# Rates

## Arithmetic: Rates

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

Use the same process for every math problem so that you are not intimidated by any question.  (1) Read the question carefully.  (2) Identify the bottom line – what is the question asking?  (3) Take the time to assess your options – which methods can you use to solve this problem most efficiently?  (4) Attack the problem and work though it logically.  (5) Loop back to make sure that your answer matches the bottom line – did you complete every step of the problem?

A train traveling 60 miles per hour for 1 hour covers the same distance as a train traveling 30 miles per hour for how many hours?

Bottom line: Make a quick note that you are solving for hours: hrs = ?

Assess Your Options:  You could try to use logic for this problem by thinking that if a train goes more slowly, it must take longer to go the same distance as it did at a faster speed.  Unfortunately, logic will not eliminate all of your answer choices.  Use the distance formula to solve this problem.

Attack the problem:  The distance formula is distance is equal to rate(speed) times time:  D = R × T.  Start with the first train and multiply the rate (60 m/hr) by the time (1 hr) to get the distance:

60 × 1 = 60

The first train traveled 60 miles.  You know that both trains traveled the same distance, so plug in 60 as the distance for the second train. You also know that the rate is 30 and the time is unknown.  That should look like:

30 × T = 60
30T = 60
T = 2

Note:  If you are good at balancing equations, there is an even faster way to do this problem.  Look at the distance equation:  D = R × T.  If the distance for a problem stays the same, but you increase the speed (rate), then you must decrease the time by the reciprocal of the speed increase.  That keeps the equation balanced.  Ex:  If you double the speed, you must halve the time.  In this particular problem you halve the speed (from 60 to 30), so you must double the time.  2 × 1 hour = 2 hours.  This reciprocal rule will always work!

Loop Back:  You solved for the time of the second train, which is already in hours, so you are ready to look at your answer choices!

(A) 3
(B) 2
(C) 1
(D)
(E)