Sometimes you will encounter a word problem that asks you to determine how long it would take two people working together to finish a job.  Solving this type of problem requires a few steps of logic.  Let’s jump straight to an example.

Example:  Jennifer can mop a warehouse in 8.3 hours.  Heather can mop the same warehouse in 11.2 hours.  Find how long it would take them if they worked together.

Solution:  We set up an equation to model Jen’s work.  We know that Jen can mop a warehouse in 8.3 hours, which means

\(\Large \frac{{1{\text{Warehouse  Mopped}}}}{{8.3{\text{  hours}}}} = 0.12{\text{Warehouse  Mopped  in  }}1{\text{  hour}}\)

That is, Jen can mop 12 percent of the warehouse in one hour.  We set up a similar equation for Heather.  We know that Heather can mop the same warehouse in 11.2 hours, which means

\(\Large \frac{{1{\text{Warehouse  Mopped}}}}{{11.2{\text{  hours}}}} = 0.09{\text{Warehouse  Mopped  in  }}1{\text{  hour}}\)

That is, Heather can mop about 9 percent of the warehouse in one hour.  Now we can find out how much of the warehouse they can mop together in one hour.  We have

\(0.12\left( {for  Jen} \right) + 0.09\left( {for  Heather} \right) = 0.21\)

That is, together they can mop 21 percent of the warehouse in 1 hour.  Let’s set up our final equation to model this word problem.  We use a simple ratio:

\(\Large \frac{{1{\text{Warehouse  Mopped}}}}{{x{\text{  hours}}}} = \Large \frac{{0.21{\text{Warehouse  Mopped}}}}{{1{\text{  hour}}}}\)

Cross multiplying gives

\(x = \Large \frac{1}{{0.21}} = 4.76{\text{   hours}}\)

Another Example:  Molly can clean an attic in 10.6 hours.  Jasmine can clean the same attic in 15 hours.  If they worked together how long would it take them?

\(\Large \frac{{1  Attic}}{{10.6  hours}} = 0.09  in  one  hour\)

For Jasmine, we have

\(\Large \frac{{1  Attic}}{{15  hours}} = 0.07  in  one  hour\)

Together, their labor yields

\(0.09 + 0.07 = 0.16  together  in  one  hour\)

Then we use another ratio to solve the problem

\(\Large \frac{{1  Attic  Cleaned}}{{x  hours}} = \Large \frac{{0.16  Cleaned}}{{1  hour}}\)

Then, by cross multiplying,

\(x = \Large \frac{1}{{0.16}} = 6.25  hours\)

Below you can download some free math worksheets and practice.