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:

**30%**  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  

**200**  =  

**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  

**680**  =  

**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 Examples

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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