## What you need to know

### Percentages

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.

In this topic we will see how you calculate percentage of an amount, percentage change, percentage increase and percentage decrease.

Covering fractions, decimals and percentage conversions will help with this topic where we will look at how to find a percentage of an amount as well as percentage increase and decrease.

Understanding this topic will help when you come to learn about reverse percentages which are considered more difficult.

### Take Note:

Percentages of an amount and percentage change (increase and decrease) can be calculated using the formulae shown.

### Example 1: Percentage of an Amount

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

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%, 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.

### Example 2: Percentage Question

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

To write a percentage as a decimal, simply divide by 100. 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 interchanged.

### Example 3: Percentage of an Amount

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.

### Example 4: Percentage Increase

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’ve seen that the multiplier for 22% is \dfrac{22}{100} or 0.22. To find the multiplier for a 22% increase, 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

### Example 5: Percentage Change

Calculate the percentage change when a car goes down in value from £8500 to £6000.

Using the equation above we can calculate the percentage change by first calculating the difference, which is £8500 – £6000 = £1500.

We then need to divide our difference by the original amount and multiply by 100 to get the percentage change

1500\div 8500 =0.1875\times100 = 18.75\%.

### Example Questions

1) Without using a calculator, work out 33% of 180.

In order to solve this question, we will need to break down the 33% into easier, and more manageable, chunks. 33% can be broken down as follows:

3 \times 10\% + 3 \times 1\%

10% is a very easy amount to calculate since all we need to do is divide by 10:

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 need to divide the total by 100:

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

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

Finally, we need to add together the 30% amount and the 3% amount, so 33% is:

54+5.4=59.4

2) Matteo scored 99 out 150 on an exam. What is his score as a percentage?

To convert anything into a percentage, it is a lot easier to write the amount as a fraction first. If Matteo scored “99 out of 150”, then we should write this as:

\dfrac{99}{150}

To convert a fraction into a percentage, you need to divide the top by the bottom (it helps if you remember that the line in a fraction means ‘divide’) and then multiply by 100.

So, Matteo’s score as a percentage can be calculated as follows:

99\div150\times 100 = 66\%

3) Petra buys a car. After 2 years of her owning it, it is now worth £10,120, which constitutes a 56% decrease from its original value. What was its original value?

If a car has decreased in value by 56%, this simply means that the car is now worth 44% of what it was worth before (100\% - 56\% = 44\%).

This means that the current value of £10,120 represents 44% of the original price.

The original price is the 100% amount, so we need to work out what 100% represents if 44% = £10,120. The easiest way to do this is to work out what 1% is:

If 44\% = \pounds10,120

then

1\% = £10,120 \div 44 = \pounds230

If 1\% = \pounds230

then

100\% = £230 \times100 = \pounds23,000

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

Another way we can view this question is by working what we would multiply the original amount by in order to work out the new value. If we are calculating a 56% decrease, we would multiply the original amount by 0.44 to work out the new value. Therefore, if we know the new amount and wish to work out the original amount, then we can simply divide the new amount by 0.44:

\pounds10,120 \div 0.44 = \pounds23,000

4) Mildred’s salary has increased from £24,600 to £25,338. By what percentage has her salary increase?

A percentage increase (or decrease) can be calculated by dividing the difference between the two amounts by the original amount and then multiplying by 100.

In this question the difference between the two salaries is \pounds25,338 - \pounds24,600 = \pounds738

The original amount (the amount before it was increased) was £24,600, so the percentage increase can be calculated as follows:

\dfrac{\pounds738}{\pounds24,600} \times 100 = 3\%

5) The price of a motorbike is reduced by 10%. In a sale, the new price is reduced further by 10%. By what percentage has the original price of the scooter been reduced in this sale?

To most people, this would appear a very easy question with an answer of 20%, but this answer is, sadly, incorrect!

The easiest thing to do to solve this question is to invent a price for the motorbike. You can invent any price you want, but it would be advisable to make the price a nice, easy number and, since this question concerns percentages, giving the motorbike a price of £100 makes life extremely easy.

If the motorbike costs £100, when if it is reduced by 10%, then its new value is £90.

If the motorbike now costs £90 and is further reduced by 10%, then we need to deduct 10% from this £90 value (and not the previous £100 value). 10% of £90 is £9, so the new value of the motorbike is £81.

So the motorbike has decreased in value from £100 to £81. Since we set the motorbike’s original price as £100, the percentage decrease here should be relatively obvious. If not, remember that to calculate a percentage decrease (or increase), you need to divide the difference between the two values by the original value and multiply by 100.

The original value of the motorbike was £100, and its new value is £81, so the percentage decrease can be calculated as follows:

\dfrac{\pounds100 - \pounds81}{\pounds 100} \times 100 = 19\%

Therefore, the motorbike has decreased in value by 19% and not 20%.

### Worksheets and Exam Questions

#### (NEW) Percentages Exam Style Questions - MME

Level 4-5#### Ratio and Percentages

Level 4-5#### Fractions of amounts and percentages

Level 1-3### Videos

#### Percentages Q1

GCSE MATHS#### Percentages Q2

GCSE MATHS#### Percentages Q3

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