In order to represent large distances in a small space, **scales** are used. If we want to know the actual distance of things on a map, we must multiply the distance they are on the map, according to the scale. So if a scale is 1:2000, we must multiply the distances by 2000. We must be careful with units – the distance may be in centimetres on a map, but it is silly to talk about a distance of 2,100,000cm on the ground. What should we say instead?

If we want to go the other way (for instance to draw a map), then we need to use the scale backwards – typically dividing the distance on the ground according to the scale to find the distance on the map. Again, we must take care with units.

**Example**

Let’s take a scale of 1:20,000 and think about what the actual distance would be of various “map distances” and how far apart on the map various actual distances would be.

**Exercise**

Let’s complete questions 1 to 6 of exercise 32 and exercise 33 from pages 107 to 109 of the core textbook:

The answers are below:

**Ratio**

If we know the ratio between two quantities, e.g. 3:2, we can then tackle various questions. For instance, suppose that a cake requires flour and sugar to be in the ratio 3:2, then we can answer two kinds of question

In the first kind of question we know the total mass of the ingredients in the cake, let’s say it’s 100g. Then we can calculate how much flour and home much sugar there will be. How do we do this?

In the second kind of question we know the total mass of sugar (for instance) in the cake, let’s say it’s 100g. Then we can calculate how much flour there will be and the total mass of the ingredients in the cake.

**Exercise**

Let’s complete exercise 34 and exercise 35 from pages 109 and 110 of the core textbook:

The answers are below:

**Extension Questions**

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5.

**Extension Answers**

- 115 women;
- 375 cashew nuts;
- 480 pixels;
- (9,5);
- £10.