Sharing in a Ratio

A ratio compares amounts in a stated order. If Ari and Bea share money in the ratio Ari : Bea =3:5, Ari gets 3 equal parts and Bea gets 5 parts of the same size.

The parts are not £1 each. Your job is to find what one part is worth.

In the exam you are usually asked to share an amount in a ratio such as 2:3:5 and work out each share, for 2 or 3 marks. Harder questions give you one share, or the difference between two shares, instead of the total. The hardest ones, for 4 marks, ask what happens after some money is spent or passed on.

• Add the numbers in the ratio. This is the total number of parts.
• Divide the amount by the total number of parts. This is one part.
• Multiply one part by each number in the ratio.
• Check that the shares add up to the amount.

If the amount has units, keep them. Change units first if the answer is wanted in a different unit: for example, 1.5 kg is 1500 g.

A ratio tells you what fraction of the whole each person gets. In the ratio 3:5 there are 8 parts, so Ari gets \frac{3}{8} of the money and Bea gets \frac{5}{8}

The top of the fraction is that person's number of parts. The bottom is the total number of parts, not the other person's number.

To get a percentage, change the fraction. In the ratio 2:3 the first share is \frac{2}{5} of the whole, which is 40\%

Read the question to see which amount is shared. If some money is spent or kept first, take it away before you share.

In the same way, if something happens to one share afterwards, work out that share first, then do the next step to that share, not to the whole amount.

Before you divide, say what the amount you know stands for. Every method still finds the value of one part first.

• A total is all the parts: divide by the sum of the ratio numbers.
• One share is only that person's parts: divide by that person's number.
• A difference between two shares is the gap between their numbers: divide by that gap.

Some questions share an amount, then one person gives some of their share to another. Work out the shares first. Then follow the money one step at a time.

When money is passed from one person to another, the total stays the same. One share goes down and the other goes up by the same amount.

• Dividing the amount by each number in the ratio. Add the ratio numbers first, then divide by that sum to find one part.
• Treating one person's share as the total. If you are given one share, divide it by that person's number of parts.
• Giving a share to the wrong person. Match each name to its number, in the order the ratio is written.
• Sharing the whole amount when only part of it is shared. Take away anything spent or kept first, then share the rest.
• Writing the fraction upside down. A share as a fraction is its parts over the total parts, so in 2:3 the first share is \frac{2}{5}