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Mark Dugopolski, Elementary and Intermediate Algebra, 4th edition, McGraw Hill, 2011

Section 2.1

89. Births to teenagers. In 2006 there were 41.8 births per 1000 females 15 to 19 years of age. This birth rate is 2/3 of the birth rate for teenagers in 1991. a) Write an equation and solve it to find the birth rate for teenagers in 1991. b) Use the accompanying graph to estimate the birth rate to teenagers in 2000.

90. World grain demand. Freeport McMoRan projects that in 2015 world grain supply will be 2.1 trillion metric tons and the supply will be only ¾ of world grain demand. What will world grain demand be in 2015?

91. Advancers and decliners. On Thursday, 2/3 of the stocks traded on the New York Stock Exchange advanced in price. If 1918 stocks advanced, then how many stocks were traded on that day?

92. In 2009, two-fifths of all births in the United States were to unmarried women. If there were 1,707,600 births to unmarried women, then how many births were there in 2009?

93. College students. At Springfield College 40% of the students are male. If there are 1200 males, then how many students are there at the college?

94. Credit card revenue. Seventy percent of the annual revenue for a credit card company comes from interest and penalties. If the amount for interest and penalties was $210 million, then what was the annual revenue?


Section 2.2


87. The practice. A lawyer charges $300 plus $65 per hour for a divorce. If the total charge for Bill’s divorce was $1405, then for what number of hours did the lawyer work on the case?

88. The plumber. Tamika paid $165 to her plumber for a service call. If her plumber charges $45 plus $40 per hour for a service call, then for how many hours did the plumber work?

89. Celsius temperature. If the air temperature in Quebec is 68° Fahrenheit, then the solution to the equation (9/5)C + 32 = 68 gives the Celsius temperature of the air. Find the Celsius temperature.

91. Rectangular patio. If a rectangular patio has a length that is 3 feet longer than its width and a perimeter of 42 feet, then the width can be found by solving the equation 2x + 2(x+3)= 42. What is the width?

92. Perimeter of a triangle. The perimeter of the triangle shown in the accompanying figure is 12 meters. Determine the values of x, x + 1, and x + 2 by solving the equation x + (x + 1) + (x + 2) = 12.

93. Cost of a car. Jane paid 9% sales tax and a $150 title and license fee when she bought her new Saturn for a total of $16,009.50. If x represents the price of the car, then x satisfies x + 0.09x + 150 = 16,009.50. Find the price of the car by solving the equation.

94. Cost of labor. An electrician charged Eunice $29.96 for a service call plus $39.96 per hour for a total of $169.82 for installing her electric dryer. If n represents the number of hours for labor, then n satisfies 39.96n + 29.96 = 169.82. Find n by solving this equation.

Section 2.3

87. Danielle sold her house through an agent who charged 8% of the selling price. After the commission was paid, Danielle received $117,760. If x is the selling price, then x satisfies x – 0.08x = 117,760. Solve this equation to find the selling price.

88. Raising rabbits. Before Roland sold two female rabbits, half of his rabbits were female. After the sale, only one-third of his rabbits were female. If x represents his original number of rabbits, then 1/2x – 2 = 1/3(x – 2). Solve this equation to find the number of rabbits that he had before the sale.

89. Eavesdropping. Reginald overheard his boss complaining that his federal income tax for 2009 was $60,531 a) Use the accompanying graph to estimate his boss’s taxable income for 2009. b) Find his boss’s exact taxable income for 2009 by solving the equation 46,742 + 0.33(x – 208,850) = 60,531

90. According to Bruce Harrell, CPA, the federal income tax for a class C corporation is found by solving a linear equation. The reason for the equation is that the amount x of federal tax is deducted before the state tax is figured, and the amount of state tax is deducted before the federal tax is figured. To find the amount of federal tax for a corporation with a taxable income of $200,000, for which the federal tax rate is 25% and the state tax rate is 10%, Bruce must solve
x = 0.25[200,000 – 0.10(200,000 – x)].

Section 2.4

68. Finding the rate. A loan of $1000 is made for 7 years. Find the interest rate for simple interest amounts of $420, $455, and $472.50.

69. Finding the time. Kathy paid $500 in simple interest on a loan of $2500. If the annual interest rate was 5%, then what was the time?

70. Finding the time. Robert paid $240 in simple interest on a loan of $1000. If the annual interest rate was 8%, then what was the time?

71. Finding the length. The area of a rectangle is 28 square yards. Find the length if the width is 2 yards, 3 yards, or 4 yards.

