7.5 Solve Applications with Rational Equations

Proportions and similar figures, shared-work and uniform motion rate problems, and direct or inverse variation models fitted from one data pair.

Intermediate Algebra 2e, 7.5 Solve Applications with Rational Equations

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67 exercises

7.5 Exercise 1

Before you get started, take this readiness quiz.

Solve: $2 \left(n - 1\right) - 4 = −10 .$
If you missed this problem, review the reference.

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7.5 Exercise 2

An express train and a charter bus leave Chicago to travel to Champaign. The express train can make the trip in two hours and the bus takes five hours for the trip. The speed of the express train is 42 miles per hour faster than the speed of the bus. Find the speed of the bus.
If you missed this problem, review the reference.

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7.5 Exercise 3

Solve $\frac{1}{3} x + \frac{1}{4} x = \frac{5}{6} .$
If you missed this problem, review the reference.

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7.5 Exercise 5

Solve the proportion: $\frac{y}{y + 55} = \frac{3}{8} .$

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7.5 Exercise 6

Solve the proportion: $\frac{z}{z - 84} = - \frac{1}{5} .$

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7.5 Exercise 8

Pediatricians prescribe 5 milliliters (ml) of acetaminophen for every 25 pounds of a child’s weight. How many milliliters of acetaminophen will the doctor prescribe for Emilia, who weighs 60 pounds?

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7.5 Exercise 9

For every 1 kilogram (kg) of a child’s weight, pediatricians prescribe 15 milligrams (mg) of a fever reducer. If Isabella weighs 12 kg, how many milligrams of the fever reducer will the pediatrician prescribe?

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7.5 Exercise 11

Find the actual distance from Seattle to Portland.

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7.5 Exercise 12

Find the actual distance from Portland to Boise.

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7.5 Exercise 14

A telephone pole casts a shadow that is 50 feet long. Nearby, an 8 foot tall traffic sign casts a shadow that is 10 feet long. How tall is the telephone pole?

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7.5 Exercise 15

A pine tree casts a shadow of 80 feet next to a 30 foot tall building which casts a 40 feet shadow. How tall is the pine tree?

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7.5 Exercise 17

Link has an electric bike which runs at a constant speed. That speed will be reduced by the amount of any headwind and increased by the amount of any tailwind. Link can ride his bike 20 miles into a 3 mph headwind in the same amount of time he can ride 30 miles with a 3 mph tailwind. What is Link’s biking speed?

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7.5 Exercise 18

On a river that flows at 7 miles per hour, Danica can take a motorboat 5 miles upstream, in the same amount of time she can take her motorboat 12 miles downstream. How fast is Danica's boat in a lake?

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7.5 Exercise 20

Dennis went cross-country skiing for 6 hours on Saturday. He skied 20 mile uphill and then 20 miles back downhill, returning to his starting point. His uphill speed was 5 mph slower than his downhill speed. What was Dennis’ speed going uphill and his speed going downhill?

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7.5 Exercise 21

Joon drove 4 hours to his home, driving 208 miles on the interstate and 40 miles on country roads. If he drove 15 mph faster on the interstate than on the country roads, what was his rate on the country roads?

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7.5 Exercise 23

Kayla rode her bike 75 miles home from college one weekend and then rode the bus back to college. It took her 2 hours less to ride back to college on the bus than it took her to ride home on her bike, and the average speed of the bus was 10 miles per hour faster than Kayla’s biking speed. Find Kayla’s biking speed.

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7.5 Exercise 24

Victoria jogs 12 miles to the park along a flat trail and then returns by jogging on an 20 mile hilly trail. She jogs 1 mile per hour slower on the hilly trail than on the flat trail, and her return trip takes her two hours longer. Find her rate of jogging on the flat trail.

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7.5 Exercise 26

One gardener can mow a golf course in 4 hours, while another gardener can mow the same golf course in 6 hours. How long would it take if the two gardeners worked together to mow the golf course?

