Percentage Change: Successive Changes, Population, Consumption and Reverse Percentage
Percentage Change: Successive Changes, Population, Consumption and Reverse Percentage
Percentage-change questions reward one habit: turn every change into a multiplying factor and multiply. Successive salary changes, population growth, area of a square, volume of a cylinder and "find the original price" are all the same calculation in different words.
1. Core Formulas
Percentage change = (New − Old)/Old × 100
Successive changes of a% and b%: net change = a + b + ab/100 (use a minus sign for a decrease)
Equal rise and fall of r%: net change = −r²/100 % (always a loss)
| Changes | Net | Why |
|---|---|---|
| +20%, −20% | −4% | 1.2 × 0.8 = 0.96 |
| +10%, +10% | +21% | 1.1 × 1.1 = 1.21 |
| +25%, −20% | 0% | 1.25 × 0.8 = 1 |
| +50%, −33⅓% | 0% | 1.5 × 2/3 = 1 |
| −10%, −10% | −19% | 0.9 × 0.9 = 0.81 |
| +10%, +20%, −25% | −1% | 1.1 × 1.2 × 0.75 = 0.99 |
For three or more changes, multiply the factors; the a + b + ab/100 formula works only two at a time.
2. Price, Consumption and Expenditure
Expenditure = Price × Consumption
If the price rises by r%, consumption must fall by [r/(100 + r)] × 100 % to keep expenditure unchanged
If the price falls by r%, consumption can rise by [r/(100 − r)] × 100 %
| Price change | Consumption change for the same expenditure |
|---|---|
| +10% | −9 1/11% |
| +20% | −16⅔% |
| +25% | −20% |
| +33⅓% | −25% |
| −20% | +25% |
The same product rule covers speed × time (same distance), length × breadth (same area) and men × days (same work).
3. Population Growth and Depreciation
Population after n years = P × (1 + r/100)ⁿ Value after n years of depreciation = V × (1 − r/100)ⁿ
Different yearly rates: multiply the separate factors, P × (1 + r₁/100)(1 + r₂/100)...
Population n years ago = Present ÷ (1 + r/100)ⁿ
Example: a machine worth ₹1,25,000 depreciating at 20% a year is worth 1,25,000 × 0.8 × 0.8 = ₹80,000 after 2 years.
4. Area and Volume Changes
| Change | Area or volume factor | Net change |
|---|---|---|
| Side of a square +r% | (1 + r/100)² | Radius +10% gives area +21% |
| Length +a%, breadth +b% | (1 + a/100)(1 + b/100) | +30% and −30% give −9% |
| Side of a cube +r% | Surface area × (1 + r/100)², volume × (1 + r/100)³ | +20% gives SA +44%, volume +72.8% |
| Cylinder radius +a%, height +b% | Volume × (1 + a/100)² × (1 + b/100) | +20% and −20% give +15.2% |
✗ Side +20% so volume +60% | ✓ Volume grows by the cube: 1.2³ = 1.728, so +72.8%
✗ Final ₹27,600 after a 15% rise, so original = 27,600 − 15% of 27,600 | ✓ Divide by the factor: 27,600 ÷ 1.15 = ₹24,000
हिंदी नोट: लगातार प्रतिशत परिवर्तन में प्रतिशत जोड़े नहीं जाते, गुणांक (factor) गुणा किए जाते हैं। मूल मान निकालने के लिए अंतिम मान को गुणांक से भाग दीजिए, प्रतिशत घटाइए नहीं।
Exam Pointer: Verified NTPC patterns: successive salary changes (June 2026 UG CBT-1), population growth at two rates (June 2025 Graduate CBT-1), area change of a square (June 2022 CBT-2), original number from the value after a percentage fall (August 2025 UG CBT-1) and cylinder volume when radius and height change (August 2025 UG CBT-1). Price-consumption had no verified NTPC shift in our check and is tagged syllabus-based.
Pariksha Pattern: Every Way NTPC Asks This Topic
Pattern 1: Successive percentage changes
[PYQ: NTPC UG CBT-1 15-Jun-2026 Shift-1]
EXAM LEVEL
Q. The price of an article is increased by 20% and then decreased by 20%. Find the net percentage change.
20 − 20 + (20 × −20)/100 = −4. The price falls by 4%. Equal rise and fall always give a net loss of r²/100%.
Answer: 4% decrease
EXAMATLAS LEVEL
Q. A price rises by 10%, then by 20%, and then falls by 25%. If the final price is ₹4,455, find the original price and the net change.
