Study tips: Percentages, proportions, ratios & fractions made easy

When I told my family and friends that I was going back to college to study accountancy, they all thought it was a joke. Let’s just say I wasn’t the best at maths. 

Percentages were my pet hate and I was still phoning my Dad, years after I’d left home, to check whether I needed to multiply or divide by 100. Then a fantastic tutor told me about the word ‘of’ and how it can be substituted for ‘divide’. In effect, she translated maths into English for me and I’ve never looked back.

Percentages, proportions, ratios and fractions are subtly different, but related enough for the same techniques to be applied and manipulated, as long as we understand the connections.

So here’s how it works.

Percentages

First we have to know that ‘per cent’ means ‘out of 100’ and that percentages are a way of working out part of a number ie. 66% means 66 out of 100.

We might need to calculate a percentage of a number, say for example 8% of £26,500. The easiest way is to calculate 1% first and then scale that up to 8%. 

As £26,500 represents 100% and we want to calculate 1% of it, we change the word ‘of’ to ‘divide’ and the calculation becomes:

£26,500 ÷ 100 = £265.

As £265 represents 1% of the whole 100%, we can ‘multiply’ it by 8 to calculate 8%:

£265 x 8 = £2,120   

So 8% of £26,500 is £2,120

Calculating an amount as a percentage of another

We might also need to calculate an amount as a percentage of another, for example £30 as a percentage of £600.  We can turn the maths into English here too, as we want to know what 30 is as a percentage of 600.  When we substitute ‘divide’ for ‘of’ the calculation becomes:

£30 ÷ £600 = 0.05

However, this is where a connection comes in, as we’ve calculated a decimal, and now need to ‘multiply’ that by 100 to convert it back into a percentage:

0.05 x 100 = 5%

We can check our calculations by re-working it the opposite way, using the first technique, as now we could reasonably expect 5% of £600 to be £30.  

But is it?

It’s worth noting that we can convert between percentages and decimals:

  • Divide by 100 to change a percentage to a decimal

eg. 15% ÷ 100 = 0.15

  • Multiply by 100 to change a decimal to a percentage

eg. 0.46 x 100 = 46%

Proportions and ratios

Let’s now look at how this relates to proportions and ratios. Firstly we need to know that when we talk about proportions, we simply mean a ‘part’, ‘share’, ‘bit’ or ‘number’ of a whole. 

In effect, 80% is a proportion of 100%.

Proportions can be described in general terms; a small proportion of enquiries are generated by referrals.

Or they can be described specifically using ratios, for example, the proportion of enquiries between newspaper advertising, the website and referrals is 10:5:1 respectively.

The word ‘respectively’ means that the order of the numbers in the ratio (10, 5 and 1) relate to the ‘parts’ in the same order. For example, newspaper advertising is 10 as a proportion of the whole, the website is 5 and referrals only make up the small proportion of 1.

Next we need to understand the relationship of each part or proportion to the ‘whole’ and for that we need to calculate what the whole is.  We do that by adding all the numbers in the ratio:

10 + 5 + 1 = 16

So in this case 16 represents the ‘whole’.

Finally, we can apply this knowledge and understanding to answer a question such as:

If we had 12,000 enquiries in total, how many were generated by each source?

We can use the same thought process as we did with percentages to find 1/16th of the total enquiries and then scale it up to the number of 16ths needed for each part. 

As 12,000 is the total and we want to calculate 1/16th of it, we change the word ‘of’ to ‘divide’ and the calculation becomes:

12,000 ÷ 16 = 750

As 750 represents 1/16th of the total, we can ‘multiply’ it by 10 to calculate the proportion generated by newspaper advertising:

750 x 10 = 7500

Again we can check our answer, in this case by calculating the other two ‘parts’ and ensuring that when all three ‘parts’ are added together the answer is the total.

Newspaper Advertising 750 x 10 7500
Website 750 x 5 3750
Referrals 750 x 1 750
     
Total    1200

Fractions 

By using the technique to calculate 16ths , we have in effect used fractions to help us calculate the proportions in the correct ratios. This is because, like percentages, proportions and ratios, a fraction is another way of expressing ‘a part of a whole’.

10/16th’s is a fraction which means that the ‘whole’ amount has been split into 16 and we are looking at 10 of those 16 bits!

Instead of using a ratio, we could have been told that the website is the source of 5/16 of total enquiries. Even if we only had this information and knew nothing about the newspaper advertising or referrals, we still could have performed the calculation above. 

Alternatively, we could read the fraction as 5 divided by 16, which would be:

5 ÷ 16 = 0.3125

As we now have a decimal we could turn it into a percentage if needed, but in this case we are trying to calculate the actual number of enquiries that 0.3125 is as a proportion of the 12,000 that were made in total. 

Therefore, we can just multiply the decimal by the total:

0.3125 x 12,000 = 3,750

This brings us full circle back to the first calculation we started with for percentages, and that’s because all of these basic maths concepts and calculations are related. 

It’s also why the techniques we’ve looked at can be applied to them all. The trick is to understand the connections and turn the maths into English or a fraction into a percentage.

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Gill Myers is a self-employed accounts consultant. She has taught AAT qualifications since 2005 and written numerous articles and e-learning resources.

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