Study Notes: Percentage
Overview
Percentage is one of the most fundamental and high-yield topics in SSC CHSL Tier 1 quantitative aptitude. Nearly every arithmetic sub-topic—profit and loss, simple and compound interest, ratio and proportion—builds upon percentage concepts.
Mastery of percentage means two things: (1) lightning-fast conversion between fractions, decimals and percentages, and (2) the ability to apply percentage increase/decrease formulas to real-world situations like price changes, population growth, salary hikes and exam score analysis. Students who can mentally compute common percentage values (like 12.5% = 1/8 or 16⅔% = 1/6) gain a significant speed advantage.
The topic is entirely formula-driven with minimal theory. Focus on recognizing problem patterns—successive percentage changes, percentage point differences, and reverse percentage calculations are the three most commonly tested variations beyond basic conversion.
Key Concepts
- Percentage means "per hundred": 45% = 45/100 = 0.45. It is a way of expressing a fraction with denominator 100.
- Conversion is bidirectional: To convert a fraction to percentage, multiply by 100. To convert percentage to fraction, divide by 100 and simplify.
- Percentage of a quantity: To find x% of N, compute (x/100) × N. This is the most common calculation in every problem.
- Percentage increase/decrease: If a value changes from A to B, percentage change = [(B − A)/A] × 100. Positive means increase, negative means decrease.
- Successive percentage changes do not add: If a value increases by 10% then decreases by 10%, the net effect is NOT zero. Use the formula: Net% = a + b + (ab/100) where a and b are the two percentage changes.
- Percentage point vs percentage change: If an exam pass rate rises from 40% to 50%, the percentage point increase is 10, but the percentage increase is (10/40) × 100 = 25%.
- Reverse percentage: If the final value after x% increase is known, the original value = Final/(1 + x/100). For decrease, use Final/(1 − x/100).
- Common fraction–percentage pairs speed up calculation: Memorize 1/2 = 50%, 1/3 = 33⅓%, 1/4 = 25%, 1/5 = 20%, 1/6 = 16⅔%, 1/8 = 12.5%, 1/10 = 10%, 3/4 = 75%, 2/5 = 40%, 3/5 = 60%.
Formulas / Key Facts
- Basic percentage: x% = x/100
- Percentage of N: x% of N = (x/100) × N
- Percentage change: [(New Value − Old Value)/Old Value] × 100
- Percentage increase: New Value = Old Value × (1 + increase%/100)
- Percentage decrease: New Value = Old Value × (1 − decrease%/100)
- Successive changes: Net% = a + b + (ab/100) for changes a% and b%
- Three successive changes: Net% = a + b + c + (ab + bc + ca)/100 + abc/10000
- Reverse calculation after increase: Original = Final/(1 + x/100)
- Reverse calculation after decrease: Original = Final/(1 − x/100)
- Fraction to percentage: Multiply by 100 and add % symbol
- Percentage to decimal: Divide by 100
- If A is x% more than B: A = B(1 + x/100), so B = A/(1 + x/100) = A × 100/(100 + x)
- If A is x% less than B: A = B(1 − x/100), so B = A/(1 − x/100) = A × 100/(100 − x)
Worked Examples
Example 1: Basic percentage of a quantity Question: What is 35% of 840? Solution: 35% of 840 = (35/100) × 840 = 35 × 8.4 = 294 Answer: 294
Example 2: Percentage increase Question: The price of rice increases from ₹40/kg to ₹50/kg. Find the percentage increase. Solution: Percentage increase = [(New − Old)/Old] × 100 = [(50 − 40)/40] × 100 = (10/40) × 100 = 25% Answer: 25%
Example 3: Successive percentage changes Question: A number is first increased by 20%, then decreased by 20%. What is the net percentage change? Solution: Using formula: Net% = a + b + (ab/100) Here a = +20, b = −20 Net% = 20 + (−20) + (20 × (−20))/100 = 0 − 400/100 = 0 − 4 = −4% Answer: 4% decrease (net loss)
Example 4: Reverse percentage Question: After a 25% increase, the price of an item becomes ₹750. What was the original price? Solution: Let original price = P P × (1 + 25/100) = 750 P × 1.25 = 750 P = 750/1.25 = 750 × (4/5) = 600 Answer: ₹600
Example 5: Comparing percentage and percentage points Question: If pass percentage increases from 60% to 75%, find (a) percentage point increase, (b) percentage increase. Solution: (a) Percentage point increase = 75 − 60 = 15 points (b) Percentage increase = [(75 − 60)/60] × 100 = (15/60) × 100 = 25% Answer: 15 percentage points; 25% increase
Common Mistakes
- Successive percentages assumed additive → Wrong: 10% increase then 10% decrease = 0% change. Correct: Net = 10 + (−10) + (10 × −10)/100 = −1%. Always use the formula.
- Confusing "x% more than" with "x% of" → Wrong: "A is 25% more than B" does NOT mean A = 25% of B. Correct: A = B + 25% of B = 1.25B.
- Reversing percentage without adjusting the base → Wrong: If salary increased 20% to ₹6000, original = 6000 − 20% of 6000. Correct: Original = 6000/1.20 = 5000. The 20% was applied on the original, not the new amount.
- Percentage point vs percentage confusion → Wrong: Stating "interest rate rose from 5% to 6%, a 1% increase." Correct: It's a 1 percentage point increase, but a 20% increase in the rate itself.
- Rounding too early in multi-step problems → Wrong: Converting 2/7 to 28% early and using 28 in formulas. Correct: Keep fractions or use at least two decimals (28.57%) until the final step to avoid cumulative rounding errors.
Quick Reference
- 50% = 1/2; 33⅓% = 1/3; 25% = 1/4; 20% = 1/5; 16⅔% = 1/6; 12.5% = 1/8
- Percentage change = [(New − Old)/Old] × 100
- Successive changes: Net = a + b + ab/100 (signs matter: increase +, decrease −)
- To remove x% increase: Divide by (1 + x/100)
- To remove x% decrease: Divide by (1 − x/100)
- Speed trick: 10% of any number = shift decimal left one place (10% of 340 = 34)