# Percentages

## Defining Percentages

Percentage literally means per 100. A percentage is a fraction out of 100.

## Interpreting percentages and percentage change as a fraction or decimal

**Percentages to fractions:**

- Make an equivalent fraction and then simplify

Eg. 25% = 25/100 = 1/4

**Percentages to decimals:**

- Divide by the number by 100

Eg. 45% = 45100

```
= 0.45
```

**Percentage Change**

Percentage change is the amount a value has gone up or down in terms of a percentage of itself.

**Example 1**

A phone decreases in value over time. If a phone initially cost £120 and it now is only worth £100 there has been a change in its value. We want to work out the % by which the phone decreased in value.

Step 1: Find the difference in the prices: £120 - £100 = £20

Step 2: Work out the percentage the difference is of the original amount.

£20/£120 x 100 = 16.67% (to 2 dp)

This is also the case when things increase in price. An antique violin increase in price from £1000 to £1250. What is the percentage change?

Step 1: Find the difference: £1250 - £1000 = £250

Step 2: Work out this difference as a percentage of the original: £250/£1000 x 100 = 25%

## Percentage Multipliers

- Percentage multipliers are decimals that we can multiply numbers by to find certain percentages.
- Remembering that 100% = 1
- Therefore to find 100% of something simply multiply by 1.
- It follows that to find 50% you can multiply by 0.5.
- Here is a list of very useful percentage multiplies. Remember if you multiply by a number more than 100% the number will increase, if it is a percentage less than 100% it will decrease.
- So if you want a quick way to find 1% of 70, simply think 70 x0.01= 0.7.

To find % =>Decimal multiplier

1% => 0.01

5% => 0.05

10% => 0.10

25% => 0.25

50% => 0.50

100% => 1.00

120% => 1.20

## One quantity as a percentage of another

- If we want to write one quantity as a percentage of another we need to think about which quantity is equal to 100%.
- This is useful for us to compare quantities.

For example, if I have 40 biscuits and I eat 4 of them, I might want to know what percentage of biscuits I have eaten..

- Because there are 40 biscuits in total 40 = 100%

By using multiplicative reasoning, we can therefore see that 4 biscuits will represent 10%.

- Alternatively, if you cannot easily see a multiplier or divisor, think about the percentage as an equivalent fraction.

We want to know the percentage that is equal to 4/40 = ?%

Therefore 4/40 x 100 = 10%

Dividing the fraction creates a decimal, multiplying by 100 make it out of 100!

## Comparing with percentages

- Percentage increase problems are usually about the new amount something will be after it has been increased by a specific percentage.
- This is incredibly important and useful! If you ever set up a bank account, take out a credit card or loan- this is a skill that will come in handy.

**The basic steps:**

Step 1: Find the percentage you want to increase by

Step 2: Add it onto the original amount

**Example:**

A vintage car that costs £3000 will increase in value by 3% each year.

After one year, how much will the car cost?

- This is asking us to work out £3000 + 3% of £3000 = ?
- Therefore we need to find 3% of £3000 and add it to our original price.

Original price: £3000

3% of original: £90 Remember find 1% by dividing by 100 or multiplying by 0.01

£3090 is the new price after one year.

**Shortcut**

- If we remember think of the increase directly in terms of percentages we can see that we have gone from 100% to 103%.
- Therefore, if we want to find 103% of something we can convert 103% into a decimal multiplier and multiply this by the original.

103% = 1.03

- Therefore to find the above percentage increase would be £3000 x 1.03 = £3090. This is a quick way to check your answer especially if you have a calculator!

**Percentage decrease problems**

Percentage decrease problems are incredibly similar to percentage increase problems, all we do is subtract instead of add!

**The basic steps:**

Step 1: Find the percentage you want to increase by

Step 2: __Subtract __it onto the original amount

**Example:**

A new phone decreases in value by 10% after the first year. It originally cost £240, how much is it worth now?

Original - Percentage = New value

£240 - £ 24 = £216 = New value

Equally we can use a multiplier.

100% - 10% = 90% therefore in this case our multiplier will be 0.9

£240 x 0.9= £216

**Reverse percentage problems**

If we want to find the original price of something we will have to reverse the percentage change.

This is *__very* __important to help you work out how much a bargain you are getting in a sale!

**Basic steps**

- Step 1: Work out how much percentage you have of the original. 100% - the percentage decrease.
- Step 2: Divide to find 1%
- Step 3: Multiply by 100 to find 100% which will be the original value.

**Example:**

Amy managed to buy a dress for £55 that was reduced by 20% in a sale. How much did it originally cost?

We want to find the value of 100% of the dress.

At the moment we know that:

100%-20% = £55

80% = 55

Therefore to find 1% we will need to divide both sides by 80

1% = 55 80

Now to go from 1% to 100% we just need to multiply both sides by 100

100% = £68.75

- A Car decreases in value from £1250 to £1000, what is the percentage change?
- Your answer should include: 25% / 25
- There are 25 biscuits in a package, 5 are custard creams. What percentage of the biscuits are custard creams?
- Your answer should include: 20% / 20
- Increase 50 by 4% using a multiplier.
- 52
- A T-shirt is reduced by 10% in a sale and now costs £36, how much did it originally cost?
- Your answer should include: £40 / 40