Grade 11 Algebraic Fractions: Short Notes

11 ශ්‍රේණිය · 1 වාරය · 7 පරිච්ඡේදය · වීජීය භාග

49%

of Grade 11 term tests include it (30 of 61)

122

marks in past papers (33 questions)

9 min

full lesson with quick checks

🎯 What you'll learn

Cancel common factors in an algebraic fraction (and know what cannot be cancelled)
Factorise numerators and denominators before simplifying
Multiply algebraic fractions and give the simplest form
Divide algebraic fractions using the reciprocal

Buns in boxes

A Kandy bakery shares (x² + 5x + 6) buns equally among (x + 2) boxes. How many per box? Factorise: (x + 2)(x + 3) ÷ (x + 2) = x + 3. Whatever x is, the answer is that simple. Algebraic fractions are ordinary fractions with letters, and factorising is the key.

Same rules as number fractions

To multiply fractions, multiply numerators and multiply denominators. Cancel common factors first to keep numbers small:6/8 × 4/9: 6 and 9 share 3, 4 and 8 share 4,so 2/2 × 1/3 = 1/3.Algebra works the same way: (2a/b) × (b²/6a) = b/3.

Cancel factors, never terms

You may cancel something only if it multiplies the whole numerator and the whole denominator.✓ 3(x + 1)/3 = x + 1 (3 is a factor)✗ (x + 3)/3 = x + 1 (3 is only a term)✗ (x + 5)/(x + 2) = 5/2 (x is a term)So factorise first, then cancel whole brackets.

Your factorising toolkit

Before cancelling, look for:• Common factor: 4x + 8 = 4(x + 2)• Difference of two squares: x² − 25 = (x − 5)(x + 5)• Trinomial: x² + x − 6 = (x + 3)(x − 2)• Both: 2x² − 18 = 2(x² − 9) = 2(x − 3)(x + 3)Always take out a common factor first.

✏️ Multiplying single terms

Simplify (3a/4b) × (8b²/9a).

1Numbers: 3 with 9 gives 1 and 3; 8 with 4 gives 2 and 1.
2Letters: a cancels with a; b² ÷ b leaves b on top.
3Result: (1 × 2b)/(1 × 3) = 2b/3

✅ 2b/3

✏️ Factorise, then multiply

Simplify (x² − 9)/(2x + 4) × (x + 2)/(x + 3).

1x² − 9 = (x − 3)(x + 3) and 2x + 4 = 2(x + 2).
2Write: (x − 3)(x + 3)/2(x + 2) × (x + 2)/(x + 3)
3Cancel (x + 3) and (x + 2), top with bottom.
4Left: (x − 3)/2

✅ (x − 3)/2

🧮 Dividing: flip the second

To divide by a fraction, multiply by its reciprocal: turn only the second fraction upside down, then factorise and cancel as usual.

a/b ÷ c/d = a/b × d/c

⚠️ Forgetting the reciprocal

O/L examiners flagged this in more than one year: candidates divide but never turn the second fraction over, or they flip the first one instead.Write the ÷ line again as × with the second fraction upside down before you factorise or cancel anything.

⚠️ Cancelling across a + sign

A very common slip: (x² + 5x + 6)/(x + 3) 'simplified' by crossing out x² and x, or 6 and 3. Nothing cancels until the top is in factors: (x + 2)(x + 3)/(x + 3) = x + 2.If you can't see brackets, you can't cancel.

📌 Remember

• Multiply: numerator × numerator, denominator × denominator.• Divide: flip the second fraction, then multiply.• Factorise everything first: common factor, difference of squares, trinomials.• Cancel whole factors (brackets), never single terms joined by + or −.• Whatever is left over is the answer; it may be just a number.

🧾 Quick revision sheet

a/b × c/d	ac/bd (cancel common factors first)
a/b ÷ c/d	a/b × d/c: multiply by the reciprocal of the second fraction
Factorise x² − 49	(x − 7)(x + 7)
Can you cancel in (x + 4)/4?	No. 4 is a term, not a factor of the numerator.
Simplify (x² − 1)/(x + 1)	x − 1
Reciprocal of (x + 2)/5x	5x/(x + 2)

Now test yourself: 4 quick checks and 2 more worked examples

The full lesson checks your answers as you go, and Gedara Sir tracks which parts you've got.

📖 Start the 9-minute lesson free ▶ Practise 5 past-paper questions