Ratio and Proportion Notes

Q: How do ratios compare shapes?

Similar shapes have corresponding lengths in the ratio 1:k, areas in 1:k^2. Scale factors compare lengths, areas and volumes, and link to similarity and trigonometric ratios.

AQA GCSE Maths: Revision notes

Key facts

Always write a ratio in its simplest form by dividing by the HCF.
One quantity can be a fraction of another, even greater than 1.
In similar shapes with scale factor $k$ , lengths are in ratio $1:k$ and areas in $1:k^2$ .
To divide in a ratio: total the shares, find one share, multiply.
Know part:part and part:whole.
For best buys compare the same unit, such as price per 100 g.

Similar triangles: scale factor 2 turns 3-4-5 into 6-8-10

Ratio notation

A ratio compares quantities and should be written in simplest form.

Ratio notation such as $3:2$ compares two or more quantities. Divide by the highest common factor to get the simplest form.

One quantity can also be a fraction of another, which may be less than 1 (a part of a whole) or greater than 1.

A multiplicative relationship such as "one is 3 times the other" can be written as a ratio or fraction. Ratios link to linear functions $y = kx$ .

8 : 12 = 2 : 3. A bar of 5 equal parts: 2 parts for the first quantity and 3 for the second.

$8 : 12$ $2 : 3$ (divide by HCF 4)

Write $18:24$ in simplest form.

Ratio and similar shapes

Similar shapes have corresponding lengths in the ratio $1:k$ , areas in $1:k^2$ .

Scale factors compare lengths, areas and volumes, and link to similarity and trigonometric ratios.

If two shapes are similar with linear scale factor $k$ , corresponding lengths are in the ratio $1:k$ . Areas are multiplied by $k^2$ , and volumes by $k^3$ .

Here $k = 2$ , so every length doubles and the area is multiplied by 4.

Similar triangles: scale factor 2 turns 3-4-5 into 6-8-10

Lengths $\times k$
Areas $\times k^2$
Volumes $\times k^3$

A shape is enlarged by scale factor 3. By what factor is its area multiplied?

Dividing in a ratio

Add the parts, find one share, then multiply.

To divide a quantity in a ratio:

Add the parts to find the total shares.
Divide the quantity by the shares to find one share.
Multiply each part by one share.

A ratio can be part:part (boys to girls) or part:whole. Be sure which one is meant.

1
Add the parts
total number of shares
2
One share
quantity divided by total shares
3
Multiply
each part times one share

Dividing in a ratio

Worked example

Share £60 in the ratio $2:3:5$ .

1
Total shares $= 2 + 3 + 5 = 10$ .
2
One share $= 60 \div 10 = 6$ .
3
$2 \times 6 = 12$ , $3 \times 6 = 18$ , $5 \times 6 = 30$ .

Share £40 in the ratio $3:5$ . What is the larger share?

Ratio in real contexts

Proportion means two ratios are equal. It is used in conversions, mixing, best buys and scale drawings.

Proportion means an equality of ratios.

For best buys, compare the same unit, such as price per 100 g, across every option.

Proportion: 200 g of flour for 4 people means 50 g per person, so the amount is directly proportional to the number of people.

Conversion

Currencies
Units

Mixing

Paint
Recipes

Best buy

Compare pack sizes
Same unit for each

Scale

Scale diagrams and maps
Geometry problems

Worked example

Which is better value: 250 g for £1.50 or 400 g for £2.20?

1
£1.50 $\div$ 2.5 = £0.60 per 100 g.
2
£2.20 $\div$ 4 = £0.55 per 100 g.

Which is better value: 500 g for £2 or 750 g for £2.70?

Try an exam question

Amir and Beth share £84 in the ratio 2:5. Work out how much each receives.

[3 marks]

That's the notes covered.

Carry on to the next subtopic.