Ratios, Proportions and Fractions

Ratios are like potatoes, you can't use them without a bit of cooking. Ratios need to be turned into fractions because that's their useful form. So if you have the ratio 1:1 the process is as follows:

• Add the numbers in the ratios (1+1=2)
• The useful fractions are the numbers divided by the total (a half and a half)

If we had a ratio of 3 things of 5:3:11 we'd use exactly the same method. This ratio means you have a total of 19 "parts" and that these are split into 5 parts, 3 parts and 11 parts. Parts aren't as useful as fractions.

• Add the number of parts in the ratio 5 + 3 + 11 = 19 parts
• The useful fractions are 5/19, 3/19 and 11/19

Now if we have £1000 and we have to split it between three people using the 5:3:11 ratio we'd simply multiply 1000 by the fractions:

• £1000 * 5 / 19 = £263.16
• £1000 * 3 / 19 = £157.89
• £1000 * 11 / 19 = £578.95

The best way to illustrate this is with more example questions, and we'll cover all the forms of ratio question that crop up in GCSE maths.

Ratios in Recipes & Ingredients

Let's imagine we're making a banana and strawberry smoothie. The ingredients are four cups of strawberries to two cups of chopped bananas to one cup of orange juice.

Amount of one ingredient given another

The simplest type of question is like this:

• You have 100 ml of orange juice. What volume of strawberries and bananas will you need?

Remember the first thing to do? We convert the ratios into fractions.

• Ratios are 4 cups strawberries : 2 cups bananas : 1 cup orange juice
• Add the numbers in the ratios: 4+2+1=7
• Fractions are strawberries 4/7, bananas 2/7 and orange juice 1/7

That means if we have 100 ml of orange juice we need twice the volume of bananas, that's 200 ml and we need four times the volume of strawberries which is 400 ml.

Total amount given, convert ratio to amount for each ingredient

We want to make 500 ml of this smoothie. How much of each ingredient will we need to the nearest millilitre?

• Strawberries 500 times 4/7 = 286 ml
• Bananas 500 times 2/7 = 143 ml
• Orange juice 500 times 1/7 = 71 ml

Simplifying ratios

This is just like algebra, you have to remember to divide and multiply all the ratios by the same amount and search for a common denominator. For example if we have the ratio 100:300 you can quickly spot that this could be simplified really easily because both are divisible by 100. We divide all the elements of the ratio by 100 to get 1:3. So 100:300 = 1:3, they're the same ratio. We can have more than two elements in the ratio, so if it was 100:300:500 the ratio would be simplified to 1:3:5.

Try a few more:

• 9:27 simplifies because both are divisible by 9, and so 9:27 = 1:3
• 15:3:18 simplifies because all are divisible by 3, and so 15:3:18 = 5:1:6
• 70:21 simplifies because both are divisible by 7, and so 70:21 = 10:3

Write ratio as 1:n or n:1

You might have some cases where you are given a ratio and asked to write it in one of those two forms. All you have to remember is to multiply or divide both sides by the same amount.

To write 5:6 in the form 1:n we need to make the 5 into a 1 and so we divide by 5. What we do to the 5 we must also do to the 6 so the ratio becomes 1:6/5, but 6/5 is 1.2 so we could also write this as a decimal 1:1.2.

If we write 5:6 in the form n:1 we need to make the 6 into a 1 so we divide by 6. What we do to the 6 we must also do to the 5 so we divide 5 by 6 to get 5/6:1 or as a decimal 0.83:1 to two significant figures.

Simplify a ratio with a decimal

If we have a ratio like 1.2:3 we can always get rid of the decimal by multiplying by ten. Each time we multiply by ten we move the decimal point to the right once. So in this case we need to multiply by ten once to change 1.2 to 12. We have to do the same to the right hand part of the ratio so we get 12:30. But you're probably thinking both are divisible by 6 and you're right. We would simplify this to 2:5.

How about 0.32:3? Now we have to multiply by 100 to make 0.32 into a whole number, so we get 32:300. This simplifies because both are divisible by 4, so we divide both by 4 to get 8:75.

Simplify a ratio with fractions

For the ratio $$\frac{3}{4}:\frac{1}{4}$$ we would multiply both sides by 4 to get rid of the fractions. $$\frac{3}{4} \times 4:\frac{1}{4} \times 4$$ and the ratio would become 3:1.

If the fractions have different denominators we could just multiply by the common denominator. For example, if the ratio was $$\frac{3}{8}:\frac{2}{3}$$ the lowest common denominator is 24, and we multiply both sides by this $$\frac{3}{8} \times 24 : \frac{2}{3} \times 24$$ and after some cancellation we would get 9:16.

Ratios with units

Some ratios ask you to compare different units. For example the ratio 30 cm : 2.1 m has different units. The trick is to convert to the smallest unit. In this case that's centimetres, and we know there are 100 cm in one metre so we multiply the metre measure by 100 to convert it to centimetres, 30cm:210cm. We can also simplify because both are divisible by 30. We would end up with the ratio 1 cm : 7 cm.

Length: 1 metre = 100 cm = 1000 mm and 1 km = 1000 m

Volume: 1 litre = 1000 ml

Difference between two parts of a ratio

A sausage is cut into three pieces in the ratio 3:5:7. The first piece is 2 cm shorter than the second piece. How long is the third piece?

First we turn the ratio components into fractions of 3+5+7=15, and the fractions are 3/15, 5/15 and 7/15.

The difference between the first and second piece as a fraction of the whole length of the sausage is $$\frac{5}{15}-\frac{3}{15}=\frac{2}{15}$$ and so we know that two fifteenths of the length of the sausage is 2 cm. That means the whole sausage must be

$$\frac{2 \mbox{ cm}}{\frac{2}{15}}=2 \mbox{ cm} \times \frac{15}{2}=15 \mbox{ cm}$$

The third piece is $$\frac{7}{15}$$ of 15 cm which is 7 cm.

Another way of thinking about this is that the different between the first and second piece is two parts of the whole sausage. Two parts is therefore 2 cm, one part must be 1 cm and so the third piece is 7 parts and 7 x 1 cm = 7 cm.

Adding a number to a ratio

The most difficult ratio questions involve solving simultaneous equations. You can spot these because the question will give a ratio, add some numbers to some or all of the components of the ratio, give you the new ratio and ask you to work out the total number of components. Let’s look at a couple of examples.

A zoo has a ratio of 7:5 female to male otters. They have a new litter of otter pups which contains 8 females and 4 males and the ratio of female to male otters changes to 5:3. How many otters does the zoo have in total?

If the number of female otters before the litter was born is $$f$$ and males is $$m$$ we know that
$$\frac{f+8}{m+4}=\frac{5}{3}$$ and $$\frac{f}{m}=\frac{7}{5}$$

We cross-multiply both equations to get rid of the fractions:
$$3(f+8)=5(m+4)$$ and $$5f=7m$$

Multiply out the first equation and we get
$$3f+24=5m+20$$

And we can substitute from the second equation
$$f=\frac{7}{5} m$$

This gives
$$3 \times \frac{7}{5} m + 24 = 5m + 20$$

Multiply both sides by 5 to get rid of the fraction
$$3 \times 7m + 24 \times 5=25m+100$$

Collect terms in m
$$25m – 21m=120-100$$

And simplifying $$4m=20$$ and so $$m=5$$ and we know that $$f=\frac{7}{5}\times 5=7$$. The total otter population was originally $$12$$ and after the litter of 12 pups it doubled to 24.

If you want to try some of these for yourself to check you understand them here are some fully worked examples of ratios.