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Percentages
CAT'26 QUANT CHEATSHEET
Every percentage formula and CAT shortcut you need for CAT 2026 — on one page.
Percentages sit underneath almost every other CAT quant topic, so the marks here compound: get the base-value habit right once, and profit & loss, interest, and data interpretation all get faster too. The exam rarely tests a plain percentage; it hides successive changes, reverse calculations, and "x% more than" comparisons that quietly swap the base value on you. This sheet lays out every formula you need for the CAT quant section: fraction-percentage conversions, increase and decrease, successive change, reverse percentage, and the population and comparison traps examiners keep reusing, each with a worked example in real numbers. Keep it open while you practise, and after a mock check where you stand on the CAT score predictor to see which idea is costing you marks.
Dev note: the canonical route
/cheatsheets/optima-learn-percentages-cheatsheet is not live yet
(no /cheatsheets hub in the current sitemap). Built production-ready as a drop-in once it ships.Percentages: every formula you need
1Core Idea
Percentage means “per hundred” — a common scale for comparison.
Percentage = (Part ÷ Whole) × 100
Example: 50 girls out of 200 students → (50/200)×100 = 25%.
CAT Hack: Convert percentages to fractions whenever possible — fractions are usually faster to compute with.
2Fraction ↔ Percentage Conversions
The common equivalents, worth memorising outright.
| Fraction | % | Fraction | % |
|---|---|---|---|
| 1/2 | 50% | 3/5 | 60% |
| 1/3 | 33.33% | 4/5 | 80% |
| 2/3 | 66.67% | 1/8 | 12.5% |
| 1/4 | 25% | 5/8 | 62.5% |
| 3/4 | 75% | 7/8 | 87.5% |
| 1/5 | 20% | 2/5 | 40% |
Example: 40% of a class of 250 = 2/5 × 250 = 100 students.
CAT Favourite: Memorise these conversions — they save real time versus computing each one by hand.
3Finding a Percentage
The direct calculation, plus quick mental shortcuts.
x% of y = (x ÷ 100) × y
Example: 15% of 240 = (15/100)×240 = 36. Shortcut: 10%=÷10, 5%=half of 10%, 1%=÷100.
4Percentage Increase
Scale the old value up by a growth multiplier.
New Value = Old Value × (1 + x/100)
Example: ₹50,000 salary up 20% → 50000×1.2 = ₹60,000 (increase = ₹10,000).
Common Mistake: A 20% increase is not a flat ₹20 — always calculate the increase from the base value.
5Percentage Decrease
Scale the old value down by a shrink multiplier.
New Value = Old Value × (1 − x/100)
Example: ₹800 price down 25% → 800×0.75 = ₹600 (reduction = ₹200).
6Successive % Change (Two Increases)
Combine two successive increases into one net change.
Net Change = a + b + (ab ÷ 100)
Example: two 10% and 20% increases → 20+10+(20·10)/100 = 32% net (multiplier 1.2×1.1=1.32).
Common Mistake: 20% + 10% is not 30% — the actual combined increase here is 32%.
7Successive Increase & Decrease
One increase, one decrease, combined in one shot.
Net Change = a − b − (ab ÷ 100)
Example: +20% then −10% → 20−10−(20·10)/100 = +8% net increase.
CAT Hack: Think in fractions, not raw percentages — 20% and 10% become 1/5 and 1/10 and multiply cleanly.
8Equivalent Percentage Changes
Percentages recast as clean multiplying fractions.
| Increase | Equivalent Fraction |
|---|---|
| 20% | 1/5 |
| 25% | 1/4 |
| 33.33% | 1/3 |
| 50% | 1/2 |
| 100% | Double |
Example: a 25% increase means the new value is 5/4 of the old value.
9Reverse Percentage
Work backward from the final value to the original.
Original = A ÷ (1 + x/100)
Example: price is ₹1000 after a 25% increase → 1000/1.25 = ₹800 original.
10Percentage Change
The standard formula for any change, expressed as a percentage.
% Change = (Change ÷ Original) × 100
Example: price rises from 100 to 120 → (20/100)×100 = 20%.
11Population / Salary / Sales
A rise and an equal-percentage fall never cancel out.
Two successive equal-magnitude opposite changes leave a net loss
Example: population 1000 → +10% → 1100 → −10% → 990: net change = −1%, not 0%.
CAT Favourite: A CAT favourite: equal percentage increase and decrease never cancel — there is always a net loss.
12Percentage Comparison
“x% more than” and “x% less than” are not symmetric.
A is x% more than B → A = B × (1 + x/100)
Example: A is 25% more than B → A=1.25B, so B is (25/125)×100 = 20% less than A, not 25%.
Common Mistake: 25% more is not the same as 25% less — always convert back through the ratio, never assume symmetry.
CAT exam shortcuts, traps & revision
13
CAT Exam Shortcuts
- 10% = ÷10; 1% = ÷100; 50% = half; 25% = one-fourth
- 12.5% = one-eighth; 33.33% = one-third; 66.67% = two-thirds
- Two increases: Net = a+b+ab/100; increase & decrease: Net = a−b−ab/100
- Reverse percentage: Original = A ÷ (1+x/100)
- Equal % increase and decrease never cancel to zero
- “x% more than” and “x% less than” use different base values
14
Most Common CAT Traps
- Adding successive percentage changes directly: 20% + 10% is not 30% (actual is 32%).
- Assuming +20% and −20% cancel out to zero (they leave a net −4%).
- Confusing “A is x% more than B” with “B is x% less than A” — the percentages differ.
- Calculating a percentage increase as a flat amount instead of from the base value.
- Forgetting to reduce a fraction from the memorised table, and computing it from scratch instead.
15
30-Second Revision Box
- Percentage = Part ÷ Whole × 100
- Memorise the fraction ↔ percentage conversions
- Increase/decrease: multiply by (1±x/100)
- Successive change: a+b+ab/100 (both up) or a−b−ab/100 (up then down)
- Reverse percentage: divide by the growth factor, don't subtract
- Equal increase & decrease ≠ zero; comparisons are not symmetric
Percentages reward thinking in fractions over raw percentage arithmetic — once 20% becomes 1/5 and 25% becomes 1/4, successive changes and comparisons multiply cleanly instead of needing careful addition. Drill this sheet until the fraction conversions and the successive-change formula are reflex, then test them on full sets and track progress with the CAT score predictor. It pairs directly with the Interest cheat sheet, since growth factors are the same idea in both. For more guides, browse the Optima Learn blog or explore every study guide, work through the CAT exam hub, and when you want mentor-led prep, book a free CAT 2026 call.
