7.6: Applications of Rational Equations

Learning Objectives

• Solve applications involving relationships between real numbers.
• Solve applications involving uniform motion (distance problems).
• Solve work-rate applications.

Number Problems

Recall that the reciprocal of a nonzero number is . For example, the reciprocal of is and . In this section, the applications will often involve the key word “reciprocal.” When this is the case, we will see that the algebraic setup results in a rational equation.

Example

A positive integer is less than another. The sum of the reciprocals of the two positive integers is . Find the two integers.

Solution:

Begin by assigning variables to the unknowns.

Let represent the larger positive integer.

Let represent the smallest positive integer.

Next, use the reciprocals and to translate the sentences into an algebraic equation.

We can solve this rational expression by multiplying both sides of the equation by the least common denominator (LCD). In this case, the LCD is .

Solve the resulting quadratic equation.

The question calls for integers and the only integer solution is . Hence disregard . Use the expression to find the smaller integer.

Answer:

The two positive integers are and . The check is left to the reader.

Example

A positive integer is less than another. If the reciprocal of the smaller integer is subtracted from twice the reciprocal of the larger, then the result is . Find the two integers.

Solution:

Let represent the larger positive integer.

Let represent the smaller positive integer.

Set up an algebraic equation.

Solve this rational expression by multiplying both sides by the LCD. The LCD is .

Here we have two viable possibilities for the larger integer. For this reason, we will we have two solutions to this problem.

If , then .

If , then .

As a check, perform the operations indicated in the problem.

Answer:

Two sets of positive integers solve this problem: {} and {}.

Exercise

The difference between the reciprocals of two consecutive positive odd integers is . Find the integers.

Answer

The integers are and .

Uniform Motion Problems

Uniform motion problems, also referred to as distance problems, involve the formula

where the distance, , is given as the product of the average rate, , and the time, , traveled at that rate. If we divide both sides by the average rate, , then we obtain the formula

For this reason, when the unknown quantity is time, the algebraic setup for distance problems often results in a rational equation. Similarly, when the unknown quantity is the rate, the setup also may result in a rational equation.

We begin any uniform motion problem by first organizing our data with a chart. Use this information to set up an algebraic equation that models the application.

Example

Mary spent the first 120 miles of her road trip in traffic. When the traffic cleared, she was able to drive twice as fast for the remaining 300 miles. If the total trip took 9 hours, then how fast was she moving in traffic?

Solution:

First, identify the unknown quantity and organize the data.

Let represent Mary's average speed (miles per hour) in traffic.

Let represent her average speed after the traffic cleared.

Figure

To avoid introducing two more variables for the time column, use the formula . Here the time for each leg of the trip is calculated as follows:

Use these expressions to complete the chart.

Figure

The algebraic setup is defined by the time column. Add the times for each leg of the trip to obtain a total of 9 hours:

We begin solving this equation by first multiplying both sides by the LCD, .

Answer:

Mary averaged 30 miles per hour in traffic.

Example

A passenger train can travel, on average, miles per hour faster than a freight train. If the passenger train covers miles in the same time it takes the freight train to cover miles, then how fast is each train?

Solution:

First, identify the unknown quantities and organize the data.

Let represent the average speed of the freight train.

Let represent the speed of the passenger train.

Next, organize the given data in a chart.

Figure

Use the formula to fill in the time column for each train.

Figure

Because the trains travel the same amount of time, finish the algebraic setup by equating the expressions that represent the times:

Solve this equation by first multiplying both sides by the LCD, .

Use to find the speed of the passenger train.

Answer:

The speed of the passenger train is miles per hour and the speed of the freight train is miles per hour.

Example

Brett lives on the river miles upstream from town. When the current is miles per hour, he can row his boat downstream to town for supplies and back in hours. What is his average rowing speed in still water?

Solution:

Let represent Brett's average rowing speed in still water.

Rowing downstream, the current increases his speed, and his rate is miles per hour. Rowing upstream, the current decreases his speed, and his rate is miles per hour. Begin by organizing the data in the following chart:

Figure

Use the formula to fill in the time column for each leg of the trip.

