# Rates

## Arithmetic: Rates

Use the same method for every math problem on the SAT.  Read the problem carefully, identify the bottom line, and assess your options for solving the problem.  Choose the most efficient method to attack the problem.  Often there will be multiple steps to a single problem, so when you have an answer, be sure to loop back and verify that it matches the bottom line.

A machine can insert letters in envelopes at the rate of 120 per minute. Another machine can stamp the envelopes at the rate of 3 per second. How many such stamping machines are needed to keep up with 18 inserting machines of this kind?

Bottom Line:  # stamping machines = ?

Assess your Options:  You could try to work backwards from the answers, but there is no need.  It will be faster just to solve the problem.

Attack the Problem:  You have been given two different units of time: minutes and seconds.  There are 60 seconds in a minute.  Changing the minutes to seconds will be easiest, so the letter inserting machine works at a rate of 120 letters per 60 seconds.  120 divided by 60 is 2 letters in envelopes per second.  If there are 18 letter inserting machines, then together they will insert 36 letters in envelopes per second (2 × 18 = 36).

You don’t know how many stamping machines you need, so use x to represent that number.  Stamping machines have a rate of 3 envelopes per second, so each machine will finish 3 envelopes in a second.  You know that the stamping machines must keep pace with the 18 letter inserting machines that finish 36 envelopes per second, so the outcome must be 36.  Write 3x = 36.  When you solve for x,  x = 12.

Loop Back:  You used x to represent the number of stamping machines, your bottom line, so you are ready to look at the answer choices.

(A) 12
(B) 16
(C) 20
(D) 22
(E) 24