Percentage — Study Notes for Railway Group D

Overview

Percentage is one of the most frequently tested topics in RRB Group D Mathematics, appearing in 5–8 questions per paper. It forms the foundation for Profit & Loss, Simple & Compound Interest, and many Data Interpretation questions. The word "percent" means "per hundred" — so 25% literally means 25 out of every 100.

Mastery requires two skills: quick conversion between fractions, decimals, and percentages; and solving real-world word problems involving percentage increase/decrease, successive changes, and population/price variation scenarios. Questions are straightforward if you memorize common fraction-percent equivalents and practice the standard formulas. Most errors come from confusion about base values (percentage "of what?") rather than calculation mistakes.

Strong command of percentage directly improves your speed on 15–20% of the entire Mathematics section, making it a high-return investment of study time.

Key Concepts

• **Definition**: Percentage expresses a number as a fraction of 100. If a quantity is x% of another, it means x/100 of that quantity.

• **Base value matters**: "30% of 200" means (30/100) × 200 = 60. Always identify what the percentage is calculated on — the base or reference value.

• **Percentage increase/decrease**: If a value V increases by r%, new value = V × (1 + r/100). If it decreases by r%, new value = V × (1 – r/100).

• **Successive percentage changes**: Two successive changes of a% and b% do NOT simply add. Net effect = a + b + (ab/100). The sign depends on increase (+) or decrease (–).

• **Reverse percentage problems**: If new value after r% increase is N, original value = N/(1 + r/100). Similarly for decrease, original = N/(1 – r/100).

• **Fraction of one quantity to another**: If A is compared to B, then A as a percentage of B = (A/B) × 100%.

• **Percentage point vs percentage change**: A change from 40% to 50% is a 10 percentage point increase but a 25% relative increase (because 10 is 25% of the original 40).

Formulas / Key Facts

1. **Basic conversion**: Percent = (Fraction × 100)% = (Decimal × 100)% Example: 3/4 = 0.75 = 75%

2. **Percentage of a number**: r% of N = (r/100) × N = rN/100

3. **Increase formula**: New Value = Original × (100 + Increase%)/100

4. **Decrease formula**: New Value = Original × (100 – Decrease%)/100

5. **Percentage change**: % Change = [(New – Old)/Old] × 100

6. **Successive changes**: Net % = a + b + ab/100 (Use + for increase, – for decrease in the formula)

7. **Common equivalents** (memorize these):

• 1/2 = 50%, 1/3 ≈ 33.33%, 1/4 = 25%, 1/5 = 20%
• 1/6 ≈ 16.67%, 1/8 = 12.5%, 1/10 = 10%
• 2/3 ≈ 66.67%, 3/4 = 75%, 4/5 = 80%

8. **Reverse percentage**: Original = (Final × 100)/(100 ± Change%)

Worked Examples

**Example 1: Basic percentage calculation** *Question*: What is 35% of 240? *Solution*: 35% of 240 = (35/100) × 240 = 35 × 2.4 = 84 **Answer: 84**

**Example 2: Percentage increase** *Question*: A worker's salary is ₹15,000. It is increased by 12%. What is the new salary? *Solution*: Increase = 12% of 15,000 = (12/100) × 15,000 = 1,800 New salary = 15,000 + 1,800 = 16,800 *Or directly*: New salary = 15,000 × (112/100) = 15,000 × 1.12 = 16,800 **Answer: ₹16,800**

**Example 3: Successive percentage changes** *Question*: The price of an item is increased by 20% and then decreased by 10%. What is the net percentage change? *Solution*: Let original price = 100 After 20% increase = 100 × 1.20 = 120 After 10% decrease = 120 × 0.90 = 108 Net change = 108 – 100 = 8 Net % change = (8/100) × 100 = 8% increase *Formula method*: a = +20, b = –10 Net % = 20 + (–10) + [20 × (–10)]/100 = 20 – 10 – 2 = 8% increase **Answer: 8% increase**

**Example 4: Reverse percentage** *Question*: After a 25% reduction, a jacket costs ₹1,200. What was its original price? *Solution*: Let original price = P After 25% decrease, price = P × (75/100) = 1,200 P = 1,200 × (100/75) = 1,200 × (4/3) = 1,600 **Answer: ₹1,600**

**Example 5: Comparing quantities** *Question*: Ram scores 45 marks out of 60. What is his percentage? *Solution*: Percentage = (Marks obtained / Total marks) × 100 = (45/60) × 100 = (3/4) × 100 = 75% **Answer: 75%**

Common Mistakes

1. **Wrong base for percentage**: Calculating "A is 20% more than B" as B + 20% of A instead of B + 20% of B. *Fix*: Always take percentage of the reference (base) value mentioned after "than" or "of."

2. **Adding successive percentages directly**: Thinking 10% increase followed by 10% decrease returns to original. *Fix*: Use the formula a + b + ab/100 or calculate step-by-step. A 10% gain then 10% loss yields 1% net loss.

3. **Confusing percentage change with percentage points**: Saying tax increased from 10% to 12% is a "2% increase" when it's actually a 20% relative increase. *Fix*: Percentage point = absolute difference; percentage change = relative to original.

4. **Reversing increase/decrease operations**: Using (100 + r%) when the problem states a decrease. *Fix*: Carefully note whether the change is an increase (+) or decrease (–) and choose the correct formula.

5. **Rounding too early**: Converting 1/3 to 33% instead of 33.33% and losing accuracy in multi-step problems. *Fix*: Keep fractions or more decimal places until the final answer, or use the given options to back-calculate.

Quick Reference

• **Percent to fraction**: Divide by 100. Example: 40% = 40/100 = 2/5
• **Fraction to percent**: Multiply by 100. Example: 7/20 = (7/20) × 100 = 35%
• **r% increase**: Multiply by (100 + r)/100
• **r% decrease**: Multiply by (100 – r)/100
• **Successive changes a%, b%**: Net = a + b + ab/100 (signs matter: + for increase, – for decrease)
• **Find original after r% increase**: Original = Final/(1 + r/100)