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. In the exam, you can expect 2–4 direct questions on percentage, plus another 5–8 questions where percentage is embedded in other topics.

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

1. **Basic percentage**: x% = x/100 2. **Percentage of N**: x% of N = (x/100) × N 3. **Percentage change**: [(New Value − Old Value)/Old Value] × 100 4. **Percentage increase**: New Value = Old Value × (1 + increase%/100) 5. **Percentage decrease**: New Value = Old Value × (1 − decrease%/100) 6. **Successive changes**: Net% = a + b + (ab/100) for changes a% and b% 7. **Three successive changes**: Net% = a + b + c + (ab + bc + ca)/100 + abc/10000 8. **Reverse calculation after increase**: Original = Final/(1 + x/100) 9. **Reverse calculation after decrease**: Original = Final/(1 − x/100) 10. **Fraction to percentage**: Multiply by 100 and add % symbol 11. **Percentage to decimal**: Divide by 100 12. **If A is x% more than B**: A = B(1 + x/100), so B = A/(1 + x/100) = A × 100/(100 + x) 13. **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

1. **Successive percentages assumed additive** → Wrong: 10% increase then 10% decrease = 0% change. Correct: Net = 10 + (−10) + (10 × −10)/100 = −1%. Always use the formula.

2. **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.

3. **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.

4. **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.

5. **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)**