Mathematics (Core)Exam code: 0580
Introduction to Fractions
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
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
Write the fraction, then cancel fully
When a question asks for a fraction of a shape, a group or a total in its simplest form, a correct uncancelled fraction such as 18/42 earns one mark on a two-mark question. The final mark, and the only mark on a one-mark question, needs it fully cancelled, here to 3/7. So write the uncancelled fraction down first, then keep dividing until the only common factor is 1.
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