In precalc 11 this week, we continued learning about rational equations, and how to apply them into real life word problems. Most word problems can be solve creating a rational equation and solving it. This blog post will explain one type of word problem I learned, which involves work and time. Here is an example:

It takes John 5 hours to clean his basement. When John and his friend Barry clean it together, it only takes 3 hours. How long would it take Barry to clean the basement by himself?

To solve this, you can create a table with John, Barry and Both as the categories on top. On the left side, write the time. Choose values of time that represent the information given in the question. Because it takes John 5 hours alone and 3 hours together, I chose to use 1 hour, 3 hours, and 5 hours. Just make sure the value of time for which they can finish the job together is one of the options (in this case, 3).

Next, add in the values you know. It takes John 5 hours to finish the job, so in 1 hour John will have cleaned 1/5 of the basement. In 3 hours, he will have done 3/5 of the job. In 5 hours, he will have finished 5/5 (which equals 1), which means all of the basement.

Because we don’t know the values for Barry, write similar fractions in the table, but use a variable. In one hour, he will have done 1/x of the work. Repeat this for 3 and 5 hours.

Finally, complete the section with both people working as a team. Add the values from each row, as shown below. In the section with 3 hours, we know they finished the job. Therefore, the two values added together equal one.

Now, just solve the equation you created:

This means that it would take Barry 7.5 hours to clean the basement on his own.

This is a different example using the same strategy:

Mary takes three times as long as her daughter to set up their Christmas decorations. When they do it together, it takes 9 hours to finish the job. How long would it take each person on their own to set up the decorations?