traveled 230 miles in 4 hours. 10. A post office truck delivered mail on a 9-mile route in 2 hours. 11. A commuter drove 48 miles to work in 11 2 hours. 12. A race-car driver traveled 31 miles in 15 minutes. (Use 15 minutes hour.) 1 4 13. If there are 240 tennis balls in 80 cans, how many tennis balls are in each can? 14. If an 11-ounce |
can of shaving cream costs 88 cents, what is the unit cost of the shaving cream in the can? 15. In a supermarket, the regular size of CleanRight cleanser contains 14 ounces and costs 49 cents. The giant size of CleanRight cleanser, which contains 20 ounces, costs 66 cents. a. Find, correct to the nearest tenth of a cent, the cost per ounce for the regular can. b. Find, correct to the nearest tenth of a cent, the cost per ounce for the giant can. c. Which is the better buy? 16. Johanna and Al use computers for word processing. Johanna can keyboard 920 words in 20 minutes, and Al can keyboard 1,290 words in 30 minutes. Who is faster at entering words on a keyboard? 17. Ronald runs 300 meters in 40 seconds. Carlos runs 200 meters in 30 seconds. Who is the faster runner for short races? 214 Ratio and Proportion 6-3 VERBAL PROBLEMS INVOLVING RATIO Any pair of numbers in the ratio 3 : 5 can be found by multiplying 3 and 5 by the same nonzero number. 3(3) 9 5(3) 15 3 : 5 9 : 15 3(7) 21 5(7) 35 3 : 5 21 : 35 3(0.3) 0.9 5(0.3) 1.5 3 : 5 0.9 : 1.5 3(x) 3x 5(x) 5x 3 : 5 3x : 5x Thus, for any nonzero number x, 3x : 5x 3 : 5. In general, when we know the ratio of two or more numbers, we can use the terms of the ratio and a nonzero variable, x, to express the numbers. Any two numbers in the ratio a : b can be written as ax and bx where x is a nonzero real number. EXAMPLE 1 The perimeter of a triangle is 60 feet. If the sides are in the ratio 3 : 4 : 5, find the length of each side of the triangle. Solution Let 3x the length of the first side, 4x the length of the second side, 5x the length of the third side. The perimeter of the triangle is 60 feet. 3x 4x 5x 60 12x 60 x 5 3x 3(5) 15 4x 4(5) 20 5x 5(5) 25 Check 15 : 20 : 25 3 : 4 : 5 ✔ 15 20 |
25 60 ✔ Answer The lengths of the sides are 15 feet, 20 feet, and 25 feet. EXAMPLE 2 Two numbers have the ratio 2 : 3. The larger is 30 more than of the smaller. Find the numbers. 1 2 Solution Let 2x the smaller number, 3x the larger number. Verbal Problems Involving Ratio 215 Check The ratio 30 : 45 in lowest terms is 2 : 3. ✔ One-half of the smaller number, 30, is 15. The larger number, 45, is 30 more than 15. ✔ The larger number is 30 more than of the smaller number. 1 2 1 2(2x) 1 30 3x 3x x 30 2x 30 x 15 2x 2(15) 30 3x 3(15) 45 Answer The numbers are 30 and 45. EXERCISES Writing About Mathematics 1. Two numbers in the ratio 2 : 3 can be written as 2x and 3x. Explain why x cannot equal zero. 2. The ratio of the length of a rectangle to its width is 7 : 4. Pete said that the ratio of the length to the perimeter is 7 : 11. Do you agree with Pete? Explain why or why not. Developing Skills 3. Two numbers are in the ratio 4 : 3. Their sum is 70. Find the numbers. 4. Find two numbers whose sum is 160 and that have the ratio 5 : 3. 5. Two numbers have the ratio 7 : 5. Their difference is 12. Find the numbers. 6. Find two numbers whose ratio is 4 : 1 and whose difference is 36. 7. The lengths of the sides of a triangle are in the ratio of 6 : 6 : 5. The perimeter of the triangle is 34 centimeters. Find the length of each side of the triangle. 8. The perimeter of a triangle is 48 centimeters. The lengths of the sides are in the ratio 3 : 4 : 5. Find the length of each side. 9. The perimeter of a rectangle is 360 centimeters. If the ratio of its length to its width is 11 : 4, find the dimensions of the rectangle. 10. The sum of the measures of two angles is 90°. The ratio of the measures of the angles is 2 : 3. Find the measure of each angle. 11. The sum of the measures of two angles is 180°. The ratio of the measures of the angles is 4 : 5. Find the measure of each angle. 216 Ratio and Proportion 12. The ratio of the measures of |
the three angles of a triangle is 2 : 2 : 5. Find the measures of each angle. 13. In a triangle, two sides have the same length. The ratio of each of these sides to the third side is 5 : 3. If the perimeter of the triangle is 65 inches, find the length of each side of the triangle. 14. Two positive numbers are in the ratio 3 : 7. The larger exceeds the smaller by 12. Find the numbers. 15. Two numbers are in the ratio 3 : 5. If 9 is added to their sum, the result is 41. Find the numbers. Applying Skills 16. A piece of wire 32 centimeters in length is divided into two parts that are in the ratio 3 : 5. Find the length of each part. 17. The ratio of the number of boys in a school to the number of girls is 11 to 10. If there are 525 pupils in the school, how many of them are boys? 18. The ratio of Carl’s money to Donald’s money is 7 : 3. If Carl gives Donald $20, the two then have equal amounts. Find the original amount that each one had. 19. In a basketball free-throw shooting contest, the points made by Sam and Wilbur were in the ratio 7 : 9. Wilbur made 6 more points than Sam. Find the number of points made by each. 20. A chemist wishes to make 121 2 liters of an acid solution by using water and acid in the ratio 3 : 2. How many liters of each should she use? 6-4 PROPORTION A proportion is an equation that states that two ratios are equal. Since the ratio 1 is equal to the ratio 1 : 5 or, we may write the proportion 4 : 20 or 5 4 20 4 : 20 1 : 5 or 4 20 5 1 5 Each of these proportions is read as “4 is to 20 as 1 is to 5.” The general form of a proportion may be written as: a : b c : d or b 5 c a d (b 0, d 0) Each of these proportions is read as “a is to b as c is to d.” There are four terms in this proportion, namely, a, b, c, and d. The outer terms, a and d, are called the extremes of the proportion. The inner terms, b and c, are the means. Proportion 217 means a : b c : d extremes or extreme |
mean ↓ ↓ a b 5 c d ↑ ↑ mean extreme In the proportion, 4 : 20 1 : 5, the product of the means, 20(1), is equal to the product of the extremes, 4(5)., the product of the means, 15(10), is equal to the In the proportion, 15 5 10 5 30 product of the extremes, 5(30). a b 5 c, we can show that the product of the means is equal In any proportion d a b 5 c to the product of the extremes, ad bc. Since is an equation, we can d multiply both members by bd, the least common denominator of the fractions in the equation. bd bd > b 5 c a d 5 bd ad bc d > B B Therefore, we have shown that the following statement is always true: In a proportion, the product of the means is equal to the product of the extremes. Notice that the end result, ad bc, is the result of multiplying the terms that are cross-wise from each other: ➤ ➤ a b 5 c d ➤ ➤ This is called cross-multiplying, which we have just shown to be valid. If the product of two cross-wise terms is called a cross product, then the fol- lowing is also true: In a proportion, the cross products are equal. If a, b, c, and d are nonzero numbers and b 5 c a d, then ad bc. There are three other proportions using a, b, c and d for which ad = bc. d c 5 b a is a proportion because 6(10) 4(15). For example, we know that 15 10 Therefore, each of the following is also a proportion. 6 15 5 4 10 4 5 15 10 6 10 15 5 4 6 218 Ratio and Proportion EXAMPLE 1 Show that 16 5 5 4 20 is a proportion. Solution Three methods are shown here. The first two use paper and pencil; the last makes use of a calculator. METHOD 1 Reduce each ratio to simplest form. 4 16 5 4 4 4 16 4 4 5 1 4 and 20 5 5 4 5 5 20 4 5 5 1 4 Since each ratio equals proportion. 1 4, the ratios are equal and 16 5 5 4 20 is a METHOD 2 Show that the cross products are equal. ➤ ➤ ➤ ➤ 16 5 5 4 20 16 5 4 |
20 80 80 Therefore, 16 5 5 4 20 is a proportion. METHOD 3 Use a calculator. Enter the proportion. If the ratios are equal, then the calculator will display 1. If the ratios are not equal, the calculator will display 0. ENTER: 4 16 2nd TEST ENTER 5 20 ENTER DISPLAY Since the calculator displays 1, the statement is true. The ratios are equal and 16 5 5 4 20 is a proportion. Answer Any one of the three methods shows that 16 5 5 4 20 is a proportion. Proportion 219 EXAMPLE 2 Solve the proportion 25 : q 5 : 2 for q. Solution Since 25 : q 5 : 2 is a proportion, the product of the means is equal to the product of the extremes. Therefore: Check Reduce each ratio to simplest form. 25 : q 5 : 2 25 : 10 5 : 2 5? 5 : 2 5 : 2 ✔ means 25 : q 5 : 2 extremes 5q 25(2) means extremes 5q 50 q 10 Answer q 10 Note: Example 2 could also have been solved by setting up the proportion q 5 5 25 and then using cross-multiplication to solve for the variable. 2 EXAMPLE 3 Solve for x: x 2 2 5 32 12 x 1 8 Solution Use the fact that the product of the means equals the product of the extremes (the cross products are equal). ➤ ➤ ➤ x 2 2 5 32 12 ➤ x 1 8 32(x – 2) 12(x 8) 32x 64 12x 96 12x 64 12x 64 160 20x 20 5 160 20x 20 x 8 Answer x 8 Check x 2 2 5 32 12 x 1 8 8 – 2 5? 32 12 8 1 8 6 5? 32 12 16 2 2 ✔ 220 Ratio and Proportion EXAMPLE 4 The denominator of a fraction exceeds the numerator by 7. If 3 is subtracted from the numerator of the fraction and the denominator is unchanged, the value of the resulting fraction becomes. Find the original fraction. 1 3 Solution Let x the numerator of original fraction, x 7 the denominator of the original fraction the original fraction. the new fraction. 1 The value of the new fraction is. (x 7) 3(x 3) x 7 3x 9 x 9 x 9 16 2x x 8 x 7 15 Answer The original fraction was 8 15. Check The original fraction was 8 15. The new fraction is 8 – 3 15 |
