The Mowing Problem

The following problem was posted on CollegeConfidential (for full thread see here):

Jan can mow her lawn in 60 minutes by herself. If she hires Wally to do it, he takes 30 minutes, while Peter can do it in 40 minutes. If Jan starts mowing her lawn for 15 minutes then decides to hire Wally and Peter to help her finish, how long will it take (in hours) from when Jan started to when the three finished together?

1. First, let's find out how much of the lawn Jan mowed. That's pretty easy. Since it takes her 60 minutes to mow the whole lawn, she mowed 15/60=1/4 of the lawn already. Therefore, 3/4 of the lawn have to be mowed by the three together.
2. We now need to find out how much the three can mow together in a certain amount of time. If we knew how much of the lawn each of the three mows in one minute, we could simply add those fractions to get the total amount of mowed lawn per minute.
3. We know that it takes Jan 60 minutes to mow the whole lawn. Therefore, she mows 1/60 of the lawn in one minute.
4. Wally and Peter can mow 1/30 and 1/40, respectively, in one minute. Now we can simply add the fractions:
   1/60 + 1/30 + 1/40 = 9/120
   That's how much the three mow in one minute.
5. They still need to mow 90/120 (=3/4). If it takes them 1 minute to mow 9/120, how much does it take them to mow 90/120?
   9/120 --> *10 --> 90/120
   1min --> *10 --> 10min
6. It takes them 10 minutes. Don't forget to add the first 15 minutes Jan mowed alone:
   15min + 10min = 25min
   Since the question asks for hours we need to convert the minutes:
   25min = 25/60h = 5/12h
   

First post!

Hi and welcome to the first post of the SAT Math Explanations Blog! Here, I'll try to offer explanations to the more difficult SAT math questions. Feel free to mail me (SAT.math.section@gmail.com) any math questions you need explanations for. I'll try to answer as many as possible. I don't care whether they're from previously administered tests, from the Official SAT Question of the Day or from any other source, as long as they present topics that are covered on the SAT. So fire away!

The following question is from the Offical SAT Question of the Day series by Collegeboard. According to their statistics 65% answered incorrectly. Let's take a look at it:

A woman drove to work at an average speed of 40 miles per hour and returned along the same route at 30 miles per hour. If her total traveling time was 1 hour, what was the total number of miles in the round trip?

1. Let's call the number of miles in one direction x. Therefore, our final solution will be 2x. That's important to remember.
2. It is given that the total time was 1 hour, so one side of our equation will be 1 hour:
   1h = ?
3. We need to have the same quantity (the same "type" of something) on the other side of the equation, i.e. we need something in hours. Therefore, we need to "convert" the mph into hours. We don't know the number of miles in one direction, but we have defined it as x. We can now express the time it took in terms of x. If we need 1h for 40miles, how long does it take when we drive x miles?
   40miles -->:40 *x --> x
   1 hour -->:40 *x --> x/40
   It takes x/40 hours. We have now expressed the time in terms of x.
4. The same holds true for 30mph, except that it now takes x/30 hours. Therefore, x/40 + x/30 is the total time:
   1h = x/40 + x/30
5. Solving the equation yields 17 1/7 for x. Therefore, the total number of miles is:
   2 * 17 1/7 = 34 2/7

This type of question is quite common on the SAT, by the way.