Net factor = 1.1 × 1.2 × 0.75 = 0.99, a 1% decrease. Original = 4,455 ÷ 0.99 = ₹4,500. Adding 10 + 20 − 25 = +5% is the trap.
Answer: ₹4,500; net 1% decrease
Pattern 2: Price rise and consumption cut
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. The price of sugar rises by 25%. By what percent must a family reduce its consumption so that its expenditure on sugar does not change?
25/(100 + 25) × 100 = 20%. Check: 1.25 × 0.8 = 1.
Answer: 20%
EXAMATLAS LEVEL
Q. The price of petrol rises by 20%, and a family's monthly expenditure on petrol rises by only 8%. By what percent did the family change its petrol consumption?
Expenditure factor = price factor × consumption factor: 1.08 = 1.2 × c, so c = 0.9. Consumption fell by 10%. Subtracting 20 − 8 = 12% ignores that factors multiply.
Answer: 10% decrease
Pattern 3: Population growth and depreciation
[PYQ: NTPC Graduate CBT-1 11-Jun-2025 Shift-2]
EXAM LEVEL
Q. The population of a town is 50,000 and grows at 10% per year. Find the population after 2 years.
50,000 × 1.1 × 1.1 = 60,500.
Answer: 60,500
EXAMATLAS LEVEL
Q. A town's population rose by 10% in the first year, fell by 10% in the second year and rose by 20% in the third year, reaching 47,520. Find the population at the start.
Net factor = 1.1 × 0.9 × 1.2 = 1.188. Start = 47,520 ÷ 1.188 = 40,000. Note that +10% then −10% is not zero; the factor is 0.99, a 1% fall.
Answer: 40,000
Pattern 4: Percentage change in area and volume
[PYQ: NTPC CBT-2 16-Jun-2022 Shift-2]
EXAM LEVEL
Q. The radius of a circle increases by 10%. By what percent does its area increase?
Area ∝ r², so the factor is 1.1² = 1.21, an increase of 21%. Formula check: 10 + 10 + 100/100 = 21.
Answer: 21%
EXAMATLAS LEVEL
Q. Each edge of a cube increases by 20%. Find the percentage increase in its total surface area and in its volume.
Surface area ∝ edge²: 1.2² = 1.44, so +44%. Volume ∝ edge³: 1.2³ = 1.728, so +72.8%. Using 20 × 3 = 60% for the volume is the trap.
Answer: Surface area +44%; volume +72.8%
Pattern 5: Original value from the final value
[PYQ: NTPC UG CBT-1 12-Aug-2025 Shift-2 | NTPC 2025-26 cycle]
EXAM LEVEL
Q. After a 15% increase, a salary becomes ₹27,600. Find the original salary.
Original = 27,600 ÷ 1.15 = ₹24,000.
Answer: ₹24,000
EXAMATLAS LEVEL
Q. A man's salary is cut by 20%, and the reduced salary is then raised by 20%. He now gets ₹384 less than his original salary. Find his original salary.
Net factor = 0.8 × 1.2 = 0.96, so he loses 4% of the original. 4% = 384 gives the original salary = ₹9,600.
Answer: ₹9,600
Pattern 6: Keeping a product constant and two-dimension changes
[PYQ: NTPC UG CBT-1 11-Aug-2025 Shift-3]
EXAM LEVEL
Q. If x increases by 25%, by what percent must y decrease so that the product xy remains the same?
The new y-factor must be 1/1.25 = 0.8, a 20% decrease. Same as the price-consumption rule: 25/125 × 100.
Answer: 20%
EXAMATLAS LEVEL
Q. The radius of a cylinder is increased by 20% and its height is decreased by 20%. Find the percentage change in its volume.
Volume ∝ r²h, so the factor = 1.2² × 0.8 = 1.44 × 0.8 = 1.152. The volume increases by 15.2%. Treating radius and height symmetrically (+20 − 20 = 0) misses that the radius is squared.
Answer: 15.2% increase
60-Second Revision
- Two changes: a + b + ab/100; three or more: multiply the factors.
- Equal rise and fall of r%: net loss of r²/100%.
- Price up r%: consumption down r/(100 + r) × 100 % for the same spend.
- Population P(1 + r/100)ⁿ; depreciation V(1 − r/100)ⁿ; past value: divide by the factor.
- Area ∝ length², volume ∝ length³; cylinder volume ∝ r²h.
- Original from final: divide by the factor, never subtract the percentage.
Next Step: Percentage done. Solve the percentage questions in the ExamAtlas RRB NTPC 2026 mock tests and topic tests next; every slip you make there will match one of the 13 patterns on these two pages.