72. Finding the width. The area of a rectangle is 60 square feet. Find the width if the length is 10 feet, 16 feet, or 18 feet.

73. Finding the length. If it takes 600 feet of wire fencing to fence a rectangular feed lot that has a width of 75 feet, then what is the length of the lot?

74. Finding the depth. If it takes 500 feet of fencing to enclose a rectangular lot that is 104 feet wide, then how deep is the lot?

75. Finding MSRP. What was the manufacturer’s suggested retail price (MSRP) for a Lexus SC 430 that sold for $54,450 after a 10% discount?

76. Finding MSRP. What was the MSRP for a Hummer H1 that sold for $107,272 after an 8% discount?

77. Finding the original price. Find the original price if there is a 15% discount and the sale price is $255.

78. Finding the list price. Find the list price if there is a 12% discount and the sale price is $4400.The area of a rectangle is 60 square feet. The length is 4 feet. Find the width.

79. Rate of discount. Find the rate of discount if the discount is $40 and the original price is $200.

80. Rate of discount. Find the rate of discount if the discount is $20 and the original price is $250.

81. Width of a football field. The perimeter of a football field in the NFL, excluding the end zones, is 920 feet. How wide is the field? See the figure on the next page.

82. Perimeter of a frame. If a picture frame is 16 inches by 20 inches, then what is its perimeter?

83. Volume of a box. A rectangular box measures 2 feet wide, 3 feet long, and 4 feet deep. What is its volume?

84. Volume of a refrigerator. The volume of a rectangular refrigerator is 20 cubic feet. If the top measures 2 feet by feet, then what is the height?

85. Radius of a pizza. If the circumference of a pizza is 8π inches, then what is the radius?

86. Diameter of a circle. If the circumference of a circle is 4π meters, then what is the diameter?

87. Height of a banner. If a banner in the shape of a triangle has an area of 16 square feet with a base of 4 feet, then what is the height of the banner?

88. Length of a leg. If a right triangle has an area of 14 square meters and one leg is 4 meters in length, then what is the length of the other leg?

89. Length of the base. A trapezoid with height 20 inches and lower base 8 inches has an area of 200 square inches. What is the length of its upper base?

90. Height of a trapezoid. The end of a flower box forms the shape of a trapezoid. The area of the trapezoid is 300 square centimeters. The bases are 16 centimeters and 24 centimeters in length. Find the height.

91. Fried’s rule. Doctors often prescribe the same drugs for children as they do for adults. The formula d = 0.08aD (Fried’s rule) expresses the child’s dosage d as a function of the adult dosage D and the child’s age a. a) If a doctor prescribes 1000 milligrams of acetaminophen for an adult, then how many milligrams would he prescribe for an 8-year-old child? b) If a doctor uses Fried’s rule to prescribe 200 milligrams of a drug to a child when he would prescribe 600 milligrams to an adult, then how old is the child? c) Use the accompanying bar graph to determine the age at which  a  child  would  get  the  same  dosage  as  an  adult.

92. Cowling’s rule. Cowling’s rule is another function for determining the child’s dosage of a drug. For this rule, the formula d = D(a+1)/24 expresses the child’s dosage d as a function of the adult dosage D and the child’s age a. a) If a doctor prescribes 1000 milligrams of acetaminophen for an adult, then how many milligrams would she prescribe for an eight-year-old child using Cowling’s rule? b) If  a  doctor  uses  Cowling’s  rule  to  prescribe  200  milligrams of a drug to a child when she would prescribe 600 milligrams to an adult, then how old is the child?

93. Administering vancomycin. A patient is to receive 750 milligrams (desired dose) of the antibiotic vancomycin. However, vancomycin comes in a solution containing 1000 milligrams (available dose) of vancomycin per 5 milliliters (quantity) of solution. The amount of solution to be given to the patient is a function of the desired dose, the available dose, and the quantity, given by the formula Amount = (desired dose)/(available dose) × quantity. Find the amount of the solution that should be administered to the patient.

94. International communications. The global investment in telecom infrastructure since 1990 can be modeled by the function I = 7.5t + 115, where I is in billions of dollars and t is the number of years since 1990. a) Use the formula to find the global investment in 2000. b) Use the accompanying graph to estimate the year in which the global investment will reach $300 billion. c) Use the formula to find the year in which the global investment will reach $300 billion.


Section 6.4

83. Traveling time.  Janet drove 120 miles at x mph before 6:00 A.M. After 6:00 A.M., she increased her speed by 5 mph and drove 195 additional miles. Use the fact that T = D/R to complete the following table. Write a rational expression for her total traveling time. Evaluate the expression for x = 60.

