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How to Add Algebraic Fractions (Mathematics Lesson)

Adding Algebraic Fractions

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

An Example Question

Add the algebraic fractions below.

x over 2 plus y over 3
Step 1

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.

Step 2

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
Step 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

Another Real Example of How to Add Algebraic Fractions

The slider below shows a real example of how to add algebraic fractions.

Interactive Test
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Here's a second test on adding algebraic fractions.
Here's a third test on adding algebraic fractions.

Note

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