Percentages: Successive Change & Base-Shift Traps
What it is
Percentage questions ask you to express one quantity as a fraction of another out of 100 — for increases, decreases, comparisons, and repeated changes. Both shapes turn up: a single flat percentage of a number, and a chain of two or more changes applied one after another.
The core method
Convert every percentage change into a multiplying factor. A rise of r% is a factor of (1 + r/100); a fall of r% is (1 - r/100). When a value changes more than once, MULTIPLY the factors in sequence — never add or subtract the percentages directly, because each change applies to the NEW base left behind by the previous one.
Worked example
A trader's sales rise by 25% in January, then fall by 20% (from the new, higher level) in February. What is the net percentage change in sales over the two months, compared to the original figure?
Assume the original sales = 100 (using 100 makes percentages read off directly). After the 25% rise: 100 × 1.25 = 125. After the 20% fall on this new base: 125 × 0.80 = 100. Net change = (100 − 100)/100 × 100 = 0%. A 25% rise followed by a 20% fall exactly cancels out — sales end up back where they started, NOT with a "5% gain" that naive subtraction (25 − 20) would suggest.
Common traps
- Adding or subtracting percentage changes directly instead of multiplying the retention factors — this only works for very small, single-step changes.
- Applying the second percentage to the ORIGINAL value instead of the value left after the first change.
- Confusing a percentage-point change (e.g., a rate moving from 20% to 25%, a rise of 5 points) with a percentage change (a 25% relative increase) — they are not the same number.
What the exam tests here
Across the 3 papers we hold, this skill was asked 2 times, not every sitting — 2 of 3.
What it actually asked:
- original sum from a chain of percentage deductions (2024)
- percentage of a number solved for the unknown part (2024)
Worked example 2 — chaining two rises into one figure
A college's fee rose by 15% in 2024 and by a further 20% in 2025. What single percentage increase is equivalent to the two together?
Take the 2023 fee as 100. After the first rise: 100 × 1.15 = 115. After the second, applied to the new base: 115 × 1.20 = 138. The equivalent single rise is 138 − 100 = 38%, not 35%.
The extra 3 percentage points are the 20% charged on the 15 rupees the first rise added. That product term is the whole subject: for two changes of a% and b%, the net is always a + b + ab/100. Learn the formula as a sanity check, but reach for the multiplying factors when the question chains three or more changes, because the formula does not extend cleanly.
Speed technique
Assume the starting value is 100 whenever the question asks only for a percentage — the arithmetic then reads off as rupees and you never carry an unknown.
Memorise the common fraction equivalents, because this exam's percentages are almost always chosen to be clean: 12.5% = 1/8, 16⅔% = 1/6, 20% = 1/5, 25% = 1/4, 33⅓% = 1/3, 37.5% = 3/8, 62.5% = 5/8, 75% = 3/4. "Increase 640 by 12.5%" becomes "add an eighth of 640" — 640 + 80 = 720 — in one step, with no decimal multiplication at all.
For an increase then an equal-percentage decrease, the answer is always a net loss of (r²/100)%. A 10% rise then a 10% fall is a 1% loss; 20% and 20% is 4%; 30% and 30% is 9%. Recognising this shape saves the whole calculation.
Check yourself
- A salary falls 20% and then rises 25%. What is the net change?
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Factors 0.80 × 1.25 = 1.00 — no change at all. - If 30% of a number is 72, what is 45% of it?
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The number is 240; 45% of 240 = 108. Faster: 45/30 = 1.5, so 72 × 1.5 = 108 without ever finding the number. - A price is increased by 25%. By what percentage must it now be cut to return to the original?
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Not 25%. You need 1/1.25 = 0.8, a cut of 20%. - In an election of two candidates, the winner polls 56% and wins by 2,160 votes. How many votes were cast?
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The margin is 56 − 44 = 12% of the total, so total = 2,160 / 0.12 = 18,000.
Exam framings you will actually see
- "Net effect of two changes" — chain the factors; the answer is never the sum.
- "A is x% more than B; B is what % less than A?" — not the same number. If A is 25% more than B, B is 20% less than A, because the base changed. The pair (x, 100x/(100+x)) is worth recognising: 25→20, 50→33⅓, 100→50.
- "Population/price after n years at r% growth" — a compound-interest question in disguise; use P(1 + r/100)ⁿ.
- "Passing marks" — a candidate scores 35% and fails by 40 marks; another scores 45% and passes by 20. The 10-point gap equals 60 marks, so the paper is out of 600 and the pass mark is 250.
- "Percentage of a percentage" — 20% of 30% of 500 is 0.2 × 0.3 × 500 = 30. Multiply the decimals, never add the percentages.
Each of these is one line once the base is identified, and unanswerable until it is. Identify the base first, every time.
Try it: Percentages questions
Real questions from the PGCET MBA bank on exactly this skill. Pick an answer to see the full solution — the intuition, the worked steps, the faster methods and the traps.
A stationery shop starts the day with 250 pens. By evening, 36% of them have been sold. How many pens remain unsold?
Show the answer and worked solution
Answer: option C
Only the unsold portion is asked, so find that percentage directly: sold = 36%, so unsold = 100% − 36% = 64% of the pens.
Unsold pens = 64% of 250 = () × 250.
() × 250 = 160 pens remain unsold.
So 160 pens are unsold out of the original 250 — option C.
An employee's monthly salary of ₹24,000 is increased by 15%. What is the new monthly salary?
Show the answer and worked solution
Answer: option A
A 15% rise means the new salary is 115% of the original salary, so first find the rupee increase.
Increase = 15% of 24000 = () × 24000 = 3600.
New salary = old salary + increase = 24000 + 3600.
24000 + 3600 = 27600, so the employee's new monthly salary is ₹27600 — option A.
The price of an item increases from ₹250 to ₹300. Find the percentage increase in price.
Show the answer and worked solution
Answer: option B
Percentage increase is always calculated on the ORIGINAL value: Increase = New value − Original value = 300 − 250 = ₹50.
Percentage increase = (Increase ÷ Original value) × 100.
Substitute the values: Percentage increase = (50 ÷ 250) × 100.
That gives 20% - option B.
The price of an item is ₹500. It is first increased by 20% and then the new price is decreased by 20%. What is the final price?
Show the answer and worked solution
Answer: option D
Successive percentage changes multiply one after another, they do not simply add or cancel out.
After a +20% rise: 500 × = 600.
After a −20% fall on the new price 600: 600 × = 480.
So the final price is ₹480, a net 4% fall rather than back to ₹500 — option D.
A person spends 60% of his monthly income and saves the remaining ₹2,000. What is his monthly income?
Show the answer and worked solution
Answer: option B
Savings are whatever is left after spending, so savings = 100% − 60% = 40% of the monthly income.
This 40% of income equals the given savings amount of ₹2000.
So income = 2000 × (), scaling the known 40% chunk up to the full 100%.
2000 × () = 5000, so his monthly income is ₹5000 — option B.
Answer above — every one shows its working.