Algebraic Fractions (Grade 9)

Simplifying algebraic fractions by factorising and cancelling, and adding, subtracting, multiplying and dividing them — including why you may only cancel factors.

An algebraic fraction is just a fraction with letters in it. Every rule you already know for 34 still applies — the only new skill is factorising first.

1Simplifying: cancel factors, never terms

Worked ExampleSimplify 6x²3x

1
Split it up: 63 × x²x.
Numbers with numbers, letters with letters.
2
= 2 × x.
6 ÷ 3 = 2 and x² ÷ x = x.
3
= 2x.
Check with x = 4: 9612 = 8 = 2(4). ✓

Worked ExampleSimplify x² + 3xx

1
Factorise the top: x(x + 3).
x is a common factor — this is the whole trick.
2
= x(x + 3)x.
3
= x + 3.
Now x is a FACTOR of the top, so it cancels.

⚠️

In x² + 3xx you may not cancel the x straight away to get x² + 3. Cancelling only works on factors (things multiplied), never on terms (things added). Factorise first — every time.

2Multiplying and dividing

ab × cd = acbd · ab ÷ cd = ab × dc

Worked ExampleSimplify 2x3 × 94x²

1
Multiply straight across: 18x12x².
2
Simplify: 1812 = 32 and xx² = 1x.
3
= 32x.

💡

To divide, flip the second fraction and multiply. Never try to divide fractions directly.

3Adding and subtracting

As with ordinary fractions, you need a common denominator before you can add.

Worked ExampleSimplify x2 + x3

1
The LCD of 2 and 3 is 6.
2
3x6 + 2x6.
Multiply the first by 33 and the second by 22.
3
= 5x6.
Check with x = 6: 3 + 2 = 5, and 5(6)6 = 5. ✓

Worked ExampleSimplify 3x + 25

1
The LCD is 5x.
Multiply the two denominators when they share no factor.
2
155x + 2x5x.
3
= 15 + 2x5x.
The top will not simplify further — 15 and 2x are not like terms.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
Simplify: 6x²3x
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2x
2
Simplify: x² + 3xx
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x(x+3)x = x + 3
3
Simplify: 10a³5a
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2a²
4
Simplify: 2x3 × 94x²
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18x12x² = 32x
5
Simplify: x4 ÷ x2
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x4 × 2x = 12
6
Simplify: x2 + x3
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5x6
7
Simplify: 3x + 25
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15 + 2x5x
8
Why can you not cancel the x in x² + 3xx before factorising?
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Because cancelling works only on factors, not on terms that are added. Factorise the top first.