# Rates

## Arithmetic: Rates

Read the following SAT test question and then select the correct answer.

Using the same method with every math problem to minimize mistakes.  Read the question carefully.  Identify the bottom line and assess your options for finding it.  Choose the most efficient method to attack the problem.  Once you have an answer, loop back to make sure it addresses the bottom line.

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?

Bottom Line: Td = ?  (Total distance)

Assess your Options: Remember that speed is really a rate.  There are 4 key rate scenarios: separation, overtake, round trip, and meet in between--this one is a round trip.  You can figure all of these out by using the distance formula (rate × time = distance), but it can be difficult to keep track of which scenario you have unless you treat all of them the same way.  Knowsys recommends that you use a chart to quickly organize your thoughts so that you can be sure that you accounted for all of the information in the problem. (Spoiler: many students make mistakes on these types of problems!  You do not get any extra points for ignoring the chart, so use it!)

Attack the Problem:  Here is the chart that you should use with all rate scenarios:

 1 2 Total Rate Time Distance

Start filling in the information that you know.  The first trip was at a rate of 40 miles per hour and the second trip was at a rate of 30 miles per hour.  The total time was 1 hour.

 Trip 1 Trip 2 Total Rate 40 30 Time 1 Distance

If you don’t know the time for the first trip, choose a variable to represent the unknown.  Put an “x” in that box.  You know that the time for the trips together must total 1 hour (x + ? = 1).  Therefore, the second trip is equal to 1 minus x

 Trip 1 Trip 2 Total Rate 40 30 Time x 1 – x 1 Distance

You already know that rate × time = distance, so multiply the two columns representing the trips.

 Trip 1 Trip 2 Total Rate 40 30 Time x 1 – x 1 Distance 40x 30(1 – x)

Before you start worrying about the total number of miles, remember that this person is using the same route each time.  That means the distance traveled each time is an equal length.  Set the distances equal to each other.

40x = 30(1 – x)
40x = 30 – 30x
70x = 30
$x=\frac{3}{7}$

If you know x, you can now find a number value for each part of your chart.  What was the bottom line?  You need to find the total number of hours.  You could plug x into both distances and add them up; however, there is an even faster method.  Take the first distance and multiply it by 2.  (Remember that the distances are the same.)

$2\times40\times \frac{3}{7}=Total\; distance$

$\frac{240}{7}=Total\; distance$

$34\frac{2}{7}=Total\; distance$

(B) $30\frac{1}{7}$
(C) $34\frac{2}{7}$