Study Notes: Percentage

Overview

Percentage is one of the most frequently tested topics in UPSSSC PET Elementary Arithmetic. It appears in 2–4 questions directly and forms the foundation for profit-loss, simple interest, and data interpretation questions. The word "percent" means "per hundred" (from Latin *per centum*), so 25% literally means 25 out of every 100.

Mastery of percentage requires three core skills: converting between fractions, decimals, and percentages; calculating percent increase or decrease; and solving word problems involving successive changes, population growth, and exam marks. Students often lose marks due to careless conversion errors or misunderstanding what quantity forms the "base" in percent change problems. Strong command here builds confidence across multiple arithmetic sections.

Focus on mental calculation shortcuts—recognizing that 10% is simply dividing by 10, 50% is half, and 25% is one-quarter can save 15–20 seconds per question. Practice both direct calculation and reverse problems (given the percentage result, find the original).

Key Concepts

• **Percentage as a fraction:** Any percentage *x*% equals *x*/100. For example, 35% = 35/100 = 7/20. Always reduce fractions for easier calculation.

• **Three quantity relationship:** In any percentage problem, three quantities exist—the base (original/whole), the part (portion), and the percentage. If any two are known, the third can be found using: Part = (Percentage/100) × Base.

• **Percent increase/decrease:** To find percent change, use: Percent Change = [(New Value − Old Value)/Old Value] × 100. Positive result indicates increase; negative indicates decrease.

• **Successive percentage changes:** Two successive changes of *a*% and *b*% do not simply add up. The net effect is: *a* + *b* + (*a*×*b*)/100. For example, a 10% increase followed by 10% decrease does not restore the original value.

• **Base reference is critical:** "A is 20% more than B" means A = 1.2B, but "A is 20% less than B" means A = 0.8B. Always identify which quantity is the base (100%).

• **Reverse percentage:** If a value after *x*% increase is *V*, the original value is *V*/(1 + *x*/100). For decrease, use *V*/(1 − *x*/100).

Formulas / Key Facts

1. **Basic conversion:** *x*% = *x*/100 = *x* ÷ 100 (decimal form)

2. **Finding percentage of a number:** *x*% of *N* = (*x*/100) × *N*

3. **Finding what percent *A* is of *B*:** (*A*/*B*) × 100 %

4. **Percent increase:** New Value = Original × (1 + Increase%/100)

5. **Percent decrease:** New Value = Original × (1 − Decrease%/100)

6. **Successive changes:** Net% = *a* + *b* + (*a*×*b*)/100, where *a* and *b* are percentage changes (use negative sign for decrease)

7. **Common fraction-percentage equivalents:** 1/2 = 50%, 1/3 = 33.33%, 1/4 = 25%, 1/5 = 20%, 1/8 = 12.5%, 2/3 = 66.67%, 3/4 = 75%

8. **Reverse calculation for increase:** Original = Final/(1 + *x*/100)

Worked Examples

**Example 1: Basic percentage calculation** *Question:* What is 18% of 450?

*Solution:* Step 1: Convert percentage to decimal: 18% = 18/100 = 0.18 Step 2: Multiply: 0.18 × 450 = 81 *Answer:* 81

*Alternative shortcut:* 10% of 450 = 45, so 1% = 4.5, therefore 18% = 18 × 4.5 = 81

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**Example 2: Percent increase** *Question:* A number increases from 80 to 92. What is the percent increase?

*Solution:* Step 1: Find absolute increase: 92 − 80 = 12 Step 2: Percent increase = (Increase/Original) × 100 = (12/80) × 100 Step 3: Simplify: (12/80) = (3/20) = 0.15 = 15% *Answer:* 15%

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**Example 3: Successive percentage changes** *Question:* The population of a town increased by 10% in the first year and decreased by 10% in the second year. If the initial population was 50,000, what is the population after two years?

*Solution:* Step 1: After 10% increase: 50,000 × 1.10 = 55,000 Step 2: After 10% decrease of new value: 55,000 × 0.90 = 49,500 *Answer:* 49,500

*Formula method:* Net change = 10 + (−10) + [10×(−10)]/100 = 0 − 1 = −1% Final population = 50,000 × (1 − 0.01) = 49,500

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**Example 4: Reverse percentage** *Question:* After a 25% reduction, the price of a shirt is ₹600. What was the original price?

*Solution:* Step 1: If price reduced by 25%, current price is 75% of original Step 2: 75% of Original = 600 Step 3: Original = 600/(75/100) = 600 × (100/75) = 800 *Answer:* ₹800

*Verification:* 25% of 800 = 200, so 800 − 200 = 600 ✓

Common Mistakes

**Mistake 1: Adding successive percentages directly** *Wrong thinking:* 20% increase then 30% increase equals 50% increase. *Correct fix:* Use net formula: 20 + 30 + (20×30)/100 = 56%. The second increase applies to the already-increased value, not the original.

**Mistake 2: Using wrong base for comparison** *Wrong thinking:* If A is 80 and B is 100, "A is 20% of B." *Correct fix:* A is 80% of B, not 20%. Or say "A is 20% less than B." Always check: percent of what?

**Mistake 3: Reversing increase/decrease formula** *Wrong thinking:* If value increases by 20%, original = new value × 1.20. *Correct fix:* Original = New value / 1.20. Multiplying further increases the value; division reverses the increase.

**Mistake 4: Forgetting to multiply by 100 in percentage calculation** *Wrong thinking:* (25/200) = 0.125 is the answer. *Correct fix:* Percentage = 0.125 × 100 = 12.5%. The decimal form is not the percentage.

**Mistake 5: Confusing "percent of" with "percent more/less than"** *Wrong thinking:* "30% of X" is the same as "30% more than X." *Correct fix:* 30% of X = 0.3X, but 30% more than X = 1.3X. Read the language carefully.

Quick Reference

• To find *x*% of *N*: multiply *N* by *x*/100
• Percent change = [(Final − Initial)/Initial] × 100
• Successive changes: Net = *a* + *b* + *a*b/100
• 50% = 1/2, 25% = 1/4, 20% = 1/5, 10% = 1/10 (use for quick mental math)
• Original value after *x*% increase = Final/(1 + *x*/100)
• Always identify the base: "percent *of* what?" determines your denominator