346 MCGRAW-HILL’S SAT
What Are Rates?
The word ratecomes from the same Latin root as the
word ratio.All rates are ratios. The most common type
of rate is speed, which is a ratio with respect to time, as
in miles per houror words per minute,but some rates
don’t involve time at all, as in miles per gallon.Rate
units always have perin their names: miles per gallon,
meters per second,etc. Per,remember, means divided
by,and is like the colon (:) or fraction bar in a ratio.
The Rate Pyramid
Watch Your UnitsWhenever you work with formulas, you can
check your work by paying attention to units.
For instance, the problem above asks how
long,so the calculation has to produce a time
unit. Check the units in the calculation:
miles
miles
hoursmileshours
miles=× =hoursLesson 4: Rate Problems
The name of any rate is equivalent to its formula.
For instance, speed is miles per hourcan be translated asor
Speed =
distance
timeSpeed =
number of miles
number of hoursSince this formula is similar to the “average” formula,
you can make a rate pyramid.
This can be a great tool for
solving rate problems. If a
problem gives you two of the
quantities, just put them in
their places in the pyramid,
and do the operation be-
tween them to find
the missing quantity.
Example:
How long will it take a car to travel 20 miles at
60 miles per hour?
Simply fill the quantities into the pyramid: 20 miles
goes in the distance spot, and 60 miles an hour goes
in the speed spot. Now what? Just do the division the
way the diagram says: 20 miles ÷60 miles per hour =
1/3 hour.
Two-Part Rate ProblemsRate problems are tougher when they involve
two parts. When a problem involves, say, two
people working together at different rates and
times, or a two-part trip, you have to analyze
the problem more carefully.Example:
Toni bicycles to work at a rate of 20 miles per
hour, then takes the bus home along the same
routeat a rate of 40 miles per hour. What is her
average speed for the entire trip?
At first glance, it might seem that you can just aver-
age the two rates: (20 +40)/2 =30 miles per hour,
since she is traveling the same distance at each of the
two speeds. But this won’t work, because she isn’t
spending the same timeat each speed, and that is
what’s important. But if that’s true, you might notice
that she spends twiceas much time going 20 miles per
hour as 40 miles per hour (since it’s half as fast), so
instead of taking the average of 20 and 40, you can
take the average of two20s and a 40:
(20 + 20 +40)/3 =26.67 miles per hour. Simple! But if
that doesn’t make sense to you, think of it this way:
Imagine, for simplicity’s sake, that her trip to work is
40 miles. (It doesn’t matter what number you pick,
and 40 is an easy number to work with here.) Now the
average speed is simply the total distance divided by
the total time (as the pyramid says). The total distance,
there and back, is 80 miles. The total time is in two
parts. Getting to work takes her 40 miles ÷20 miles per
hour =2 hours. Getting home takes her 40 miles ÷ 40
miles per hour =1 hour. So the totaltime of the trip
is 3 hours. The average speed, then, must be 80 miles
÷3 hours =26.67 miles per hour!speed timedistance×