Figure

The algebraic setup is defined by the time column. Add the times for each leg of the trip to obtain a total of 3 hours:

Solve this equation by first multiplying both sides by the LCD, .

Next, solve the resulting quadratic equation.

Use only the positive solution, miles per hour.

Answer:

His rowing speed is miles per hour.

Exercise

Dwayne drove miles to the airport to pick up his father and then returned home. On the return trip he was able to drive an average of miles per hour faster than he did on the trip there. If the total driving time was hour, then what was his average speed driving to the airport?

Answer

His average speed driving to the airport was miles per hour.

Work-Rate Problems

The rate at which a task can be performed is called a work rate.

Example

For example, if a painter can paint a room in 8 hours, then the task is to paint the room, and we can write

In other words, the painter can complete of the task per hour.

a. Calculate the fraction of the task that one painter can accomplish in 2, 4 and 8 hours

b. If an apprentice would take 10 hours to paint the room, how long would it take for the apprentice and the master painter to paint the room.

Solution

a. If he works for less than 8 hours, then he will perform a fraction of the task. For example,

Obtain the amount of the task completed by multiplying the work rate by the amount of time the painter works. Typically, work-rate problems involve people working together to complete tasks. When this is the case, we can organize the data in a chart, just as we have done with distance problems.

b. Suppose an apprentice painter can paint the same room by himself in hours. Then we say that he can complete of the task per hour.

Let represent the time it takes both of the painters, working together, to paint the room.

Figure

To complete the chart, multiply the work rate by the time for each person.

The portion of the room each can paint adds to a total of task completed.

This is represented by the equation obtained from the first column of the chart:

Figure

This setup results in a rational equation that can be solved for by multiplying both sides by the LCD, .

Therefore, the two painters, working together, complete the task in hours.

In general, we have the following work-rate formula:

Here and are the individual work rates and is the time it takes to complete one task working together. If we factor out the time, , and then divide both sides by , we obtain an equivalent work-rate formula:

In summary, we have the following equivalent work-rate formulas:

Example

Working alone, Billy’s dad can complete the yard work in 3 hours. If Billy helps his dad, then the yard work takes 2 hours. How long would it take Billy working alone to complete the yard work?

Solution:

The given information tells us that Billy’s dad has an individual work rate of task per hour. If we let represent the time it takes Billy working alone to complete the yard work, then Billy’s individual work rate is , and we can write

Figure

Working together, they can complete the task in hours. Multiply the individual work rates by hours to fill in the chart.

Figure

The amount of the task each completes will total 1 completed task. To solve for , we first multiply both sides by the LCD, .

Answer:

It takes Billy hours to complete the yard work alone.

Of course, the unit of time for the work rate need not always be in hours.

Example

Working together, two construction crews can build a shed in days. Working separately, the less experienced crew takes twice as long to build a shed than the more experienced crew. Working separately, how long does it take each crew to build a shed?

Solution:

Let represent the time it takes the more experienced crew to build a shed.

Let represent the time it takes the less experienced crew to build a shed.

Working together, the job is completed in days. This gives the following setup:

Figure

The first column in the chart gives us an algebraic equation that models the problem:

Solve the equation by multiplying both sides by .

To determine the time it takes the less experienced crew, we use :

Answer:

Working separately, the experienced crew takes days to build a shed, and the less experienced crew takes days to build a shed.

Exercise

Joe’s garden hose fills the pool in hours. His neighbor has a thinner hose that fills the pool in hours. How long will it take to fill the pool using both hoses?

Answer

It will take both hoses hours to fill the pool.

Key Takeaways

• In this section, all of the steps outlined for solving general word problems apply. Look for the new key word “reciprocal,” which indicates that you should write the quantity in the denominator of a fraction with numerator .
• When solving distance problems where the time element is unknown, use the equivalent form of the uniform motion formula, , to avoid introducing more variables.
• When solving work-rate problems, multiply the individual work rate by the time to obtain the portion of the task completed. The sum of the portions of the task results in the total amount of work completed.