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7.5 Exercise 27

Daria can weed the garden in 7 hours, while her mother can do it in 3. How long will it take the two of them working together?

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7.5 Exercise 29

Alice can paint a room in 6 hours. If Kristina helps her it takes them 4 hours to paint the room. How long would it take Kristina to paint the room by herself?

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7.5 Exercise 30

Tracy can lay a slab of concrete in 3 hours, with Jordan’s help they can do it in 2 hours. If Jordan works alone, how long will it take?

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7.5 Exercise 32

The number of calories, c, burned varies directly with the amount of time, t, spent exercising. Arnold burned 312 calories in 65 minutes exercising.

ⓐ Write the equation that relates c and t. ⓑ How many calories would he burn if he exercises for 90 minutes?

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7.5 Exercise 33

The distance a moving body travels, d, varies directly with time, t, it moves. A train travels 100 miles in 2 hours

ⓐ Write the equation that relates d and t. ⓑ How many miles would it travel in 5 hours?

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7.5 Exercise 35

The number of hours it takes for ice to melt varies inversely with the air temperature. Suppose a block of ice melts in 2 hours when the temperature is 65 degrees Celsius.

ⓐ Write the equation of variation. ⓑ How many hours would it take for the same block of ice to melt if the temperature was 78 degrees?

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7.5 Exercise 36

Xander’s new business found that the daily demand for its product was inversely proportional to the price, $p .$ When the price is $5, the demand is 700 units.

ⓐ Write the equation of variation. ⓑ What is the demand if the price is raised to $7?

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7.5 Exercise 37

Solve each proportion.

$\frac{x}{56} = \frac{7}{8}$

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7.5 Exercise 39

Solve each proportion.

$\frac{98}{154} = \frac{−7}{p}$

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7.5 Exercise 41

Solve each proportion.

$\frac{a}{a + 12} = \frac{4}{7}$

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7.5 Exercise 43

Solve each proportion.

$\frac{m + 90}{25} = \frac{m + 30}{15}$

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7.5 Exercise 45

Solve each proportion.

$\frac{2 p + 4}{8} = \frac{p + 18}{6}$

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7.5 Exercise 47

Solve.

Kevin wants to keep his heart rate at 160 beats per minute while training. During his workout he counts 27 beats in 10 seconds.

ⓐ How many beats per minute is this? ⓑ Has Kevin met his target heart rate?

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7.5 Exercise 49

Solve.

Pediatricians prescribe 5 milliliters (ml) of acetaminophen for every 25 pounds of a child’s weight. How many milliliters of acetaminophen will the doctor prescribe for Jocelyn, who weighs 45 pounds?

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7.5 Exercise 51

Solve.

A new energy drink advertises 106 calories for 8 ounces. How many calories are in 12 ounces of the drink?

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7.5 Exercise 53

Solve.

Kyra is traveling to Canada and will change $250 US dollars into Canadian dollars. At the current exchange rate, $1 US is equal to $1.3 Canadian. How many Canadian dollars will she get for her trip?

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7.5 Exercise 55

Solve.

Ronald needs a morning breakfast drink that will give him at least 390 calories. Orange juice has 130 calories in one cup. How many cups does he need to drink to reach his calorie goal?

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7.5 Exercise 57

Solve.

Phil wants to fertilize his lawn. Each bag of fertilizer covers about 4,000 square feet of lawn. Phil’s lawn is approximately 13,500 square feet. How many bags of fertilizer will he have to buy?

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7.5 Exercise 61

Find the actual distance from New York to Memphis.

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7.5 Exercise 63

Find the actual distance from New Orleans to Miami.

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7.5 Exercise 65

Answer each question.

A 2-foot-tall dog casts a 3-foot shadow at the same time a cat casts a one foot shadow. How tall is the cat ?

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7.5 Exercise 67

Answer each question.

The tower portion of a windmill is 212 feet tall. A six foot tall person standing next to the tower casts a seven-foot shadow. How long is the windmill’s shadow?

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