5 5 15 5 1 3 ✔ EXERCISES Writing About Mathematics 1. Jeremy said that if the means and the extremes of a proportion are interchanged, the result- ing ratios form a proportion. Do you agree with Jeremy? Explain why or why not. 2. Mike said that if the same number is added to each term of a proportion, the resulting ratios form a proportion. Do you agree with Mike? Explain why or why not. Developing Skills In 3–8, state, in each case, whether the given ratios may form a proportion. 3. 3 4, 30 40 4. 2 3, 10 5 5. 4 5, 16 25 6. 2 5, 5 2 7. 14 18, 28 36 8. 36 30, 18 15 In 9–16, find the missing term in each proportion. 2 5? 1 9. 8 13. 4 :? 12 : 60 3 5 5 18 10.? 14.? : 9 35 : 63 4 5 6 1 11.? 15.? : 60 6 : 10 6 5? 4 12. 42 16. 16 :? 12 : 9 In 17–25, solve each equation and check the solution. Proportion 221 17. 20. 23. 60 5 3 x 20 5 15 5 x x 1 8 3x 1 3 3 5 7x 2 1 5 18. 5 4 5 x 12 x 12 2 x 5 10 30 24. 12 : 15 x : 45 21. 19. 30 4x 5 10 24 16 8 5 21 2 x 25. 5 : x 2 4 : x 22. x In 26–28, in each case solve for x in terms of the other variables. 26. a : b c : x 27. 2r : s x : 3s 28. 2x : m 4r : s Applying Skills In 29–36, use a proportion to solve each problem. 29. 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 30. The denominator of a fraction exceeds twice the numerator of the fraction by 10. The value of the fraction is. Find the fraction. 5 12 31. The denominator of a fraction is 30 more than the numerator of the fraction. If 10 is added to the numerator of the fraction and the denominator is unchanged, the value of the 3 resulting fraction becomes. Find the original fraction. 5 32. The numerator of a certain fraction is |
3 times the denominator. If the numerator is decreased by 1 and the denominator is increased by 2, the value of the resulting fraction is. Find the original fraction. 5 2 33. What number must be added to both the numerator and denominator of the fraction make the resulting fraction equal to? 3 4 7 19 to 34. The numerator of a fraction exceeds the denominator by 3. If 3 is added to the numerator and 3 is subtracted from the denominator, the resulting fraction is equal to. Find the original fraction. 5 2 35. The numerator of a fraction is 7 less than the denominator. If 3 is added to the numerator and 9 is subtracted from the denominator, the new fraction is equal to. Find the original fraction. 3 2 36. Slim Johnson was usually the best free-throw shooter on his basketball team. Early in the season, however, he had made only 9 of 20 shots. By the end of the season, he had made all the additional shots he had taken, thereby ending with a season record of 3 : 4. How many additional shots had he taken? 222 Ratio and Proportion 6-5 DIRECT VARIATION If the length of a side, s, of a square is 1 inch, then the perimeter, P, of the square is 4 inches. Also, if s is 2 inches, P is 8 inches; if s is 3 inches, P is 12 inches. These pairs of values are shown in the table at the right. s P 1 4 2 8 3 12 From the table, we observe that, as s varies, P also varies. Comparing each value of P to its corresponding value of s, we notice that all three sets of values result in the same ratio when reduced to lowest terms 12 3 If a relationship exists between two variables so that their ratio is a constant, that relationship between the variables is called a direct variation. In every direct variation, we say that one variable varies directly as the other, or that one variable is directly proportional to the other. The constant ratio is called a constant of variation. It is important to indicate the order in which the variables are being com- pared before stating the constant of variation. For example: • In comparing P to s, • In comparing s to P,. The constant of variation is 4. 1. The constant of variation is. Note that each proportion, the perimeter of a square. and s P 5 1 4, becomes P 4s, |
the formula for In a direct variation, the value of each term of the ratio increases when we multiply each variable by a factor greater than 1; the value of each term of the ratio decreases when we divide each variable by a factor greater than 1, as shown below 12 4 3 EXAMPLE 1 If x varies directly as y, and x 1.2 when y 7.2, find the constant of variation by comparing x to y. Solution Answer 1 6 Constant of variation x y 5 1.2 7.2 5 1.2 4 1.2 7.2 4 1.2 5 1 6 EXAMPLE 2 Direct Variation 223 The table gives pairs of values for the variables x and y. a. Show that one variable varies directly as the other. b. Find the constant of variation by comparing y to x. c. Express the relationship between the variables as a formula. d. Find the values missing in the table. x y 1 8 2 16 3 24 10?? 1,600 Solution a. x y 5 2 16 5 2 4 2 x y 5 1 8 Since all the given pairs of values have the same ratio, x and y vary directly. 24 5 3 4 3 16 4 2 5 1 8 24. Constant of variation 8. 8 1 c. Write a proportion that includes both variables and the constant of variation: x y 5 1 8 y 8x Cross multiply to obtain the formula. d. Substitute the known value in the equation written in b. Solve the equation. When x 10, find y. y 8x y 8(10) y 80 When y 1,600, find x. y 8x 1,600 8x 200 x Answers a. The variables vary directly because the ratio of each pair is the same constant. b. 8 c. y 8x d. When x 10, y 80; when y 1,600, x 200. 224 Ratio and Proportion EXAMPLE 3 There are about 90 calories in 20 grams of a cheese. Reggie ate 70 grams of this cheese. About how many calories were there in the cheese she ate if the number of calories varies directly as the weight of the cheese? Solution Let x number of calories in 70 grams of cheese. number of grams of cheese 5 90 20 70 5 90 x 20 20x 90(70) 20x 6,300 x 315 number of calories Answer There were about 315 calories in 70 grams of the cheese. EXERCISES Writing About Mathematics 1. On a cross-country |
trip, Natasha drives at an average speed of 65 miles per hour. She says that each day, her driving time and the distance that she travels are directly proportional. Do you agree with Natasha? Explain why or why not. 2. The cost of parking at the Center City Parking Garage is $5.50 for the first hour or part of an hour and $2.75 for each additional half hour or part of a half hour. The maximum cost for 24 hours is $50. Does the cost of parking vary directly as the number of hours? Explain your answer. Developing Skills In 3–11, in each case one value is given for each of two variables that vary directly. Find the constant of variation. 3. x 12, y 3 6. P 12.8, s 3.2 9. s 88, t 110 4. d 120, t 3 7. t 12, n 8 10. A 212, P 200 5. y 2, z 18 8. I 51, t 6 11. r 87, s 58 In 12–17, tell, in each case, whether one variable varies directly as the other. If it does, express the relation between the variables by means of a formula. 12. P s 3 1 6 2 9 3 13. n c 3 6 4 8 5 10 14. x y 4 6 5 8 6 10 15. t d 1 20 2 40 3 60 16. 2 3 x y 6 9 12 4 Direct Variation 225 17. x y 1 1 2 4 3 9 In 18–20, in each case one variable varies directly as the other. Write the formula that relates the variables and find the missing numbers. 18. h A 1 5 2?? 25 19. h S 4 6 8?? 15 20. l w 2 1 8?? 7 In 21–24, state whether the relation between the variables in each equation is a direct variation. In each case, give a reason for your answer. 22. 15T D 21. R + T 80 25. C 7n is a formula for the cost of n articles that sell for $7 each. e 23. 20 i 24. bh 36 a. How do C and n vary? b. How will the cost of nine articles compare with the cost of three articles? c. If n is doubled, what change takes place in C? 26. A 12l is a formula for the area of any rectangle whose width is 12. a. Describe how A and l |
