Percentage — Study Notes for UP Police Constable
Overview
Percentage is one of the highest-scoring and most frequently tested topics in the Numerical & Mental Ability section of UP Police Constable exam. The word "percent" means "per hundred" — so 25% literally means 25 out of every 100. This concept appears both as direct calculation questions (convert fractions to percentages, find x% of y) and in applied word problems involving marks, population, consumption, elections, and price changes.
Mastery of percentage is essential not just for standalone questions but also as a foundation for profit-loss, discount, simple interest, and data interpretation problems. Expect 4–6 direct or indirect questions on this topic. The key to success is speed in mental calculation and recognizing common fraction-percentage equivalents. Most questions are solvable within 30–45 seconds if you know the shortcuts.
Students often lose marks by making conversion errors or misinterpreting "percentage increase" versus "percentage of original." Clear conceptual understanding and practice with varied problem types will secure these easy marks.
Key Concepts
- **Definition**: Percentage expresses a number as a fraction of 100. If a quantity is x out of y, then percentage = (x/y) × 100%.
- **Conversion clarity**: To convert percentage to fraction, divide by 100. To convert fraction to percentage, multiply by 100. For example, 3/4 = (3/4) × 100 = 75%.
- **Percentage increase/decrease**: If a value changes from A to B, percentage change = [(B - A)/A] × 100%. Positive result means increase, negative means decrease.
- **Successive percentage changes**: When two percentage changes apply consecutively, the net effect is NOT their simple sum. Use the formula: Net% = a + b + (ab/100), where a and b are the two percentage changes.
- **Percentage of a percentage**: "x% of y%" means (x/100) × (y/100) × original value. This is common in population or consumption problems.
- **Reversing percentage changes**: If a value increases by x%, to bring it back to original, it must decrease by [x/(100+x)] × 100%, not by x%.
- **Expressing one quantity as percentage of another**: If A is to be expressed as percentage of B, calculate (A/B) × 100%. Order matters — "A as % of B" differs from "B as % of A".
- **Base value identification**: Always identify what is the base (denominator) for percentage calculation. "Marks increased by 20%" means 20% of original marks, not 20% of new marks.
Formulas / Key Facts
1. **Basic conversion**: Percentage = (Part/Whole) × 100; Part = (Percentage/100) × Whole 2. **Percentage increase**: New Value = Original × (1 + r/100), where r is percentage increase 3. **Percentage decrease**: New Value = Original × (1 - r/100), where r is percentage decrease 4. **Successive changes**: Net% = a + b + (ab/100) for changes of a% and b% 5. **Reverse calculation**: If value becomes x after r% increase, Original = x/(1 + r/100) 6. **Common equivalents**: 1/2 = 50%, 1/3 = 33.33%, 1/4 = 25%, 1/5 = 20%, 1/8 = 12.5%, 2/5 = 40%, 3/4 = 75% 7. **Passing marks formula**: If pass percentage is P% and student scored S marks but failed by F marks, Maximum marks = (S + F) × 100/P 8. **Election formula**: If winner gets W% and wins by M votes, Total votes = M × 100/(2W - 100), when only two candidates
Worked Examples
**Example 1: Basic percentage calculation** *Question*: What is 35% of 840?
*Solution*: 35% of 840 = (35/100) × 840 = 35 × 8.4 = 294 **Answer: 294**
*Shortcut*: 10% of 840 = 84, so 30% = 252, and 5% = 42. Total = 252 + 42 = 294.
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**Example 2: Percentage increase** *Question*: The price of sugar increases from ₹40/kg to ₹50/kg. What is the percentage increase?
*Solution*: Increase = 50 - 40 = ₹10 Percentage increase = (10/40) × 100 = 25% **Answer: 25%**
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**Example 3: Successive percentage changes** *Question*: A number is first increased by 10%, then decreased by 10%. What is the net percentage change?
*Solution*: Using formula: Net% = 10 + (-10) + [10 × (-10)/100] = 0 + (-1) = -1% **Answer: 1% decrease**
*Important*: The net change is NOT zero. After 10% increase, the base becomes larger, so 10% decrease on the new base is more than the original increase.
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**Example 4: Passing marks problem** *Question*: A student scored 280 marks and failed by 20 marks. If passing percentage is 40%, what are the maximum marks?
*Solution*: Student needed = 280 + 20 = 300 marks to pass This 300 is 40% of maximum marks Maximum marks = (300/40) × 100 = 750 **Answer: 750 marks**
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**Example 5: Population problem** *Question*: The population of a town is 50,000. If it increases by 10% in the first year and 20% in the second year, what is the population after two years?
*Solution*: After first year = 50,000 × (1 + 10/100) = 50,000 × 1.1 = 55,000 After second year = 55,000 × (1 + 20/100) = 55,000 × 1.2 = 66,000 **Answer: 66,000**
*Alternative*: Using successive change — Net% = 10 + 20 + (10×20/100) = 30 + 2 = 32% Final population = 50,000 × 1.32 = 66,000
Common Mistakes
1. **Confusing base for percentage change**: Students calculate percentage of new value instead of original. *Fix*: Percentage change is always with respect to the ORIGINAL value unless stated otherwise.
2. **Adding successive percentages directly**: Thinking 10% increase followed by 20% increase = 30% increase. *Fix*: Use the formula a + b + (ab/100). The correct answer is 32%, not 30%.
3. **Reversing percentage incorrectly**: Believing that a 25% increase can be reversed by 25% decrease. *Fix*: If increased by 25%, to reverse, decrease by [25/125 × 100] = 20%.
4. **Decimal confusion in conversion**: Writing 0.45 as 0.45% instead of 45%. *Fix*: To convert decimal to percentage, multiply by 100. So 0.45 = 45%.
5. **Misreading "A is what percent more/less than B"**: Calculating (A/B) × 100 instead of [(A-B)/B] × 100. *Fix*: "More than" or "less than" requires calculating the DIFFERENCE as percentage of the base.
Quick Reference
- **50% = 1/2, 33.33% = 1/3, 25% = 1/4, 20% = 1/5, 12.5% = 1/8** — memorize these for instant conversion.
- **Successive changes formula**: Net% = a + b + ab/100 (divide product by 100).
- **Percentage increase**: New = Old × (1 + r/100); Percentage decrease: New = Old × (1 - r/100).
- **To find original after x% increase**: Original = New/(1 + x/100).
- **Passing marks**: Max = (Scored + Shortfall) × 100/Pass%.
- **For quick 10% calculations**: Shift decimal one place left. 10% of 650 = 65.