Percentage — RRB NTPC Mathematics Study Notes

Overview

Percentage is one of the highest-weightage topics in RRB NTPC Mathematics. Every exam contains 3–5 direct questions on percentage conversions, percentage change, and application problems. This topic is foundational — profit-loss, simple interest, discount, and even data interpretation questions are built on percentage concepts.

The core idea is simple: "per cent" means "per hundred." Expressing a quantity as a percentage allows easy comparison and calculation. Examiners test your ability to convert between fractions, decimals, and percentages, calculate percentage increase or decrease, and solve word problems involving successive changes, population growth, salary hikes, and marks-based scenarios. Speed and accuracy in basic percentage calculations can save 2–3 minutes per question.

Master the standard conversions (1/2 = 50%, 1/4 = 25%, etc.), the percentage change formula, and the method for handling successive percentage changes. These form 80% of the questions you'll encounter.

Key Concepts

• **Definition**: Percentage means "out of 100." x% = x/100. It is a ratio or fraction with denominator 100.
• **Conversion**: Fraction to percentage: multiply by 100. Decimal to percentage: shift decimal point two places right. Percentage to fraction: divide by 100 and simplify.
• **Percentage of a quantity**: x% of N = (x/100) × N. To find "what percent A is of B," calculate (A/B) × 100.
• **Percentage change formula**: Percentage change = [(New Value − Old Value) / Old Value] × 100. Positive result = increase, negative = decrease.
• **Successive changes**: Two successive changes of a% and b% do not simply add. Net effect = a + b + (ab/100). Sign matters: increase is +, decrease is −.
• **Reverse percentage problems**: If x% of a number gives result R, the number is R/(x/100) = (R × 100)/x.
• **Comparison and ratios**: If A is x% more than B, then B is [x/(100+x)] × 100% less than A. Memorize this relationship to save time.

Formulas / Key Facts

1. **Basic conversion**: x% = x/100 (as a fraction) = 0.01x (as a decimal). 2. **Percentage of N**: x% of N = (x × N)/100. 3. **What percent A is of B**: (A/B) × 100. 4. **Percentage increase**: [(Increase)/Original] × 100. 5. **Percentage decrease**: [(Decrease)/Original] × 100. 6. **Net change after two successive percentage changes a% and b%**: a + b + (ab/100). For successive decreases, use negative signs. 7. **If A is x% more than B**: A = B[1 + x/100], hence B = A/[1 + x/100]. 8. **If A is x% less than B**: A = B[1 − x/100], hence B = A/[1 − x/100]. 9. **Common fractions to percentages**: 1/2 = 50%, 1/3 = 33.33%, 1/4 = 25%, 1/5 = 20%, 1/8 = 12.5%, 1/10 = 10%, 2/5 = 40%, 3/4 = 75%.

Worked Examples

**Example 1: Basic Percentage of a Quantity**

*Question*: What is 18% of 450?

*Solution*: 18% of 450 = (18/100) × 450 = 18 × 4.5 = 81.

**Answer**: 81.

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**Example 2: Percentage Increase**

*Question*: A salary increases from ₹25,000 to ₹28,000. Find the percentage increase.

*Solution*: Increase = 28,000 − 25,000 = 3,000. Percentage increase = (3,000/25,000) × 100 = (3/25) × 100 = 12%.

**Answer**: 12%.

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**Example 3: Successive Percentage Changes**

*Question*: The price of a commodity first increases by 20% and then decreases by 10%. What is the net percentage change?

*Solution*: Use formula: a + b + (ab/100), where a = +20, b = −10. Net change = 20 + (−10) + [20 × (−10)]/100 = 20 − 10 − 2 = 8%.

**Answer**: Net increase of 8%.

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**Example 4: Reverse Percentage Problem**

*Question*: 15% of a number is 60. Find the number.

*Solution*: Let the number be N. 15% of N = 60 ⇒ (15/100) × N = 60 ⇒ N = 60 × (100/15) = 60 × 20/3 = 400.

**Answer**: 400.

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**Example 5: Comparative Percentage**

*Question*: A's salary is 25% more than B's salary. By what percentage is B's salary less than A's?

*Solution*: If B's salary = 100, then A's salary = 125. Difference = 25. B is less than A by: (25/125) × 100 = 20%.

**Answer**: 20%.

Common Mistakes

• **Adding successive percentages directly**: Students often add 20% increase and 10% decrease to get 10% net increase. **Fix**: Use the formula a + b + (ab/100). The compounding effect changes the result.
• **Using wrong base for percentage change**: Computing percentage increase/decrease using the new value instead of the original. **Fix**: Always divide the change by the *original* value, not the new one.
• **Confusing "x% of y" with "y% of x"**: While mathematically x% of y = y% of x, in word problems context matters. Read carefully which quantity is the base. **Fix**: Identify the base (the quantity after "of") clearly.
• **Misapplying comparative percentage formulas**: If A is 20% more than B, thinking B is 20% less than A. **Fix**: Use the exact formula. B is [x/(100+x)] × 100% less than A. Here, B is ~16.67% less than A, not 20%.
• **Forgetting to convert percentage to fraction before calculation**: Directly multiplying the percentage number instead of dividing by 100 first. **Fix**: Always write x% as x/100 in calculations. Example: 25% of 80 is (25/100) × 80, not 25 × 80.

Quick Reference

• **x% = x/100**. To find x% of N: (x × N)/100.
• **Percentage change** = [(New − Old)/Old] × 100.
• **Successive changes** of a% and b%: Net = a + b + ab/100.
• **If A is x% more than B**, then **B is [x/(100+x)] × 100% less than A**.
• **Reverse calculation**: If x% of N = R, then N = (R × 100)/x.
• **Standard fractions**: 1/4 = 25%, 1/5 = 20%, 1/8 = 12.5%, 2/3 = 66.67%, 3/5 = 60%.