Adding Algebraic Fractions
(KS3, Year 8)

The Lesson

Algebraic fractions can be added. Imagine you wanted to add ab and cd.

a over b plus c over d

How to Add Algebraic Fractions

To add algebraic fractions, use the rule:

a over b plus c over d equals a d plus b c over c d

A Real Example of How to Add Algebraic Fractions


Add the algebraic fractions below.
x over 2 plus y over 3



Compare the fractions you are adding with the rule shown above.

compare a over b plus c over d and x over 2 plus y over 3 By comparing, we see that a = x, b = 2, c = y, d = 3.


Use the rule, with a = x, b = 2, c = y, d = 3:

a d plus bc over bd, replacing a with x, b with 2, c with y and d with 3


Calculate the terms in the rule. Where we have written two numbers or letters in brackets together, multiply them together:
(x)(3) = x × 3 = 3x (2)(y) = 2 × y = 2y (2)(3) = 2 × 3 = 6


We have added x2 and y3:

x over 2 plus y over 3 equals 3 x plus 2 y over 6

Lesson Slides

The slider below shows a real example of how to add algebraic fractions. Open the slider in a new tab

Understanding The Rule

a over b plus c over d equals a d plus b c over b d The letters written next to each other denotes that they are multiplying each other. The rule works when the a, b, c and d are numbers, letters, terms or expressions. Make sure you can:

Why Does This Work?

When adding fractions (algebraic or not) all of the fractions must have a common denominator. If initially the denominators are not the same...

a over b plus c over d ...multiplying the denominators together will make a common denominator.

b times d equals b d But having multiplied the denominator of each fraction, the numerator must be multiplied by the same value if we are not to change the fraction.

multiply the numerator by the same value as the denominator This gives us the rule for adding algebraic fractions:

a d plus b c over b d
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See Also

What is algebra? What is an algebraic fraction? What is a numerator? What is a denominator? Multiplying letters Multiplying terms Multiplying expressions