Percentage

• Introduction

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.

* For example

40% (read as 40 percent) is equal to 40/100 or 0.4 or 2/5 OR can say that 40 parts out of 100 parts.

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.

• Expressing a quantity as a percentage of another

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%

Example :

Express 200 gm as a percentage of 1 kg.
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
1000 gm = 100% (initial value)
200 gm = ?
Percentage = (200 ÷ 1000) × 100% = 20%

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.

Example :

If we remove 20 marbles from each bags calculate the percentage of marbles we remove from each bags.
 For bag 1 100 marbles = 100% If we remove 20 marbles from Bag 1 then 20 marbles = (20 ÷ 100) × 100% = 20% For bag 2 80 marbles = 100% If we remove 20 marbles from Bag 2 then 20 marbles = (20 ÷ 80) × 100% = 25%

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)

• Relation between, Fraction, Decimal , Percentage

* Fraction as a Percentage

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

 ¼ of pizza = ¼ × 100% = 25%

* Percentage as a fraction

 50% of pizza = 50% ÷ 100% = ½

* Decimal express as a Percentage

To convert a decimal as a percentage, multiply it by 100%
0.4 => 0.4 × 100% = 40%

* Percentage express as a Decimal

To convert a percentage as a decimal , divide it by 100%
60% => 60% ÷ 100% - 0.6

• Comparison formula

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

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 1st statement initial value is FB
for 2nd statement initial value is What's App

• Finding the whole from the given part of it

* 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

• Percentage increase or decrease

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%

• Some daily life usage of Percentage

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%

• Formulas to calculate selling price, cost price, profit, loss and percentage of profit/ loss

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