The Relationship between Work, Rate, and Time
If it takes x–number of hours to complete a job, we can describe the rate as (1 job) / (x hours), or as the rational expression , with the unit being jobs per hour. We can also express the relationship between work, rate, and time using the equation , where w is work, r is rate, and t is time. When two people work together, the total work is the sum of their individual work. We will use these relationships to answer some questions involving work, rate, and time.
Modeling a Work, Rate, and Time Problem
Abby and Bobby are maintenance workers for a building management company. They share the task of mowing the lawn and trimming the hedges at the different complexes the company manages. If Abby were to work alone, it would take 30 minutes of her to complete the work at one building. If her coworker Bobby helps, they can complete the same amount of work in only 18 minutes. How long would it take Bobby to complete the work by himself?
To model this scenario, we can create a work, rate and time chart that organizes known and unknown information. Our chart will have columns for Work, Rate, and Time, and rows for Abby and Bobby:
We know information about the rate of Abby's work if she works alone, as well as the time it takes for Abby and Bobby to complete the work together. We will describe Abby's rate as 1 job per 30 minutes. We can also enter 18 minutes in the time column for both Abby and Bobby to show that if they work together, they can complete a job in 18 minutes:
If we multiply Abby's rate by time, we can get the proportion of one job that she completes on her own in 18 minutes:
Note: 18 over 30 simplifies to 3 over 5; and the units of minutes cancel, which confirms that our unit for work is jobs.
We don't know how long it takes Bobby to complete one job by himself, so we will use the variable x to describe the number of minutes it takes him to complete a job by himself. We can use the rate 1 over x, and then multiply it by 18 minutes to show the proportion of work Bobby can do in 18 minutes:
Solving a Work, Rate, and Time Problem using a Rational Equation
Using the fact that when two people work together, their total work (1 job) is the sum of their individual works, we have the rational equation:
To solve this equation, one strategy is to rewrite every term so that it has a common denominator. If all terms have a common denominator, then we can create an equation with just the numerators, which makes our equation easier to solve:
This means that if Bobby were to work alone, it would take him 45 minutes to complete one job.