Percentages

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%