vary. b. How will the area of a rectangle whose length is 8 inches compare with the area of a rectangle whose length is 4 inches? c. If l is tripled, what change takes place in A? 27. The variable d varies directly as t. If d 520 when t 13, find d when t 9. 28. Y varies directly as x. If Y 35 when x 5, find Y when x 20. 29. A varies directly as h. A 48 when h 4. Find h when A 36. 30. N varies directly as d. N 10 when d 8. Find N when d 12. Applying Skills In 31–48, the quantities vary directly. Solve algebraically. 31. If 3 pounds of apples cost $0.89, what is the cost of 15 pounds of apples at the same rate? 32. If four tickets to a show cost $17.60, what is the cost of seven such tickets? 33. If pound of meat sells for $3.50, how much meat can be bought for $8.75? 1 2 34. Willis scores an average of 7 foul shots in every 10 attempts. At the same rate, how many shots would he score in 200 attempts? 35. There are about 60 calories in 30 grams of canned salmon. About how many calories are there in a 210-gram can? 226 Ratio and Proportion 36. There are 81 calories in a slice of bread that weighs 30 grams. How many calories are there in a loaf of this bread that weighs 600 grams? 37. There are about 17 calories in three medium-size shelled peanuts. Joan ate 30 such peanuts. How many calories were there in the peanuts she ate? 38. A train traveled 90 miles in 11 2 330 miles? hours. At the same rate, how long will the train take to travel 39. The weight of 20 meters of copper wire is 0.9 kilograms. Find the weight of 170 meters of the same wire. 40. A recipe calls for 11 2 cups of sugar for a 3-pound cake. How many cups of sugar should be used for a 5-pound cake? 41. In a certain concrete mixture, the ratio of cement to sand is 1 : 4. How many bags of cement would be used with 100 bags of sand? 42. The owner of a house that is assessed for $12,000 pays $960 in realty taxes. At the same rate, what should be the realty tax on a house assessed for $ |
16,500? 43. The scale on a map is given as 5 centimeters to 3.5 kilometers. How far apart are two towns if the distance between these two towns on the map is 8 centimeters? 44. David received $8.75 in dividends on 25 shares of a stock. How much should Marie receive in dividends on 60 shares of the same stock? 45. A picture 21 8 inches. What will be the width of the enlarged picture? inches long and 31 4 61 2 inches wide is to be enlarged so that its length will become 46. An 11-pound turkey costs $9.79. At this rate, find: a. the cost of a 14.4-pound turkey, rounded to the nearest cent. b. the cost of a 17.5-pound turkey, rounded to the nearest cent. c. the price per pound at which the turkeys are sold. d. the largest size turkey, to the nearest tenth of a pound, that can be bought for $20 or less. 47. If a man can buy p kilograms of candy for d dollars, represent the cost of n kilograms of this candy. 48. If a family consumes q liters of milk in d days, represent the amount of milk consumed in h days. Percent and Percentage Problems 227 6-6 PERCENT AND PERCENTAGE PROBLEMS Base, Rate, and Percent Problems dealing with discounts, commissions, and taxes involve percents. A percent, which is a ratio of a number to 100, is also called a rate. Here, the word rate is treated as a comparison of a quantity to the whole. For example, 8% (read 8 as 8 percent) is the ratio of 8 to 100, or. A percent can be expressed as a frac100 tion or as a decimal: 8% 8 100 0.08 If an item is taxed at a rate of 8%, then a $50 pair of jeans will cost an addi- tional $4 for tax. Here, three quantities are involved. 1. The base, or the sum of money being taxed, is $50. 2. The rate, or the rate of tax, is 8% or 0.08 or 8 100. 3. The percentage, or the amount of tax being charged, is $4. These three related terms may be written as a proportion or as a formula: As a proportion percentage rate base For example: 50 5 8 4 100 4 50 or 0.08 As a formula base rate percentage |
For example: 50 4 or 50 0.08 4 8 100 Just as we have seen two ways to look at this problem involving sales tax, we will see more than one approach to every percentage problem. Note that when we calculate using percent, we always use the fraction or decimal form of the percent. Percent of Error When we use a measuring device such as a ruler to obtain a measurement, the accuracy and precision of the measure is dependent on the type of instrument used and the care with which it is used. Error is the absolute value of the difference between a value found experimentally and the true theoretical value. For example, when the length and width of a rectangle are 13 inches and 84 inches, the true length of the diagonal, found by using the Pythagorean Theorem, is 85 inches. A student drew this rectangle and, using a ruler, found the inches. The error of measurement would be measure of the diagonal to be 847 8 228 Ratio and Proportion 85 2 847 1 8 8 value, written as a percent. or inches. The percent of error or is the ratio of the error to the true Percent of error zmeasured value2true valuez true value 3 100% In the example above, the percent of error is EXAMPLE 1 Solution 1 8 85 = 8 4 85 5 1 1 8 3 1 85 5 1 680 < 0.001470588 < 0.15% Note: The relative error is simply the percent of error written as a decimal. Find the amount of tax on a $60 radio when the tax rate is 8%. METHOD 1 Use the proportion: Let t the percentage or amount of tax. amount of tax base percentage base rate. 5 8 100 60 5 8 t 100 100t 480 t 4.80 The tax is $4.80. METHOD 2 Use the formula: base rate percentage. Let t percentage or amount of tax. Change 8% to a fraction. base rate percentage Change 8% to a decimal. base rate percentage 60 8% t 60 3 8 t 100 480 t 100 4.8 t 60 8% t 60 0.08 t 4.8 t Whether the fraction or the decimal form of 8% is used, the tax is $4.80. Answer The tax is $4.80. Percent and Percentage Problems 229 EXAMPLE 2 During a sale, a store offers a discount of 25% off any purchase. What is the regular price of a dress that a customer purchased for $73.50? Solution The |
rate of the discount is 25%. Therefore the customer paid (100 25)% or 75% of the regular price. The percentage is given as $73.50, and the base is not known. Let n the regular price, or base. METHOD 1 Use the proportion. percentage base 5 rate n 5 75 73.50 100 75n 7,350 n 98 Check If 25% of 98 is subtracted from 98 does the difference equal 73.50? 0.25 98 24.50 98 24.50 73.50 ✔ METHOD 2 Use the formula. base rate percentage n 75% 73.50 Use fractions n 3 75 n 3 75 73.50 100 100 3 100 75 5 73.50 3 100 75 n 98 Use decimals n 0.75 73.50 0.75 5 73.50 0.75n 0.75 n 98 Answer The regular price of the dress was $98. Alternative Solution Let n the regular price of the dress. Then, 0.25n the discount. The price of the dress minus the discount is the amount the customer paid. |___________________________| |____________________| ↓ ↓ 73.50 n ↓ ↓ |___________| ↓ 0.25n n 0.25n 73.50 1.00n 0.25n 73.50 0.75n 73.50 0.75 5 73.50 0.75n 0.75 n 98 The check is the same as that shown for Method 1. Answer The regular price of the dress was $98. 230 Ratio and Proportion Percent of Increase or Decrease A percent of increase or decrease gives the ratio of the amount of increase or decrease to the original amount. A sales tax is a percent of increase on the cost of a purchase. A discount is a percent of decrease on the regular price of a purchase. To find the percent of increase or decrease, find the difference between the original amount and the new amount. The original amount is the base, the absolute value of the difference is the percentage, and the percent of increase or decrease is the rate. Percent of increase or decrease zoriginal amount 2 new amountz original amount 100% EXAMPLE 3 Last year Marisa’s rent was $600 per month. This year, her rent increased to $630 per month. What was the percent of increase in her rent? Solution The original rent was $600. The new rent was $630. The amount of increase was |$600 – $630 |
| $30. Percent of increase 30 600 0.05 1 20 Change 0.05 to a percent: 0.05 5% Answer The percent of increase is 5%. EXAMPLE 4 Solution A store reduced the price of a television from $840 to $504. What was the percent of decrease in the price of the television? Original price $840 New price $504 Amount of decrease |$840 – $504| $336 Percent of decrease 0.4 336 840 Change 0.4 to a percent: 0.4 40% Answer The percent of decrease was 40%. Percent and Percentage Problems 231 EXERCISES Writing About Mathematics 1. Callie said that two decimal places can be used in place of the percent sign. Therefore, 3.6% can be written as 0.36. Do you agree with Callie. Explain why or why not. 2. If Ms. Edwards salary was increased by 4%, her current salary is what percent of her salary before the increase? Explain your answer. Developing Skills In 3–11, find each indicated percentage. 3. 2% of 36 6. 2.5% of 400 9. 121 2% of 128 4. 6% of 150 7. 60% of 56 10. 331 3% of 72 In 12–19, find each number or base. 5. 15% of 48 8. 100% of 7.5 11. 150% of 18 12. 20 is 10% of what number? 13. 64 is 80% of what number? 14. 8% of what number is 16? 16. 125% of what number is 45? 18. 662 3% of what number is 54? 15. 72 is 100% of what number? 17. 371 2% of what number is 60? 19. 3% of what number is 1.86? In 20–27, find each percent. 20. 6 is what percent of 12? 22. What percent of 10 is 6? 24. 5 is what percent of 15? 26. 18 is what percent of 12? Applying Skills 21. 9 is what percent of 30? 23. What percent of 35 is 28? 25. 22 is what percent of 22? 27. 2 is what percent of 400? 28. A newspaper has 80 pages. If 20 of the 80 pages are devoted to advertising, what percent of the newspaper consists of advertising? 29. A test was passed by 90% of a class. If 27 students passed the test, how many students are in the |
