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# Percentages

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Percentages are used in much the same way as decimals and fractions in that they show a link between two values. The word ‘per cent’ really just means ‘out of 100’ so we know that a percentage is always related to 100.

The percentage sign is written as %. So if we wanted to denote 50 per cent we would write 50%, which is the same as 50 out of 100. This then shows that we can very easily convert a percentage into a fraction by simply writing the number over 100. For example, 40% is equal to .

It is fairly straightforward to convert a percentage to a decimal also; we simply have to divide the percentage by 100. So if we were to change 89% to a decimal we simply must calculate . This is the same method used for any other percentage that we want to change to a decimal.

### Converting from fractions to a percentage

If we wish to convert from a fraction to a percentage, we must do a little bit more work than if we were to convert the other way around. This is because a percentage must be ‘out of 100’, therefore we must convert the fraction to have a denominator of 100 before we can change it to a percentage. This is done by the rules that we explored in the last module on equivalent fractions.

#### Example

Convert  into a percentage.

To do this we must find a fraction that is equivalent to  with a denominator of 100. This can be done by multiplying both the top and bottom of the fraction by 5:  now we can simply see that  is equal to 85% since % means ‘out of 100’.

### Converting from a decimal to a percentage

If we were to solve a problem where we had to convert from a decimal to a percentage we can simply reverse the process that was outlined earlier for converting from a percentage to a decimal. Instead of dividing the number by 100 we must simply multiply by 100.

Therefore, a decimal of 0.45 converts to a fraction by multiplying by 100, so we get 0.45 100 = 45%. The same method is used for any decimal.

#### Example

Convert the following decimals to percentages

a) 1.83   b) 0.29   c) 0.725

a) Multiplying 1.83 by 100 gives the answer of 183%. This is ok as a percentage does not have to be less than 100.

b) Multiplying 0.29 by 100 gives the answer as 29%.

c) 0.725 multiplied by 100 is equal to 72.5%. This is ok as a percentage does not have to be a whole number.

Below is a table that shows some conversions between decimals, fractions and percentages. The rules outlined above can clearly be seen.

Fraction Percentage Decimal
50% 0.5
10% 0.1
1% 0.01
85% 0.85
57.5% 0.575

Fractions shown here are not necessarily in their lowest forms.

Finding a percentage of a number

When it comes to finding a percentage of a given number or quantity, it is easiest to find a certain amount and then use this to find the percentage needed. For example, we could work out the corresponding value for 1% and then multiply this by any number to get the percentage that we need. To find 1% we must simply divide by 100.

#### Example

Find 65% of 400

To do this we must first find 1% of 400, so by dividing 400 by 100 we get 1% to be 4. Now since we require 65% we can simply times 4 by 65 to get this amount. .

Another way of finding a percentage of a quantity is to convert it into a fraction or decimal and then simply proceed as before. Using the above example again, we would see that 65% is equal to  or 0.65 and we can just multiply either of these numbers by 400 to get the answer for 65%.

e we should take note as to which one is needed. An increase should result in an answer that is more than the original value given, whereas a decrease should be less. By doing this we will stop any silly mistakes.

To do the actual conversion we need to recognise that the entire number before a conversion is equal to 100%. So if we want an increase of 10%, then we need to work out the amount for the original 100% plus a further 10%. Therefore, we need to find 110% of the number given. To do this we can simply use the rules from earlier and convert 110% to a fraction or decimal ( or 1.1) and then multiply the quantity by this.

For a decrease in the amount we again recognise that the entire original amount is worth 100% and then we subtract the decrease from this. So a decrease of 10% is the same as finding 100% minus 10% which is equal to 90%. Then again we can use the rules from earlier to convert this to  or 0.9 and multiply by the original to work out how much 90% is.

So all you need to remember is that an increase by a percentage is added to 100% and a decrease is taken away from 100%.

#### Example

How much do we need to find, as a percentage, with the following changes:

a) An increase by 45%

b) A decrease by 80%

c) A decrease by 100%

d) An increase by 150%

a) 100% plus 45% is equal to 145%

b) 100% minus 80% is 20%

c) 100% minus 100% is 0%, which will always be equal to 0

d) 100% plus 150% equals 250%

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