Exercise Number Problems

Use algebra to solve the following applications.

1. A positive integer is twice another. The sum of the reciprocals of the two positive integers is . Find the two integers.
2. A positive integer is twice another. The sum of the reciprocals of the two positive integers is . Find the two integers.
3. A positive integer is twice another. The difference of the reciprocals of the two positive integers is . Find the two integers.
4. A positive integer is twice another. The difference of the reciprocals of the two positive integers is . Find the two integers.
5. A positive integer is less than another. If the sum of the reciprocal of the smaller and twice the reciprocal of the larger is , then find the two integers.
6. A positive integer is more than another. If the sum of the reciprocal of the smaller and twice the reciprocal of the larger is , then find the two integers.
7. The sum of the reciprocals of two consecutive positive even integers is . Find the two even integers.
8. The sum of the reciprocals of two consecutive positive odd integers is . Find the integers.
9. The difference of the reciprocals of two consecutive positive even integers is . Find the two even integers.
10. The difference of the reciprocals of two consecutive positive odd integers is . Find the integers.
11. If times the reciprocal of the larger of two consecutive integers is subtracted from times the reciprocal of the smaller, then the result is . Find the two integers.
12. If times the reciprocal of the smaller of two consecutive integers is subtracted from times the reciprocal of the larger, then the result is . Find the two integers.
13. A positive integer is less than another. If the reciprocal of the smaller integer is subtracted from times the reciprocal of the larger, then the result is . Find the two integers.
14. A positive integer is less than another. If the reciprocal of the smaller integer is subtracted from times the reciprocal of the larger, then the result is . Find the two integers.

Answer

1. {}

3. {}

5. {}

7. {}

9. {}

11. {} or {}

13. {} or {}

Exercise Uniform Motion Problems

Use algebra to solve the following applications.

1. James can jog twice as fast as he can walk. He was able to jog the first miles to his grandmother’s house, but then he tired and walked the remaining miles. If the total trip took hours, then what was his average jogging speed?
2. On a business trip, an executive traveled miles by jet aircraft and then another miles by helicopter. If the jet averaged times the speed of the helicopter and the total trip took hours, then what was the average speed of the jet?
3. Sally was able to drive an average of miles per hour faster in her car after the traffic cleared. She drove miles in traffic before it cleared and then drove another miles. If the total trip took hours, then what was her average speed in traffic?
4. Harry traveled miles on the bus and then another miles on a train. If the train was miles per hour faster than the bus and the total trip took hours, then what was the average speed of the train?
5. A bus averages miles per hour faster than a trolley. If the bus travels miles in the same time it takes the trolley to travel miles, then what is the speed of each?
6. A passenger car averages miles per hour faster than the bus. If the bus travels miles in the same time it takes the passenger car to travel miles, then what is the speed of each?
7. A light aircraft travels miles per hour less than twice as fast as a passenger car. If the passenger car can travel miles in the same time it takes the aircraft to travel miles, then what is the average speed of each?
8. Mary can run mile per hour more than twice as fast as Bill can walk. If Bill can walk miles in the same time it takes Mary to run miles, then what is Bill’s average walking speed?
9. An airplane traveling with a -mile-per-hour tailwind covers miles. On the return trip against the wind, it covers miles in the same amount of time. What is the speed of the airplane in still air?
10. A jet airliner traveling with a -mile-per-hour tailwind covers miles in the same amount of time it is able to travel miles after the tailwind eases to miles per hour. What is the speed of the airliner in still air?
11. A boat averages miles per hour in still water. With the current, the boat can travel miles in the same time it travels miles against it. What is the speed of the current?
12. A river tour boat averages miles per hour in still water. If the total -mile tour down river and miles back takes hours, then how fast is the river current?
13. If the river current flows at an average miles per hour, then a tour boat makes the -mile tour downstream with the current and back the miles against the current in hours. What is the average speed of the boat in still water?
14. Jane rowed her canoe against a -mile-per-hour current upstream miles and then returned the miles back downstream. If the total trip took hours, then at what speed can Jane row in still water?
15. Jose drove miles to pick up his sister and then returned home. On the return trip, he was able to average miles per hour faster than he did on the trip to pick her up. If the total trip took hour, then what was Jose’s average speed on the return trip?
16. Barry drove the miles to town and then back in hour. On the return trip, he was able to average miles per hour faster than he averaged on the trip to town. What was his average speed on the trip to town?
17. Jerry paddled his kayak upstream against a -mile-per-hour current for miles. The return trip downstream with the -mile-per-hour current took hour less time. How fast can Jerry paddle the kayak in still water?
18. It takes a light aircraft hour more time to fly miles against a - mile-per-hour head wind than it does to fly the same distance with it. What is the speed of the aircraft in calm air?

