Adding and subtracting algebraic fractions

Covers AS91261

Why you need a common denominator

You can only add or subtract fractions when the parts are the same size — that is, when the denominators match. This is the same rule as for ordinary number fractions.
With algebra, the easiest common denominator is usually the product of the two denominators (they rarely share a factor).
Once the denominators match, you add or subtract the numerators only and keep the common denominator.

The method

Rewrite each fraction over the common denominator (multiply top and bottom of each by whatever it is missing).
Combine into a single fraction, adding or subtracting the numerators.
Expand and simplify the numerator, then factorise it to check whether anything cancels.

$\frac{2}{x} + \frac{3}{x + 1} = \frac{2(x + 1) + 3x}{x(x + 1)} = \frac{5x + 2}{x(x + 1)}$

Subtracting: watch the sign

When subtracting, the minus sign applies to the whole second numerator, not just its first term.
Bracket the second numerator before removing it, so every sign flips correctly.

Worked ExampleAdding algebraic fractions

Write $\dfrac{2}{x + 3} + \dfrac{1}{x}$ as a single fraction.

Step 1 — Find the common denominator

The denominators $x + 3$ and $x$ share no factor, so the common denominator is their product, $x(x + 3)$ .

Step 2 — Rewrite each fraction over it

Multiply top and bottom of each so the denominator becomes $x(x + 3)$ :

$\frac{2 \cdot x}{x(x + 3)} + \frac{1 \cdot (x + 3)}{x(x + 3)} = \frac{2x + (x + 3)}{x(x + 3)}$

Step 3 — Simplify the numerator, then check for a factor

$\frac{2x + x + 3}{x(x + 3)} = \frac{3x + 3}{x(x + 3)} = \frac{3(x + 1)}{x(x + 3)}$

Worked ExampleSubtracting algebraic fractions

Write $\dfrac{3}{x - 2} - \dfrac{1}{x + 1}$ as a single fraction.

Step 1 — Common denominator

The common denominator is $(x - 2)(x + 1)$ . Rewrite each fraction over it:

$\frac{3(x + 1)}{(x - 2)(x + 1)} - \frac{1(x - 2)}{(x - 2)(x + 1)}$

Step 2 — Combine over one denominator, bracketing the subtraction

$= \frac{3(x + 1) - (x - 2)}{(x - 2)(x + 1)}$

Step 3 — Expand and simplify the numerator

$3(x + 1) - (x - 2) = 3x + 3 - x + 2 = 2x + 5$

$= \frac{2x + 5}{(x - 2)(x + 1)}$

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

Write $\dfrac{x}{3} + \dfrac{x}{4}$ as a single fraction.

Merit

Write $\dfrac{2}{x + 3} + \dfrac{1}{x}$ as a single fraction.

Excellence

Write $\dfrac{4}{x - 1} - \dfrac{2}{x + 2}$ as a single simplified fraction.

Adding and subtracting algebraic fractions

Covers AS91261

Why you need a common denominator

You can only add or subtract fractions when the parts are the same size — that is, when the denominators match. This is the same rule as for ordinary number fractions.
With algebra, the easiest common denominator is usually the product of the two denominators (they rarely share a factor).
Once the denominators match, you add or subtract the numerators only and keep the common denominator.

The method

Rewrite each fraction over the common denominator (multiply top and bottom of each by whatever it is missing).
Combine into a single fraction, adding or subtracting the numerators.
Expand and simplify the numerator, then factorise it to check whether anything cancels.

$\frac{2}{x} + \frac{3}{x + 1} = \frac{2(x + 1) + 3x}{x(x + 1)} = \frac{5x + 2}{x(x + 1)}$

Subtracting: watch the sign

When subtracting, the minus sign applies to the whole second numerator, not just its first term.
Bracket the second numerator before removing it, so every sign flips correctly.

Worked ExampleAdding algebraic fractions

Write $\dfrac{2}{x + 3} + \dfrac{1}{x}$ as a single fraction.

Step 1 — Find the common denominator

The denominators $x + 3$ and $x$ share no factor, so the common denominator is their product, $x(x + 3)$ .

Step 2 — Rewrite each fraction over it

Multiply top and bottom of each so the denominator becomes $x(x + 3)$ :

$\frac{2 \cdot x}{x(x + 3)} + \frac{1 \cdot (x + 3)}{x(x + 3)} = \frac{2x + (x + 3)}{x(x + 3)}$

Step 3 — Simplify the numerator, then check for a factor

$\frac{2x + x + 3}{x(x + 3)} = \frac{3x + 3}{x(x + 3)} = \frac{3(x + 1)}{x(x + 3)}$

Worked ExampleSubtracting algebraic fractions

Write $\dfrac{3}{x - 2} - \dfrac{1}{x + 1}$ as a single fraction.

Step 1 — Common denominator

The common denominator is $(x - 2)(x + 1)$ . Rewrite each fraction over it:

$\frac{3(x + 1)}{(x - 2)(x + 1)} - \frac{1(x - 2)}{(x - 2)(x + 1)}$

Step 2 — Combine over one denominator, bracketing the subtraction

$= \frac{3(x + 1) - (x - 2)}{(x - 2)(x + 1)}$

Step 3 — Expand and simplify the numerator

$3(x + 1) - (x - 2) = 3x + 3 - x + 2 = 2x + 5$

$= \frac{2x + 5}{(x - 2)(x + 1)}$

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

Write $\dfrac{x}{3} + \dfrac{x}{4}$ as a single fraction.

Merit

Write $\dfrac{2}{x + 3} + \dfrac{1}{x}$ as a single fraction.

Excellence

Write $\dfrac{4}{x - 1} - \dfrac{2}{x + 2}$ as a single simplified fraction.