Ratios
Ratios look like fractions and are related to fractions,
but they don’t quack like fractions. Whereas a fraction describes
a part of a whole, a ratio compares one part to another part.
A ratio can be written in a variety of ways. Mathematically,
it can appear as 3/
1 or as 3:1.
In words, it would be written out as the ratio of 3 to 1.
Each of these three forms of the ratio 3:1 mean the
same thing, that there are three of one thing for every one
of another. If you have three red alligators and one blue alligator,
then you would have a ratio of 3:1 for red alligators
to blue alligators. For the SAT, you must remember that ratios compare parts
to parts rather than parts to a whole. Why do you have to
remember that? Because of questions like this:
|
|
|
For
every 40 games a baseball team plays, it loses 12 games. What is
the ratio of the team’s losses to wins? |
| (A) |
3:10 |
| (B) |
7:10 |
| (C) |
3:7 |
| (D) |
7:3 |
| (E) |
10:3 |
|
The question says that the team loses 12 of
every 40 games, but it asks you for the ratio of losses
to wins, not losses to games.
So the first thing you have to do is find out how many games the
team wins in 40 games:
The team wins 28 games for every 40.
So for every 12 losses, the team wins 28 games,
for a ratio of 12:28. You can reduce this ratio by
dividing both sides by 4 to get 3 losses
for every 7 wins, or 3:7. Answer C is
correct. If you didn’t realize that the losses to games was a part
to whole, you might have just reduced the ratio 12:40 to 3:10,
and then chosen answer A. And there is no question
that on ratio problems, the SAT will include an incorrect part: whole answer
to try to trip you up.
Proportions
Just because you have a ratio of three red
alligators to one blue alligator doesn’t mean that
you can only have three red alligators
and one blue one. It could also mean that you have
six red and two blue alligators or that
you have 240 red and 80 blue alligators.
Ratios compare only relative magnitude. In order
to know how many of each color alligator you actually have, in addition
to knowing the ratios, you also need to know how many total alligators
there are.
The SAT often asks questions testing your ability to figure
out an answer based on the ratio between items and the total number
of all items:
|
|
|
Egbert
has red, blue, and green marbles in the ratio of 5:4:3, and he has
a total of 36 marbles. How many blue marbles does Egbert have? |
|
For each group of 5 red marbles, you have
a group of 4 blue marbles and a group of 3 green
marbles. The ratio therefore tells you that out of every 12 marbles
(since 5 + 4 + 3 = 12), 4 marbles will
be blue.
The question also tells you that you have 36 total
marbles, and since the ratio of blue marbles to total marbles will
not change no matter how many marbles you have,
you can solve this problem by setting up a proportion, which is
an equation that states that two ratios are equal. In this case,
you can set equal 4:12 and x:36, with x standing
in for the number of blue marbles that you’d have out of a total
of 36. To do math with proportions, it’s most useful
to set up proportions in fractional form:
Now isolate x by cross-multiplying,
and then you can solve.
