7.5 Solve Applications with Rational Equations

Learning objectives

Learning objectives

• Solve proportions

• Solve similar figure applications

• Solve uniform motion applications

• Solve work applications

• Solve direct variation problems

• Solve inverse variation problems

Solve Proportions

When two rational expressions are equal, the equation relating them is called a proportion.

The equation $\frac{1}{2}=\frac{4}{8}$ is a proportion because the two fractions are equal. The proportion $\frac{1}{2}=\frac{4}{8}$ is read “1 is to 2 as 4 is to 8.”

Since a proportion is an equation with rational expressions, we will solve proportions the same way we solved rational equations. We’ll multiply both sides of the equation by the LCD to clear the fractions and then solve the resulting equation.

Example 7.43

Solution

Notice in the last example that when we cleared the fractions by multiplying by the LCD, the result is the same as if we had cross-multiplied.

For any proportion, $\frac{a}{b}=\frac{c}{d},$ we get the same result when we clear the fractions by multiplying by the LCD as when we cross-multiply.

To solve applications with proportions, we will follow our usual strategy for solving applications But when we set up the proportion, we must make sure to have the units correct—the units in the numerators must match each other and the units in the denominators must also match each other.

Example 7.44

Solution

Solve similar figure applications

When you shrink or enlarge a photo on a phone or tablet, figure out a distance on a map, or use a pattern to build a bookcase or sew a dress, you are working with similar figures. If two figures have exactly the same shape, but different sizes, they are said to be similar. One is a scale model of the other. All their corresponding angles have the same measures and their corresponding sides have the same ratio.

For example, the two triangles in Figure 7.2 are similar. Each side of $\Delta{}ABC$ is four times the length of the corresponding side of $\Delta{}XYZ.$

This is summed up in the Property of Similar Triangles.

To solve applications with similar figures we will follow the Problem-Solving Strategy for Geometry Applications we used earlier.

Example 7.45

Solution

Since the triangles are similar, the corresponding sides are proportional.

On the map, Seattle, Portland, and Boise form a triangle. The distance between the cities is measured in inches. The figure on the left below represents the triangle formed by the cities on the map. The actual distance from Seattle to Boise is 400 miles.

We can use similar figures to find heights that we cannot directly measure.

Example 7.46

Solution

Solve Uniform Motion Applications

We have solved uniform motion problems using the formula $D=rt$ in previous chapters. We used a table like the one below to organize the information and lead us to the equation.

The formula $D=rt$ assumes we know r and t and use them to find D. If we know D and r and need to find t, we would solve the equation for t and get the formula $t=\frac{D}{r}.$

We have also explained how flying with or against the wind affects the speed of a plane. We will revisit that idea in the next example.

Example 7.47

Solution

This is a uniform motion situation. A diagram will help us visualize the situation.

In the next example, we will know the total time resulting from travelling different distances at different speeds.

Example 7.48

Solution

This is a uniform motion situation. A diagram will help us visualize the situation.

Once again, we will use the uniform motion formula solved for the variable $t.$

Example 7.49

Solution

This is a uniform motion situation. A diagram will help us visualize the situation.

Solve Work Applications

The weekly gossip magazine has a big story about the Princess’ baby and the editor wants the magazine to be printed as soon as possible. She has asked the printer to run an extra printing press to get the printing done more quickly. Press $1 takes 6 hours to do the job and Press $2 takes 12 hours to do the job. How long will it take the printer to get the magazine printed with both presses running together?

This is a typical ‘work’ application. There are three quantities involved here—the time it would take each of the two presses to do the job alone and the time it would take for them to do the job together.

If Press $1 can complete the job in 6 hours, in one hour it would complete $\frac{1}{6}$ of the job.

If Press $2 can complete the job in 12 hours, in one hour it would complete $\frac{1}{12}$ of the job.

We will let t be the number of hours it would take the presses to print the magazines with both presses running together. So in 1 hour working together they have completed $\frac{1}{t}$ of the job.

We can model this with the word equation and then translate to a rational equation. To find the time it would take the presses to complete the job if they worked together, we solve for $t.$

A chart will help us organize the information. We are looking for how many hours it would take to complete the job with both presses running together.

Keep in mind, it should take less time for two presses to complete a job working together than for either press to do it alone.

Example 7.50

Solution

This is a ‘work’ application. A chart will help us organize the information. We are looking for the numbers of hours it will take them to paint the room together.

In one hour Pete did $\frac{1}{10}$ of the job. Alicia did $\frac{1}{8}$ of the job. And together they did $\frac{1}{t}$ of the job.

