Study Notes: Percentage

Overview

Percentage is one of the most frequently tested topics in SSC MTS Paper 1, appearing in 3–5 questions every year. The word "percent" means "per hundred," so 25% literally means 25 out of 100 or 25/100. Mastering percentage is critical because it forms the foundation for profit and loss, simple interest, data interpretation, and discount problems.

In SSC MTS, percentage questions test three core skills: converting between fractions/decimals/percentages, calculating percentage increases or decreases, and solving word problems involving successive changes, population changes, exam pass percentages, or salary adjustments. The questions are straightforward but require speed and accuracy. Students who master the basic formulas and practice mental calculation techniques can solve most percentage problems in under 60 seconds.

The key to success is understanding that percentage is just another way to express ratios and fractions, combined with the ability to quickly apply the standard formulas for percentage change and successive percentage changes without confusion.

Key Concepts

• **Percentage as a fraction**: x% means x/100. To convert any 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, calculate (x/100) × N. This is the base formula for all percentage calculations.

• **Percentage change formula**: Percentage change = [(New Value − Original Value) / Original Value] × 100. Positive result means increase, negative means decrease.

• **Reverse percentage problem**: If x% of a number is given and you need the number itself, use: Number = (Given Value × 100) / x.

• **Successive percentage changes**: When two percentage changes happen one after another, they don't simply add. Use the formula: Net change = a + b + (ab/100), where a and b are the two percentage changes (use negative sign for decrease).

• **Comparing quantities using percentage**: "A is what percent more/less than B?" = [(A − B) / B] × 100. Always divide by the reference value (the one mentioned after "than").

Formulas / Key Facts

1. **Basic conversion**: x% = x/100 = 0.0x in decimal form 2. **Percentage of N**: x% of N = (x/100) × N = xN/100 3. **Increase by x%**: New Value = Original × (1 + x/100) = Original × (100 + x)/100 4. **Decrease by x%**: New Value = Original × (1 − x/100) = Original × (100 − x)/100 5. **Percentage increase**: [(Increase) / Original] × 100 6. **Percentage decrease**: [(Decrease) / Original] × 100 7. **Successive changes**: If increased by a% then by b%, net = a + b + (ab/100)% 8. **Common fraction-percentage conversions**: 1/2 = 50%, 1/4 = 25%, 1/5 = 20%, 1/3 = 33.33%, 2/3 = 66.67%, 1/8 = 12.5%, 3/4 = 75%

Worked Examples

**Example 1: Basic percentage calculation**

*Question*: What is 35% of 840?

*Solution*:

• 35% of 840 = (35/100) × 840
• = 35 × 8.4
• = 294

Quick method: 10% of 840 = 84, so 30% = 252 and 5% = 42. Total = 252 + 42 = 294.

**Example 2: Percentage increase**

*Question*: The price of petrol increased from ₹80 per liter to ₹92 per liter. What is the percentage increase?

*Solution*:

• Increase = 92 − 80 = ₹12
• Percentage increase = (Increase / Original) × 100
• = (12/80) × 100
• = 15%

**Example 3: Successive percentage changes**

*Question*: A number is first increased by 20% and then decreased by 20%. What is the net percentage change?

*Solution*: Using the formula: Net change = a + b + (ab/100) Here a = +20, b = −20

• Net change = 20 + (−20) + (20 × (−20)/100)
• = 0 + (−400/100)
• = −4%

The number decreases by 4%. (Common trap: Students think the changes cancel out to 0%!)

Alternative method: Assume original = 100

• After 20% increase: 100 × 1.20 = 120
• After 20% decrease: 120 × 0.80 = 96
• Net change = 96 − 100 = −4, so −4%

**Example 4: Reverse percentage problem**

*Question*: 15% of a number is 72. Find the number.

*Solution*:

• Let the number be N
• 15% of N = 72
• (15/100) × N = 72
• N = 72 × (100/15)
• N = 72 × 20/3 = 24 × 20 = 480

Common Mistakes

**Mistake 1: Adding successive percentages directly** Wrong thinking: If a value increases by 10% then 20%, total increase is 30%. Correct fix: Use the formula a + b + (ab/100). Here: 10 + 20 + (10×20/100) = 30 + 2 = 32% net increase.

**Mistake 2: Wrong reference value in comparison** Wrong thinking: If A = 80 and B = 100, "A is what % less than B?" = (20/80) × 100 = 25%. Correct fix: Always divide by the reference value (after "than"). Answer = (20/100) × 100 = 20%.

**Mistake 3: Percentage decrease followed by equal increase doesn't restore original** Wrong thinking: Decrease by 25% then increase by 25% = back to original. Correct fix: Let original = 100. After −25%: 75. After +25% on 75: 75 × 1.25 = 93.75, not 100.

**Mistake 4: Converting fractions incorrectly** Wrong thinking: 3/8 as percentage = 3.8%. Correct fix: (3/8) × 100 = 37.5%.

**Mistake 5: Confusion between "percentage increase" and "what percent one value is of another"** Wrong thinking: Using the same formula for both types of questions. Correct fix: "Percentage increase" = [(New − Old)/Old] × 100. "A is what % of B" = (A/B) × 100.

Quick Reference

• **50% = half, 25% = quarter, 20% = one-fifth** — use these for mental math shortcuts
• **Successive changes never simply add** — always use a + b + ab/100
• **To find 15%**: Find 10% + 5% (half of 10%)
• **Increase by x% then decrease by x% → net decrease** of x²/100 %
• **Reference value in comparisons** = the quantity mentioned after "than"
• **Memorize common fractions** — 1/3 = 33.33%, 1/6 = 16.67%, 1/8 = 12.5%, 3/8 = 37.5%