Calculate the time it takes two people to complete a task together in a given scenario

Question

Jane and Bill are cleaning up a park. If she had to clean up the park herself, it would take Jane six hours to complete the job. If Bill alone were to clean the park, it would take him three hours.

Working together, how long will it take Jane and Bill to clean the park?



Answer 1

Lorie, these “shared work” problems are no harder than most word problems, but they do introduce new concepts: Work and Rate.

Here’s a brief background on Work and Rate problems, so that you can see how to deal with them in the future. I promise I WILL get around to your problem.

Work is any amount of stuff to get done: 14 pies to bake, 3 dogs to groom, 1 house to paint, 1 email job, etc.

Rate is how much work is done in one unit of time, like 3 pies baked per hour, .8 dogs groomed per hour, 0.05 house painted per day, .03 email job per minute, etc.

These are always related by the formula

Work = Rate * Time

word problems involving Work and Rate generally involve two agents working together on a job. In this case, the formula looks like

W = T * (R_a + R_b)

What is tedious about these problems is that they usually give TIMES rather than RATES. as a result, you usually have to play with reciprocals. That is, a Rate is usually 1/(a time).

Finally I’m getting to your problem at hand.

We are told Jane’s Time to clean the park is 6 hours, so we say

T_J = 6

we also know bill’s Time To clean the same park is 3 hours, so...

T_B = 3

But we really want know their Rates so we can add them together in that formula above.

Here is where understanding the concept comes in. The formula W = RT shows how they are all related. In particular we can see that if cleaning the park is 1 quantity of Work. Then

1 = R_J * T_J

If you know R_J , then you know T_J = 1 / R_J, And you know the same relationship holds for Bill.

So, you know their times, but we want to know their Rates, and plug them into the shared work equation

W = T * (R_a + R_b)

I’ll relabel to represent Jane and Bill

W = T * (R_J + R_B)

now replace the rates with the reciprocals of the times

W = T * (1/T_J + 1/T_B)

Now solve for T (I promised it would be tedious)

T = W / (1/T_J + 1/T_B)

Now put the fractions over a common denominator

T = W / ((T_B+T_J)/(T_BT_J))

now since dividing by a fraction is the same as inverting the fraction and multiplying,

T = (W * T_BT_J) / (T_B+ T_J)

Finally, fill in the numbers you know

T = (1 * 6 * 3) / (6 + 3) = 18 / 9 = 2 hours

if you have any questions, please ask