Example 7.51

Solution

This is a work problem. A chart will help us organize the information.

We are looking for how many hours it would take Ra’shon’s sister to complete the job by herself.

Solve Direct Variation Problems

When two quantities are related by a proportion, we say they are proportional to each other. Another way to express this relation is to talk about the variation of the two quantities. We will discuss direct variation and inverse variation in this section.

Lindsay gets paid $15 per hour at her job. If we let s be her salary and h be the number of hours she has worked, we could model this situation with the equation

Lindsay’s salary is the product of a constant, 15, and the number of hours she works. We say that Lindsay’s salary varies directly with the number of hours she works. Two variables vary directly if one is the product of a constant and the other.

In applications using direct variation, generally we will know values of one pair of the variables and will be asked to find the equation that relates x and y. Then we can use that equation to find values of y for other values of x.

We’ll list the steps here.

Now we’ll solve an application of direct variation.

Example 7.52

Solution

ⓐ

ⓑ

Solve Inverse Variation Problems

Many applications involve two variable that vary inversely. As one variable increases, the other decreases. The equation that relates them is $y=\frac{k}{x}.$

The word ‘inverse’ in inverse variation refers to the multiplicative inverse. The multiplicative inverse of x is $\frac{1}{x}.$

We solve inverse variation problems in the same way we solved direct variation problems. Only the general form of the equation has changed. We will copy the procedure box here and just change ‘direct’ to ‘inverse’.

Example 7.53

Solution

ⓐ

ⓑ

$\,$Find $f$ when $L=20.$

Write the equation that relates $f\mathrm{and}\,L.$ · $\,f=\frac{11,440}{L}$ Substitute the given value for $L.$ · $\,f=\frac{11,440}{20}$ Simplify. · $\,f=572$

$\,$A 20″-guitar string has frequency 572 $\,$vibrations per second.

Section 7.5 Exercises

Section 7.5 Exercises

Solve Proportions

In the following exercises, solve each proportion.

In the following exercises, solve.

Solve Similar Figure Applications

In the following exercises, the triangles are similar. Find the length of the indicated side.

In the following exercises, use the map shown. On the map, New York City, Chicago, and Memphis form a triangle. The actual distance from New York to Chicago is 800 miles.

In the following exercises, use the map shown. On the map, Atlanta, Miami, and New Orleans form a triangle. The actual distance from Atlanta to New Orleans is 420 miles.

In the following exercises, answer each question.

Solve Uniform Motion Applications

In the following exercises, solve the application problem provided.

Solve Work Applications

Solve Direct Variation Problems

In the following exercises, solve.

Solve Inverse Variation Problems

In the following exercises, solve.

In the following exercises, write an inverse variation equation to solve the following problems.

Writing Exercises

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

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

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

3. 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?

4. 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?

5. Find the actual distance from Seattle to Portland.

1. Find the actual distance from Portland to Boise.

2. 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?

3. 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?

4. 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?

5. 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?

1. 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?

2. 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?

3. 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?

4. 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?

5. 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?

1. 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?

2. 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?

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

4. $\frac{98}{154}=\frac{-7}{p}$

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

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

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

3. 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?

4. 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?

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

1. 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?

2. 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?

3. 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?

4. ⓐ side x ⓑ side b

5. Find the actual distance from New York to Memphis.

1. Find the actual distance from New Orleans to Miami.

2. 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 ?

3. 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?

4. Mary takes a sightseeing tour on a helicopter that can fly 450 miles against a 35-mph headwind in the same amount of time it can travel 702 miles with a 35-mph tailwind. Find the speed of the helicopter.

5. A boat travels 140 miles downstream in the same time as it travels 92 miles upstream. The speed of the current is 6mph. What is the speed of the boat?

1. Josephine can correct her students test papers in 5 hours, but if her teacher’s assistant helps, it would take them 3 hours. How long would it take the assistant to do it alone?

2. At the end of the day Dodie can clean her hair salon in 15 minutes. Ann, who works with her, can clean the salon in 30 minutes. How long would it take them to clean the shop if they work together?

3. If $y$ varies directly as $x$ and $y=14,$ when $x=3.$ find the equation that relates $x$ and $y.$

4. If $p$ varies directly as $q$ and $p=9.6,$ when $q=3.$ find the equation that relates $p$ and $q.$

5. The price, $P,$ that Eric pays for gas varies directly with the number of gallons, $g,$ he buys. It costs him $50 to buy 20 gallons of gas.

ⓐ Write the equation that relates $P$ and $g.$ ⓑ How much would 33 gallons cost Eric?

