Percentage

Percentage

Concept Explanation

Percentage is just a way of expressing a fraction with a denominator of 100. Instead of saying “35 out of every 100 things,” we say 35%. This makes comparisons easier because everything is on the same scale — you’re always talking about parts of one whole, expressed as hundredths.

The key mental shift is that “percent” and “fraction” are two faces of the same coin. When you see 15%, think 15/100 or 0.15. Going the other way, the fraction 3/4 becomes 75% because 3 divided by 4 equals 0.75, and 0.75 times 100 gives you 75. Once this clicks, percentage problems become straightforward multiplication and division.

Percentage change comes up constantly in real life — salary hikes, price reductions, marks improvement. The formula measures how much something has grown or shrunk relative to where it started. The denominator is always the original value, which is why people sometimes forget to use the old number as the base.

Key Formulas

Symbol	Meaning
X% of Y	Y × X/100
X as % of Y	(X/Y) × 100
% change	((New − Old)/Old) × 100
A% increase then B% decrease	Net = (1 + A/100)(1 − B/100)

Step-by-Step Example

Q: A shirt costs Rs. 800. Its price is first increased by 15%, then decreased by 10%. What is the final price?

Step 1: Apply 15% increase: 800 × (115/100) = 920

Step 2: Apply 10% decrease on 920: 920 × (90/100) = 828

Answer: Rs. 828

Common Mistakes

• Confusing original value with intermediate value → Always use the immediate previous value as base for sequential percentage changes
• Converting X% to decimal incorrectly (25% = 0.25, not 0.025) → Remember: percent means divide by 100

Quick Test (2 Qs)

1. Q: If the population of a town grows from 50,000 to 60,000, what is the percentage increase? Options: 10%, 15%, 20%, 25%. Ans: 20% (Reason: (60000-50000)/50000 × 100 = 20%)
2. Q: 40% of a number is 120. What is 60% of the same number? Options: 160, 170, 180, 200. Ans: 180 (Reason: Number = 120 × 100/40 = 300; 60% of 300 = 180)