Ratios
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.
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