In Mathematics, percentage refers to the number or ratio expressed as a fraction of 100. The unit is given as %, which is read as 'per cent' or the abbreviation 'pct'. The word ' per cent ' means ' out of 100' or per 100.

Percentage are used to express how large or small one quantity is relative to another quantity.

It shows relative comparison between two quantities or old and new values of same quantity.

To express a quantity as a percentage of another, the two quantities must be in the same units, and we take the other quantity (initial value) to be 100%

To express X as a Percentage of Y. (here Y is initial value)

**Percentage = ( X ÷ Y) × 100%**

First of all unit should be equal that's why either convert 200 gm into kg or 1 kg into gm

1 kg = 1000 gm

200 gm = ?

Sometimes due to inconsistent usage, it is not clear from the context what a percentage is relative to and which quantity is the initial value. However, change in initial value gives different answers.

For bag 1 |
For bag 2 |

In above given example , same number of marbles (20) have been removed from both the bags even though when we convert them into percentage we can see the change in output (example 1: 20% and example 2: 25%) The change in percentage is due to difference in initial value of both the quantity (ex 1 : 100 and ex 2 : 80)

To convert a fraction as a percentage, multiply it by 100%

¼ of pizza = ¼ × 100% = 25% |

50% of pizza = 50% ÷ 100% = ½ |

To convert a decimal as a percentage, multiply it by 100%

0.4 => 0.4 × 100% = 40%

To convert a percentage as a decimal , divide it by 100%

60% => 60% ÷ 100% - 0.6

To understand various comparison relations study below given examples carefully

* Example 1 : What's App has 40% **as much users as **Facebook. If suppose Facebook has 1050 users then find out the number of users What's App has.

It means Facebook has 100% users

and what's App has 40% users

we can say that 100 % FB users = 1050

40 % What's App users = (40% ÷ 100%) × 1050 = 420 users

* Example 2 :
Facebook has 60% **more users than **What's App. If suppose What's App has 420 users then find out the number of users Facebook has.

It means What's App has 100% users

and Facebook has 100% + 60% = 160% users

So we can say that 100% What's App users = 420

160% Facebook users = (160 ÷ 100) × 420 = 672 users

(Important Note : Remember as we said earlier, the value of initial quantity is very important, and from above given all examples it gets very clear. As one can see in above given both the examples FB has 60% more users than What's App plus What's App has 420 users. Even though it turned out that the number of users FB has in both the examples are different and it is due to the change of value of initial quantity ,ex 1 : initial value 1050 and ex 2 : initial value 420)

* Example 3 : What's App has 60% **less users than **Facebook. If suppose Facebook has 1050 users then calculate the number of users What's App has.

So Facebook has 100% users

What's App has 100% - 60% = 40% users

100% Facebook users = 1050

40% What's App users = (40 ÷ 100) × 1050 = 420 users

* Example 4 :What's App has **60% less users than** FB

and Instagram has **10% more users than** What's App

Here FB = 100%

What's App = 100% - 60% = 40%

Instagram = 40% + 10% = 50%

Here, for 1^{st} statement initial value is FB

for 2^{nd} statement initial value is What's App

* Example : James ate 30% of his total sweets. If he ate 15 sweets then total how many sweets he has ?

We can say that , he ate 30% = 15 sweets

so total 100 % = (100 ÷ 30) × 15 = 50 sweets

To find the percentage increase or decrease we always take the original amount to be 100%

Example : Last month James' weight was 80 kg. This month he weight decreased by 4 kg. Calculate the percentage of his weight loss.

decrease = 4 kg

original => 80 kg = 100%

so percentage decrease => 4kg = (4 ÷ 80) × 100% = 5%

To calculate Tax, Selling price, Cost price, Discount, Loss, Profit, Percentage of profit & loss.

To find out the amount of goods tax or discount, the original selling price is taken as 100%.

If 100% = Original selling price

then given discount/tax percentage = discounted/tax amount

discounted/tax amount = (% discount/ tax ÷ 100%) × original selling price

=> Same formula applied to calculate amount of profit or loss

amount of profit /loss = (% profit/loss ÷ 100%) × Original price

Example : In Singapore goods and service tax is 7%. How much James has to pay in tax if he wants to buy TV that cost S$500

Original selling price 100% = S$500

Goods and Service Tax 7% = (7% ÷ 100%) × S$500 = S$35

**Same example in reverse order , if we want to calculate % of tax , discount or profit loss**

If James paid $35 in tax for the $500 TV set, calculate the % of tax he had to pay

Selling price $500 = 100%

Tax amount $35 = ($35 ÷ $500) × 100% = 7%

1) Selling price = Cost price + Profit

2) Profit = Selling price - Cost price

3) Cost price = Selling price + Loss

4)Loss = Cost price - Selling price

(Note : Here cost price is mostly the price shopkeeper or owner's original buying price and Selling price is the price at which they sell things to their customers & that's why the cost price is taken as 100%)

Examples :

=> If Profit is 20%, then cost price is 100% and

selling price is 100% + 20% = 120%

=> If Loss is 15%, then cost price is 100% and

Selling price is 100% - 15% = 85%

=> If Selling price is 130%, then Cost price is 100% and

Profit = 130% - 100% = 30%

=> If discount of 25% is given on Selling price then the discounted Selling price is 85% of the marked price.

=> If a shopkeeper A made a15% profit of $30 selling Tea Set. Calculate the cost price and Selling price of the Tea Set.

As we know that, for shopkeeper cost price is the original price he had paid and it is taken as 100% and profit is the amount he earned when he sold Tea Set to customer.

Profit 15% = $30

Cost price 100% = (100% ÷ 15%) × $30 = $200

Selling price = $200 + $30 = $230

So we can say that ,
Shop keeper A paid $200 to supplier for Tea Set and Sold it for $230 to customer.

=> 250 women and 130 men attended concert. Calculate how many percentage more women than men attended the concert

Women - men => 250 - 130 = 120 more women

Original ( Women) => 250 = 100%

more women => 120 = (120 ÷ 250) × 100% = 48% more