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# Adding and Subtracting Algebraic Fractions
- URL: https://www.esheets.io/adding-and-subtracting-algebraic-fractions/
- Published: 2026-10-07T16:46:31.000Z
- Updated: 2026-10-07T16:46:31.000Z
- Author: Richard Linnington
- Tags: Maths, Algebra

Practise adding and subtracting algebraic fractions by finding the lowest sensible common denominator, forming equivalent fractions and simplifying the result. The eight questions progress from matching denominators to expressions whose denominators must be factorised first. [Jump to the questions](#practise-now)

## Practise now

Worksheet preview and key skills 

### Worksheet preview

Answer eight self-marking questions on adding and subtracting two algebraic fractions, with one mark for each fully simplified result.

The difficulty rises from matching denominators and numerical LCMs to shared factors and factorised quadratic denominators.

#### What you’ll practise

- Building equivalent fractions with the lowest sensible common denominator.
- Expanding and combining complete numerator expressions, including careful subtraction.
- Factorising denominators and simplifying the final algebraic fraction.

Use the interactive worksheet below, or read the [Topic guide](#topic-guide) for the method and worked example.

## Adding and Subtracting Algebraic Fractions

Add or subtract each pair and simplify fully. Enter the numerator and denominator separately; use denominator 1 for a non-fraction answer. Use ², ³ or ⁴ for powers and brackets where needed.

## Topic guide

Algebraic fractions can be added or subtracted only after their denominators match. Choose the lowest sensible common denominator, multiply the numerator and denominator of each fraction by the same factor, combine the numerators, then simplify fully.

### When the denominators already match

Keep the denominator and combine only the numerators:

(2x + 5)/(x + 3) − (x − 1)/(x + 3) \= (2x + 5 − (x − 1))/(x + 3) \= (x + 6)/(x + 3).

When subtracting, the subtraction applies to the whole second numerator. Brackets help prevent sign errors.

### Numerical and monomial denominators

For numerical denominators, use their LCM. For example, the LCM of 3 and 4 is 12:

x/3 + x/4 = 4x/12 + 3x/12 = 7x/12.

For monomials, include the LCM of the coefficients and the highest required power of x. Thus the lowest common denominator of 6x and 4x2 is 12x2. Remember that creating an equivalent fraction means multiplying both its numerator and denominator by the same expression.

### Linear denominator factors

For two distinct factors, such as x + 1 and x + 4, use both complete factors:

2/(x + 1) + 3/(x + 4) = \[2(x + 4) + 3(x + 1)\]/\[(x + 1)(x + 4)\] = (5x + 11)/\[(x + 1)(x + 4)\].

Multiplying the displayed denominators works here because the factors are distinct. If denominators share a factor, use it only once. For example, the lowest common denominator of x + 1 and (x + 1)(x + 2) is (x + 1)(x + 2), not their raw product.

### Factorise before finding the LCM

A quadratic denominator can hide a shared factor. Factorise first:

x2 − 9 = (x − 3)(x + 3), so 2/(x + 3) + 1/(x2 − 9) = \[2(x − 3) + 1\]/\[(x − 3)(x + 3)\] = (2x − 5)/\[(x − 3)(x + 3)\].

The same idea applies to a monic quadratic such as x2 \+ 7x + 12 = (x + 3)(x + 4), and to a simple extracted factor such as 2x + 6 = 2(x + 3).

### Simplify and check

- Expand and collect the numerator carefully.
- Cancel only complete common factors of the numerator and denominator. Terms separated by + or − cannot be cancelled individually.
- Do not add or subtract the denominators.
- Do not multiply only a denominator when forming an equivalent fraction.
- Check whether the final numerator and denominator still share a numerical or polynomial factor.

Original denominators must be non-zero. Any excluded values from the original expression remain excluded even if a factor later cancels.

Solving equations that contain algebraic fractions is a separate topic and is not practised here.