Important Formula Percentages - CSAT Preparation - UPSC PDF Download

It is represented by the % symbol. In all fractions where the denominator is 100, we remove the denominator and put the % sign.

Calculating the Percentage of a Number

To calculate the percentage of a number, we use the formula:

P% of Number = X

where X is the required percentage value.

If we remove the % sign, the formula becomes:

P / 100 × Number = X

Let us understand this with some examples.

Example: Calculate 10% of 80.
Solution: Let 10% of 80 = X

Formula: P / 100 × Number = X

So, 10 / 100 × 80 = X

X = 8

Example: The number 450 is what percentage of 1200?
Solution:

Percentage = (450 / 1200) × 100

By simplifying the fraction, we get

= (3 / 8) × 100

= 0.375 × 100

= 37.5%

So, the number 450 is 37.5% of 1200.

Example: Convert 0.32 to percentage.
Solution: Multiplying 0.32 by 100, we get 0.32 × 100 = 32%.

Example: Convert the fraction 23/25 to percentage.
Solution: Converting 23/25 to decimal, we get 0.92.

So, 0.92 × 100 = 92%.

Percentage Formulas

(i) Percentage Increase

The percentage increase is equal to the difference between the new number and the original number, divided by the original number and multiplied by 100.

% increase = [(New number – Original number) / Original number] × 100

where increase in number = New number – Original number

(ii) Percentage Decrease

Similarly, percentage decrease is the difference between the original number and the new number, divided by the original number and multiplied by 100.

% decrease = [(Original number – New number) / Original number] × 100

where decrease in number = Original number – New number

(iii) Percentage Formula in terms of a Fraction

To convert a fraction into a percentage, divide the numerator by the denominator and multiply the result by 100%.

(iv) Percentage Change Formula

The percentage change, which represents increase or decrease in any quantity, is given by the formula:

How to Convert Fractions to Percentages?

A fraction represented by x/y can be converted into percentage using the formula: (x / y) × 100

Therefore, to convert a fraction to percentage, multiply the fraction by 100.

Steps to Convert a Fraction into a Percentage

Step 1: Convert the Fraction to a Decimal

Divide the numerator (top number) by the denominator (bottom number).
Example: 3 ÷ 4 = 0.75

Step 2: Multiply the Decimal by 100
0.75 × 100 = 75

Step 3: Add the Percent Symbol
The final answer is 75%

Note: To convert percentages into fractions, divide by 100.
Example: 25% = 25/100 = ¼
To convert a fraction into percentage, multiply it by 100.

Percentage Table

Some common fractions and their percentage equivalents are given below.

Solved Examples

Question 1: Peaches are now 50% more expensive at Reliance Smart Point. By what percentage must consumption be reduced so that total expenditure on peaches remains unchanged?
(a) 35% 
(b) 35.50% 
(c) 33.33% 
(d) 32.50%

View Answer  

Correct option: C (33.33%)
Required reduction = [ r / (100 + r) ] × 100%
= 50 / 150 × 100 = 33⅓ %.

Question 2: Jason Statham used to save 10% of his income. Two years later his income rose by 20%, yet the rupee amount he saved stayed the same. By what percentage did his expenditure rise?
(a) 22 2⁄9% 
(b) 21 5⁄7%
(c) 22 1⁄4% 
(d) 23 1⁄9%

View Answer  

Correct option: A (22 2⁄9%)
Old income = 100 ⇒ savings = 10 ⇒ expenditure = 90.
New income = 120 ⇒ savings still = 10 ⇒ expenditure = 110.
Increase = 20; percentage increase = 20 / 90 × 100 = 22 2⁄9%.

Question 3: To pass the NDA Recruitment Test a candidate must score 290 marks in aggregate. A candidate who scored 203 marks was declared unsuccessful as he fell short by 12% of the maximum aggregate. What is the maximum aggregate?
(a) 750 
(b) 725
(c) 700
(d) 675

View Answer  

Correct option: B (725)
Let maximum aggregate = x.
203 + 0.12x = 290 ⇒ 0.12x = 87 ⇒ x = 87 / 0.12 = 725.

Question 4: Robert Pattinson spends 12% of his monthly income on electricity, 25% on food, 20% on his children’s education and 30% on rent. The remaining ₹1,040 is spent on other items. What is his monthly income?
(a) ₹7,500
(b) ₹7,700
(c) ₹7,900
(d) ₹8,000

View Answer  

Correct option: D (₹8,000)
Percentage already allocated = 12 + 25 + 20 + 30 = 87%.
Remaining = 13% = ₹1,040.
Income = 1,040 / 0.13 = ₹8,000.

Question 5: Yasir’s income is 20% less than Xavier’s. By what percentage does Yasir’s salary fall short of Xavier’s?

(a) 20% 
(b) 15.50% 
(c) 14.45% 
(d) 13.45%

View Answer  

Correct option: A (20%)
Assume Yasir’s income = 100. For Yasir to be “20% less”, Xavier’s income must be 120.
Shortfall = 120 − 100 = 20.
Percentage shortfall relative to Xavier = 20 / 100 × 100 = 20%