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Ratio & Proportion · Higher

Simplifying & using ratios

At Higher tier, ratio problems involve more complex contexts: algebraic ratios, ratios involving fractions and decimals, combining ratios, and solving problems where the difference between shares is given.

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Key facts to remember

  • 1To simplify a ratio, divide all parts by their highest common factor (HCF).
  • 2To simplify ratios with fractions: multiply every part by the LCM of all denominators.
  • 3To combine ratios: make the shared quantity the same in both ratios.
  • 4If a:b = m and b:c = n, express a:b:c by making b the same in both.
  • 5The unitary method: find the value of one part, then multiply for other parts.
  • 6Algebraic ratio: if a:b = 3:5 and a + b = 40, form an equation.
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Worked examples

Example 1

A:B = 3:5 and A:C = 2:7. Find A:B:C.

Working

  1. A:B = 3:5 and A:C = 2:7.
  2. Make A the same in both: LCM of 3 and 2 is 6.
  3. A:B = 6:10 (multiply by 2) and A:C = 6:21 (multiply by 3).
  4. So A:B:C = 6:10:21.
AnswerA:B:C = 6:10:21
Example 2

Jamie and Kira share money in the ratio 4:7. Kira gets £18 more than Jamie. How much does each person receive?

Working

  1. Difference in ratio parts: 7 − 4 = 3 parts.
  2. 3 parts = £18, so 1 part = £6.
  3. Jamie: 4 × £6 = £24.
  4. Kira: 7 × £6 = £42.
  5. Check: £42 − £24 = £18 ✓
AnswerJamie receives £24, Kira receives £42.
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Common mistakes

✗Only simplifying some parts of the ratio, not all.
✗When combining ratios, not making the shared quantity equal first.
✗Dividing the total by the wrong number of parts.
✗Forgetting to check the answer satisfies the original conditions.
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Exam tips

✓Always check your final ratio cannot be simplified further.
✓If the question gives a difference between shares, find the value of one part first.
✓For combining three-part ratios, make the shared value equal by finding the LCM.
✓Write out each step clearly — ratio questions are easy to lose marks through arithmetic slips.

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