Algebraic fractions: simplifying, adding, multiplying and solving

What are algebraic fractions?

An algebraic fraction is a fraction with a letter in the numerator, the denominator or both. 3x over 4, 5 over x and x + 2 over x − 3 are all algebraic fractions. Fractions in algebraic expressions work just like the fractions children have used since primary school, so everything they know about number fractions still applies.

Algebraic fractions are in the key stage 4 programme of study, which covers Years 10 and 11 without saying which year each topic is taught in. At GCSE, the DfE marks algebraic fractions as content for the more highly attaining students only, so this is a higher tier topic. The school's scheme of work will show when it comes up.

Two earlier skills do most of the work: simplifying fractions and factorising, including factorising quadratics.

How to simplify algebraic fractions

To simplify an algebraic fraction, factorise the numerator and the denominator, then cancel any factor they share. It's the same as simplifying ⁶⁄₉ to ²⁄₃ by dividing the top and bottom by 3.

Worked example: simplify a single term

6x over 9 simplifies to 2x over 3, because 3 is a common factor of 6 and 9. Similarly, 4x² over 6x is 2x over 3, because 2x is a common factor of the top and the bottom.

Worked example: factorise first

Simplify x² + 3x over x² − 9.

1. Factorise the top: x² + 3x is x(x + 3).
2. Factorise the bottom. It's a difference of two squares: (x + 3)(x − 3).
3. Cancel the common factor (x + 3).

The answer is x over x − 3.

Check with x = 5: the question gives 40 over 16, and 40 ÷ 16 = 2.5. The answer gives 5 over 2, which is also 2.5. A second check with x = 4: the question gives 28 over 7, which is 4, and the answer gives 4 over 1, which is 4.

How to reduce algebraic fractions: only cancel factors

You can only cancel something that multiplies the whole of the top and the whole of the bottom. x + 6 over x + 3 doesn't simplify to 2, and the x's don't cancel, because they're added, not multiplied. Try x = 1: it gives ⁷⁄₄, not 2.

How to multiply algebraic fractions

To multiply algebraic fractions (sometimes called timesing them), multiply the numerators together and the denominators together, then simplify. It's usually easier to factorise and cancel first.

Worked example: multiplying single terms

3x over 4 × 8 over 9x: the 3 and the 9 share a factor of 3, the 4 and the 8 share a factor of 4, and the x on top cancels with the x underneath. What's left is 2 over 3.

Check with x = 1: ³⁄₄ × ⁸⁄₉ = ²⁄₃.

Worked example: factorise, then multiply

Multiply x + 1 over x − 2 × x² − 4 over x² + x.

1. Factorise: x² − 4 is (x + 2)(x − 2), and x² + x is x(x + 1).
2. Now the top has (x + 1)(x + 2)(x − 2) and the bottom has (x − 2) × x × (x + 1).
3. Cancel (x + 1) and (x − 2).

The answer is x + 2 over x.

Check with x = 3: the question gives 4 over 1, times 5 over 12. That's ⁴⁄₁ × ⁵⁄₁₂ = ⁵⁄₃. The answer gives 5 over 3. With x = 4: ⁵⁄₂ × ¹²⁄₂₀ = ³⁄₂, and the answer gives 6 over 4, which is also ³⁄₂.

Dividing algebraic fractions

To divide, turn the second fraction upside down and multiply, as in dividing fractions. 2x over 3 ÷ x over 6 is 2x over 3 × 6 over x, which is 12x over 3x, or 4. Check with x = 3: 2 ÷ 0.5 = 4.

How to add and subtract algebraic fractions

To add or subtract algebraic fractions, write them over a common denominator, then add or subtract the numerators. It's the same as adding and subtracting fractions with numbers.

Worked example: number denominators

x over 3 + x over 4: the lowest common denominator is 12. x over 3 is 4x over 12 and x over 4 is 3x over 12, so the answer is 7x over 12.

Check with x = 12: the question gives 4 + 3 = 7, and the answer gives 84 over 12, which is 7.

Worked example: letters in the denominators

Add 2 over x + 1 + 3 over x − 2.

1. The common denominator is (x + 1)(x − 2).
2. Multiply the top and bottom of each fraction by the other denominator: 2(x − 2) over (x + 1)(x − 2) + 3(x + 1) over (x + 1)(x − 2).
3. Add the numerators: 2(x − 2) + 3(x + 1) is 2x − 4 + 3x + 3, which is 5x − 1.

