Percentage Calculation and Change Questions for UP POLICE CONSTABLE

Basic Percentage Concept

Percentage means 'per hundred'. It's a way to express a part of a whole as a fraction of 100. Think of it as a universal scale: if you score 75 out of 100, that's 75%. If you score 75 out of 150, you convert it to the same scale: (75/150) × 100 = 50%. The key is always converting to a denominator of 100 for easy comparison.

• Percentage = (Part / Whole) × 100
• To find Part: Part = (Percentage × Whole) / 100
• To find Whole: Whole = (Part × 100) / Percentage
• Always identify what is the 'whole' (base) — this is the most common mistake
• Percentage is dimensionless — it's just a ratio expressed as a number out of 100

Percentage Increase and Decrease

This is where most UP Police questions focus. When something increases or decreases, you calculate the change relative to the original value (the base). The base is always the starting value, not the ending value. A 50% increase followed by a 50% decrease does NOT bring you back to the original — this is the classic trap.

• Percentage Increase = ((New Value - Old Value) / Old Value) × 100
• Percentage Decrease = ((Old Value - New Value) / Old Value) × 100
• Base is ALWAYS the original/starting value, never the final value
• Successive percentage changes multiply, not add: if +20% then +25%, result is 1.20 × 1.25 = 1.50 (i.e., +50% total)
• A +50% increase followed by -50% decrease gives: 1.50 × 0.50 = 0.75 (i.e., 25% net decrease)

Real-World Percentage Scenarios

UP Police exams love practical questions: population growth, profit-loss, discount-markup, tax, etc. The method is always the same: identify the base (original value), calculate the change, apply the percentage formula. Common scenarios include salary hikes, price changes, population increase, and compound percentage changes over years.

• Profit/Loss: Base is always the Cost Price (CP), not Selling Price (SP)
• Discount: Base is always the Marked Price (MP), not Selling Price
• Population growth: Base is the initial population, not the final
• Tax/GST: Base is the original amount before tax
• Read the question twice to identify what is the 'base' — this prevents 80% of errors

Common Percentage Shortcuts

In a timed exam, knowing quick conversions saves 30-60 seconds per question. Memorize common fractions as percentages: 1/2 = 50%, 1/4 = 25%, 1/5 = 20%, 1/3 ≈ 33.33%, 1/6 ≈ 16.67%, 3/4 = 75%, 2/5 = 40%, 3/5 = 60%. Also, for quick percentage calculations, use the multiplier method: +20% means multiply by 1.20, -15% means multiply by 0.85.

• 1/2 = 50%, 1/4 = 25%, 1/5 = 20%, 1/3 = 33.33%, 1/6 = 16.67%, 1/10 = 10%
• 3/4 = 75%, 2/5 = 40%, 3/5 = 60%, 4/5 = 80%
• Multiplier method: +p% means multiply by (1 + p/100); -p% means multiply by (1 - p/100)
• For quick mental math: 10% of any number = divide by 10; 5% = half of 10%; 20% = double of 10%

• Confusing the base: Calculating percentage decrease using the new value as base instead of the old value. Example: If price drops from Rs 100 to Rs 80, the decrease is 20% (not 25%). Always use the original/starting value as the denominator.
• Adding percentages instead of multiplying: If salary increases 20% then 25%, the total is NOT 45%. It's 1.20 × 1.25 = 1.50, which is 50% increase. Successive changes multiply, never add.
• Forgetting to convert back to percentage: After calculating (Part/Whole), forgetting to multiply by 100. You'll get a decimal like 0.75 instead of 75%.
• Misidentifying the base in word problems: In 'A shirt costs Rs 500 and is sold at 20% profit', the base for profit is the Cost Price (500), not the Selling Price. Profit = 20% of 500 = 100, so SP = 600.
• Percentage of percentage errors: If 60% of a class is girls, and 50% of girls play sports, the percentage of the whole class that are girls playing sports is 60% × 50% = 30%, not 110%.

• BASE RULE: In any percentage problem, ask 'What is the 100%?' — that's your base. Always use the original/starting value as the denominator.
• MULTIPLY FOR SUCCESSIVE: Multiple percentage changes? Use multipliers. +20% then +25% = 1.20 × 1.25 = 1.50 (50% total). Never add percentages.
• FRACTION ANCHOR: Memorize 1/5 = 20% and 1/4 = 25%. Most exam percentages are multiples of these. 40% = 2/5, 75% = 3/4.
• PROFIT/LOSS/DISCOUNT: CP is base for profit/loss. MP is base for discount. Always identify which value is the 'whole' before calculating.

• UP Police typically asks 2-3 percentage questions in the Numerical section. Expect 1 straightforward calculation (basic percentage or increase/decrease) and 1 word problem (salary, price, population).
• Time management: Percentage questions should take 1-2 minutes max. If you're stuck on identifying the base, re-read the question and underline the original/starting value.
• Recent pattern: Questions often combine percentage with other concepts (e.g., 'If a person's salary increases 15% and then decreases 10%, and final salary is Rs 10,350, find original salary'). Practice reverse calculations.
• Avoid calculator dependency: Learn mental math for common percentages (10%, 20%, 25%, 50%). The exam hall may have time pressure; quick mental shortcuts are faster.
• Watch for trap options: If the correct answer is 20% increase, options might include 20% decrease, or the reverse calculation (80% of original). Read the question direction carefully.