GRE Quantitative: Ratios and Proportions

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Ratios and proportions are favorites of most standardized tests, and the GRE Math section is no exception. They may be a bit intimidating if you are unfamiliar with how to approach them, but once you learn the basics, you’ll learn that ratio and proportions problems require only simple algebra.
A ratio is a kind of fraction that measures two or more quantities in a group.
 

Ratio Example:

A ratio of boys to girls (Boys: Girls) at a party is 3:4. You can also write this ratio as 3/4. That means that for every 3 boys at the party, there are 4 girls. This does not mean that 3/4 of the party goers are boys, nor does it mean that 4/3 of the party goers are girls.


If you do want to find out what proportion or percentage of the party goers are boys or girls, respectively, you add the numerator and the denominator, and then form a fraction in which this sum is the denominator.
Proportion Example:

If I want to find out what proportion of the party goers are boys, I add 3 and 4 (=7), and then I take the ratio quantity of boys, 3, and form the fraction 3/7.
3/7, or 42.9 %, of the party goers are boys.
4/7, or 57.1%, of the party goers are girls.


Stated algebraically, if we have a ratio x:y, then x / x+y and y /x+y express the proportions of x to the group and y to the group, respectively.
Let’s see what an example using this rule might look like:

At a party, 40% of the party goers are male. What is the ratio of male to female party goers?

This question uses the aforementioned rule, but reverses the process. We now have to find the ratio.
If 40% of the party is male, then 4/10 or 2/5 of the party is male and 3/5 of the party is female.
Since we have two proportions expressed with the same denominator (2/5 and 3/5), we can simply express the ratio of males to females as 2/3 or 2:3.
Note: If the question had asked for the proportion of females to males, the answer would be 3:2.


 
 
Though we cannot find the total number of items in a group (e.g. number of people at the party) if we are given a ratio, we can deduce some important information about the number of items. A GRE question testing this rule may look like this:

If the ratio of men to women at a party is 4:7, which of the following could be the number of people at the party?
A. 50
B. 64
C. 66
D. 70
E. 78

At first, you may think that you do not have enough information to answer this question, but you do.
To answer a problem like this, just add the coefficient x to each quantity and add: 4x+7x=11x.
We know that the sum of the quantities, 11, represents a fraction of the total number of party goers, so our answer MUST be a multiple of 11.
The only multiple of 11 in our choices is C. 66.
Extra Credit: If there were 66 people at the party, how many males and females would be there?
If 11x=66, then x = 6.
4x= 4*6= 24 men
7x=7*6=42 women


 
 
That last example was pretty simple, but how about a tougher one that uses the same concept:

In a right triangle, the two acute angles have a ratio of 1:5. What’s the measure of the larger acute angle?

Before we apply the same method, let’s write down some important info. If this is a right triangle, then the largest angle is 90 degrees. Since the sum of the angles of a triangle equal 180 degrees, then the two acute angles must equal 90 degrees.
so: 1x+5x =90
6x=90
x=15
The larger angle is 5x, so 5*15 = 75


 
 

GRE Time Management


Time management is crucial for great scores on the GRE Test, and one way to improve your pacing is to become faster at some of the more accessible skill tags. Ratios and proportions are the basics of algebra, and better scores with this concept will help you get harder GRE Quantitative questions correct!
 

  • A ratio is a comparison between two quantities.

    It is usually expression as a fraction (x/y) or with a colon (x:y), or in a word problems (“the ratio of apples to oranges”). Typically, whatever follows the word “of” is in the numerator, and whatever follows the word “to” is in the denominator.

  • A proportion is a set of ratios set equal to each other.

    A proportion is basically an equation with two fractions, such as 4/x = y/7. You can always solve a proportion by cross-multiplying the numerator of one fraction by the denominator of the other. 4/x = y/7 would become 28 = xy after we cross-multiplied.


Ratios are usually expressed as part:whole or part:part. Making that distinction is important, especially in complex GRE word problems. Ratios are always reduced to the simplest form, but you can multiply them by any integer to increase the numerator/denominator values, as long as you do the same thing to the top and the bottom of the fraction.
 

Remember:

When given a part:whole ratio and at least one “real-world” number, you can solve for the other “real-world” value.


Practice


To examine this strategy in more detail, try the following example problem.

if the ratio of girls to total students in a class is 3:5, and there are 8 boys in the class, how many girls are in the class?

We know the ratio of boys:total students must be 2:5, since there are 3 girls out of 5. Let’s set up a proportion to solve: 2/5 = 8/x. There are 20 students total in the class, so there must be 12 girls.


 
When working with proportions, make sure to carefully look for any change in units. This especially occurs in questions involving time. Don’t forget – there are 60 seconds in 1 minute, and 60 minutes in 1 hour.