1. The mass of a liquid varies directly with its volume. A liquid with mass 16 kilograms has a volume of 2 liters.

ⓐ Write the equation that relates the mass to the volume. ⓑ What is the volume of this liquid if its mass is 128 kilograms?

2. The maximum load a beam will support varies directly with the square of the diagonal of the beam’s cross-section. A beam with diagonal 6 inch will support a maximum load of 108 pounds.

ⓐ Write the equation that relates the load to the diagonal of the cross-section. ⓑ What load will a beam with a 10-inch diagonal support?

3. If $y$ varies inversely with $x$ and $y=5$ when $x=4,$ find the equation that relates $x$ and $y.$

4. If $v$ varies inversely with $w$ and $v=6$ when $w=\frac{1}{2},$ find the equation that relates $v$ and $w.$

5. The fuel consumption (mpg) of a car varies inversely with its weight. A Toyota Corolla weighs 2800 pounds getting 33 mpg on the highway.

ⓐ Write the equation that relates the mpg to the car’s weight. ⓑ What would the fuel consumption be for a Toyota Sequoia that weighs 5500 pounds?

1. 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?

2. 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?

3. 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.

4. 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.

5. 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?

1. Jane spent 2 hours exploring a mountain with a dirt bike. First, she rode 40 miles uphill. After she reached the peak she rode for 12 miles along the summit. While going uphill, she went 5 mph slower than when she was on the summit. What was her rate along the summit?

2. Byron wanted to try out different water craft. He went 62 miles downstream in a motor boat and 27 miles downstream on a jet ski. His speed on the jet ski was 10 mph faster than in the motor boat. Bill spent a total of 4 hours on the water. What was his rate of speed in the motor boat?

3. Chester rode his bike uphill 24 miles and then back downhill at 2 mph faster than his uphill. If it took him 2 hours longer to ride uphill than downhill, what was his uphill rate?

4. Hudson travels 1080 miles in a jet and then 240 miles by car to get to a business meeting. The jet goes 300 mph faster than the rate of the car, and the car ride takes 1 hour longer than the jet. What is the speed of the car?

5. John can fly his airplane 2800 miles with a wind speed of 50 mph in the same time he can travel 2400 miles against the wind. If the speed of the wind is 50 mph, find the speed of his airplane.

1. Hazel needs to get to her granddaughter’s house by taking an airplane and a rental car. She travels 900 miles by plane and 250 miles by car. The plane travels 250 mph faster than the car. If she drives the rental car for 2 hours more than she rode the plane, find the speed of the car.

2. When driving the 9-hour trip home, Sharon drove 390 miles on the interstate and 150 miles on country roads. Her speed on the interstate was 15 more than on country roads. What was her speed on country roads?

3. Dana enjoys taking her dog for a walk, but sometimes her dog gets away, and she has to run after him. Dana walked her dog for 7 miles but then had to run for 1 mile, spending a total time of 2.5 hours with her dog. Her running speed was 3 mph faster than her walking speed. Find her walking speed.

4. Mike, an experienced bricklayer, can build a wall in 3 hours, while his son, who is learning, can do the job in 6 hours. How long does it take for them to build a wall together?

5. Mia can clean her apartment in 6 hours while her roommate can clean the apartment in 5 hours. If they work together, how long would it take them to clean the apartment?

1. The time required to empty a tank varies inversely as the rate of pumping. It took Ada 5 hours to pump her flooded basement using a pump that was rated at 200 gpm (gallons per minute).

ⓐ Write the equation that relates the number of hours to the pump rate. ⓑ How long would it take Ada to pump her basement if she used a pump rated at 400 gpm?

2. Paul, a dentist, determined that the number of cavities that develops in his patient’s mouth each year varies inversely to the number of minutes spent brushing each night. His patient, Lori, had four cavities when brushing her teeth 30 seconds (0.5 minutes) each night.

ⓐ Write the equation that relates the number of cavities to the time spent brushing. ⓑ How many cavities would Paul expect Lori to have if she had brushed her teeth for 2 minutes each night?

3. The cost of a ride service varies directly with the distance traveled. It costs $35 for a ride from the city center to the airport, 14 miles away.

ⓐ Write the equation that relates the cost, c, with the number of miles, $m.$ ⓑ What would it cost to travel 22 miles with this service?

4. Marisol solves the proportion $\frac{144}{a}=\frac{9}{4}$ by ‘cross multiplying,’ so her first step looks like $4\cdot{}144=9\cdot{}a.$ Explain how this differs from the method of solution shown in Example 7.44.

5. In your own words, explain the difference between direct variation and inverse variation.