How to Pass Numerical Reasoning (11 page)

Fraction operations

Adding fractions

Adding fractions with the same denominator

Worked example

To add fractions that share a common denominator, simply add the numerators together.

Adding fractions with different denominators

To add fractions with different denominators, find a common denominator for all the fractions you need to add together. For example:

Method 1: Multiply the denominators to find a common denominator

When the numbers are simple, you can simply multiply the denominators by each other to find a common denominator. Note that this method will not necessarily always give you the lowest common denominator.

3 × 5 = 15, so you know that a common denominator of both fractions is 15.

Tip:
If you find it easier to find a common denominator by multiplying the denominators together than to find the lowest common denominator at the start, this approach is fine. Just remember to reduce the fraction to its lowest terms as your final step in the calculation.

Set up the fractions with a common denominator of 15:

To find the equivalent numerators, multiply the numerator of each fraction by the same number as you multiplied the denominator. You multiplied the denominator (3) by 5 in the first fraction to find the common denominator (15), so you must also multiply the numerator (2) by 5.

2 × 5 = 10

Now substitute the numerator into the equation.

Now do the same to the second fraction. You multiplied the denominator (5) by 3 to find a common denominator, so multiply the numerator (4) by 3 as well.

4 × 3 = 12

Substitute the numerator into the equation.

Now you can simply add the numerators:

The final step is to reduce the fraction to its lowest terms:

Method 2: Find the lowest common denominator

First, find the lowest common multiple of the denominators. The lowest common multiple of 6 and 9 is the product of the prime factorization of each number.

6 = 2 × 3

9 = 3 × 3

Common prime factor

The lowest common multiple is therefore 2 × 3 × 3 = 18. Rewrite each of the fractions with a denominator of 18:

Now add the numerators:

The fraction cannot be reduced any further.

Worked example

Adding mixed fractions

A mixed fraction is a fraction consisting of an integer and a fraction. To add mixed fractions, first ensure that the fractions are set up with common denominators and then add the fractions in the usual way.

First convert the mixed fractions to improper fractions:

Now find the lowest common multiple of 4 and 9:

4 = 2 × 2

9 = 3 × 3

There are no common prime factors, so the lowest common multiple is a product of the prime factors: 2 × 2 × 3 × 3 = 36. Now set up each fraction with a denominator of 36: