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Fractions Toolkit

Number

Equivalent fractions

  • Equivalent fractions look different but represent the same amount
  • Multiplying numerator and denominator by the same non-zero number leaves the value unchanged
    • 4/7 = 8/14 = 12/21 = 16/28
  • Every fraction therefore has infinitely many equivalent forms
  • Equivalence is easiest to see by regrouping: 21 small squares arranged as 7 columns of 3 means shading 15 squares shades exactly 5 whole columns, so 15/21 and 5/7 cover the same area
A rectangle divided into 21 small squares, arranged as 7 columns of 3, with the 15 squares of the left-hand 5 columns shaded green, showing that 15 out of 21 covers the same area as 5 out of 7
Source: Introduction to fractions by Save My Exams

Simplifying

  • Dividing numerator and denominator by a common factor produces a smaller equivalent fraction, a process also called cancelling
  • A fraction is in its simplest form when the only common factor left is 1
  • Dividing by the highest common factor reaches simplest form in one step
    • 24/36 has highest common factor 12, so 24/36 = 2/3
    • 30/48 has highest common factor 6, so 30/48 = 5/8
  • Repeated cancelling by smaller factors reaches the same place, only more slowly
Exam tip

"In its simplest form" is asked for in 33 of the 43 papers, and 42 of the 43 mark schemes mark these as a final answer — whatever stands last on the line is what counts. A correct fraction rewritten uncancelled below it scores nothing.

Worked example

Simplifying a fraction fully

Write 84/126 in its simplest form.

Solution:

  • Look for a common factor of the numerator and the denominator
  • Both are even, so divide each by 2: 84/126 = 42/63
  • Both digit sums are multiples of 3 (4 + 2 = 6 and 6 + 3 = 9), so divide by 3: 42/63 = 14/21
  • Both are multiples of 7: 14/21 = 2/3
  • Check nothing is left: 2 and 3 share no factor above 1, so this is fully simplified
  • Dividing by the highest common factor of 42 in one step gives the same answer, and either route earns the mark