4.2 Percentages
Key Takeaways
- Percentage = (part / whole) × 100; percentage change = (change / original) × 100 — always divide by the ORIGINAL value, not the new one.
- A rise of x% followed by a fall of x% never returns to the start: net change = −(x²/100)%. A 20% rise then 20% fall yields a 4% net decrease.
- Successive percentage changes multiply as factors: (1 + r₁)(1 + r₂)… — a 10% rise then 20% rise gives 1.10 × 1.20 = 1.32, a 32% increase.
- Reverse percentage: original = new value / (1 ± change fraction). If a price after 25% increase is Rs 250, original = 250 / 1.25 = Rs 200.
- Memorise the fraction-equivalent table — 12.5% = 1/8, 20% = 1/5, 25% = 1/4, 50% = 1/2 — to convert percentages to fractions in seconds.
4.2 Percentages
Quick Answer: A percentage is a fraction out of 100. For any change question, divide by the original value, not the new one. Successive changes multiply as decimal factors; a rise then an equal fall always nets a loss.
Percentage problems appear in roughly one out of every six verbal-intelligence items on the PAF GD Pilot Initial Test. They are quick wins if you internalise three moves: the basic percentage of a quantity, the successive-change multiplier, and the reverse-percentage reset.
1. Percentage of a Quantity
To find x% of N: value = (x / 100) × N.
Example: 15% of 240 = (15/100) × 240 = 0.15 × 240 = 36.
Use the fraction table below to convert percentages to mental fractions: 25% of 240 = 240/4 = 60 — no multiplication needed.
2. Percentage Increase and Decrease
percentage change = (change / original) × 100. The denominator is always the original value.
Example: A candidate's score rises from 80 to 92. Change = 12. Percentage increase = (12 / 80) × 100 = 15%. The common error is dividing by 92 — that gives 13.0%, which is wrong.
3. Successive Percentage Changes — the Multiplier Method
Convert each change to a decimal factor and multiply:
- 10% increase → factor 1.10
- 10% decrease → factor 0.90
- 25% increase → factor 1.25
Example: A salary rises 10%, then rises 20% more. Net factor = 1.10 × 1.20 = 1.32 → 32% net increase, not 30%.
4. The Rise-Then-Fall Trap
A rise of x% followed by a fall of x% (same percentage) does not return to the original. The net factor is (1 + x/100)(1 − x/100) = 1 − x²/10000, a net decrease of x²/100 percent.
Example: A price rises 20% then falls 20%. Net factor = 1.20 × 0.80 = 0.96 → 4% net decrease. A Rs 100 item ends at Rs 96, not Rs 100.
5. Reverse Percentage — Finding the Original
Given the new value after a known change, recover the original by dividing by the factor.
Example: After a 25% increase, a watch costs Rs 250. Original = 250 / 1.25 = Rs 200.
Example: After a 20% discount, a jacket sells for Rs 1600. Original = 1600 / 0.80 = Rs 2000.
Fraction-Percentage Equivalents (Memorise)
| Fraction | Percentage | Decimal |
|---|---|---|
| 1/2 | 50% | 0.50 |
| 1/3 | 33.33% | 0.333… |
| 1/4 | 25% | 0.25 |
| 1/5 | 20% | 0.20 |
| 1/6 | 16.67% | 0.166… |
| 1/8 | 12.5% | 0.125 |
| 1/10 | 10% | 0.10 |
| 3/8 | 37.5% | 0.375 |
| 5/8 | 62.5% | 0.625 |
With these nine rows in your head, most PAF percentage items become one-step mental arithmetic.
Worked Example: Successive Discounts
A store marks a shirt at Rs 1200, then offers 20% off, then an additional 25% off the discounted price. Final price?
- After 20% off: 1200 × 0.80 = Rs 960.
- After 25% off: 960 × 0.75 = Rs 720.
Net factor = 0.80 × 0.75 = 0.60 → a 40% total discount. Rs 1200 × 0.60 = Rs 720. Note: 20% + 25% = 45% is wrong — successive discounts multiply, they do not add.
Worked Example: Price–Consumption–Expenditure
A household's monthly expenditure on flour is Rs 2400 at Rs 60/kg. If the price rises by 20% and the household keeps expenditure constant, what is the new consumption?
Step 1 — Original consumption = expenditure / price = 2400 / 60 = 40 kg. Step 2 — New price = 60 × 1.20 = Rs 72/kg. Step 3 — New consumption = 2400 / 72 = 33.33 kg (a 16.67% drop).
The trap: candidates compute 20% of 40 kg = 8 kg and subtract to get 32 kg — wrong, because consumption falls by the reciprocal factor, not by the same percentage. When expenditure is constant, consumption ∝ 1/price; a 20% price rise (factor 1.20) cuts consumption by 1 − 1/1.20 = 1/6 ≈ 16.67%.
Percentage Points vs Percentage Change
If a pass rate rises from 40% to 55%, the change is 15 percentage points, but the percentage change is (15/40) × 100 = 37.5%. The PAF exam may offer both "15%" and "37.5%" as options to catch candidates who conflate the two.
Worked Example: Two Successive Reverse Changes
A cadet's test score after a 20% increase is 84. It is then adjusted downward by 10%. What was the original score, and what is the adjusted score?
- Original = 84 / 1.20 = 70.
- After 10% downward adjustment: 84 × 0.90 = 75.6.
- Cross-check from original: 70 × 1.20 × 0.90 = 70 × 1.08 = 75.6 ✓.
The net effect of +20% then −10% is the factor 1.20 × 0.90 = 1.08, an 8% increase on the original — not +10% (the arithmetic of +20 − 10).
Common Traps
- Adding percentages. Two successive 50% increases give 1.5 × 1.5 = 2.25, a 125% increase — not 100%.
- Dividing by the new value. The denominator for percentage change is the original.
- Confusing "of" with "off". "20% of 200" = 40; "20% off 200" = 160.
- Reverse-percentage shortcut temptation. Taking 20% of 250 (= 50) and subtracting gives 200, which happens to work for an increase, but adding 20% of 250 (= 50) to recover the original of a discounted price is wrong — always divide by the factor.
- Successive discount ≠ single equivalent discount added. A 30% discount followed by 20% gives factor 0.70 × 0.80 = 0.56, a 44% total discount — not 50%. Always multiply factors, never add the percentages.
A pilot's monthly flight hours increase by 25% in January, then decrease by 20% in February. What is the net percentage change from the original?
An engine's output rises from 800 N to 920 N. What is the percentage increase?
After a 10% discount, a helmet costs Rs 2700. What was the original price?
A train's speed is increased by 20% and then by a further 25%. What is the single equivalent percentage increase?