84. Traveling time.  After leaving Moose Jaw, Hanson drove 200 kilometers at x km/hr and then decreased his speed by 20 km/hr and drove 240 additional kilometers. Make a table like the one in Exercise 83. Write a rational expression for his total traveling time. Evaluate the expression for x = 100.

85. House painting.  Kent can paint a certain house by himself in x days. His helper Keith can paint the same house by himself in x = 3 days. Suppose that they work together on the job for 2 days. To complete the table on the next page, use the fact that the work completed is the product of the rate and the time. Write a rational expression for the fraction of the house that they complete by working together for 2 days. Evaluate the expression for x = 6.

86. Barn painting.  Melanie can paint a certain barn by herself in x days. Her helper Melissa can paint the same barn by herself in 2x days. Write a rational expression for the fraction of the barn that they complete in one day by working together. Evaluate the expression for x = 5.

Section 6.5


61. Sophomore math.  A survey of college sophomores showed that 5/6 of the males were taking a mathematics class and ¾ of the females were taking a mathematics class. One-third of the males were enrolled in calculus, and 1/5 of the females were enrolled in calculus. If just as many males as females were surveyed, then what fraction of the surveyed students taking mathematics were enrolled in calculus? Rework this problem assuming that the number of females in the survey was twice the number of males.

62. Commuting students.  At a well-known university, ¼ of the undergraduate students commute, and 1/3 of the graduate students commute. One-tenth of the undergraduate students drive more than 40 miles daily, and 1/6 of the graduate students drive more than 40 miles daily. If there are twice as many undergraduate students as there are graduate students, then what fraction of the commuters drive more than 40 miles daily?

Section 6.6


63. Lens equation. The focal length f for a camera lens is related to the object distance o and the image distance i by the formula 1/f = 1/o + 1/i. See the accompanying figure. The image is in focus at distance i from the lens. For an object that is 600 mm from a 50-mm lens, use f = 50 mm and o = 600 mm to findi.

64. Telephoto lens.  Use the formula from Exercise 63 to find the image distance i for an object that is 2,000,000 mm from a 250-mm telephoto lens.


Section 6.7


17. Men and women.  Find the ratio of men to women in a bowling league containing 12 men and 8 women.

18. Coffee drinkers.  Among 100 coffee drinkers, 36 said that they preferred their coffee black and the rest did not prefer their coffee black. Find the ratio of those who prefer black coffee to those who prefer nonblack coffee.

19. Smokers.  A life insurance company found that among its last 200 claims, there were six dozen smokers. What is the ratio of smokers to nonsmokers in this group of claimants?

20. Hits and misses.  A woman threw 60 darts and hit the target a dozen times. What is her ratio of hits to misses?

21. While watching television for one week, a consumer group counted 1240 acts of violence and 40 acts of kindness. What is the violence to kindness ratio for television, according to this group?

22. Length to width. What is the ratio of length to width for the rectangle shown?

23. Rise to run.  What is the ratio of rise to run for the stairway shown in the figure?

39. New shows and reruns.  The ratio of new shows to reruns on cable TV is 2 to 27. If Frank counted only eight new shows one evening, then how many reruns were there?

40. Fast food.  If four out of five doctors prefer fast food, then at a convention of 445 doctors, how many prefer fast food?

41. Voting.  If 220 out of 500 voters surveyed said that they would vote for the incumbent, then how many votes could the incumbent expect out of the 400,000 voters in the state?

42. New product. A taste test with 200 randomly selected people found that only three of them said that they would buy a box of new Sweet Wheats cereal. How many boxes could the manufacturer expect to sell in a country of 280 million people?

43. Basketball blowout.  As the final buzzer signaled the end of the basketball game, the Lions were 34 points ahead of the Tigers. If the Lions scored 5 points for every 3 scored by the Tigers, then what was the final score?

44. The golden ratio.  The ancient Greeks thought that the most pleasing shape for a rectangle was one for which the ratio of the length to the width was approximately 8 to 5, the golden ratio. If the length of a rectangular painting is 2 ft longer than its width, then for what dimensions would the length and width have the golden ratio?

45. Automobile sales. The ratio of sports cars to luxury cars sold in Wentworth one month was 3 to 2. If 20 more sports cars were sold than luxury cars, then how many of each were sold that month?

46. Foxes and rabbits.  The ratio of foxes to rabbits in the Deerfield Forest Preserve is 2 to 9. If there are 35 fewer foxes than rabbits, then how many of each are there?

