Algebraic Fractions — Grade 9 Maths (CAPS)

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

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Grade 10–11 · Video lessonAdding & Subtracting Algebraic Fractions

An algebraic fraction is just a fraction with letters in it. Every rule you already know for 3/4 still applies — the only new skill is factorising first.

1Simplifying: cancel factors, never terms

Worked ExampleSimplify 6x²/3x

1
Split it up: 6/3 × x²/x.
Numbers with numbers, letters with letters.
2
= 2 × x.
6 ÷ 3 = 2 and x² ÷ x = x.
3
= 2x, x ≠ 0.
Check with x = 4: 96/12 = 8 = 2(4). ✓ The bottom 3x would be 0 at x = 0, so x = 0 is not allowed.

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A denominator may never be zero. If a letter appears in a denominator, say which value it may not take — before or after simplifying, but say it. In 6x²/3x the bottom is 0 at x = 0, so the answer is 2x, x ≠ 0 — even though the simplified answer looks fine at x = 0.

Worked ExampleSimplify (x² + 3x)/x

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, x ≠ 0.
Now x is a FACTOR of the top, so it cancels — but the ORIGINAL bottom was x, so x = 0 stays excluded.

⚠️

In (x² + 3x)/x 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

a/b × c/d = ac/bd · a/b ÷ c/d = a/b × d/c
b, c, d ≠ 0 — no denominator may be 0, and when dividing, c becomes a denominator too.

Worked ExampleSimplify 2x/3 × 9/4x²

1
Multiply straight across: 18x/12x².
2
Simplify: 18/12 = 3/2 and x/x² = 1/x.
3
= 3/2x, x ≠ 0.
The original bottom 4x² is 0 when x = 0.

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To divide, flip the second fraction and multiply — the reliable method every time.

3Adding and subtracting

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

Worked ExampleSimplify x/2 + x/3

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

Worked ExampleSimplify 3/x + 2/5

1
The LCD is 5x.
Multiply the two denominators when they share no factor.
2
15/5x + 2x/5x.
3
= (15 + 2x)/5x, x ≠ 0.
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: 6x²/3x
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2x, x ≠ 0
2
Simplify: (x² + 3x)/x
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(x(x + 3))/x = x + 3, x ≠ 0
3
Simplify: 10a³/5a
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2a², a ≠ 0
4
Simplify: 2x/3 × 9/4x²
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18x/12x² = 3/2x, x ≠ 0
5
Simplify: x/4 ÷ x/2
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x/4 × 2/x = 1/2, x ≠ 0
6
Simplify: x/2 + x/3
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5x/6
7
Simplify: 3/x + 2/5
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(15 + 2x)/5x, x ≠ 0
8
Why can you not cancel the x in (x² + 3x)/x before factorising?
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Because cancelling works only on factors, not on terms that are added. Factorise the top first.