Concept

Unlike fractions are fractions with different denominators. 

A fraction is said to be in the simplest form if its numerator and denominator have no common factors other than 1.

When looking at all the multiples of two or more numbers, some multiples are the same (or common). The smallest of these common multiples is the least common multiple (L.C.M).  

Rule

To add fractions with unlike denominators,
1. Find an equivalent fraction with a common denominator.
To do this, find the least common multiple of the two denominators.
2. Rewrite the numerators with the common denominator.
3. Add all numerators. The denominator stays the same. 
4. Simplify, if needed, by dividing the numerator and the denominator by their greatest common factor. 

Example

Add. 

Write in the simplest form. 

$\frac{1}{3}\mathrm{ + }\frac{3}{6}\mathrm{ = }$

Solution

1. If the denominators are not the same, find an equivalent fraction with a common denominator. (The denominators in this problem are 3 and 6.)
To do this, find the least common multiple of the two denominators.

Some multiples for the number 3 are:
3, 6, 9, 12

Some multiples for the number 6 are:
6, 12, 18, 24

The least common multiple for 3 and 6 is 6. 

2. Change the numerators with the common denominator 6.

In order to rewrite $\frac{1}{3}$ so it has the common denominator 6, multiply the numerator and denominator by the factor 2.

$\frac{\mathrm{1\; \times \; 2}}{\mathrm{3\; \times \; 2}} \mathrm{=} \frac{2}{6}$

$\frac{3}{6}$ already has the common denominator 6, so it doesn’t need to be rewritten.

3. Add all the numerators. The denominator stays the same.

$\frac{2}{6} \mathrm{+} \frac{3}{6} \mathrm{=} \frac{5}{6}$

4. Simplify, if needed. The fraction $\frac{5}{6}$ does not need to be simplified.

Pre-requisite Skills

Simplifying fractions (5.10.6)

LCM (5.10.7)