class? 232 Ratio and Proportion 30. Marie bought a dress that was marked $24. The sales tax is 8%. a. Find the sales tax. b. Find the total amount Marie had to pay. 31. There were 120 planes on an airfield. If 75% of the planes took off for a flight, how many planes took off? 32. One year, the Ace Manufacturing Company made a profit of $480,000. This represented 6% of the volume of business for the year. What was the volume of business for the year? 33. The price of a new motorcycle that Mr. Klein bought was $5,430. Mr. Klein made a down payment of 15% of the price of the motorcycle and arranged to pay the rest in installments. How much was his down payment? 34. How much silver is in 75 kilograms of an alloy that is 8% silver? 35. In a factory, 54,650 parts were made. When they were tested, 4% were found to be defective. How many parts were good? 36. A baseball team won 9 games, which was 60% of the total number of games the team played. How many games did the team play? 37. The regular price of a sweater is $40. The sale price of the sweater is $34. What is the per- cent of decrease in the price? 38. A businessman is required to collect an 8% sales tax. One day, he collected $280 in taxes. Find the total amount of sales he made that day. 39. A merchant sold a stereo speaker for $150, which was 25% above its cost to her. Find the cost of the stereo speaker to the merchant. 40. Bill bought a wooden chess set at a sale. The original price was $120; the sale price was $90. What was the percent of decrease in the price? 41. If the sales tax on $150 is $7.50, what is the percent of the sales tax? 42. Mr. Taylor took a 2% discount on a bill. He paid the balance with a check for $76.44. What was the original amount of the bill? 43. Mrs. Sims bought some stock for $2,250 and sold the stock for $2,520. What was the percent increase in the value of the stock? 44. When Sharon sold a vacuum cleaner for $220, she received a commission of $17.60. What was the rate of commission? 45. On |
the first day of a sale, a camera was reduced by $8. This represented 10% of the original price. On the last day of the sale, the camera was sold for 75% of the original price. What was the final selling price of the camera? Percent and Percentage Problems 233 46. The regular ticketed prices of four items at Grumbell’s Clothier are as follows: coat, $139.99; blouse, $43.99; shoes, $89.99; jeans, $32.99. a. These four items were placed on sale at 20% off the regular price. Find, correct to the nearest cent, the sale price of each of these four items. b. Describe two different ways to find the sale prices. 47. At Relli’s Natural Goods, all items are being sold today at 30% off their regular prices. However, customers must still pay an 8% tax on these items. Edie, a good-natured owner, allows each customer to choose one of two plans at this sale: Plan 1. Deduct 30% of the cost of all items, then add 8% tax to the bill. Plan 2. Add 8% tax to the cost of all items, then deduct 30% of this total. Which plan if either, saves the customer more money? Explain why. 48. In early March, Phil Kalb bought shares of stocks in two different companies. Stock ABC rose 10% in value in March, then decreased 10% in April. Stock XYZ fell 10% in value in March, then rose 10% in April. What percent of its original price is each of these stocks now worth? 49. A dairy sells milk in gallon containers. The containers are filled by machine and the amount of milk may vary slightly. A quality control employee selects a container at random and makes an accurate measure of the amount of milk as 16.25 cups. Find the percent of error to the nearest tenth of a percent. 50. A carpenter measures the length of a board as 50.5 centimeters. The exact measure of the length was 50.1 centimeters. Find the percent of error in the carpenter’s measure to the nearest tenth of a percent. 51. A 5-pound weight is placed on a gymnasium scale. The scale dial displayed 51 2 pounds. If the scale is consistently off by the same percentage, how much does an athlete weigh, to the nearest tenth of a pound, |
if his weight displayed on this scale is 144 pounds? 52. Isaiah answered 80% of the questions correctly on the math midterm, and 90% of the questions correctly on the math final. Can you conclude that he answered 85% of all the questions correctly (the average of 80% and 90%)? Justify your answer or give a counterexample. 53. In January, Amy bought shares of stocks in two different companies. By the end of the year, shares of the first company had gone up by 12% while shares of the second company had gone up by 8%. Did Amy gain a total of 12% 8% 20% in her investments? Explain why or why not. 234 Ratio and Proportion 6-7 CHANGING UNITS OF MEASURE The weight and dimensions of a physical object are expressed in terms of units of measure. In applications, it is often necessary to change from one unit of measure to another by a process called dimensional analysis. To do this, we multiply by a fraction whose numerator and denominator are equal measures in two different units so that, in effect, we are multiplying by the identity element, 1. For example, since 100 centimeters and 1 meter are equal measures: 1 m 5 100 cm 100 cm 100 cm 5 1 1 m 100 cm 5 1 m 1 m 5 1 To change 4.25 meters to centimeters,. multiply by 100 cm 1 m 4.25 m 4.25 m 3 100 cm 1 m 425 cm To change 75 centimeters to meters, multiply by 1 m. 100 cm 75 cm 3 1 m 75 cm 100 cm 75 m 100 0.75 m Note that in each case, the fraction was chosen so that the given unit of measure occurred in the denominator and could be “cancelled” leaving just the unit of measure that we wanted in the result. Sometimes it is necessary to use more than one fraction to change to the required unit. For example, if we want to change 3.26 feet to centimeters and know that 1 foot 12 inches and that 1 inch 2.54 centimeters, it will be necessary to first use the fraction to change feet to inches. 12 in. 1 ft 3.26 ft 3.26 ft 3 12 in. 1 ft 39.12 1 in. 39.12 in. Then use the fraction to change inches to centimeters 2.54 cm 1 in. 39.12 in. 3 2.54 cm 1 in. 39.12 in. 99.3648 1 cm 99.364 |
8 cm This answer, rounded to the nearest tenth, can be expressed as 99.4 centimeters. EXAMPLE 1 Solution If there are 5,280 feet in a mile, find, to the nearest hundredth, the number of miles in 1,200 feet. How to Proceed (1) Write a fraction equal to 1 with the required unit in the numerator and the given unit in the denominator: 1 mi 5,280 ft Changing Units of Measure 235 1,200 ft 1,200 ft 3 1 mi 5,280 ft mi 1,200 5,280 0.227 mi 1,200 ft 0.23 mi (2) Multiply the given measure by the fraction written in step 1: (3) Round the answer to the nearest hundredth: Answer 0.23 mi EXAMPLE 2 In France, apples cost 4.25 euros per kilogram. In the United States, apples cost $1.29 per pound. If the currency exchange rate is 0.95 euros for 1 dollar, in which country are apples more expensive? Solution Recall that “per” indicates division, that is, 4.25 euros per kilogram can be and $1.29 per pound as written as. 1.29 dollars 1 pound 4.25 euros 1 kilogram (1) Change euros in euros per kilogram to dollars. Use 1 dollar 0.95 euros, a fraction equal to 1. 4.25 euros 1 kilogram 3 1 dollar 0.95 euros 5 4.25 dollars 0.95 kilograms (2) Now change kilograms in dollars per kilogram to pounds. One pound equals 0.454 kilograms. Since kilograms is in the denominator, use the fraction with kilograms in the numerator. 4.25 dollars 0.95 kilograms 3 0.454 kilogram 1 pound 5 4.25(0.454) dollars 0.95(1) pounds (3) Use a calculator for the computation. (4) The number in the display, 2.031052632, is the cost of apples in France in dollars per pound. Round the number in the display to the nearest cent: $2.03 (5) Compare the cost of apples in France ($2.03 per pound) to the cost of apples in the United States ($1.29 per pound). Answer Apples are more expensive in France. 236 Ratio and Proportion EXAMPLE 3 Change 60 miles per hour to feet per second. Solution Use dimensional analysis to change the unit of measure in the |
given rate to the required unit of measure. (1) Write 60 miles per hour as a fraction: (2) Change miles to feet. Multiply by a ratio with miles in the denominator to cancel miles in the numerator: (3) Change hours to minutes. Multiply by a ratio with hours in the numerator to cancel hours in the denominator: (4) Change minutes to seconds. Multiply by a ratio with minutes in the numerator to cancel minutes in the denominator. 60 mi 1 hr 5 60 mi 1 hr 3 5,280 ft 1 mi 5 60(5,280) ft 1 hr 3 1 hr 60 min 5 60(5,280) ft 60 min 3 1 min 60 sec Alternative Solution (5) Compute and simplify: 60(60) sec 5 88 ft Write the ratios in one expression and compute on a calculator. 1 sec 5 60(5,280) ft 60 mi 1 hr 3 5,280 ft 1 mi 3 1 hr 60 min 3 1 min 60 sec 5 60(5,280) ft 60(60) sec Answer 60 miles per hour 88 feet per second. EXERCISES Writing About Mathematics 1. Sid cannot remember how many yards there are in a mile but knows that there are 5,280 feet in a mile and 3 feet in a yard. Explain how Sid can find the number of yards in a mile. 3 8 2. A recipe uses of a cup of butter. Abigail wants to use tablespoons to measure the butter and knows that 4 tablespoons equals cup. Explain how Abigail can find the number of tablespoons of butter needed for her recipe. 1 4 Changing Units of Measure 237 Developing Skills In 3–16: a. write, in each case, the fraction that can be used to change the given units of measure, b. find the indicated unit of measure. 3. Change 27 inches to feet. 4. Change 175 centimeters to meters. 5. Change 40 ounces to pounds. 6. Change 7,920 feet to miles. 7. Change 850 millimeters to centimeters. 8. Change 12 pints to gallons. 9. Change 10.5 yards to inches 4 11. Change yard to feet. 3 13. Change 1.2 pounds to ounces. 15. Change 44 centimeters to millimeters. Applying Skills 10. Change 31 2 feet to inches. 12. Change 1.5 meters to centimeters. 14. Change 2.5 miles to feet. 21 2 16. Change gallons to |
quarts. 17. Miranda needs boards 0.8 meters long for a building project. The boards available at the local lumberyard are 2 feet, 3 feet, and 4 feet long. a. Express the length, to the nearest hundredth of a foot, of the boards that Miranda needs to buy. b. Which size board should Miranda buy? Explain your answer. 18. Carlos needs 24 inches of fabric for a pillow that he is making. The fabric store has a piece of material of a yard long that is already cut that he can buy for $5.50. If he has the exact size piece he needs cut from a bolt of fabric, it will cost $8.98 a yard. 3 4 a. Is the piece of material that is already cut large enough for his pillow? b. What would be the cost of having exactly 24 inches of fabric cut? c. Which is the better buy for Carlos? 19. A highway sign in Canada gives the speed limit as 100 kilometers per hour. Tracy is driving at 62 miles per hour. One mile is approximately equal to 1.6 kilometers. a. Is Tracy exceeding the speed limit? b. What is the difference between the speed limit and Tracy’s speed in miles per hour? 20. Taylor has a painting for which she paid 1 million yen when she was traveling in Japan. At that time, the exchange rate was 1 dollar for 126 yen. A friend has offered her $2,000 for the painting. a. Is the price offered larger or smaller than the purchase price? b. If she sells the painting, what will be her profit or loss, in dollars? c. Express the profit or loss as a percent of increase or decrease in the price of the painting. 238 Ratio and Proportion CHAPTER SUMMARY A ratio, which is a comparison of two numbers by division, is the quotient obtained when the first number is divided by a second, nonzero number. Quantities in a ratio are expressed in the same unit of measure before the quotient is found. Ratio of a to b: a b or a : b A rate is a comparison of two quantities that may have different units of measure, such as a rate of speed in miles per hour. A rate that has a denominator of 1 is called a unit rate. A proportion is an equation stating that two ratios are equal. Standard ways to write a proportion are shown below. In a proportion a : b c : d, the outer terms are called the |
extremes, and the inner terms are the means. Proportion: means a : b c : d extremes or extreme mean ↓ ↓ a b 5 c d ↑ ↑ mean extreme In a proportion, the product of the means is equal to the product of the extremes, or alternatively, the cross products are equal. This process is also called cross-multiplication. A direct variation is a relation between two variables such that their ratio is always the same value, called the constant of variation. For example, the diameter of a circle is always twice the radius, so 2 shows a direct variation between d and r with a constant of variation 2. d r A percent (%), which is a ratio of a number to 100, is also called a rate. Here, the word rate is treated as a comparison of a quantity to a whole. In basic formulas, such as those used with discounts and taxes, the base and percentage are numbers, and the rate is a percent. percentage base rate or base rate percentage The percent of error is the ratio of the absolute value of the difference between a measured value and a true value to the true value, expressed as a percent. Percent of error zmeasured value 2 true valuez true value 100% The relative error is the percent of error expressed as a decimal. Review Exercises 239 The percent of increase or decrease is the ratio of the absolute value of the difference between the original value and the new value to the original value. Percent of increase or decrease zoriginal value 2 new valuez original value 100% VOCABULARY 6-1 Ratio • Equivalent ratios • Simplest form • Continued ratio 6-2 Rate • Lowest terms • Unit rate • Unit price 6-4 Proportion • Extremes • Means • Cross-multiplying • Cross product 6-5 Direct variation • Directly proportional • Constant of variation 6-6 Percent • Base • Rate • Percentage • Error • Percent of error • Relative error • Percent increase • Percent decrease 6-7 Dimensional analysis REVIEW EXERCISES 1. Can an 8 inch by 12 inch photograph be reduced to a 3 inch by 5 inch pho- tograph? Explain why or why not? 2. Karen has a coupon for an additional 20% off the sale price of any dress. She wants to buy a dress that is on sale for 15% off of the original price. Will the original price of the dress be reduced by 35%? Explain why or why not. In 3–6, express each |
ratio in simplest form. 3. 30 : 35 4. 8w to 12w 6. 75 millimeters : 15 centimeters 5. 3 8 to 5 8 In 7–9, in each case solve for x and check. 7. 8 2x 5 12 9 8. x x 1 5 5 1 2 9. x 5 6 4 x 1 3 10. The ratio of two numbers is 1 : 4, and the sum of these numbers is 40. Find the numbers. 240 Ratio and Proportion In 11–13, in each case, select the numeral preceding the choice that makes the statement true. 11. In a class of 9 boys and 12 girls, the ratio of the number of girls to the number of students in the class is (1) 3 : 4 (2) 4 : 3 (3) 4 : 7 (4) 7 : 4 12. The perimeter of a triangle is 45 centimeters, and the lengths of its sides are in the ratio 2 : 3 : 4. The length of the longest side is (1) 5 cm (3) 20 cm (2) 10 cm 13. If a : x b : c, then x equals (4) 30 cm (1) ac b (2) bc a (3) ac – b (4) bc – a 14. Seven percent of what number is 21? 15. What percent of 36 is 45? 16. The sales tax collected on each sale varies directly as the amount of the sale. What is the constant of variation if a sales tax of $0.63 is collected on a sale of $9.00? 17. If 10 paper clips weigh 11 grams, what is the weight in grams of 150 paper clips? 18. Thelma can type 150 words in 3 minutes. At this rate, how many words can she type in 10 minutes? 19. What is the ratio of six nickels to four dimes? 20. On a stormy February day, 28% of the students enrolled at Southside High School were absent. How many students are enrolled at Southside High School if 476 students were absent? 21. After a 5-inch-by-7-inch photograph is enlarged, the shorter side of the enlargement measures 15 inches. Find the length in inches of its longer side. 22. A student who is 5 feet tall casts an 8-foot shadow. At the same time, a tree casts a 40-foot shadow. How many feet tall is the tree? 23. If |
