# 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:

1. Make an equivalent fraction and then simplify

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

Percentages to decimals:

1. 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

1. Percentage multipliers are decimals that we can multiply numbers by to find certain percentages.
2. Remembering that 100% = 1
3. Therefore to find 100% of something simply multiply by 1.
4. It follows that to find 50% you can multiply by 0.5.
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.
6. 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

1. If we want to write one quantity as a percentage of another we need to think about which quantity is equal to 100%.
2. 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..

1. Because there are 40 biscuits in total 40 = 100%

By using multiplicative reasoning, we can therefore see that 4 biscuits will represent 10%. 1. 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

1. Percentage increase problems are usually about the new amount something will be after it has been increased by a specific percentage.
2. 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?

1. This is asking us to work out £3000 + 3% of £3000 = ?
2. 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

1. If we remember think of the increase directly in terms of percentages we can see that we have gone from 100% to 103%.
2. 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

1. 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

1. Step 1: Work out how much percentage you have of the original. 100% - the percentage decrease.
2. Step 2: Divide to find 1%
3. 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?