Percentages are everywhere in aptitude — profit/loss, interest, data interpretation, you name it. The person who has percentage-fraction equivalents memorized and knows the successive change formula has a massive advantage. Let’s build that toolkit.
What Is a Percentage?
A percentage is just a fraction with denominator 100. “Per cent” literally means “per hundred.”
25% = 25/100 = 1/4
In simple language, if we say “25% of students passed,” we mean 25 out of every 100 students.
Percentage-Fraction Equivalents — The Power Table
This table is the single biggest time-saver in aptitude. Memorize it and we’ll solve problems twice as fast.
How to use these: Instead of calculating “12.5% of 480,” we think “1/8 of 480 = 60.” Instant.
Instead of “find 37.5% of 640,” we think “3/8 of 640 = 3 × 80 = 240.” Done in 3 seconds.
Basic Percentage Calculations
Common mistake alert: “A is what percent more than B” → base is B. “A is what percent less than B” → base is still B. Always figure out what the “compared to” value is — that’s our denominator.
Successive Percentage Changes
When two percentage changes happen one after another, we DON’T just add them.
Formula: If there are two successive changes of a% and b%, the effective change is:
a + b + (ab/100) %
Example: Price increases by 20%, then decreases by 10%. Net change?
- = 20 + (-10) + (20 × -10)/100
- = 20 - 10 - 2
- = 8% increase
Not 10%! The extra -2% is because the 10% decrease applied to the already-increased amount.
Super useful case: If something increases by x% and then decreases by x%, the net result is always a DECREASE of (x²/100)%.
- Increase 10%, decrease 10% → net decrease of 1%
- Increase 20%, decrease 20% → net decrease of 4%
Reverse Percentage Problems
These are the ones that trip people up: “After a 20% increase, the price is 600. What was the original?”
The mistake: People calculate 20% of 600 = 120, then say original = 480. WRONG.
The right way: After 20% increase, new = 120% of original = 1.2 × original.
- 1.2 × original = 600
- Original = 600/1.2 = 500
Shortcut with fractions: 20% increase means the new price is 6/5 of the original.
- Original = 600 × 5/6 = 500
This fraction approach is faster. Here’s the pattern:
- 10% increase → multiply by 11/10 → to reverse, multiply by 10/11
- 20% increase → multiply by 6/5 → to reverse, multiply by 5/6
- 25% increase → multiply by 5/4 → to reverse, multiply by 4/5
- 33.33% increase → multiply by 4/3 → to reverse, multiply by 3/4
Population / Depreciation Problems
These are compound percentage change problems.
Example: A town’s population is 50,000 and grows at 10% per year. What will it be in 3 years?
- = 50000 × (1.1)³
- = 50000 × 1.331
- = 66,550
Worked Examples
Example 1: In an exam, a student scores 280 out of 400. What percentage did they score?
(280/400) × 100 = 70%. Or even faster: 280/4 = 70%
Example 2: A’s salary is 20% more than B’s. B’s salary is what percent less than A’s?
Let B’s salary = 100. Then A’s = 120. B is less than A by: (120 - 100)/120 × 100 = 20/120 × 100 = 16.67% (= 1/6)
Key insight: “20% more” and “16.67% less” describe the same relationship. They’re NOT the same number because the base changes.
Example 3: The price of sugar increases by 25%. By what percent should consumption be reduced so that expenditure remains the same?
Expenditure = Price × Consumption. If price becomes 5/4, consumption must become 4/5. Reduction = 1 - 4/5 = 1/5 = 20%
Shortcut formula: If price increases by r%, reduce consumption by r/(100+r) × 100%. = 25/125 × 100 = 20%
Example 4: In an election, candidate A gets 55% of votes. If A wins by 2400 votes, what’s the total number of votes?
A gets 55%, B gets 45%. Difference = 10% of total = 2400. Total = 2400/0.10 = 24,000
Example 5: The population of a city was 200,000 two years ago. It increased by 10% in the first year and decreased by 5% in the second year. What is the current population?
- After year 1: 200000 × 1.10 = 220000
- After year 2: 220000 × 0.95 = 209,000
Or using successive change: net change = 10 + (-5) + (10×-5)/100 = 10 - 5 - 0.5 = 4.5% increase. 200000 × 1.045 = 209,000 ✓
Common Exam Patterns
- “A is what % more/less than B?” → Find difference, divide by the BASE (B, not A)
- “After x% increase, value is Y. Find original” → Divide Y by (1 + x/100), or use fraction equivalents
- “Two successive changes” → Use a + b + ab/100 formula
- “Same expenditure, price changes” → New consumption = Old × (old price/new price)
- “Population/depreciation after n years” → Use (1 ± r/100)ⁿ formula
- “x% of A = y% of B” → A/B = y/x (the percentages flip!)
Practice Problems
Q1: If the price of an item is reduced by 30%, by what percent must consumption increase so that total expenditure increases by 5%?
Q2: Two successive discounts of 20% and 15% are equivalent to a single discount of what percent?
Q3: In an exam with 80 questions, a student answers 65 correctly. If each correct answer gives 4 marks and each wrong answer deducts 1 mark, what percentage of maximum marks did the student score?
Answers:
A1: New price = 0.7 × Old. New expenditure = 1.05 × Old expenditure. New consumption = New expenditure / New price = (1.05 × Old exp) / (0.7 × Old price) = 1.05/0.7 × Old consumption = 1.5 × Old consumption. Increase = 50%. Answer: 50%
A2: Using successive change: -20 + (-15) + (-20)(-15)/100 = -20 - 15 + 3 = -32%. Equivalent single discount = 32%.
A3: Correct = 65, Wrong = 80-65 = 15. Marks = 65×4 - 15×1 = 260 - 15 = 245. Max marks = 80×4 = 320. Percentage = 245/320 × 100 = 76.5625% ≈ 76.56%