## Arithmetic Concepts in GMAT Quant

# Simplifying fractions on the GMAT

Fractions can often appear difficult to deal with in GMAT Quant questions. However, if you simplify fractions, things become easier.

Consider the following question from GMAT Tutor:

*If a and b are integers, and 3 ^{5}a=5^{3}b, which of the following must be true?*

*A) b / 125 is an integerB) a / (125 • 3 ^{5}) is an integerC) b / 27 is an integerD) a / 3 is an integerE) a / 250 is an integer*

## How to simplify

Let’s rearrange the equation in the question to find an expression for b:

- 3
^{5}a=5^{3}b - b = 3
^{5}a / 5^{3}

Both sides of this latest equation must be integers, as we have been told that b is an integer. For 3^{5}a / 5^{3} to be an integer, the 5^{3} in the denominator must be balanced by at least 5^{3} in the numerator. Let’s balance it with exactly 5^{3} by making a = 5^{3}:

- b = 3
^{5}5^{3}/ 5^{3} - b = 3
^{5}

If you look at Option C, the 27, which is (3^{3}), will be canceled by the presence of 3^{3} in 3^{5}, leaving an integer.

**C is therefore our answer.**

Alternatively, find an expression for a:

- 3
^{5}a=5^{3}b - a = 5
^{3}b / 3^{5}

Again, we know both sides must be integers, as a is an integer. For 5^{3}b / 3^{5} to be an integer, the 3^{5} in the denominator must be balanced by at least 3^{5} in the numerator. Thus, b must be at least 3^{5}• 3^{5} / 27 is an integer. Using this method, we have further proof that Option C is correct.

This question really just asks you to cancel elements that are present in both numerator and denominator. As long as you do this carefully, you will be able to solve such questions quickly.