## What you need to know

Percentage means “number of parts per one hundred” and is denoted by the % sign. For example, 50% of a number means 50 parts of it out of a total of 100, and since 50 is one half of 100, 50% means half of the total amount. In this topic we’ll look at one example for each type of scenario you’ll come across. Firstly, we will look at finding a percentage of a given number.

Example: Work out 16% of 60 (without a calculator).

16% means “16 parts out of 100”. To find 16% of 60, we can either split 60 into 100 parts and multiply that by 16, or we can break it down into more digestible parts. We’ll do that here.

We can split 16% into 15% and 1%. Firstly, we observe that 15% is 3 lots of 5%. To work out 5%, we consider this: 5% means 5 parts out of 100, but how many times does 5 go into 100? 20 times, so if we split 100% into 20 equal parts, we get 5%. Therefore, if we divide 60 by 20, we get 5% of 60:

$60 \div 20 = 3\text{, therefore }5\%\text{ of } 60\text{ is }3$.

So, since 15% is 3 lots of 5%, we get that 15% of 60 is $3\times 3 = 9$. Then, to find the missing 1% (1 part out of 100), we divide 60 by 100:

$1\%\text{ of }60=60\div 100 = 0.6$.

16% is the sum of 15% and 1%, so 16% of 60 is $9+0.6=9.6$.

Next, we will see how to convert between percentages, decimals, and fractions.

Example: Write 28% as a decimal and as a fraction in its simplest form.

To write a percentage as a decimal, simply divide by 100 (aka, move the decimal point left two spaces) So, 28% becomes 0.28. Some more examples: 89.6% becomes 0.896, and 5% becomes 0.05.

Now, we must convert it to a fraction. Given that 28% means “28 parts out of 100”, we can write it as $\frac{28}{100}$. Then, simplifying this fraction we get

$28\%=\dfrac{28}{100}=\dfrac{7}{25}$

This cannot get any simpler, so we are done.

What about converting from fractions to percentages? To do this, firstly work out what the fraction is as a decimal by doing whatever kind or short/long division you prefer. Then, to turn the decimal into a percentage, simply times by 100 (move the decimal point right two spaces). As you will continue to see, decimals and percentages are very commonly (and usefully) interchanged.

From here on, we will be looking at working with percentages on a calculator. Calculators allow us to do more things with percentages whilst also reducing the amount of effort needed.

Example: Work out 132 as a percentage of 480.

This question is asking what 132 is as a proportion of 480, and it wants you to represent that answer as a percentage. So, all we need to do is represent “132 out of 480” as a fraction, and then convert it to a percentage. To do this, we divide 132 by 480 and then multiplying the resulting decimal by 100:

$\dfrac{132}{480}\times 100 = 27.5\%$

Before going further, we must introduce multipliers. Any percentage value has a corresponding multiplier which we find by converting it to decimal/fraction form. So, the multiplier for 34% is 0.34 or $\frac{34}{100}$. When we want to find a percentage of some number, all we have to do is multiply that number by the percentage multiplier. For example, if we wanted to find 34% of 520 using a calculator, all we have to do is identify the multiplier to be 0.34 and multiply by it, so

$34\%\text{ of }520=0.34\times 520=176.8$.

We can extend the idea of multipliers further to include percentage increases/decreases.

Example: In 2016 Jane deposited £1,360 into her bank account. 1 year later, this amount had increased by 22%. How much did she have in her account in 2017?

So, by 2017 the new total is equal to the original total plus 22% of the original total. We could work out what 22% of 1,360 is and then add it to the original amount – and this is by no means a bad or slow method – but we’re going to look at a method that is more useful and can be even quicker.

We’ve seen that the multiplier for 22% is $\dfrac{22}{100}$ or 0.22. To find the multiplier for a 22% increase, we all we have to do is add 1 to the multiplier for 22%. We get that

$1 + 0.22 = 1.22\text{ is the multiplier for a }22\%\text{ decrease}$.

Therefore, in 2017 Jane’s bank account totalled

$1,360 \times 1.22 = \pounds 1,659.20$.

Really, this is equivalent to finding 122% of the original value. It seems contradictory to have percentages greater than 100, but it’s just the result of extending the idea further to account for all kinds of situations, including the very real situation in this example. Percentages greater than 100 can, and will, occur all over the place

What about if it had decreased by 22% instead? To find the multiplier for a 22% decrease, all we have to do is subtract the 22% multiplier from 1 (as opposed to adding it). We get that

$1 - 0.22 = 0.78\text{ is the multiplier for a }22\%\text{ decrease}$

and so, by 2017, her account would total

$1,360 \times 0.78 = \pounds 1,060.80$

In this next example, we’re going to look at reverse percentages. This will include scenarios like the one above, except you are given the outcome and have to work backwards.

Example: Rogelio’s new record for the 100m sprint is 10.8 seconds. This is 5.8% shorter than his previous record. What was his previous record (to 2dp)?

As said, we have to work backwards. Firstly, the multiplier for a 5.8% decrease is

$1 - \dfrac{5.8}{100} = 0.942$

Normally, we multiply this by the original value to get the new value, but here we know the new value. So, since 10.8 is the result of multiplying his old record by 0.942, to find his old record we must do the opposite and divide by 0.942. So, his previous record is

$10.8 \div 0.942 = 11.46\text{ seconds (2dp)}$.

## Example Questions

We’re going to split the 33% up into 30% and 3%. To find 30%, we will first find 10% and then multiply the result by 3. 10 goes into 100 ten times, so to find 10% of 180 we will divide it by 10. Doing so, we get

$10\%\text{ of }180 = 180\div 10 = 18$

Therefore, 30% of 180 is $3\times 18=54$.

Next, we will find 3% by first finding 1% and multiplying the answer by 3. To find 1% of 180, we divide it by 100:

$1\%\text{ of }180 = 180\div 10 = 1.8$

Therefore, 3% of 180 is $3\times 1.8 = 5.4$.

Finally, we add the results together, and get that

$33\%\text{ of }180 = 54+5.4=59.4$.

To find Mateo’s score as a percentage, we must turn “99 out of 150” into a fraction and convert to a percentage, i.e. we will divide 99 by 150 and then multiply by 100. So, Mateo’s score as a percentage is

$\dfrac{99}{150}\times 100 = 66\%$.

This is a reverse percentage question, so we will need to work backwards. To do so, we need first determine what the multiplier for a 56% decrease would be. We do this by writing 56% as a fraction and subtracting the result from 1:

$1-\dfrac{56}{100} = 0.44$.

So, the original value of Petra’s car must have been multiplied by 0.44 to give its new value of £10,120. Therefore, to get the original value, we must go in the opposite direction and divide the new value by 0.44. Putting this into the calculator, we get

$10,120 \div 0.44 = \pounds 23,000$

So, the original price of Petra’s car was £23,000.

Whether you are a GCSE Maths tutor in York or Leeds or even London, you will find plenty of percentage resources and worksheets on this Maths Made Easy percentage page. All questions are relevant to each major exam board including AQA, Edexcel and OCR and they have been designed with the 9-1 course in mind. For more exceptional GCSE Maths resources… think Maths Made Easy.