Percentage: Basic Concepts, Formulas

What do you mean by Percentage?

• Percent is derived from the Latin word ‘per centum’ which means ‘per hundred.
• A percentage is a way of comparing quantities.
• Percentages are basically numerators of fractions with denominator 100.
• Percent is represented by the symbol – ‘%’.

Ex- 1% means 1 out of 100 or one-hundredths.

1% = 1/100 = 0.01

Interpreting Percentages

Example – What do you mean by this statement.

Let say – “Raj saves 25% of his income.

Interpretation – It means 25 parts out of 100, or it means Raj is saving 25 rupees out of every 100 rupees that he earns.

Important Formulas

1. % Change – (Increase or decrease)

℅change = \frac { \left( Final\quad Value-Initial\quad Value \right) }{ Initial\quad Value } \times 100

• Remember Percentage change is always calculated wrt the initial value. So, take the initial value as the reference value for finding %change.
• Note- Always take ‘+’ for an increase and ‘-‘ for a decrease.

2. If the price of an article increases or decreases by ‘x℅’, then the reduction in consumption, and expenditure remaining same.

\frac { \left( 100\times x \right) }{ 100+x } %

3. If the price of an article decreases by ‘x℅’, then the reduction in consumption, and expenditure remaining same

\frac { \left( 100\times x \right) }{ 100-x } %

4. If the population of a town is ‘P’ and the annual increase is x%, then population after ‘t’ years.

P{ \left( 1+\frac { r }{ 100 } \right) }^{ t }

5. If the population of a town is ‘P’ and the annual decrease is x%, then population after ‘t’ years.

P{ \left( 1-\frac { r }{ 100 } \right) }^{ t }

6. If first value is ‘x%’ more than the 2nd value, then the 2nd value –

\frac { 100x }{ 100+x } % Less than the first value.

7. If the first value is ‘x%’ less than the 2nd value, then the 2nd value –

\frac { 100x }{ 100-x } % more than the first value

8. If a number is first increased by x℅ and then again decreased by x% then the net result will be – 

\frac { { x }^{ 2 } }{ 100 } % decrease (always)

9. If a number is first increased by x℅ and then again decreased by y% then the net result will be –

(x-y-\frac { xy }{ 100 }) % decrease or increase, depending upon the sign.

10. If a number is successively increased by x% and y% then the net increase is –

(x+y+\frac { xy }{ 100 }) %

11. If the length of a rectangle is increased by x% and its breadth is increased by y%, then the change in its area –

(x+y+\frac { xy }{ 100 }) %

(if the result is +ve then increase and if -ve then decrease)