# Ratio and Proportion

Ratio is a mathematical term used for comparing the size of one part to another part.

Proportion compares one part to the whole.

You need to have an understanding of these mathematical concepts more often that you would expect, such as when:

• Converting between one currency and another when travelling abroad
• Measuring quantities in a recipe
• Comparing prices in the supermarket
• Using a scale, such as on a map or when making a model
• Working out food and drink you need for a party
• Calculating your likely winnings when you place a bet

## What is a Ratio?

You will usually see ratios used to compare two numbers, but they are often used to compare several quantities.

Ratios are usually shown as two or more numbers separated with a colon, for example, 7:5, 1:8 or 5:2:1

They are also often shown in a form similar to a fraction, e.g. 7/5 or 1/8

Sometimes they are simply expressed in words and numbers, such as ‘7 to 5’ or ‘one to eight’.

If you have an understanding of how Fractions work, then you will see that ratios work in a very similar way, but there is an important difference, illustrated in the following example.

Looking at the row of 10 boxes below, you can see that 7 of them are white and three of them are purple.

The ratio of purple to white is therefore 3:7

However, the fraction of purple boxes is 3/10 (or 30%, when expressed as a percentage).

The fraction is expressed in relation to the whole, whereas the ratio is expressed as a comparison between two (or more) parts of the whole.

### Reducing and Multiplying Ratios

Example 1:

Dave is ordering take-away lunches for himself and some friends. For every 4 packs of sandwiches he buys, he gets a free drink. If he buys 12 packs of sandwiches, how many free drinks does he get?

The ratio is four sandwiches to one drink, which is written 4:1

Dave buys 12 sandwiches, which is 3 lots of 4. To find out how many drinks he will get, you multiply both sides of the ratio by the same amount:

3 × 4 = 12 sandwiches

3 × 1 = 3 free drinks

Example 2:

James is sorting out the office stationery order. He has received 36 year planners and 3 free packs of marker pens. How many year planners were required to get one free pack of pens?

The ratio of planners to pens is 36:3

The ratio can be reduced or simplified by dividing both sides by a common factor. This is the same as the method used for simplifying fractions.

In this case, the ratio is reduced by dividing both sides of the ratio by three, giving the answer: 12:1

1 pack of pens is received for every 12 planners ordered.

When you are working with fractions, the numerator and denominator (top and bottom numbers) must always be whole numbers.

However when you are working with ratios, it is perfectly correct to use a decimal. For example, the ratio 5:12 can be expressed as 1:2.4

### Scaling Ratios

Ratios are especially useful when we need to scale an amount, i.e. increasing or decreasing a quantity or size of something.

The most common examples are maps or scale models, where areas of many kilometres in size are accurately represented on a small map, or a large steam locomotive, for example, is translated into much a smaller but precise representation of itself.

The ability to scale a ratio is also a very useful skill when increasing or decreasing the amount of ingredients in a recipe.

Ratios can be scaled up or down by multiplying both parts of the ratio by the same number, in the same way as in the examples above.

For example, a map scale of 1:25000 means that every 1mm on the map represents 25000mm (or 25m) on the ground.

A 1:12 scale model car means that every 1 inch on the model is equivalent to 12 inches on the full-size vehicle.

In the map and car examples above, the units are given as millimetres and inches. However, they could be anything as long as they are the same on both sides of the ratio.

The map scale of 1:25000 could be 1 inch on the map and 25000 inches on the ground, but it cannot be 1 inch on the map to 25000 cm on the ground as the units are not equivalent.

The model car scale of 1:12 could be 1 cm on the model to 12 cm on the vehicle, but it cannot be 1 cm on the model to 12 metres on the vehicle, because the units are not consistent.

The only exception is if the units are given on both sides. For example, Ordnance Survey maps in the UK used to be ‘One Inch to One Mile’. This is fine because the units for both sides were provided.

Example 3:

You need to make 20 cupcakes, but the quantity in the recipe below is only enough for 12. You could double the ingredients and make 24 cupcakes, having four left over for yourself! However, if you don’t have quite enough ingredients for 24, you can use ratio to calculate how much of each ingredient is needed to make 20 cupcakes.

120g butter
120g caster sugar
3 eggs
1 tsp vanilla extract
120g self-raising flour
1 tbsp milk

You need to scale the recipe from 12 to 20, so the scale ratio is 12:20

However, the ratio isn’t in its simplest form, so you can reduce it to make the calculation easier. Both 12 and 20 can be divided equally by 2 or by 4. Dividing both sides by 4 will reduce the ratio to its simplest form: 3:5

The next step requires some abstract thinking! You need to think of the original recipe as three units and the amount you need as 5 units.

The method for converting the recipe is therefore to divide all the original quantities by three, to give the amounts for 1 unit, then multiply by 5.

The amounts of butter, sugar and flour are all the same, so you only need to do one calculation for all of these:

120g ÷ 3 = 40g butter/sugar/flour
and
3 eggs ÷ 3 = 1 egg

In order to calculate the quantity of milk, first convert the units from tablespoons (tbsp) to millilitres (ml) to make it easier.

1tbsp milk = 15ml
15ml ÷ 3 = 5ml milk

One teaspoon (tsp) of vanilla extract is a bit more tricky but, similarly, convert the units to millilitres: one teaspoon is equivalent to 5ml. You therefore end up with 5/3ml of vanilla for this part of the calculation!

To calculate the quantities for 20 cupcakes, you need to multiply the quantities for ‘1 unit’ by 5.

40g × 5 = 200g butter/sugar/flour
1 egg × 5 = 5 eggs
5ml milk × 5 = 25ml milk
5/3ml of vanilla × 5 = 8.33 ml vanilla (this will require a little bit of estimation when you are measuring! However, this is often the way in real life.)

Finally, take care with the order of the ratio!

Always check you have read the ratio the right way round. A ratio of 4 cockerels to 15 hens should be written 4:15, not 15:4.

## Proportion

Let’s look again at the white and purple boxes.

You now know that the ratio of purple to white is 3:7

However, the fraction of purple boxes is 3/10

Proportion compares the part to the whole, in the same way as fractions. The proportion of purple boxes is therefore 3 in 10.

Even if you have multiple lines of boxes identical to the line above, no matter how many you have, the ratio of purple to white remains 3:7 and the proportion of purple to white remains 3 in every 10.

Example 4:

Pam keeps tropical fish in an aquarium at home. She has 6 Tetra, 15 Minnow, 5 Platy and 4 Guppy.

What proportion of her fish are Minnow?

There are 30 fish in total and 15 of them are Minnow. So the proportion of fish are that are Minnow is 15 in 30, which is the same as 1 in 2. Since proportion is related to fractions, you can say that 1/ (half) of Pam’s fish are Minnow.

Similarly, 5 in 30 fish are Platy, which is the same as 1 in 6.

We can use this example to look at ratios as well.

The ratio of Minnow to other fish is 15:15, i.e. 1:1.

The ratio of Tetra to other fish is 6:24, i.e. 1:4

And the ratio of Tetra to Minnow to Platy to Guppy is 6:15:5:4!

### Conclusion

Ratio and proportion are mathematical concepts that compare an amount to another amount. They can be tricky to understand, but work in a similar way to fractions. They can be useful in many everyday situations, especially if you need to scale a recipe.

TOP