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Dugopolski M Elementary and Intermediate Algebra
Sophomore math. A survey of college sophomores showed that 5/6 of the males were taking a mathematics class and ? 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.
Commuting students. At a well-known university, ? 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?
Men and women. Find the ratio of men to women in a bowling league containing 12 men and 8 women.
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.
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?
Hits and misses. A woman threw 60 darts and hit the target a dozen times. What is her ratio of hits to misses?
Violence and kindness. 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?
Length to width. What is the ratio of length to width for the rectangle shown?
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?
Fast food. If four out of five doctors prefer fast food, then at a convention of 445 doctors, how many prefer fast food?
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?
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?
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?
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?
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?
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?
Inches and feet. If there are 12 inches in 1 foot, then how many inches are there in 7 feet?
Feet and yards. If there are 3 feet in 1 yard, then how many yards are there in 28 feet?
Minutes and hours. If there are 60 minutes in 1 hour, then how many minutes are there in 0.25 hour?
Meters and kilometers. If there are 1000 meters in 1 kilometer, then how many meters are there in
Miles and hours. If Alonzo travels 230 miles in 3 hours, then how many miles does he travel in 7 hours?
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?
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.
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?
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.
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?
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?
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?
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?
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?
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?
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?
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 ?
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 ? hours. How long would it take Laura to do the job by herself ?
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?
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?
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?
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?
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?
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?
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?
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?
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?
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?
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?
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?
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?
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?
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.
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?
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?
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?
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?
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?
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?
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?
Taxis in Times Square. The ratio of taxis to private automobiles in Times Square at 6:00 P.M. on New Year's Eve was estimated to be 15 to 2. If there were 60 taxis, then how many private automobiles were there?
Student-teacher ratio. The student-teacher ratio for Washington High was reported to be 27.5 to 1. If there are 42 teachers, then how many students are there?
Water and rice. At Wong's Chinese Restaurant the secret recipe for white rice calls for a 2 to 1 ratio of water to rice. In one batch the chef used 28 more cups of water than rice. How many cups of each did he use?
Oil and gas. An outboard motor calls for a fuel mixture that has a gasoline-to-oil ratio of 50 to 1. How many pints of oil should be added to 6 gallons of gasoline?
Making a puzzle. Tracy, Stacy, and Fred assembled a very large puzzle together in 40 hours. If Stacy worked twice as fast as Fred and Tracy worked just as fast as Stacy, then how long would it have taken Fred to assemble the puzzle alone?
Going skiing. Leon drove 270 miles to the lodge in the same time as Pat drove 330 miles to the lodge. If Pat drove 10 miles per hour faster than Leon, then how fast did each of them drive?
Merging automobiles. When Bert and Ernie merged their automobile dealerships, Bert had 10 more cars than Ernie. While 36% of Ernie's stock consisted of new cars, only 25% of Bert's stock consisted of new cars. If they had 33 new cars on the lot after the merger, then how many cars did each one have before the merger?
Magazine sales. A company specializing in magazine sales over the telephone found that in 2500 phone calls, 360 resulted in sales and were made by male callers, and 480 resulted in sales and were made by female callers. If the company gets twice as many sales per call with a woman's voice than with a man's voice, then how many of the 2500 calls were made by females?
Distribution of waste. The accompanying figure shows the distribution of the total municipal solid waste into various categories in 2000 (U.S. Environmental Protection Agency, www.epa.gov). If the paper waste was 59.8 million tons greater than the yard waste, then what was the amount of yard waste generated?
Total waste. Use the information given in Exercise 81 to find the total waste generated in 2000 and the amount of food waste.