# 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 &nbsp100%.

Say you had &nbsp56&nbsp biscuits in a tin,

then &nbsp100%&nbsp of that tin is &nbsp56&nbsp 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% &nbsp=&nbsp \bf{\frac{20}{100}} &nbsp &nbsp , &nbsp &nbsp 70% &nbsp=&nbsp \bf{\frac{70}{100}} &nbsp &nbsp , &nbsp &nbsp 34% &nbsp=&nbsp \bf{\frac{34}{100}} &nbsp &nbsp , &nbsp &nbsp 9% &nbsp=&nbsp \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}} &nbsp=&nbsp 0.2 &nbsp &nbsp , &nbsp &nbsp \bf{\frac{70}{100}} &nbsp=&nbsp 0.7 &nbsp &nbsp , &nbsp &nbsp \bf{\frac{34}{100}} &nbsp=&nbsp 0.34 &nbsp &nbsp , &nbsp &nbsp \bf{\frac{9}{100}} &nbsp=&nbsp 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 &nbsp300:

30%&nbsp of &nbsp300 &nbsp = &nbsp \bf{\frac{20}{100}} × 300 &nbsp = &nbsp \bf{\frac{6000}{100}} &nbsp = &nbsp60 &nbsp &nbsp or &nbsp &nbsp 0.2 × 300 &nbsp=&nbsp 60

34%&nbsp of &nbsp300 &nbsp = &nbsp 0.34 × 300 &nbsp=&nbsp 102

9%&nbsp of &nbsp200 &nbsp = &nbsp 0.09 × 300 &nbsp=&nbsp 27

Examples &nbsp &nbsp

(1.1)&nbsp

In a shoe shop a pair of boots has &nbsp32%&nbsp reduced from the usual price of &nbsp$80. What amount was taken off the price of &nbsp$80?

Solution&nbsp &nbsp

32%&nbsp of &nbsp680 &nbsp = &nbsp 0.32 × 80 &nbsp=&nbsp 25.6

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

(1.2)&nbsp

A motorbike is initially worth &nbsp$14’000,&nbsp then lowers in value by &nbsp18%&nbsp over 3 years. How much is the motorbike worth after &nbsp3&nbsp years? Solution&nbsp &nbsp A reduction of &nbsp18%,&nbsp means that the bike will be &nbsp82%&nbsp of the initial value. 0.82 × 14’000 &nbsp=&nbsp 11’480 After &nbsp3&nbsp years the motorbike is worth a value of &nbsp$11’480.

(1.3)&nbsp

A painting bought for &nbsp$16’000,&nbsp has increased in value since it was purchased by &nbsp>12%,&nbsp what is the new value? Solution So it’s the original piano price,&nbsp 100%,&nbsp plus an additional &nbsp12%&nbsp of the original price. 100% &nbsp=&nbsp \bf{\frac{100}{100}} &nbsp&nbsp , &nbsp&nbsp 12% &nbsp=&nbsp \bf{\frac{12}{100}} 112% &nbsp=&nbsp 100% + 12% &nbsp=&nbsp \bf{\frac{100}{100}} + \bf{\frac{12}{100}} &nbsp=&nbsp \bf{\frac{112}{100}} &nbsp=&nbsp 1.12 1.12 × 16’000 &nbsp=&nbsp 17’920 The new amount the painting is valued at is &nbsp$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 &nbsp &nbsp

(2.1)&nbsp

A clothes shop raises the price of a jumper by &nbsp20%.&nbsp The new price is &nbsp$48. What was the price of the jumper before the raise? Solution&nbsp &nbsp The fraction for adding an extra &nbsp20%&nbsp on is &nbsp\bf{\frac{120}{100}}. Flipping this fraction gives &nbsp\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 &nbsp=&nbsp 40 The jumper was &nbsp$40&nbsp before the clothes shop raised the price by &nbsp20%.

(2.2)&nbsp

The clothes shop also has a jacket which they have decreased the price of by &nbsp18%.&nbsp The new price is &nbsp$88.56. What was the original price of this TV before the decrease? Solution&nbsp &nbsp The fraction for taking an extra &nbsp18%&nbsp off is &nbsp\bf{\frac{82}{100}}. Flipping this fraction gives &nbsp\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 &nbsp=&nbsp 108 The jacket was &nbsp$108&nbsp before the clothes shop decreased the price by &nbsp18%.

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