47. Inches and feet. If there are 12 inches in 1 foot, then how many inches are there in 7 feet?

48. Feet and yards. If there are 3 feet in 1 yard, then how many yards are there in 28 feet?

49. Minutes and hours. If there are 60 minutes in 1 hour, then how many minutes are there in 0.25 hour?

50. Meters and kilometers. If there are 1000 meters in 1 kilometer, then how many meters are there in 2.33 kilometers?

51. Miles and hours. If Alonzo travels 230 miles in 3 hours, then how many miles does he travel in 7 hours?

52. Hiking time. If Evangelica can hike 19 miles in 2 days on the Appalachian Trail, then how many days will it take her to hike 63 miles?

53. Force on basketball shoes. The force exerted on shoe soles in a jump shot is proportional to the weight of the person jumping. If a 70-pound boy exerts a force of 980 pounds on his shoe soles when he returns to the court after a jump, then what force does a 6 ft 8 in. professional ball player weighing 280 pounds exert on the soles of his shoes when he returns to the court after a jump? Use the accompanying graph to estimate the force for a 150-pound player.

54. Force on running shoes. The ratio of the force on the shoe soles to the weight of a runner is 3 to 1. What force does a 130-pound jogger exert on the soles of her shoes?

55. Capture-recapture. To estimate the number of trout in Trout Lake, rangers used the capture-recapture method. They caught, tagged, and released 200 trout. One week later, they caught a sample of 150 trout and found that 5 of them were tagged. Assuming that the ratio of tagged trout to the total number of trout in the lake is the same as the ratio of tagged trout in the sample to the number of trout in the sample, find the number of trout in the lake.

56. Bear population. To estimate the size of the bear population on the Keweenaw Peninsula, conservationists captured, tagged, and released 50 bears. One year later, a random sample of 100 bears included only 2 tagged bears. What is the conservationist’s estimate of the size of the bear population?

57. Fast-food waste. The accompanying figure shows the typical distribution of waste at a fast-food restaurant. What is the ratio of customer waste to food waste? b) If a typical McDonald’s generates 67 more pounds of food waste than customer waste per day, then how many pounds of customer waste does it generate?

58. Corrugated waste. Use the accompanying figure to find the ratio of waste from corrugated shipping boxes to waste not from corrugated shipping boxes. If a typical McDonald’s generates 81 pounds of waste per day from corrugated shipping boxes, then how many pounds of waste per day does it generate that is not from corrugated shipping boxes?


Section 6.8

35. Fast walking. Marcie can walk 8 miles in the same time as Frank walks 6 miles. If Marcie walks 1 mile per hour faster than Frank, then how fast does each person walk?

36. Upstream, downstream. Junior’s boat will go 15 miles per hour in still water. If he can go 12 miles downstream in the same amount of time as it takes to go 9 miles upstream, then what is the speed of the current?

37. Delivery routes. Pat travels 70 miles on her milk route, and Bob travels 75 miles on his route. Pat travels 5 miles per hour slower than Bob, and her route takes her one-half hour longer than Bob’s. How fast is each one traveling?

38. Ride the peaks. Smith bicycled 45 miles going east from Durango, and Jones bicycled 70 miles. Jones averaged 5 miles per hour more than Smith, and his trip took one-half hour longer than Smith’s. How fast was each one traveling?

39. Walking and running. Raffaele ran 8 miles and then walked 6 miles. If he ran 5 miles per hour faster than he walked and the total time was 2 hours, then how fast did he walk?

40. Triathlon. Luisa participated in a triathlon in which she swam 3 miles, ran 5 miles, and then bicycled 10 miles. Luisa ran twice as fast as she swam, and she cycled three times as fast as she swam. If her total time for the triathlon was 1 hour and 46 minutes, then how fast did she swim?

41. Fence painting. Kiyoshi can paint a certain fence in 3 hours by himself. If Red helps, the job takes only 2 hours. How long would it take Red to paint the fence by himself ?

42. Envelope stuffing. Every week, Linda must stuff 1000 envelopes. She can do the job by herself in 6 hours. If Laura helps, they get the job done in 5 1/2 hours. How long would it take Laura to do the job by herself?

43. Garden destroying. Mr. McGregor has discovered that a large dog can destroy his entire garden in 2 hours and that a small boy can do the same job in 1 hour. How long would it take the large dog and the small boy working together to destroy Mr. McGregor’s garden?

