# GMAT Math: Comparing Fractions

A GMAT question may ask you to choose the largest or smallest of five fractions or to say in which interval a simple fraction is located. Success in solving such problems is based on understanding fractions—how the numerator and denominator are related—and on the ability to convert fractions to a common denominator quickly and correctly.

The basic concepts connected with fractions are simple and familiar to us since childhood. Let's review them.

**What is a fraction?**

The common fraction is a number represented in the form , where *a* and *b* are certain integers. The number *a* is called the numerator, and *b* the denominator of the fraction. The denominator shows how many parts a whole has been divided into. The numerator shows how many parts of a whole are included in the fraction. For example, from the cake in Picture 1, which has been divided into four parts, one part, , has been removed and three parts, , remain. Picture 2 shows a segment of five parts with three parts, , marked.

**Comparing fractions**

Let's compare three pairs of fractions:

The fractions in the first pair (a) have the same denominator. That means that the smaller of the two will be the one with the smaller numerator. So, .

The fractions in the second pair (b) have the same numerator. Of two fractions with the same numerator, the greater is the one with the smaller denominator. In this case, .

Finally, in the third pair (c), neither denominators nor numerators match. You must convert the fractions to a common denominator, making them fit the conditions in case (a). The common denominator here is 36. Thus, and . Finally, so, .

When solving GMAT problems, it is helpful to convert fractions to a common denominator, but it is easier and quicker to compare them in pairs.

Consider some examples:

1. Which of the following fractions is the smallest?

(A)

(B)

(C)

(D)

(E)

Explanation:

Let's compare each of the other answer choices with Choice A, .

(B): × 2 = , which is less than (larger denominator).

(C): again is less than (larger denominator).

(D): = , which is less than (smaller numerator).

(E): = , which is less than (larger denominator).

The smallest fraction is given in answer Choice A.

Try to solve the following problem yourself:

1. Which of the following fractions has the greatest value?

Explanation:

The common denominator of all these fractions will be 5 (9!). Let's convert all fractions to the common denominator:

The largest fraction is given in Choice D.

Try to solve the similar problem yourself:

Which of the following numbers is the greatest?

Explanation:

These fractions have neither common denominator nor common numerator. To search them is completely pointless. Note that the the numerator and denominator of each fraction differ by 3. Then we can present data of the fraction as follows:

So, the answer is C.

Illustration:

Represent these fractions on the segment. The denominator is the number of parts into which we divide the segment. The larger it is, the greater the number of parts. The greater the number of parts, the shorter their length. The fact that the numerators and denominators differ by 3 means that we select all the parts except three:

Obviously, the larger the denominator is, the greater the length of the selected area, which corresponds to the value of the fraction.

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