Percentages: Complete Master Guide for UPSC/MPSC CSAT (English)

Introduction and Importance of Percentages

The term ‘Percent’ is derived from the Latin word ‘per centum’, which translates to ‘by the hundred’. In mathematics, a percentage is a number or ratio expressed as a fraction of 100. It is universally denoted using the percent sign, “%”. For instance, when we say 45%, it essentially means 45 per 100, or the fraction 45/100, which can also be written in decimal form as 0.45.

For aspirants preparing for competitive examinations such as UPSC Civil Services (CSAT) and MPSC, mastering percentages is not just a choice, but a necessity. The concept of percentage acts as the backbone for a plethora of other mathematical topics. Without a solid grip on percentages, one would find it extremely difficult to tackle problems in Profit and Loss, Simple and Compound Interest, Partnership, Mixtures and Alligations, and especially Data Interpretation (DI), which frequently features questions involving percentage changes, growth rates, and comparisons.

Core Concepts, Formulas, and Quick Tricks

Understanding the core concepts rather than memorizing formulas is the key to solving percentage problems quickly and accurately.

1. Basic Formulas

• To express x as a percentage of y: (x / y) × 100%
• To find x% of y: (x / 100) × y
• Percentage Increase: (Increase in Value / Original Value) × 100%
• Percentage Decrease: (Decrease in Value / Original Value) × 100%

2. Successive Percentage Change

If a value undergoes two successive changes of x% and y%, the net percentage change is given by the formula:

Net Change = x + y + (xy / 100) %

Note: Use a positive sign for an increase and a negative sign for a decrease. If the final result is positive, it indicates a net increase; if negative, a net decrease.

3. Fraction to Percentage Conversion

One of the most powerful tricks to solve percentage questions in seconds is memorizing the fraction-to-percentage equivalent table. Here are the most commonly used fractions:

• 1 = 100%
• 1/2 = 50%
• 1/3 = 33.33%
• 1/4 = 25%
• 1/5 = 20%
• 1/6 = 16.66%
• 1/7 = 14.28%
• 1/8 = 12.5%
• 1/9 = 11.11%
• 1/10 = 10%
• 1/11 = 9.09%
• 1/12 = 8.33%

4. The A = B Concept

If A is x% more than B, then B is less than A by: [x / (100 + x)] × 100%

If A is x% less than B, then B is more than A by: [x / (100 – x)] × 100%

Solved Examples with Step-by-Step Explanations

Example 1: Basic Percentage Calculation

Question: In a class of 80 students, 60% are girls. How many boys are there in the class?

Solution:

Step 1: Understand that the total percentage of students is 100%.

Step 2: If girls are 60%, then the percentage of boys = 100% – 60% = 40%.

Step 3: Calculate 40% of the total number of students (80).

Step 4: Number of boys = (40 / 100) × 80 = 32.

Answer: There are 32 boys in the class.

Example 2: Percentage Increase/Decrease

Question: The price of petrol increases from ₹80 per liter to ₹96 per liter. What is the percentage increase in the price?

Solution:

Step 1: Find the actual increase in price. Increase = New Price – Original Price = 96 – 80 = ₹16.

Step 2: Use the percentage increase formula: (Increase / Original Value) × 100.

Step 3: Percentage Increase = (16 / 80) × 100 = (1/5) × 100 = 20%.

Answer: The price of petrol increased by 20%.

Example 3: Successive Changes

Question: The population of a town increased by 10% in the first year and then decreased by 5% in the second year. What is the net percentage change in the population?

Solution:

Step 1: Identify x and y. Here, x = +10 (increase) and y = -5 (decrease).

Step 2: Apply the successive change formula: Net Change = x + y + (xy / 100).

Step 3: Net Change = 10 – 5 + ((10 × -5) / 100).

Step 4: Net Change = 5 – (50 / 100) = 5 – 0.5 = +4.5%.

Answer: The net population increased by 4.5%.

Example 4: The A = B Concept (Consumption and Expenditure)

Question: If the price of sugar increases by 25%, by what percentage must a household reduce its consumption so that the expenditure on sugar remains the same?

Solution:

Step 1: Recognize the formula. Reduction % = [x / (100 + x)] × 100.

Step 2: Here, x = 25.

Step 3: Reduction % = [25 / (100 + 25)] × 100 = (25 / 125) × 100 = (1/5) × 100 = 20%.

Answer: The household must reduce its consumption by 20%.

Pro-tips to Avoid Common Mistakes

• Base Value Confusion: Always be extremely careful about the ‘base’ value. When asked “A is what percent of B?”, the base is B. The formula is (A/B) × 100. A common mistake is taking A as the denominator.
• Percentage Change Base: For percentage increase or decrease, the denominator must ALWAYS be the initial/original value, never the final value.
• Successive vs. Simple Addition: A 10% increase followed by a 10% decrease does NOT mean the net change is 0%. Applying the successive formula (10 – 10 – 1) gives a 1% decrease.
• Don’t overcalculate: Use fraction equivalents instead of calculating percentages manually. For example, to find 12.5% of 640, do not multiply 12.5 by 640 and divide by 100. Instead, simply use the fraction 1/8. (1/8) × 640 = 80.

Practice Questions

1. If 20% of a number is 45, what is 80% of that number?
2. In an examination, a candidate needs 40% marks to pass. If a candidate scores 220 marks and fails by 20 marks, what are the maximum marks of the examination?
3. The price of an article is first increased by 20% and then decreased by 25%. What is the overall percentage change in its price?
4. If A’s salary is 20% less than B’s salary, by what percentage is B’s salary more than A’s?
5. A student multiplies a number by 3/5 instead of 5/3. What is the percentage error in the calculation?