Solving Word Problems on Rational Equations

Introduction

Now that you understand how to solve rational equations, let's take it a step further by applying this knowledge to real-world scenarios.

In this lesson, we'' explore how rational equations can be used to solve word problems. These problems often involve rates, work, distance and other practical situations where rational expressions naturally arise.

Worked Example #1

James, a carwash boy, can finish washing a car in 30 minutes, while his co-worker, Joshua, can do the same work in 20 minutes. If they work together, how long would it take them to finish washing a car?

Solutions: Let be the number of minutes it takes them to finish the work together.

With this, the requires equation that represent the problem is

Note that the Least Common Denominator (LCD) of 30 and 20, and x, which is 60x.

To verify if the answer is correct, we will substitute x=12 to the crafted rational algebraic equation:

Thus, it will take \bf 12 minutes to finish the job together.

Worked Example #2

One pump can fill a tank in 6 hours, while a second pump can do the same job in 10 hours. If both pumps are turned on at the same time, how long will it take to completely fill the tank?

Solution: Let x be the number of hours it takes to fill the tank when both pumps are working together.

With this, the requires equation that represents the problem is

Note that the Least Common Denominator (LCD) 0f 6, \space 10, \space \text{ and } x \text{ is } 30x.

To verify if the answer is correct, we will substitute x=\dfrac{15}{4} into the crafted rational algebraic equation:

Hence, it will take \bf\dfrac{15}{4}hours to completely fill the tank if both pipes are turned on.

Worked Example #3

Kenneth can clean the house in 30 minutes, and Felix can complete the task in 1 hour. If they team up and work together, how much time will they need to clean the house?

Solution: Let x be the number of minutes it takes them to finish cleaning the house together.

With this, the requires equation that represents the problem is

Note that the Least Common Denominator (LCD) of 30, \space 60, \text{ and } x \text{ is } 60x.

To verify if the answers is correct, we will substitute x=20 into the crafted rational algebraic equation:

Thus, it will take \bf 20minutes to clean the house together.

Worked Example #4

An inlet pipe fills a pool in 12 hours, while an outlet pipe can drain it in 15 hours. If the pool starts out empty and both pipes are opened at the same time, how long will it take to completely fill the pool?

Solution: Let x be the number of hours it takes to completely fill the pool when both pipes are open.

Thus, the equation that models the situation is

Note that the Least Common Denominator (LCD) of 12, \space 15, \text{ and } x \text{ is } 60x

To verify if the answer is correct, substitute x=60 into the original equation:

Thus, it will take \bf 60 hours to completely fill the pool when both pipes are open.

Worked Example #5

Joe can finish mowing their backyard lawn in 40 minutes, while her sister Althea can do the same work in 50 minutes. How long would it take them to finish mowing the lawn together?

Solution: Let x be the number of minutes it takes them to finish the work together.

The equation that represents the problem is

Note that the Least Common Denominator (LCD) of 40, \space 50, \text{ and } x \text{ is } 200x.

To verify if the answer is correct, substitute x= \dfrac{200}{9} into the original equation:

Thus, it will take them \bf \dfrac{200}{9} minutes to finish mowing the lawn together.