Answer

1. miles per hour

3. miles per hour

5. Trolley: miles per hour; bus: miles per hour

7. Passenger car: miles per hour; aircraft: miles per hour

9. miles per hour

11. miles per hour

13. miles per hour

15. miles per hour

17. miles per hour

Exercise Work-Rate Problems

Use algebra to solve the following applications.

1. James can paint the office by himself in hours. Manny paints the office in hours. How long will it take them to paint the office working together?
2. Barry can lay a brick driveway by himself in hours. Robert does the same job in hours. How long will it take them to lay the brick driveway working together?
3. Jerry can detail a car by himself in minutes. Sally does the same job in hour. How long will it take them to detail a car working together?
4. Jose can build a small shed by himself in hours. Alex builds the same small shed in days. How long would it take them to build the shed working together?
5. Allison can complete a sales route by herself in hours. Working with an associate, she completes the route in hours. How long would it take her associate to complete the route by herself?
6. James can prepare and paint a house by himself in days. Working with his brother, Bryan, they can do it in days. How long would it take Bryan to prepare and paint the house by himself?
7. Joe can assemble a computer by himself in hour. Working with an assistant, he can assemble a computer in minutes. How long would it take his assistant to assemble a computer working alone?
8. The teacher’s assistant can grade class homework assignments by herself in hour. If the teacher helps, then the grading can be completed in minutes. How long would it take the teacher to grade the papers working alone?
9. A larger pipe fills a water tank twice as fast as a smaller pipe. When both pipes are used, they fill the tank in hours. If the larger pipe is left off, then how long would it take the smaller pipe to fill the tank?
10. A newer printer can print twice as fast as an older printer. If both printers working together can print a batch of flyers in minutes, then how long would it take the newer printer to print the batch working alone?
11. Working alone, Henry takes hours longer than Mary to clean the carpets in the entire office. Working together, they clean the carpets in hours. How long would it take Mary to clean the office carpets if Henry were not there to help?
12. Working alone, Monique takes hours longer than Audrey to record the inventory of the entire shop. Working together, they take inventory in hours. How long would it take Audrey to record the inventory working alone?
13. Jerry can lay a tile floor in hours less time than Jake. If they work together, the floor takes hours. How long would it take Jerry to lay the floor by himself?
14. Jeremy can build a model airplane in hours less time than his brother. Working together, they need hours to build the plane. How long would it take Jeremy to build the model airplane working alone?
15. Harry can paint a shed by himself in hours. Jeremy can paint the same shed by himself in hours. How long will it take them to paint two sheds working together?
16. Joe assembles a computer by himself in hour. Working with an assistant, he can assemble computers in hours. How long would it take his assistant to assemble computer working alone?
17. Jerry can lay a tile floor in hours, and his assistant can do the same job in hours. If Jerry starts the job and his assistant joins him 1 hour later, then how long will it take to lay the floor?
18. Working alone, Monique takes hours to record the inventory of the entire shop, while it takes Audrey only hours to do the same job. How long will it take them working together if Monique leaves hours early?

Answer

1. hours

3. minutes

5. hours

7. hours

9. hours

11. hours

13. hours

15. hours

17. hours