four carpenters can build four tables in 4 days, how long will it take one carpenter to build one table? 24. How many girls would have to leave a room in which there are 99 girls and 1 boy in order that 98% of the remaining persons would be girls? 25. On an Australian highway, the speed limit was 110 kilometers per hour. A motorist was going 70 miles per hour. (Use 1.6 kilometers 1 mile) a. Should the motorist be stopped for speeding? b. How far over or under the speed limit was the motorist traveling? Review Exercises 241 26. The speed of light is 3.00 105 kilometers per second. Find the speed of light in miles per hour. Use 1.61 kilometers 1 mile. Write your answer in scientific notation with three significant digits. 27. Which offer, described in the chapter opener on page 207, is the better buy? Does the answer depend on the price of a package of frozen vegetables? Explain. 28. A proposal was made in the state senate to raise the minimum wage from $6.75 to $7.15 an hour. What is the proposed percent of increase to the nearest tenth of a percent? 29. In a chemistry lab, a student measured 1.0 cubic centimeters of acid to use in an experiment. The actual amount of acid that the student used was 0.95 cubic centimeters. What was the percent of error in the student’s measurement? Give your answer to the nearest tenth of a percent. Exploration a. Mark has two saving accounts at two different banks. In the first bank with a yearly interest rate of 2%, he invests $585. In the second bank with a yearly interest rate of 1.5%, he invests $360. Mark claims that at the end of one year, he will make a total of 2% 1.5% 3.5% on his investments. Is Mark correct? b. Two different clothing items, each costing $30, were on sale for 10% off the ticketed price. The manager of the store claims that if you buy both items, you will save a total of 20%. Is the manager correct? c. In a class of 505 graduating seniors, 59% were involved in some kind of after-school club and 41% played in a sport. The principal of the school claims that 59% 41% 100% of the graduating seniors were involved in some kind of after-school activity. Is the principal correct? d. A certain |
movie is shown in two versions, the original and the director’s cut. However, movie theatres can play only one of the versions. A journalist for XYZ News, reports that since 30% of theatres are showing the director’s cut and 60% are showing the original, the movie is playing in 90% of all movie theatres. Is the reporter correct? e. Based on your answers for parts a through d, write a rule stating when it makes sense to add percents. 242 Ratio and Proportion CUMULATIVE REVIEW CHAPTERS 1–6 Part I Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. 1. When –4ab is subtracted from ab, the difference is (1) –3ab (2) –4 (3) 5ab (4) –5ab 2. If n – 4 represents an odd integer, the next larger odd integer is (1) n – 2 (2) n – 3 (3) n – 5 (4) n – 6 3. The expression 2.3 10–3 is equal to (1) 230 (2) 2,300 (3) 0.0023 (4) 0.023 4. The product 2x3y(–3xy4) is equal to (1) 6x3y4 (2) 6x4y5 (3) 6x4y5 (4) 6x3y4 5. What is the multiplicative inverse of? 3 2 (1) 23 2 (2) 2 3 6. If 0.2x 4 x 0.6, then x equals (1) 46 (2) 460 7. The product 34 33 (3) 22 3 (4) 1 (3) 5.75 (4) 0.575 (1) 37 (2) 37 (3) 97 (4) 97 8. The diameter of a circle whose area is 144p square centimeters is (1) 24p cm (2) 6 cm (3) 12 cm (4) 24 cm 9. Jeannine paid $88 for a jacket that was on sale for 20% off the original price. The original price of the jacket was (1) $105.60 (2) $110.00 (3) $440.00 10. When a 2 and b 5, a2 ab equals (1) 14 (2) 14 (3) 6 |
Part II (4) $70.40 (4) 6 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. Find two integers, x and x 1, whose squares differ by 25. 12. Solve and check: 4(2x 1) 5x 5 Cumulative Review 243 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. Rectangle ABCD is separated into three squares as shown in the diagram at the right. The length of DA is 20 centimeters. a. Find the measure of AB. b. Find the ratio of the area of square AEFD to the area of rectangle ABCD. D 20 cm A F G E C H B 14. Sam drove a distance of 410 miles in 7 hours. For the first part of the trip his average speed was 40 miles per hour and for the remainder of the trip his average speed was 60 miles per hour. How long did he travel at each speed? 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. Three friends started a part-time business. They plan, each month, to share the profits in the ratio of the number of hours that each worked. During the first month, Rita worked 18 hours, Fred worked 30 hours, and Glen worked 12 hours. a. Express, in lowest terms, the ratio of the times that they worked during the first month. b. Find the amount each should receive if the profit for this first month was $540. c. Find the amount each should receive if the profit for this first month was $1,270. 244 Ratio and Proportion 16. A skating rink is in the form of a rectangle with a semicircle at each end as shown in the diagram. The rectangle is 150 feet long and 64 feet wide. Scott skates around the rink 2.5 feet from |
the edge. 64 ft 150 ft a. Scott skates once around the rink. Find, to the nearest ten feet, the dis- tance that he skated. b. Scott wants to skate at least 5 miles. What is the smallest number of complete trips around the rink that he must make? GEOMETRIC FIGURES, AREAS, AND VOLUMES A carpenter is building a deck on the back of a house. As he works, he follows a plan that he made in the form of a drawing or blueprint. His blueprint is a model of the deck that he is building. He begins by driving stakes into the ground to locate corners of the deck. Between each pair of stakes, he stretches a cord to indicate the edges of the deck. On the blueprint, the stakes are shown as points and the cords as segments of straight lines as shown in the sketch. • • • • • • At each corner, the edges of the deck meet at an angle that can be classified according to its size. Geometry combines points, lines, and planes to model the world in which we live. An understanding of geometry enables us to understand relationships involving the sizes of physical objects and the magnitude and direction of the forces that interact in daily life. In this chapter, you will review some of the information that you already know to describe angles and apply this information to learn more about geometry. CHAPTER 7 CHAPTER TABLE OF CONTENTS 7-1 Points, Lines, and Planes 7-2 Pairs of Angles 7-3 Angles and Parallel Lines 7-4 Triangles 7-5 Quadrilaterals 7-6 Areas of Irregular Polygons 7-7 Surface Areas of Solids 7-8 Volumes of Solids Chapter Summary Vocabulary Review Exercises Cumulative Review 245 246 Geometric Figures, Areas, and Volumes 7-1 POINTS, LINES, AND PLANES Undefined Terms We ordinarily define a word by using a simpler term. The simpler term can be defined by using one or more still simpler terms. But this process cannot go on endlessly; there comes a time when the definition must use a term whose meaning is assumed to be clear to all people. Because the meaning is accepted without definition, such a term is called an undefined term. In geometry, we use such ideas as point, line, and plane. Since we cannot give satisfactory definitions of these words by using simpler defined words, we will consider them to be undefined terms. Although point, line, and |
plane are undefined words, we must have a clear understanding of what they mean. Knowing the properties and characteristics they possess helps us to achieve this understanding. A point indicates a place or position. It has no length, width, or thickness. A point is usually indicated by a dot and named with a capital letter. For example, point A is shown on the left. A line is a set of points. The set of points may form a curved line, a broken line, or a straight line. A straight line is a line that is suggested by a stretched string but that extends without end in both directions. A Curved line Broken line A B l Straight line Unless otherwise stated, in this discussion the term line will mean straight line. A line is named by any two points on the line. For example, the straight line g BA shown above is line AB or line BA, usually written as. A line can also be named by one lowercase letter, for example, line l shown above. The arrows remind us that the line continues beyond what is drawn in the diagram. g AB or P A plane is a set of points suggested by a flat surface. A plane extends endlessly in all directions. A plane may be named by a single letter, as plane P shown on the right. A plane can also be named by three points of the plane, as plane ABC in the diagram on the right. A C B Facts About Straight Lines A statement that is accepted as true without proof is called an axiom or a postulate. If we examine the three accompanying figures pictured on the next page, we see that it is reasonable to accept the following three statements as postulates: S R Points, Lines, and Planes 247 1. In a plane, an infinite number of straight lines can be drawn through a given point. 2. One and only one straight line can be drawn that contains two given points. (Two points determine a straight line.) 3. In a plane, two different nonparallel straight lines will intersect at only one point. Line Segments The undefined terms, point, line, and plane, are used to define other geometric terms. A line segment or segment is a part of a line consisting of two points called endpoints and all points on the line between these endpoints. At the left is pictured a line segment whose endpoints are points R and S. We use these endpoints to name this segment as segment RS, which may be written as SR. RS or Recall that the measure of a line segment |
