Percentage: Master Guide for SSC, Banking & UPSC
1. Introduction
Percentage (denoted by %) means "per hundred" or "out of 100". It is a fraction with a denominator of 100. For competitive exams, percentage is the backbone of Quantitative Aptitude, directly linking to Profit & Loss, Simple/Compound Interest, and Data Interpretation (DI).
Basic Formula:
Percentage = (Given Value / Total Value) × 100
Percentage = (Given Value / Total Value) × 100
2. Concept Building
Fraction ↔ Percentage Conversion (Crucial for Speed)
Memorizing these fractional values is non-negotiable for SSC and Banking exams. It eliminates calculation time.
| Fraction | Percentage | Fraction | Percentage |
|---|---|---|---|
| 1/2 | 50% | 1/9 | 11.11% or 11 1/9% |
| 1/3 | 33.33% or 33 1/3% | 1/10 | 10% |
| 1/4 | 25% | 1/11 | 9.09% or 9 1/11% |
| 1/5 | 20% | 1/12 | 8.33% or 8 1/3% |
| 1/6 | 16.66% or 16 2/3% | 1/14 | 7.14% or 7 1/7% |
| 1/7 | 14.28% or 14 2/7% | 1/15 | 6.66% or 6 2/3% |
| 1/8 | 12.5% or 12 1/2% | 1/16 | 6.25% or 6 1/4% |
Percentage Increase and Decrease
- % Increase: (Increase / Original Value) × 100
- % Decrease: (Decrease / Original Value) × 100
- Multiplier Concept: A 20% increase means the new value is 120% of the original (or Original × 1.2). A 20% decrease means 80% of the original (or Original × 0.8).
3. Important Formulas & Short Tricks
- The Reversibility Trick: x% of y = y% of x.
(e.g., 64% of 25 is hard, but 25% of 64 is just 64/4 = 16). - Successive Percentage Change: If a number is changed by x% and then by y%, net change = x + y + (xy/100) %.
(Use '+' for increase, '-' for decrease). - Price & Consumption: If the price of a commodity increases by R%, the reduction in consumption to keep expenditure same = [R / (100 + R)] × 100 %.
- A is more/less than B: If A's income is R% more than B, B's income is less than A by = [R / (100 + R)] × 100 %.
- Population Growth: If current population is P and grows at R% p.a., population after n years = P(1 + R/100)n.
4. Solved Questions (From Easy to Advanced)
Type A: Basic Calculation & Conversions
Q1. Evaluate: 16.66% of 216 + 12.5% of 640
View Solution
- Convert to fractions: 16.66% = 1/6, and 12.5% = 1/8.
- (1/6) × 216 + (1/8) × 640
- = 36 + 80 = 116
Q2. 80% of A = 50% of B and B = x% of A. Find x.
View Solution
- 80% of A = 50% of B ⇒ 8A = 5B ⇒ B = (8/5)A
- Convert (8/5) to percentage: (8/5) × 100 = 160%
- Therefore, B = 160% of A. x = 160.
Q3. If 20% of a number is added to itself, the result becomes 480. Find the number.
View Solution
- 20% = 1/5. Adding it to itself means the new number is 1 + 1/5 = 6/5 of the original.
- (6/5) × x = 480 ⇒ x = 480 × (5/6) = 400.
Type B: Successive Changes & Price/Consumption
Q4. The price of sugar is increased by 25%. By what percentage must a family reduce its consumption so that the expenditure remains the same?
View Solution
- Using Formula: [R / (100 + R)] × 100
- [25 / (100 + 25)] × 100 = (25 / 125) × 100 = 1/5 × 100 = 20%.
Q5. The length of a rectangle is increased by 10% and breadth decreased by 10%. What is the % change in area?
View Solution
- Use successive formula: x + y + (xy/100)
- Here x = +10, y = -10.
- +10 - 10 + [(10)(-10)/100] = 0 - 1 = -1% (1% decrease).
- Shortcut: When same % increase and decrease happens, there is always a loss of (x2/100)%.
Type C: Elections, Exams & Demographics
Q6. In Riddhi Classes, 60% of the students are boys. If the number of girls is 120, find the total number of students.
View Solution
- If boys = 60%, then girls = 40% of total students.
- 40% of Total = 120 ⇒ (2/5) × Total = 120.
- Total = 120 × (5/2) = 300 students.
Q7. The monthly traffic on the educational portal Sarkari Watch increased by 25% to reach 50,000 visitors. What was the original traffic?
View Solution
- Let original traffic be x. 25% increase means new traffic is 125% of x.
- 125% of x = 50,000 ⇒ (5/4)x = 50,000.
- x = 50,000 × (4/5) = 40,000 visitors.
Type D: Income, Ratios & Mixtures
Q8. A man spends 75% of his income. His income increases by 20% and his expenditure increases by 10%. Find the % increase in his savings.
View Solution
- Let original Income = 100. Expenditure = 75. Savings = 25.
- New Income = 120. New Expenditure = 75 × 1.10 = 82.5.
- New Savings = 120 - 82.5 = 37.5.
- Increase in savings = 37.5 - 25 = 12.5.
- % Increase = (12.5 / 25) × 100 = 50%.
Q9. A 30 litre mixture of acid and water contains 20% acid. How much water must be added to make the acid 15% in the new mixture?
View Solution
- Initial Acid = 20% of 30 = 6 litres.
- Let x litres of water be added. Acid amount remains the same (6L).
- New total volume = 30 + x.
- 15% of (30 + x) = 6 ⇒ (15/100) × (30 + x) = 6
- 30 + x = 40 ⇒ x = 10 litres.
Type E: Advanced & Tricky Scenarios
Q10. Due to a 20% reduction in the price of wheat, a man can buy 5 kg more for Rs. 320. Find the original rate of wheat per kg.
View Solution
- Let original price be Rs. x/kg. New price = 0.8x.
- (320 / 0.8x) - (320 / x) = 5.
- (400 / x) - (320 / x) = 5 ⇒ 80 / x = 5 ⇒ x = Rs. 16/kg.
- Shortcut: 20% of Rs. 320 = Rs. 64. He gets 5kg for Rs. 64. Reduced price = 64/5 = Rs. 12.8/kg. Original price = 12.8 / 0.8 = Rs. 16.
5. Unsolved Practice Section
Test your speed with these common exam-level questions. Attempt them without pen and paper if possible!
- Evaluate: 37.5% of 800 + 14.28% of 343.
- If A is 40% less than B, how much percentage is B more than A?
- A student has to secure 35% marks to pass. He gets 80 marks and fails by 25 marks. Find maximum marks.
- The price of petrol increased by 15%. What should be the percentage drop in travel to keep the fuel budget constant?
- By selling an article at 80% of its marked price, a trader makes a loss of 10%. What will be the profit % if sold at marked price?
6. Exam Strategy Tips for Percentages
- Fractions are your friends: Never calculate "16.66 × something". Immediately write 1/6. Memorize fractions up to 1/20.
- Use the "100" Assumption Rule: In problems without absolute values (only percentages are given), always assume the initial value or Cost Price as 100. It prevents messy algebra.
- Base Errors: The most common mistake is taking the wrong base. If a question says "Profit is 20% on Selling Price", your denominator is SP, not CP. Always ask yourself: "Percentage of WHAT?"
