MathematicsExam code: 0580
Algebraic Fractions
Algebra and Sequences
Multiplying
- Factorise every top and every bottom first, because that is what exposes the cancelling
- Cancel across the multiplication sign as well as within a single fraction, since everything is being multiplied
- Then multiply the tops together and the bottoms together
- Leave the answer in factorised form and check once more for anything that still cancels
Worked example
Cancelling across a product
Express (x² − 36) ÷ (3x + 6) × (x + 2) ÷ (x − 6) as a single fraction in its simplest form.
Solution:
- The top of the first fraction is a difference of two squares, giving (x − 6)(x + 6)
- The bottom of the first fraction has a common factor of 3, giving 3(x + 2)
- The product is now [(x − 6)(x + 6)] ÷ [3(x + 2)] × (x + 2) ÷ (x − 6)
- The bracket (x − 6) sits above and below, so it divides out
- The bracket (x + 2) also sits above and below, so it divides out too
- What survives is (x + 6) ÷ 3
Dividing
- Turn the second fraction upside down and change the division into a multiplication
- Only the fraction after the division sign is inverted
- Once inverted, carry on exactly as for a product
- So 5b ÷ 12 divided by 15b² ÷ 8 becomes 5b ÷ 12 × 8 ÷ 15b², which is 40b ÷ 180b², and that reduces to 2 ÷ 9b