# Concept of Percent Change

## Percent Change

Here base is xPercentage Change (increase or decrease) in a quantity,

p% = $\frac{Change \hspace{1ex} in \hspace{1ex} quantity}{Original \hspace{1ex} quantity}$ × 100% = $\frac{y − x}{x}$ × 100%

p% = $\frac{Change \hspace{1ex} in \hspace{1ex} quantity}{Original \hspace{1ex} quantity}$ × 100% = $\frac{y − x}{x}$ × 100%

###### Q. If price of sugar increases from Rs 18/kg to Rs 30/kg, then what is the percent increase?

Explanation:

Change in price = 30 – 18 = Rs 12 (Plus sign denotes an increase)

So, Percent change = $\frac{Change \hspace{1ex} in \hspace{1ex} quantity}{Original \hspace{1ex} quantity}$ × 100%

= ($\frac{12}{18}$ × 100)% = ($\frac{2}{3}$ × 100)% = 66.67%

(Plus sign denotes a percentage increase)

So, Percent change = $\frac{Change \hspace{1ex} in \hspace{1ex} quantity}{Original \hspace{1ex} quantity}$ × 100%

= ($\frac{12}{18}$ × 100)% = ($\frac{2}{3}$ × 100)% = 66.67%

(Plus sign denotes a percentage increase)

###### Q. If price of sugar decreases from Rs 25/kg to Rs 15/kg, then what is the percent decrease?

Explanation:

Change in price = 15 – 25 = Rs -10 (Minus sign denotes a decrease)

So, Percent change = $\frac{Change \hspace{1ex} in \hspace{1ex} quantity}{Original \hspace{1ex} quantity}$ × 100%

= ($\frac{-10}{25}$ × 100)% = ($\frac{-2}{5}$ × 100)% = -40%

(Minus sign denotes a percentage decrease)

So, Percent change = $\frac{Change \hspace{1ex} in \hspace{1ex} quantity}{Original \hspace{1ex} quantity}$ × 100%

= ($\frac{-10}{25}$ × 100)% = ($\frac{-2}{5}$ × 100)% = -40%

(Minus sign denotes a percentage decrease)

## Finding new quantity

When a quantity x is increased/decreased by p%, then:

New quantity, y = Base ± Absolute change in Base

= x ± p% of x = x ± x(p/100) = x (1 ± p/100)

## Concept of Multiplying Factor

New quantity, y = x ± p% of x = x (1 ± p%)

So, Multiplying factor : (1 ± p%)

Multiplying Factor is the same as (1 + p/100) for p% increase

E.g. Multiplying Factor for 15% increase = 1.15

Multiplying Factor for 8% increase = 1.08Multiplying Factor is the same as (1 - p/100) for p% decrease

Multiplying Factor for 35% decrease = 0.65

Multiplying Factor for 5% decrease = 0.95

###### Q. If 60 is increased by 25%, find the new number.

Explanations :

Explanation 1:

Here base is the number itself, i.e. 60

So, new number = original number (i.e. our base) + Percentage increase over this base

So, new number = 60 + 25% of 60

So, new number = 60 + 15 = 75

Here base is the number itself, i.e. 60

So, new number = original number (i.e. our base) + Percentage increase over this base

So, new number = 60 + 25% of 60

So, new number = 60 + 15 = 75

Explanation 2: Using the concept of Multiplying Factors

Here base is the number itself, i.e. 60

So, new number = original number (i.e. our base) × Multiplying Factor

Multiplying Factor = 1 + p% = 1 + 25% = 1 + 0.25 = 1.25

So, new number = 60 × 1.25 = 75

Here base is the number itself, i.e. 60

So, new number = original number (i.e. our base) × Multiplying Factor

Multiplying Factor = 1 + p% = 1 + 25% = 1 + 0.25 = 1.25

So, new number = 60 × 1.25 = 75

###### Q. If 60 is decreased by 25%, find the new number.

Explanations :

Explanation 1:

Here base is the number itself, i.e. 60

So, new number = original number (i.e. our base) - Percentage decrease over this base

So, new number = 60 - 25% of 60

So, new number = 60 - 15 = 45

Here base is the number itself, i.e. 60

So, new number = original number (i.e. our base) - Percentage decrease over this base

So, new number = 60 - 25% of 60

So, new number = 60 - 15 = 45

Explanation 2: Using the concept of Multiplying Factors

Here base is the number itself, i.e. 60

So, new number = original number (i.e. our base) × Multiplying Factor

Multiplying Factor = 1 - p% = 1 - 25% = 1 - 0.25 = 0.75

So, new number = 60 × 0.75 = 45

Here base is the number itself, i.e. 60

So, new number = original number (i.e. our base) × Multiplying Factor

Multiplying Factor = 1 - p% = 1 - 25% = 1 - 0.25 = 0.75

So, new number = 60 × 0.75 = 45

###### Q. By how much percent is 36 more than 27?

Explanations :

Explanation 1:

Here base is 27

Difference = 36 – 27 = 9

Required percentage = (9/27) × 100 = 1/3 × 100 = 33.33%

Here base is 27

Difference = 36 – 27 = 9

Required percentage = (9/27) × 100 = 1/3 × 100 = 33.33%

Explanation 2: Using the concept of Multiplying Factors

Here base is 27

36/27 = 4/3 = 1.33

Multiplying Factor = 1.33, which is equivalent to 33% increase.

So, 36 is 33.33% more than 27.

Here base is 27

36/27 = 4/3 = 1.33

Multiplying Factor = 1.33, which is equivalent to 33% increase.

So, 36 is 33.33% more than 27.

###### Q. By how much percent is 40 less than 60?

Explanations :

Explanation 1:

Here base is 60

Difference = 60 – 40 = 20

Required percentage = (20/60) × 100 = 1/3 × 100 = 33.33%

So, 40 is 33.33% less than 60.

Here base is 60

Difference = 60 – 40 = 20

Required percentage = (20/60) × 100 = 1/3 × 100 = 33.33%

So, 40 is 33.33% less than 60.

Explanation 2: Using the concept of Multiplying Factors

Here base is 60

40/60 = 2/3 = 0.67

Multiplying Factor = 0.67, which is equivalent to 33% decrease.

So, 40 is 33.33% less than 60.

Here base is 60

40/60 = 2/3 = 0.67

Multiplying Factor = 0.67, which is equivalent to 33% decrease.

So, 40 is 33.33% less than 60.

###### Q. Two numbers are 12.5% and 25% more than a third number respectively. The first number must be how much percent more/less than the second number?

Explanation :

Let the third number be 100.

Then, first number = 100 + 12.5% of 100 = 112.5

And, second number = 100 + 25% of 100 = 125

Required percentage = (12.5/125) × 100 = 10% (less)

Then, first number = 100 + 12.5% of 100 = 112.5

And, second number = 100 + 25% of 100 = 125

Required percentage = (12.5/125) × 100 = 10% (less)