Ratio, Proportion and Simple Formulae

This lesson covers DfE content statements L1.5 and L1.17 — working with simple ratios and direct proportion; and using simple formulae expressed in words for one-step and two-step calculations.

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What Is a Ratio?

A ratio compares two (or more) quantities. It tells you how much of one thing there is compared to another.

Example: If a paint mix uses 2 parts blue to 3 parts white, the ratio of blue to white is 2 : 3.

Writing Ratios

• Always write ratios in the order stated in the question.
• Separate numbers with a colon (:).
• Use whole numbers — no fractions or decimals.

Simplifying Ratios

Simplify a ratio by dividing all parts by their highest common factor, just like simplifying a fraction.

Original Ratio	÷ by	Simplified
4 : 6	2	2 : 3
10 : 25	5	2 : 5
15 : 9	3	5 : 3
8 : 12 : 4	4	2 : 3 : 1

Worked Example — Simplifying a Ratio

Scenario: A class has 12 men and 18 women. Write the ratio of men to women in its simplest form.

12 : 18 → divide both by 6 → 2 : 3

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Sharing in a Ratio

This comes up a lot in exams. The method is:

1. Add up the parts of the ratio.
2. Divide the total amount by the number of parts.
3. Multiply each part of the ratio by the value of one part.

Worked Example — Sharing Money

Scenario: Two friends, Ali and Ben, share £60 in the ratio 1 : 2. How much does each person get?

Step 1: Total parts = 1 + 2 = 3 Step 2: Value of one part = £60 ÷ 3 = £20 Step 3:

• Ali gets 1 × £20 = £20
• Ben gets 2 × £20 = £40

Check: £20 + £40 = £60 ✓

Worked Example — Sharing in Three Parts

Scenario: Three departments share a budget of £15,000 in the ratio 2 : 3 : 5. How much does each department get?

Step 1: Total parts = 2 + 3 + 5 = 10 Step 2: Value of one part = £15,000 ÷ 10 = £1,500 Step 3:

• Department A: 2 × £1,500 = £3,000
• Department B: 3 × £1,500 = £4,500
• Department C: 5 × £1,500 = £7,500

Check: £3,000 + £4,500 + £7,500 = £15,000 ✓

Exam Tip: After sharing in a ratio, always add up the shares and check they equal the original total. This is a quick and reliable way to catch mistakes.

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Direct Proportion

Two quantities are in direct proportion when they increase (or decrease) at the same rate. If one doubles, the other doubles.

Everyday examples:

• If 1 tin of paint covers 5 m², then 3 tins cover 15 m².
• If 1 hour of work costs £12, then 4 hours cost £48.

The Unitary Method

The unitary method means finding the value of one unit first, then scaling up or down.

Worked Example — Shopping

Scenario: 5 notebooks cost £8.00. How much do 3 notebooks cost?

Step 1: Cost of 1 notebook = £8.00 ÷ 5 = £1.60 Step 2: Cost of 3 notebooks = £1.60 × 3 = £4.80

Worked Example — Scaling a Recipe

Scenario: A recipe for 4 people uses:

Ingredient	Amount for 4
Flour	200 g
Sugar	80 g
Butter	100 g
Eggs	2

How much of each ingredient do you need for 6 people?

Step 1: Find the amount for 1 person (divide by 4):

Ingredient	For 1 person
Flour	50 g
Sugar	20 g
Butter	25 g
Eggs	0.5

Step 2: Multiply by 6:

Ingredient	For 6 people
Flour	300 g
Sugar	120 g
Butter	150 g
Eggs	3

Worked Example — Best Buy

Scenario: A supermarket sells orange juice in two sizes:

• Small: 500 ml for £1.20
• Large: 1.5 litres for £3.00

Which is better value?

Small: £1.20 ÷ 500 = 0.24p per ml Large: £3.00 ÷ 1,500 = 0.20p per ml

Answer: The large bottle is better value (cheaper per ml).

Exam Tip: For "best buy" questions, always work out the price per unit (e.g., per ml, per gram, per item). The lowest price per unit is the best value. Show your working clearly.

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Simple Formulae (in Words)

At Level 1 you work with formulae written in words (not letters and symbols). You substitute numbers into the formula and calculate the answer.

Worked Example — One-Step Formula

Scenario: A formula says: "Total cost = number of items × price per item". You buy 8 items at £4.50 each. What is the total cost?

Total cost = 8 × £4.50 = £36.00