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§ Ratios & Proportions

Ratios & Proportions

CCSS.6.RPCCSS.7.RP3 min read

Year 10 students often struggle with proportion problems until they grasp the cross-multiplication method. The key breakthrough happens when pupils realise that 3:4 = 15:20 follows the same pattern as solving 3/4 = 15/x.

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§ 01

Why it matters

Ratios and proportions appear in countless real-world scenarios that Year 10 students encounter daily. When comparing mobile phone contracts, a student might need to determine that £25 for 10GB represents better value than £40 for 15GB by calculating unit rates. In geography lessons, map scales like 1:50,000 require proportion skills to calculate that 3cm on the map represents 1.5km in reality. Recipe calculations become essential life skills when doubling a recipe for 8 people that originally served 4, requiring students to scale ingredients proportionally. GCSE Foundation papers regularly test these applications through questions about paint mixing (5 litres of red to 3 litres of yellow), currency exchange rates (£1 = €1.17), and speed-distance-time problems where maintaining proportional relationships determines correct answers.

§ 02

How to solve ratios & proportions

Ratios & Proportions

  • A ratio compares two quantities (a:b or a/b).
  • To solve a proportion a/b = c/d: cross-multiply (a×d = b×c).
  • Simplify ratios by dividing both by their GCF.

Example: 23 = x/12 → 2×12 = 3x → x = 8.

§ 03

Worked examples

Beginner§ 01

Simplify the ratio 6:5.

Answer: 6:5

  1. Find GCF of 6 and 5 → GCF = 1 — Divide both by the GCF.
  2. Divide → 6÷1 : 5÷1 = 6:5 — Simplified ratio.
Easy§ 02

If 24 = 4/?, find the missing value.

Answer: 8

  1. Cross-multiply → 2 × ? = 4 × 4 — Set up the cross-product.
  2. Solve → ? = 16 ÷ 2 = 8 — Divide both sides.
  3. Verify → 2/4 = 4/8 ✓ — Both simplify to the same ratio.
Medium§ 03

You need 4 eggs to make 40 cookies. How many eggs do you need for 46 cookies?

Answer: 4.6

  1. Set up proportion → 4/40 = ?/46 — Eggs to cookies ratio.
  2. Cross-multiply and solve → ? = 4 × 46 ÷ 40 = 4.6 — Solve for the unknown.
§ 04

Common mistakes

  • Adding numerators and denominators incorrectly when finding equivalent ratios, writing 2:3 = 4:6 as 6:9 instead of recognising that 2×2 = 4 and 3×2 = 6
  • Confusing the order when setting up proportions, placing 3 eggs for 12 biscuits as 12/3 = x/24 instead of 3/12 = x/24, leading to x = 96 rather than x = 6
  • Cross-multiplying incorrectly by multiplying across the same side rather than diagonally, solving 4/5 = x/10 as 4×x = 5×10 instead of 4×10 = 5×x
  • Forgetting to simplify ratios at the start, working with 12:18 throughout a problem instead of simplifying to 2:3 first, making calculations unnecessarily complex
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§ 05

Frequently asked questions

What's the difference between a ratio and a proportion?
A ratio compares two quantities (like 3:4), whilst a proportion states that two ratios are equal (3:4 = 6:8). Think of ratios as individual comparisons and proportions as equations showing equivalent relationships. Both use the same solving techniques through cross-multiplication.
When should pupils use cross-multiplication versus scaling up?
Cross-multiplication works best for finding one missing value in a proportion (2/3 = x/12). Scaling up suits problems where you multiply both parts by the same factor (doubling a recipe from 2:3 to 4:6). Both methods reach the same answer.
How do students check their proportion answers?
Substitute the answer back into the original proportion and verify both ratios simplify to the same fraction. For 2/3 = 8/12, check that 2÷1 = 2 and 3÷1 = 3, whilst 8÷4 = 2 and 12÷4 = 3. Both give 2:3.
Why do map scale problems confuse students?
Students often mix up which measurement represents reality versus the map. Emphasise that 1:25,000 means 1cm on the map equals 25,000cm (250m) in real life. Always identify the map measurement first, then scale up to find the actual distance.
What's the best way to teach ratio simplification?
Start with finding the highest common factor of both numbers using factor trees or division methods. Then divide both parts of the ratio by this HCF. For 15:25, the HCF is 5, so 15÷5 : 25÷5 gives the simplified ratio 3:5.
§ 06

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