or the length of a line segment is the distance between its endpoints. We use a number line to associate a number with each endpoint. Since the coordinate of A is 0 and the coordinate of B is 5, the length of AB is 5 0 or AB 5 Note: The segment is written as endpoints. The length of the segment is written as AB, with no bar over the letters that name the endpoints., with a bar over the letters that name the AB Half-Lines and Rays When we choose any point A on a line, the two sets of points that lie on opposite sides of A are called half-lines. Note that point A is not part of the half-line. In the diagram below, point A separates into two half-lines. g CD CB BD The two points B and D belong to the same half. Points B and C, howline since A is not a point of ever, do not belong to the same half-line since A is a point of. All points in the same half-line are said to be on the same side of A. We often talk about rays of sunlight, that is, the sun and the path that the sunlight travels to the earth. The ray of sunlight can be thought of as a point and a half-line. D B A C In geometry, a ray is a part of a line that consists of a point on the line, called an endpoint, and all the points on one side of the endpoint. To name a ray we use two capital letters and an arrow with one arrowhead. The first letter must be 248 Geometric Figures, Areas, and Volumes the letter that names the endpoint. The second letter is the name of any other point on the ray. The figure on the right shows ray AB, which is writ. This ray could also be called ray AC, written h AB ten as h AC. as Two rays are called opposite rays if they are rays of the same line that have a common endpoint but no other points in common. In the diagram on the left, h PQ are opposite rays. h PR and A C B Angles An angle is a set of points that is the union of two rays having the same endpoint. The common endpoint of the two rays is the vertex of the angle. The two rays forming the angle are also called the sides of the angle. In the diagram on the left, we can think of TOS as having been formed by h OT. The in a counterclockwise |
direction about O to the position rotating, that have the common endpoint O, is TOS. union of the two rays, Note that when three letters are used to name an angle, the letter that names the vertex is always in the middle. Since are the only rays in the diagram that have the common endpoint O, the angle could also have been called O. h OT h OT h OS h OS h OS and and Q P R S O T Measuring Angles To measure an angle means to determine the number of units of measure it contains. A common standard unit of measure of an angle is the degree; 1 degree is of the sum of all of the distinct angles about a point. written as 1°. A degree is In other words, if we think of an angle as having been formed by rotating a ray around its endpoint, then rotating a ray consecutively 360 times in 1-degree increments will result in one complete rotation. 1 360 A C E D Types of Angles Angles are classified according to their measures. g CD A right angle is an angle whose measure is 90°. In the diagram on the left, g AB intersect at E so that the four angles formed are right angles. The and measure of each angle is 90° and the sum of the angles about point E is 360°. We can say that B mAEC mCEB mBED mDEA 90. Note that the symbol mAEC is read “the measure of angle AEC.” In this book, the angle measure will always be given in degrees and the degree symbol will be omitted when using the symbol “m” to designate measure. I h H G Acute angle Points, Lines, and Planes 249 An acute angle is an angle whose measure is greater than 0° and less than 90°, that is, the measure of an acute angle is between 0° and 90°. Angle GHI, which can also be called h, is an acute angle (0 mh 90). An obtuse angle is an angle whose measure is greater than 90° but less than 180°; that is, its measure is between 90° and 180°. Angle LKN is an obtuse angle where 90 mLKN 180. A straight angle is an angle whose measure is 180°. Angle RST is a straight angle where mRST 180. A straight angle is the union of two opposite rays and forms a straight line. Here are three important facts about angles: N k K |
L Obtuse angle 180° T s S Straight angle R 1. The measure of an angle depends only on the amount of rotation, not on the pictured lengths of the rays forming the angle. 2. Since every right angle measures 90°, all right angles are equal in measure. 3. Since every straight angle measures 180°, all straight angles are equal in measure. Perpendicularity Two lines are perpendicular if and only if the two lines intersect to form right angles. The symbol for perpendicularity is ⊥. g PR is perpendicular to g AB P bolized as In the diagram, g PR, sym— is used in a diagram to indicate that a right angle exists where the. The symbol g'AB A B R perpendicular lines intersect. Segments of perpendicular lines that contain the point of intersection of the lines are also perpendicular. In the diagram on the left, g ST'CD. S C T D EXERCISES Writing About Mathematics 1. Explain how the symbols AB, AB, and g AB differ in meaning. 2. Explain the difference between a half-line and a ray. 250 Geometric Figures, Areas, and Volumes Developing Skills In 3–8, tell whether each angle appears to be acute, right obtuse, or straight. 3. 6. 4. 7. 5. 8. 9. For the figure on the right: a. Name x by using three capital letters. b. Give the shorter name for COB. c. Name one acute angle. d. Name one obtuse angle. C B y x O A Applying Skills In 10–13, find the number of degrees in the angle formed by the hands of a clock at each given time. 10. 1 P.M. 11. 4 P.M. 12. 6 P.M. 13. 5:30 P.M. 14. At what time do the hands of the clock form an angle of 0°? 15. At what times do the hands of a clock form a right angle? 7-2 PAIRS OF ANGLES Adjacent Angles D B C A Adjacent angles are two angles in the same plane that have a common vertex and a common side but do not have any interior points in common. In the figure on the left, ABC and CBD are adjacent angles. Complementary Angles Two angles are complementary angles if and only if the sum of their measures is 90°. Each angle is the complement of the other |
. Pairs of Angles 251 In the figures shown below, because mCAB mFDE 25 65 90, CAB and FDE are complementary angles. Also, because mHGI mIGJ 53 37 90, HGI and IGJ are complementary angles. C 25° B A E F 65° D J I 37° 53° G H If the measure of an angle is 50°, the measure of its complement is (90 50)°, or 40°. In general, If the measure of an angle is x°, the measure of its complement is (90 x)°. Supplementary Angles Two angles are supplementary angles if and only if the sum of their measures is 180°. Each angle is the supplement of the other. As shown in the figures below, because mLKM mONP 50 130 180, LKM and ONP are supplementary angles. Also, RQS and SQT are supplementary angles because mRQS mSQT 115 65 180. M L P S 50° K 130° 65° 115° N O T Q R If the measure of an angle is 70°, the measure of its supplement is (180 70)°, or 110°. In general, If the measure of an angle is x°, the measure of its supplement is (180 x)°. 252 Geometric Figures, Areas, and Volumes Linear Pair DA. Draw and any Through two points, D and A, draw a line. Choose any point B on. The adjacent angles formed, DBC and CBA, point C not on are called a linear pair. A linear pair of angles are adjacent angles that are supplementary. The two sides that they do not share in common are opposite rays. The term linear tells us that a line is used to form this pair. Since the angles are supplementary, if mDBC x, then mCBA (180 x). h BC DA Vertical Angles g AB g CD intersect at E, then x and y share a If two straight lines such as common vertex at E but do not share a common side. Angles x and y are a pair of vertical angles. Vertical angles are two nonadjacent angles formed by two intersecting lines. In the diagram on the left, angles a and b are another pair of vertical angles. and If two lines intersect, four angles are formed that have no common interior intersect at E. There are four linear pairs of g CD g AB and point. In the diagram, |
