GMAT Exam: A Simple Guide To Solving Word Problems - Ratios and Proportions

Ratios
A ratio is a multiplicative comparison of two quantities.

If two quantities are in the ratio a to b and you know their sum is c, solve ax + bx = c for x, then compute ax or bx, whichever one is needed.

Example: The ratio of boys to girls in a classroom of 21 students is 3 to 4. How many girls are in the classroom?
Solve 3x + 4x = 21.

There are 12 girls in the classroom.
You can extend this strategy to three or more quantities.

Proportions
A proportion is a mathematical statement that two ratios are equal.

The statement is a proportion and is read “a is to b as c is to d.”

The fundamental property of proportions is that if and only if ad = bc.

The numbers a, b, c, and d are the terms of the proportion.

The products ad and bc are the cross products (illustrated below).

If the values of three of the four terms of a proportion are known, the value of the fourth term can be determined by using the fundamental property of proportions.

Find a cross product that results in a numerical value, and then divide by the numerical term you didn’t use.

For example,

When you have a word problem involving proportions, look for a sentence or phrase in the problem that provides the information you need for the left ratio of the proportion, and then look for another sentence or phrase that gives you the information you need for the right ratio of the proportion.


Example: On a map, the distance between two cities is 13.5 inches. The scale on the map shows that 0.5 inches represents 20 miles. What is the distance, in miles, between the two cities?
Let d = the distance, in miles between the two cities.

Information for the left ratio of the proportion is in the first sentence of the problem (13.5 inches represents d miles on the map), and information for the right ratio of the proportion is in the second sentence (0.5 inches represents 20 miles on the map).

Write the proportion.

Notice that the units in the left ratio match up with the units in the right ratio. You have miles in the numerator and inches in the denominator on the left, and you have miles in the numerator and inches in the denominator on the right. If the units in the left and right ratios don’t match up, your proportion is incorrect.

Solve the proportion, omitting the units for convenience.

The distance between the two cities is 540 miles.