A ratio compares parts with each other, not with a whole

A ratio compares two quantities directly against each other, with no whole in sight. That is exactly why it survives being scaled.

Writing and simplifying

12 : 18 becomes 2 : 3. But units must match first: 50 cm to 2 m is 50 : 200, which is 1 : 4.

2 : 3 does not mean two thirds

In the ratio 2 : 3 there are 5 parts altogether, so the shares are two fifths and three fifths.

Sharing in a given ratio

Add the parts, divide to find one part, then multiply out. Adding the shares back should always return the original amount.

Proportion

Two ratios are in proportion when they are equal. If 4 pens cost 60, then 4 : 60 = 7 : x, so x is 105.

Not everything is directly proportional

6 workers take 10 days, so 12 workers take 5, not 20. More workers means less time, which is inverse proportion.

Type	What happens	What stays fixed
Direct	Double one, the other doubles	Their ratio: cost per pen
Inverse	Double one, the other halves	Their product: total worker-days

Ratios must be simplified in the same units

50 cm to 2 m written as 50 : 2 gives 25 : 1, which is the wrong way round. Convert to the same unit first, and it is 1 : 4.

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Worked Example 1

Rs 2000 is shared between two brothers in the ratio 2 : 3.

Count the parts first2 + 3 = 5 parts altogether

Find what one part is worth2000 ÷ 5 = Rs 400 per part

Multiply outRs 800 and Rs 1200

Check800 + 1200 = 2000. Note that 800 is two FIFTHS of the money, not two thirds.

Worked Example 2

If 4 pens cost Rs 60, what would 7 pens cost?

Direct or inverse?More pens means more cost, so direct proportion

Set the two ratios equal4 : 60 = 7 : x, so 4x = 420

SolveRs 105

The unitary checkOne pen is 60 ÷ 4 = Rs 15, so 7 pens cost 7 × 15 = Rs 105. Two routes agree.

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Activity 17.2

Scale up a recipe

What you needA real recipe from home that feeds a known number of people, paper and pencil.

MethodWork out the amounts needed to feed the whole class, and then to feed just two people.

Then look for the odd oneCheck whether every quantity scales the same way, including the cooking time.

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A ratio compares two quantities with each other, not with a whole.

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1What is the difference between a ratio and a fraction, and why does it matter?

2How do you share an amount in a given ratio, and how do you check the answer?

3Explain the difference between direct and inverse proportion with an example of each.

4Why must a ratio be written in the same units before it is simplified?

5When scaling up a recipe, why does the cooking time not scale with the ingredients?

Question 1 of 10

1A ratio compares

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Unit 17 · Ratio and Proportion

Ratio and Proportion

Mathematics · Grade 7

What You Will Be Able To Do

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Write and simplify a ratio, with the units matched first

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Share any amount in a given ratio, and check your answer

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Solve for a missing quantity when two ratios are equal

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Tell direct from inverse proportion before you start calculating

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A ratio compares two quantities directly against each other, with no whole in sight. That is exactly why it survives being scaled up or down.

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Presenter notes: Open with a mixing question. Ask how a mason knows how much cement to use when the size of the job changes every day. The answer is that they do not remember amounts, they remember a ratio: two parts cement to three parts sand, which works for a bucket or a lorryload. Last lesson compared a part with a whole; today compares two parts with each other, and that small change is what makes recipes, maps and mixtures work.