44. Draining the vat. With only the small valve open, all of the liquid can be drained from a large vat in 4 hours. With only the large valve open, all of the liquid can be drained from the same vat in 2 hours. How long would it take to drain the vat with both valves open?

45. Cleaning sidewalks. Edgar can blow the leaves off the sidewalks around the capitol building in 2 hours using a gasoline-powered blower. Ellen can do the same job in 8 hours using a broom. How long would it take them working together?

46. Computer time. It takes a computer 8 days to print all of the personalized letters for a national sweepstakes. A new computer is purchased that can do the same job in 5 days. How long would it take to do the job with both computers working on it?

47. Repair work. Sally received a bill for a total of 8 hours labor on the repair of her bulldozer. She paid $50 to the master mechanic and $90 to his apprentice. If the master mechanic gets $10 more per hour than his apprentice, then how many hours did each work on the bulldozer?

48. Running backs. In the playoff game the ball was carried by either Anderson or Brown on 21 plays. Anderson gained 36 yards, and Brown gained 54 yards. If Brown averaged twice as many yards per carry as Anderson, then on how many plays did Anderson carry the ball?

49. Apples and bananas. Bertha bought 18 pounds of fruit consisting of apples and bananas. She paid $9 for the apples and $2.40 for the bananas. If the price per pound of the apples was 3 times that of the bananas, then how many pounds of each type of fruit did she buy?

50. Fuel efficiency. Last week, Joe’s Electric Service used 110 gallons of gasoline in its two trucks. The large truck was driven 800 miles, and the small truck was driven 600 miles. If the small truck gets twice as many miles per gallon as the large truck, then how many gallons of gasoline did the large truck use?

51. Small plane. It took a small plane 1 hour longer to fly 480 miles against the wind than it took the plane to fly the same distance with the wind. If the wind speed was 20 mph, then what is the speed of the plane in calm air?

52. Fast boat. A motorboat at full throttle takes two hours longer to travel 75 miles against the current than it takes to travel the same distance with the current. If the rate of the current is 5 mph, then what is the speed of the boat at full throttle in still water?

53. Light plane. At full throttle a light plane flies 275 miles against the wind in the same time as it flies 325 miles with the wind. If the plane flies at 120 mph at full throttle in still air, then what is the wind speed?

54. Big plane. A six-passenger plane cruises at 180 mph in calm air. If the plane flies 7 miles with the wind in the same amount of time as it flies 5 miles against the wind, then what is the wind speed?

55. Two cyclists. Ben and Jerry start from the same point and ride their bicycles in opposite directions. If Ben rides twice as fast as Jerry and they are 90 miles apart after four hours, then what is the speed of each rider?

56. Catching up. A sailboat leaves port and travels due south at an average speed of 9 mph. Four hours later a motorboat leaves the same port and travels due south at an average speed of 21 mph. How long will it take the motorboat to catch the sailboat?

57. Road trip. The Griswalds averaged 45 mph on their way to Las Vegas and 60 mph on the way back home using the same route. Find the distance from their home to Las Vegas if the total driving time was 70 hours.

58. Meeting cyclists. Tanya and Lebron start at the same time from opposite ends of a bicycle trail that is 81 miles long. Tanya averages 12 mph and Lebron averages 15 mph. How long does it take for them to meet?

59. Filling a fountain. Pete’s fountain can be filled using a pipe or a hose. The fountain can be filled using the pipe in 6 hours or the hose in 12 hours. How long will it take to fill the fountain using both the pipe and the hose?

60. Mowing a lawn. Albert can mow a lawn in 40 minutes, while his cousin Vinnie can mow the same lawn in one hour. How long would it take to mow the lawn if Albert and Vinnie work together?

61. Printing a report. Debra plans to use two computers to print all of the copies of the annual report that are needed for the year-end meeting. The new computer can do the whole job in 2 hours while the old computer can do the whole job in 3 hours. How long will it take to get the job done using both computers simultaneously?

62. Installing a dishwasher. A plumber can install a dishwasher in 50 min. If the plumber brings his apprentice to help, the job takes 40 minutes. How long would it take the apprentice working alone to install the dishwasher?

63. Filling a tub. Using the hot and cold water faucets together, a bathtub fills in 8 minutes. Using the hot water faucet alone, the tub fills in 12 minutes. How long does it take to fill the tub using only the cold water faucet?

64. Filling a tank. A water tank has an inlet pipe and a drain pipe. A full tank can be emptied in 30 minutes if the drain is opened and an empty tank can be filled in 45 minutes with the inlet pipe opened. If both pipes are accidentally opened when the tank is full, then how long will it take to empty the tank?


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