0580

Working with Ratios

Number

Equivalent ratios

  • Multiplying or dividing every part by the same number leaves the comparison unchanged
    • 2 : 3 is equivalent to 8 : 12 and to 200 : 300
  • Scaling exists to make the numbers meaningful: a recipe in the ratio 3 : 2 is useful as 300 g : 200 g

Simplest form

  • A ratio is in its simplest form once no whole number above 1 divides every part
  • Divide through by the highest common factor of all the parts
    • 6 : 9 has highest common factor 3, giving 2 : 3
    • 18 : 24 : 30 has highest common factor 6, giving 3 : 4 : 5
  • With three parts the factor must divide all three, not merely two of them
Exam tip

Cancel by a factor common to every part, not just two of three, and where the units differ convert them before simplifying — a ratio compares like with like. Never turn a ratio into a fraction on the answer line: 2 : 3 is not 2/3, and a fraction where a ratio was asked for scores nothing.

Worked example

Simplifying a ratio, including one with units

Write 45 : 60 : 75 in its simplest form, and write 400 m : 1.2 km in its simplest form.

Solution:

  • For the first, find a factor that divides all three parts
  • 15 divides 45, 60 and 75
  • 45 ÷ 15 = 3, 60 ÷ 15 = 4, 75 ÷ 15 = 5, giving 3 : 4 : 5
  • Check nothing more divides all three: 3, 4 and 5 share no factor above 1
  • For the second, the units differ, so convert to the same unit first
  • 1.2 km = 1200 m, so the ratio is 400 : 1200
  • Divide both parts by 400: 1 : 3
  • Simplifying before converting would have given 400 : 1.2, which compares metres with kilometres and is meaningless