GRE Exam: A Simple Guide To Part-to-Whole Relationships - Ratios & Proportions

Ratios
A ratio represents a relationship between two or more quantities.

Examples of ratios are:
There are 3 boys for every 4 girls.
The preceding can be expressed as “The ratio of boys to girls is 3 to 4.”
A recipe requires 3 parts sugar, 2 parts salt, and 4 parts water.
The above can be expressed as: “The ratio of sugar to salt to water is 3 to 2 to 4.”

Ratios can be represented in a few different ways:

The Elements of a Ratio
The relationship that a ratio specifies will be either part-to-part or part-to-whole.

An example of a part-to-part ratio is “In a school, there are 7 boys for every 3 girls.”

An example of a part-to-whole ratio is “In a school, there are 7 boys for every 10 students.”

It is important to note that you can derive a part-to-whole ratio from a part-to-part ratio, and vice versa.

If the ratio of boys to girls in a school is 7:3, then the ratio of boys to all students is 7:10.

Ratios Do Not Specify Amounts!
It is important to remember that a ratio only provides a relationship between quantities, not the actual amounts of each quantity. If the ratio of boys to girls is 7:3, there can be any number of values for boys and girls, as long as those values satisfy the given ratio.

Expressing Ratios
As discussed previously, there are several ways of expressing ratios.

For the purpose of the GRE, the best way to express a ratio between two quantities is as a fraction.

But before doing so, you must label the units.

If you are told that the ratio of men to women is 1:3, then on your paper, you should label which quantity corresponds with the numerator of the fraction and which quantity corresponds with the denominator of the fraction:

To make the quantities easier to work with algebraically, it’s even better to write:


Proportions
The simplest types of ratio questions will give you a ratio and the value for one quantity and will ask you to find a value for the other quantity. Whenever you have a ratio with the value for one quantity, set up a proportion.

Example: At a certain university, the ratio of doctoral students to master’s students is 3 to 5. If there are 9,000 doctoral students, how many master’s students are there?

Step 1: Set up a proportion. Remember to label your units!

Step 2: Cross-multiply to solve for m:


Representing Ratios Algebraically
Recall from the guide on fractions that the denominator of a fraction specifies the number of slices a pie is cut into, and the numerator represents the number of slices of that pie. In a certain sense, ratios can be thought of in the same way.

If you are given a part:part ratio, such as “The ratio of boys to girls is 3 to 2,” you can think of the number of boys as 3 slices of a pie and the number of girls as 2 slices of the same pie. Since you are not told the value of the slice of the pie, let x represent the quantity of 1 slice.

Thus if the ratio of boys to girls is 3 to 2, then:

Representing ratios in this way is helpful in the following situations:
Situation 1: When you know values for the parts of a ratio and want to determine the whole, or vice versa.

Example: In a certain business’s budget, the dollar amounts allocated for marketing, product development, and administration are in the ratio 5:3:2. If the business’s budget is $200,000, how much money is allocated for marketing?
Step 1: Since you are not given a value for any of the parts, use x to represent one part. Thus
5x = dollars allocated for marketing
3x = dollars allocated for product development
2x = dollars allocated for administration

Step 2: Identify an algebraic relationship:
The $200,000 budget is split among the three segments, so the sum of the amounts for the three segments will equal $200,000:
5x + 3x + 2x = $200,000

Step 3: Solve for the unknown: you are asked to solve for the amount of money allocated for marketing, so you must solve for 5x.

Using the preceding equation:

 

Situation 2: When you do not have quantities for either part of a ratio, but have information about how the ratio changes.
 

Example: In a certain school, the ratio of students to teachers is 20 to 1. If the school adds 1,000 students and 100 teachers, the new ratio of students to teachers will be 15 to 1. How many students are currently at the school?

Step 1: Set up the original ratio with x representing one slice of the pie

Step 2: Use the quantities from Step 1 to create a proportion

Step 3: Cross-multiply and solve for 20x (the original number of students)

 

Multiple Ratios with a Common Element
On some GRE questions, you will be given two ratios that share a common part and be asked to solve for the ratio of the other parts.

The most efficient way to answer these questions is to manipulate the two ratios so that the common element has the same value in both. How will you do this? By finding the least common multiple of the common element.

Example: At a certain pet store, there are 3 dogs for every 2 cats and 5 cats for every 7 birds. What is the ratio of the number of dogs to the number of birds?

Step 1: Set up the given ratios:

Step 2: Manipulate the ratios so that the quantity for the overlapping element is the same in both ratios.
In the first ratio, there are 2 parts cats, and in the second ratio, there are 5 parts cats. To make the ratios comparable, manipulate the two fractions so that cats will be 5 × 2 = 10 in both fractions.

Now that both ratios are in terms of the same number of cats, the ratio of dogs to cats to birds is 15:10:14. The ratio of dogs to birds is thus 15:14.