The answer is 5x − 1 over (x + 1)(x − 2). Leave the bottom in brackets: it's simpler that way.

Check with x = 3: the question gives ²⁄₄ + ³⁄₁, which is 3.5. The answer gives 14 over 4, which is 3.5. With x = 0: the question gives 2 + −1.5, which is 0.5, and the answer gives −1 over −2, which is 0.5.

Worked example: subtracting

Subtract x + 3 over 2 − x − 1 over 3.

1. The common denominator is 6: 3(x + 3) over 6 − 2(x − 1) over 6.
2. Subtract the whole of the second numerator: 3x + 9 − 2x + 2. The minus sign changes −2 to +2.
3. Collect like terms: x + 11.

The answer is x + 11 over 6. Check with x = 1: the question gives 2 − 0 = 2, and the answer gives 12 over 6, which is 2. With x = 3: 3 − ²⁄₃ = ⁷⁄₃, and the answer gives ¹⁴⁄₆, which is ⁷⁄₃ too.

How to solve equations with algebraic fractions

To solve an equation with algebraic fractions, multiply every term by the common denominator to clear the fractions, then solve as normal. The solving equations guide covers the next steps.

Worked example: solve (x + 2) over 3 + (x − 1) over 2 = 6

x + 2 over 3 + x − 1 over 2 = 6

1. Multiply every term by 6: 2(x + 2) + 3(x − 1) = 36.
2. Expand: 2x + 4 + 3x − 3 = 36.
3. Simplify: 5x + 1 = 36, so 5x = 35.
4. x = 7.

Check: 9 over 3 is 3, and 6 over 2 is 3, and 3 + 3 = 6.

Worked example: an equation that becomes a quadratic

1 over x + 1 over x + 2 = 3 over 4

1. Multiply every term by 4x(x + 2): 4(x + 2) + 4x = 3x(x + 2).
2. Expand: 8x + 8 = 3x² + 6x.
3. Rearrange so one side is 0: 3x² − 2x − 8 = 0.
4. Factorise: (3x + 4)(x − 2) = 0, so x = 2 or x = −4 over 3.

Check x = 2: ¹⁄₂ + ¹⁄₄ = ³⁄₄. Check x = −4 over 3: 1 over x is −3 over 4, and x + 2 is ²⁄₃, so 1 over (x + 2) is ³⁄₂. Then ³⁄₂ − ³⁄₄ = ³⁄₄. Both work. The guide to solving quadratic equations covers this last step.

Common mistakes with algebraic fractions

• Cancelling terms instead of factors. In x + 6 over x + 3, nothing cancels. Factorise first, then cancel whole brackets only.
• Adding the denominators. x over 3 + x over 4 is not 2x over 7. Find a common denominator.
• Losing a minus sign when subtracting. The minus applies to the whole of the second numerator.
• Forgetting a term when clearing fractions. Every term on both sides gets multiplied, including whole numbers.
• Using a value that makes the bottom 0. In 3 over x − 2, x can't be 2. If a solution does that, it doesn't count.

How to help at home

• Go back to numbers. If an algebraic fraction step is confusing, ask your child to do the same step with ¹⁄₃ + ¹⁄₄. The method is identical.
• Ask "can you factorise it?" Most simplifying goes wrong because the factorising was skipped.
• Check with a number. Put x = 1, 2 or 3 into the question and the answer. If they don't match, look again.
• Write every line. Algebraic fractions reward careful working, one step at a time.

Algebraic fractions questions to try

1. Simplify 8x over 12.
2. Simplify x² − 4 over x + 2.
3. Multiply 2 over x × x² over 6.
4. Add x over 2 + x over 5.
5. Subtract 3 over x − 1 over x + 1.
6. Solve x over 3 + x over 6 = 5.

Answers

1. 2x over 3
2. x − 2, because x² − 4 is (x + 2)(x − 2)
3. x over 3, because 2x² over 6x simplifies by 2x
4. 7x over 10
5. 2x + 3 over x(x + 1), because 3(x + 1) − x is 2x + 3
6. Multiply every term by 6: 2x + x = 30, so 3x = 30 and x = 10

For the rest of GCSE maths, see the GCSE maths foundation and GCSE maths higher guides.