angles: AED and DEB DEB and BEC BEC and CEA CEA and AED The angles of each linear pair are supplementary. • If mAED 130, then mDEB 180 130 50. • If mDEB 50, then mBEC 180 50 130. Therefore, mAED mBEC. A C 130° E D 50° 130° B DEFINITION When two angles have equal measures, they are congruent. We use the symbol to represent the phrase “is congruent to.” Since mAED mBEC, we can write AED BEC, read as “angle AED is congruent to angle BEC.” There are different correct ways to indicate angles with equal measures: 1. The angle measures are equal: mBEC mAED 2. The angles are congruent: BEC AED It would not be correct to say that the angles are equal, or that the angle measures are congruent. Notice that we have just shown that the two vertical angles BEC and AED are congruent. If we were to draw and measure additional pairs of vertical angles, we would find in each case that the vertical angles would be equal Pairs of Angles 253 in measure. No matter how many examples of a given situation we consider, however, we cannot assume that a conclusion that we draw from these examples will always be true. We must prove the conclusion. Statements that we prove are called theorems. We will use algebraic expressions and properties to write an informal proof of the following statement. If two lines intersect, the vertical angles formed are equal in measure; that is, they are congruent. (1) If g AB and g CD intersect at E, then AEB is a straight angle whose measure is 180°. Therefore, mAEC mCEB 180. (2) Let mAEC x. Then mCEB 180 x. (3) Likewise, CED is a straight angle whose measure is 180°. Therefore, mCEB mBED 180. (4) Since mCEB 180 x, then mBED 180 (180 x) x. C x 180 – x A E x B D (5) Since both mAEC x and mBED x, then mAEC mBED; that is, AEC BED. EXAMPLE 1 The measure of the complement of an angle is 4 times |
the measure of the angle. Find the measure of the angle. Solution Let x measure of angle. Then 4x measure of complement of angle. The sum of the measures of an angle and its complement is 90°. x 4x 90 5x 90 x 18 Check The measure of the first angle is 18°. The measure of the second angle is 4(18°) 72°. The sum of the measures of the angles is 18° 72° 90°. Thus, the angles are complementary. ✔ Answer The measure of the angle is 18°. 254 Geometric Figures, Areas, and Volumes Note: The unit of measure is very important in the solution of a problem. While it is not necessary to include the unit of measure in each step of the solution, each term in an equation must represent the same unit and the unit of measure must be included in the answer. EXAMPLE 2 The measure of an angle is 40° more than the measure of its supplement. Find the measure of the angle. Solution Let x the measure of the supplement of the angle. Then x 40 the measure of the angle. The sum of the measures of an angle and its supplement is 180°. x (x 40) 180 2x 40 180 2x 140 x 70 x 40 110 Check The sum of the measures is 110° 70° 180° and 110° is 40° more than 70°. ✔ Answer The measure of the angle is 110°. EXAMPLE 3 The algebraic expressions 5w 20 and 2w 16 represent the measures in degrees of a pair of vertical angles. a. Find the value of w. b. Find the measure of each angle. Solution a. Vertical angles are equal in measure. 5w 20 2w 16 3w 20 16 3w 36 w 12 b. 5w 20 5(12) 20 60 20 40 2w 16 2(12) 16 24 16 40 Check Since each angle has a measure of 40°, the vertical angles are equal in measure. ✔ Answers a. w 12 b. The measure of each angle is 40°. Pairs of Angles 255 EXERCISES Writing About Mathematics 1. Show that supplementary angles are always two right angles or an acute angle and an obtuse angle. 2. The measures of three angles are 15°, 26°, and 49°. Are these angles complementary? Explain why or why not. Developing Skills In 3–6, answer each of the following questions for an angle with the given measure. a |
. What is the measure of the complement of the angle? b. What is the measure of the supplement of the angle? c. The measure of the supplement of the angle is how much larger than the measure of its complement? 3. 15° 4. 37° 5. 67° 6. x° In 7–10, A and B are complementary. Find the measure of each angle if the measures of the two angles are represented by the given expressions. Solve the problem algebraically using an equation. 7. mA x, mB 5x 9. mA x, mB x 40 8. mA x, mB x 20 10. mA y, mB 2y 30 In 11–14, ABD and DBC are supplementary. Find the measure of each angle if the measures of the two angles are represented by the given expressions. Solve the problem algebraically using an equation. 11. mABD x, mDBC 3x 12. mABD x, mDBC x 80 13. mDBC x, mABD x 30 14. mDBC y, mABD 1 4y D A B C In 15–24, solve each problem algebraically using an equation. 15. Two angles are supplementary. The measure of one angle is twice as large as the measure of the other. Find the number of degrees in each angle. 16. The complement of an angle is 14 times as large as the angle. Find the measure of the com- plement. 17. The measure of the supplement of an angle is 40° more than the measure of the angle. Find the number of degrees in the supplement. 256 Geometric Figures, Areas, and Volumes 18. Two angles are complementary. One angle is twice as large as the other. Find the number of degrees in each angle. 19. The measure of the complement of an angle is one-ninth the measure of the angle. Find the measure of the angle. 20. Find the number of degrees in the measure of an angle that is 20° less than 4 times the mea- sure of its supplement. 21. The difference between the measures of two supplementary angles is 80°. Find the measure of the larger of the two angles. 22. The complement of an angle measures 20° more than the angle. Find the number of degrees in the angle. 23. Find the number of degrees in an angle that measures 10° more than its supplement. 24. Find the number |
of degrees in an angle that measures 8° less than its complement. 25. The supplement of the complement of an acute angle is always: (1) an acute angle (2) a right angle (3) an obtuse angle (4) a straight angle In 26–28, g MN and g RS intersect at T. 26. If mRTM 5x and mNTS 3x 10, find mRTM. 27. If mMTS 4x 60 and mNTR 2x, find mMTS. 28. If mRTM 7x 16 and mNTS 3x 48, find mNTS. In 29–34, find the measure of each angle named, based on the given information. 29. Given: g EF h'GH ; mEGI 62. H I 62° E G F a. Find mFGH. b. Find mHGI. 30. Given: g JK mNLM 48. h'LM ; g NLO is a line; M N 48° J L K O a. Find mJLN. b. Find mMLK. c. Find mKLO. d. Find mJLO. 31. Given: GKH and HKI are a linear pair; h KH h'KJ ; mIKJ 34. mPMN 40. Pairs of Angles 257 32. Given: h MO h'MP g ; LMN is a line; H I G K 34° J a. Find mHKI. b. Find mHKG. c. Find mGKJ. 33. Given: g ABC ABD DBE. is a line; mEBC 40; D 40° A B a. Find mABD. b. Find mDBC. E C O P 40° L M N a. Find mPMO. b. Find mOML. 34. Given: g FI intersects g JH at K; mHKI 40; mFKG mFKJ. G H 40° F K I J a. Find mFKJ. b. Find mFKG. c. Find mGKH. d. Find mJKI. In 35–38, sketch and label a diagram in each case and find the measure of each angle named. 35. intersects g AB a. mCEB g CD at E; mAED 20. Find: b. mBED 36. PQR and R |
QS are complementary; mPQR 30; a. mRQS b. mSQT c. mCEA g RQT c. mPQT is a line. Find: 258 Geometric Figures, Areas, and Volumes at R. The measure of LRQ is 80 more than mLRP. Find: b. mLRQ c. mPRM at E. Point F is in the interior of CEB. The measure of CEF is 8 times the 37. 38. g PQ intersects g LM a. mLRP g g'AB CD measure of FEB. Find: a. mFEB b. mCEF c. mAEF 39. The angles, ABD and DBC, form a linear pair and are congruent. What must be true about g ABC and g BD? 7-3 ANGLES AND PARALLEL LINES A C B D In the figure on the left, g AB Not all lines in the same plane intersect. Two or more lines are called parallel lines if and only if the lines lie in the same plane but do not intersect. g g AB CD and is parallel to lie in the same plane but do not interg CD sect. Hence, we say that. Using the symbol || for is parallel. When we speak of two parallel lines, we will mean two to, we write distinct lines. (In more advanced courses, you will see that a line is parallel to itself.) Line segments and rays are parallel if the lines that contain them are parlie in the same plane, they must be either allel. If two lines such as intersecting lines or parallel lines, as shown in the following figures. g i CD g AB g CD g AB and C A B D g g AB intersects CD. C D B A g g AB is parallel to CD. g AB g CD When two lines such as are parallel, they have no points in common. We can think of each line as a set of points. Hence, the intersection set of g AB and When two lines are cut by a third line, called a transversal, two sets of is the empty set symbolized as g d CD 5. g CD g AB and 1 2 43 5 6 7 8 angles, each containing four angles, are formed. In the figure on the left: • Angles 3, 4, 5, 6 are called interior angles. • Angles 1, 2, 7, 8 |
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