Ratio, scale factor and sharing

What is a ratio?

A ratio compares two (or more) quantities. If a necklace has 2 red beads for every 3 blue beads, the ratio of red to blue is 2 : 3. The order matters: red to blue is 2 : 3, but blue to red is 3 : 2.

The key idea is that a ratio describes parts. In 2 : 3 there are 5 parts altogether, so red beads are ⅖ of the total and blue beads are ⅗. That link between ratio and fractions is where many children get stuck, and where the bar model helps most.

Sharing in a ratio

To share an amount in a ratio, find the total number of parts, work out the value of one part, then multiply.

• £30 shared 2 : 3 → 5 parts → 1 part = £6 → £12 and £18Total parts first.
• £30 shared 2 : 3 → £2 and £3Treating the ratio numbers as the answers.
• £30 shared 2 : 3 → £30 ÷ 2 and £30 ÷ 3Dividing by each part instead of the total.

Drawing it as a bar split into 2 parts and 3 parts, all the same size, shows exactly why we divide by 5.

Ratio from a part

Many SATs questions give one quantity and ask for another. “For every 2 red beads there are 3 blue. There are 12 red beads. How many blue?” The bar model shows that 2 parts = 12, so 1 part = 6, and blue = 3 × 6 = 18. The same method finds the total (5 × 6 = 30).

Scale factors

A scale factor tells you how many times larger or smaller something becomes. If a shape is enlarged by a scale factor of 3, every length is multiplied by 3. Scale drawings and maps use the same idea: a scale of 1 : 100 means every 1 cm on the drawing is 100 cm (1 m) in real life.

• A 4 cm side, scale factor 3 → 12 cmMultiply every length.
• A 4 cm side, scale factor 3 → 7 cmAdding instead of multiplying.

Proportion and recipes

Proportion problems change quantities while keeping the relationship the same. Recipes are the classic context: if 4 people need 300 g of pasta, 6 people need 450 g. Finding the amount for one (75 g) and then multiplying is a reliable method that works for any number of people.

In the SATs

Ratio appears on the reasoning papers, usually in context: sharing money or sweets, mixing paint, beads on a necklace, recipes, or scale drawings and maps. Questions often give one part and ask for the other or for the total, and some ask children to explain their reasoning. Bar models are an accepted and very effective way to show working.

Common mistakes

• Confusing the parts of a ratio with the total number of parts.
• Writing the ratio in the wrong order.
• Adding instead of multiplying for scale factors.
• Mixing ratio and fraction: in 2 : 3, red is ⅖ of the total, not ⅔.
• Forgetting to convert units when using a map scale.

Teaching ideas

Multilink ratios

Build ratios with two colours of cubes: 2 red, 3 blue, then repeat the pattern. Children see that the total always goes up in fives, which explains sharing in a ratio.

Bar model routine

For every ratio problem, draw the bar first, label the parts, and write the value of one part. It's the single most reliable method for KS2 ratio.

Paint mixing

Mix paint (or coloured water) in different ratios and compare the shades. 1 : 2 and 2 : 4 look the same, which shows equivalent ratios in a way children remember.

Map work

Use a local map with a scale. Pupils measure distances between places they know and convert to real distances, checking whether the answers seem sensible.

Try it: five quick questions

SATs-style questions to use as a starter or a quick check. Tap to see each answer.

1. Share 42 sweets in the ratio 1 : 5.

   Show answer

   7 and 35

2. For every 3 boys there are 4 girls. There are 15 boys. How many girls?

   Show answer

   20

3. A rectangle 5 cm by 2 cm is enlarged by scale factor 4. What are the new lengths?

   Show answer

   20 cm by 8 cm

4. On a 1 : 50 drawing, a wall is 6 cm long. How long is the real wall?

   Show answer

   300 cm, or 3 m

5. Paint is mixed red to white 2 : 7. What fraction is red?

   Show answer

   2/9