Ratios
Ratios can look a lot like fractions, and they are related
to fractions, but they differ in important ways. Whereas a fraction
describes a part out of a whole, a ratio compares two separate parts
of the same whole.
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 marbles and one blue marble,
then you would have a ratio of 3:1 for red marbles
to blue marbles. For the SAT, you must remember that ratios compare parts
to parts, rather than parts to a whole. The SAT will ask
you questions like this:
|
|
|
Of
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 key to this sort of ratio question is to see that
the question is stated in terms of a whole to a part but asks for
a part to part answer. The questions tells you the team loses 12 of every 40 games,
but it asks you for the ratio of losses : wins not
losses : 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
or 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 chose
answer (A). You can bet that on this sort of ratio problem the SAT
will include the incorrect part:whole answer
to try to trip you up.
Proportions
If you have a ratio of 3 red marbles to 1 blue,
that doesn’t necessarily mean that you have exactly 3 red
marbles and 1 blue one. It could also mean that you
have 6 red and 2 blue marbles or that
you have 240 red and 80 blue marbles.
In other words, ratios compare only relative size.
In order to know how many of each color marble you actually have,
in addition to knowing the ratios, you also need to know how many
total marbles there are.
The SAT will often ask questions testing your ability
to figure out a quantity based on the given information of a ratio
between items and the total number of all items. For example:
|
|
|
You
have red, blue, and green marbles in the ratio of 5:4:3, and you
have a total of 36 marbles. How many blue marbles do you have? |
|
The information given states that for each group of 5 red
marbles, you have a corresponding 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 of
them will be blue. The question also tells you that you have 36 total
marbles.
Since the ratio of blue marbles will not change no matter
how many marbles you have, we can solve this problem by setting
up a proportion, which is an equation that states that two ratios
are equal. In this case, we are going to set equal 4:12 and x:36,
with x being the number of blue marbles
that we would have if we had 36 total marbles. To do
math with proportions, it is most useful to set up proportions in
fraction form:
Now you just need to isolate x by
cross-multiplying, and then you can solve.
When you are dealing with ratio questions of this sort,
you should always set up an equation of equivalent fractions and
cross-multiply.
