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Fractions Toolkit

Number

Multiplying

  • Cancel any common factor between a numerator and a denominator before multiplying, taking factors from either fraction
  • Multiply the numerators together, then multiply the denominators together
  • Cancelling first keeps the numbers small, which matters on the non-calculator papers

Dividing

  • Dividing by a fraction is the same as multiplying by its reciprocal, so invert the second fraction and change the sign to multiplication
  • Cancel and multiply exactly as above
  • Only the second fraction is inverted, and only after the division sign has been changed

Mixed numbers

  • Convert any mixed number to an improper fraction before multiplying or dividing
  • Attempting to multiply the whole parts and the fraction parts separately does not work
Worked example

Multiplying fractions by cancelling first

Work out 9/20 × 8/15, giving the answer in its simplest form.

Solution:

  • Cancel 9 against 15 using their common factor 3, leaving 3/20 × 8/5
  • Cancel 8 against 20 using their common factor 4, leaving 3/5 × 2/5
  • Multiply the numerators: 3 × 2 = 6
  • Multiply the denominators: 5 × 5 = 25
  • The answer is 6/25, which has no remaining common factor
Exam tip

Invert the divisor only, and change the sign to × at the same moment so the two edits cannot come apart. Show that inverted line: it is where the method mark sits. Any mixed number must go top-heavy first.

Worked example

Dividing by a fraction

Work out 2 1/3 ÷ 7/9.

Solution:

  • Convert the mixed number: 2 1/3 = 7/3
  • Invert the second fraction and change the operation: 7/3 × 9/7
  • Cancel 7 against 7, leaving 1/3 × 9/1
  • Cancel 9 against 3, leaving 1/1 × 3/1
  • The answer is 3