Direct Proportion: Scale Both Together

R5Direct Proportion

Double one, double the other.

In direct proportion, two quantities scale together — multiply both the same way.

3 pencils cost 60p. What do 6 pencils cost?

3 pencils → 60p

× 2 on both sides

6 pencils → 120p (£1.20)

Whatever multiplies one quantity multiplies the other — that's direct proportion.

Learn the 3-Step Method

1Spot the multiplier.

From 3 pencils to 6 pencils is × 2.

× 2

2Apply it to the other quantity.

60p × 2 = 120p.

60 × 2 = 120

3Awkward jumps: go through 1.

3 → 5 pencils? Find ONE pencil first: 60 ÷ 3 = 20p, then × 5 = 100p.

via 1: 20p each

💡 Key IdeaWhen the multiplier isn't obvious, drop to 1 first — the unitary method never fails (and it's the whole of lesson R6).

Worked Example

4 tickets cost £18. How much do 10 tickets cost?

One ticket

18 ÷ 4 = 4.50

£4.50

→

Ten tickets

4.50 × 10 = 45

→

✓ Answer

£45

Sense: 10 is 2.5 × 4, and 2.5 × £18 = £45 ✓

Watch Out! Common Mistakes

Adding instead of multiplying.

3 pencils 60p → 6 pencils = 60 + 3 = 63p ✕

Scale, don’t add! Twice the pencils costs twice the money: 120p.

Multiplying only one side.

6 pencils → still 60p ✕

Both together! The quantities move in step — double pencils, double price.

Skipping the sense check.

Estimate! 10 tickets should cost a bit over double 4 tickets (£36) — £45 fits; £450 or £4.50 wouldn’t.

⬇ Cheat Sheet

DIRECT PROPORTION

Find the multiplier between the two amounts.
Apply the SAME multiplier to the other quantity.
Awkward numbers: find the value of 1 first.
Sense-check the size of the answer.

Examples: 3→6 pencils: 60p→120p · 4 tickets £18 → 10 tickets £45.

📖 Continue Your Journey

✓ R1 What Is Ratio?✓ R2 Simplifying Ratios ✓ R3 Sharing in a Ratio ✓ R4 Ratio and Fractions R5 Direct Proportion R6 Best Value R7 Scale Drawings and Maps R8 Recipes and Scaling R9 Ratio and Percentages R10 Ratio Problem Solving

→

Next Lesson: R6 Best ValuePrice per one — then compare fairly.

📄 Free printable: Download the Direct Proportion cheat sheet (PDF) — keep it by your child's desk.

Quick summary

Direct proportion: both quantities multiply by the same number.
3 pencils at 60p → 6 pencils at 120p (× 2 on both).
For awkward jumps, find the value of 1 first: 60 ÷ 3 = 20p each.
Sense-check answers against a rough estimate.

Frequently asked questions

What is direct proportion?

When two quantities increase together at the same rate — doubling one doubles the other, like pencils bought and money spent.

How do you solve proportion problems with awkward numbers?

Go through 1: divide to find the value of a single item, then multiply up. 4 tickets at £18 → £4.50 each → 10 tickets £45.

How is proportion different from ratio?

Ratio compares two parts of one whole (3 : 2); proportion links two different quantities that scale together (pencils and pence).

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