Percentage — Study Notes for SSC GD

Overview

Percentage is one of the most scoring and frequently tested topics in the SSC GD Elementary Mathematics section. The word "per cent" literally means "per hundred" or "out of 100", making it a standardised way to compare fractions and express proportions. You can expect 2–4 direct questions on percentage in your exam, and many other topics like Profit & Loss, Simple Interest, and Ratio & Proportion build directly on percentage concepts.

Mastery of percentage requires two core skills: converting between fractions, decimals and percentages fluently, and solving word problems involving percentage increase, decrease, and successive changes. The questions are usually straightforward if you know the formulas and avoid calculation errors. Speed and accuracy matter — practice mental conversion of common fractions (like 1/4 = 25%, 1/5 = 20%) to save precious seconds during the exam.

This topic rewards methodical practice. Once you understand the base formula and a few standard problem types, you can tackle almost any percentage question confidently.

Key Concepts

• **Definition**: x% means x/100. To convert a percentage to a fraction or decimal, divide by 100. To convert a fraction or decimal to percentage, multiply by 100.
• **Base Value**: The quantity on which percentage is calculated is called the base or reference value. "20% of 500" means base = 500. Always identify the base correctly.
• **Percentage Increase/Decrease**: If a value V increases by r%, the new value = V × (1 + r/100). If it decreases by r%, new value = V × (1 - r/100).
• **Percentage Change Formula**: Percentage change = [(New Value - Original Value) / Original Value] × 100. Positive result = increase, negative = decrease.
• **Successive Percentage Changes**: When two percentage changes happen one after another, you cannot simply add them. Use the formula: Net effect = a + b + (ab/100), where a and b are the two percentage changes (use negative sign for decrease).
• **Reverse Percentage**: If final value after a% increase is F, then original value = F / (1 + a/100). Similarly for decrease, original = F / (1 - a/100).
• **Percentage to Fraction Shortcuts**: Memorise common conversions: 50% = 1/2, 25% = 1/4, 20% = 1/5, 12.5% = 1/8, 10% = 1/10, 33.33% = 1/3, 66.66% = 2/3.

Formulas / Key Facts

1. **Basic Conversion**: x% = x/100 2. **Finding Percentage of a Quantity**: x% of N = (x/100) × N 3. **Finding What Percentage**: If A is what percent of B, then percentage = (A/B) × 100 4. **Percentage Increase**: New Value = Original × (100 + Increase%)/100 5. **Percentage Decrease**: New Value = Original × (100 - Decrease%)/100 6. **Percentage Change**: [(Change in Value) / Original Value] × 100 7. **Successive Changes**: If first change is a% and second is b%, net change = a + b + (ab/100) 8. **Reverse Calculation (Increase)**: If new value after r% increase is N, original = N × 100/(100 + r) 9. **Reverse Calculation (Decrease)**: If new value after r% decrease is N, original = N × 100/(100 - r) 10. **When Base Changes**: Always recalculate from the new base, not the original.

Worked Examples

**Example 1: Basic Conversion and Calculation**

*Question*: What is 15% of 240?

*Solution*: Step 1: Write the formula: x% of N = (x/100) × N Step 2: Substitute values: 15% of 240 = (15/100) × 240 Step 3: Simplify: = 15 × 240/100 = 15 × 2.4 = 36 **Answer**: 36

**Example 2: Percentage Increase**

*Question*: The population of a town is 50,000. If it increases by 20%, what is the new population?

*Solution*: Step 1: Original population = 50,000 Step 2: Increase = 20% of 50,000 = (20/100) × 50,000 = 10,000 Step 3: New population = 50,000 + 10,000 = 60,000 *Alternate method*: New value = 50,000 × (100 + 20)/100 = 50,000 × 1.2 = 60,000 **Answer**: 60,000

**Example 3: Successive Percentage Changes**

*Question*: A number is increased by 10% and then decreased by 10%. What is the net percentage change?

*Solution*: Step 1: Let original number = 100 (convenient choice) Step 2: After 10% increase: 100 × 1.1 = 110 Step 3: After 10% decrease on 110: 110 × 0.9 = 99 Step 4: Net change = 99 - 100 = -1 (decrease of 1 out of 100) Step 5: Percentage change = -1% *Formula method*: a = +10, b = -10 Net = 10 + (-10) + (10 × -10)/100 = 0 - 1 = -1% **Answer**: 1% decrease

**Example 4: Reverse Percentage**

*Question*: After a 25% increase, a salary becomes ₹15,000. What was the original salary?

*Solution*: Step 1: Let original salary = x Step 2: After 25% increase: x × (125/100) = 15,000 Step 3: x × 1.25 = 15,000 Step 4: x = 15,000 / 1.25 = 12,000 *Formula method*: Original = 15,000 × 100/125 = 15,000 × 4/5 = 12,000 **Answer**: ₹12,000

Common Mistakes

1. **Adding successive percentages directly** → Wrong: 10% increase + 10% decrease = 0% change. Correct: Use formula a + b + ab/100 or calculate step-by-step. The base changes after first operation.

2. **Confusing base value** → Wrong: "A is 20% more than B" does NOT mean "B is 20% less than A". Correct: If A = B × 1.2, then B = A/1.2 = A × (100/120) = 83.33% of A, meaning B is 16.67% less than A.

3. **Calculation errors with decimals** → Wrong: 12.5% of 80 = 12.5 × 80 = 1000. Correct: Remember to divide by 100: (12.5/100) × 80 = 10.

4. **Using wrong original value in percentage change** → Wrong: Price increased from 50 to 60; some students calculate (10/60) × 100. Correct: Percentage change uses original value in denominator: (10/50) × 100 = 20%.

5. **Forgetting the negative sign in decrease** → When using the successive change formula with a decrease, use negative value. If first is +10% and second is -20%, use a = 10, b = -20.

Quick Reference

• **1% shortcut**: To find 1% of any number, divide by 100 (shift decimal two places left). Then multiply for other percentages.
• **50% = half, 25% = quarter, 10% = one-tenth** — use these to split calculations mentally.
• **Increase + Decrease of same %**: Always results in a net decrease. For r% each, net decrease = r²/100%.
• **Common conversions**: 1/8 = 12.5%, 1/6 = 16.67%, 1/3 = 33.33%, 2/3 = 66.67%.
• **Reverse formula memory**: After r% increase → divide by (1 + r/100); after r% decrease → divide by (1 - r/100).
• **Practice mental math**: 20% of 150 = (1/5) × 150 = 30 is faster than (20/100) × 150.