# How to Work out Percentages

Knowing how to work out percentages is an important thing to be able to do in Statistics and Math in general.
A good starting point with percentages is to have a solid understanding of both  fractions  and  decimals.

The main logic of percentages in Math shouldn’t be hard to get your head around with a good handling of those topics.

With percentages, an entire amount of a group or quantity is classed as  100%.

Say you had  56  biscuits in a tin,

then  100%  of that tin is  56  biscuits, the whole amount of biscuits in the tin.

## Percentages as Fractions and Decimals

One way of representing percentages is as a fraction.

For example.

20%  =  \bf{\frac{20}{100}}     ,     70%  =  \bf{\frac{70}{100}}     ,     34%  =  \bf{\frac{34}{100}}     ,     9%  =  \bf{\frac{9}{100}}

There is also a decimal form for such fractions, and the decimal form can also be used to represent percentages.

\bf{\frac{20}{100}}  =  0.2     ,     \bf{\frac{70}{100}}  =  0.7     ,     \bf{\frac{34}{100}}  =  0.34     ,     \bf{\frac{9}{100}}  =  0.09

### How to Work out Percentages:

When we want to establish a specific percentage of an initial quantity, we look to multiply the quantity by the fraction or the decimal form of the percentage.

If we wanted to find percentages of the number  300:

20%  of  300   =   \bf{\frac{20}{100}} × 300   =   \bf{\frac{6000}{100}}   =  60     or     0.2 × 300  =  60

34%  of  300   =   0.34 × 300  =  102

9%  of  300   =   0.09 × 300  =  27

Examples

(1.1)

In a shoe shop a pair of boots has  32%  reduced from the usual price of  $80. What amount was taken off the price of$80?

Solution

32%  of  80   =   0.32 × 80  =  25.6

$25.60 was reduced from the usual price of$80.

(1.2)

A motorbike is initially worth  $14’000, then lowers in value by 18% over 3 years. How much is the motorbike worth after 3 years? Solution A reduction of 18%, means that the bike will be 82% of the initial value. 0.82 × 14’000 = 11’480 After 3 years the motorbike is worth a value of$11’480.

(1.3)

A painting bought for  $16’000, has increased in value since it was purchased by 12%, what is the new value? Solution So it’s the original piano price, 100%, plus an additional 12% of the original price. 100% = \bf{\frac{100}{100}} , 12% = \bf{\frac{12}{100}} 112% = 100% + 12% = \bf{\frac{100}{100}} + \bf{\frac{12}{100}} = \bf{\frac{112}{100}} = 1.12 1.12 × 16’000 = 17’920 The new amount the painting is valued at is$17’920.

## How to Work out Percentages,Further

When learning how to work out percentages in Math, it’s handy to know how to go the other way with percentage amounts.

That is, when you know the newer value of say a car or house, but you would like to work out the original value before a percentage increase or decrease.

When this is the case, what you want to do is to flip upside down the percentage fraction that would’ve been used on the original value to give the new amount.

Then multiply the new amount you know by this new flipped fraction.

Examples

(2.1)

A clothes shop raises the price of a jumper by  20%.  The new price is  $48. What was the price of the jumper before the raise? Solution The fraction for adding an extra 20% on is \bf{\frac{120}{100}}. Flipping this fraction gives \bf{\frac{100}{120}}. Now we multiply this fraction by the raised value of the jumper, to obtain the original price. \bf{\frac{100}{120}} × 48 = 40 The jumper was$40  before the clothes shop raised the price by  20%.

(2.2)

The clothes shop also has a jacket which they have decreased the price of by  18%.  The new price is  $88.56. What was the original price of this TV before the decrease? Solution The fraction for taking an extra 18% off is \bf{\frac{82}{100}}. Flipping this fraction gives \bf{\frac{100}{82}}. Again we can multiply this fraction by the new lower price of thee jacket, to get the price before the reduction. \bf{\frac{100}{82}} × 88.56 = 108 The jacket was$108  before the clothes shop decreased the price by  18%.

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