Financial Analysis puzzles, solved step by step

Puzzles: 100
Traced to a firm: 47
Topics: 13
Hard: 30

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009A company's EBITDA margin moves from 8% to 10%. Is that up 2% or up 25%? And a bank's gross NPA ratio falls from 4% to 3%: how would you describe that change in a results note?Percentages and averagesWarm upCorporate FP&ARating agencies
Try it first
Which phrase belongs in a results note about the margin moving from 8% to 10%?
Show the worked solution
Both are true, but say 2 percentage points first. A margin is already a percentage, so a change in it is measured in points: 8% to 10% is up 2 points, or 200 basis points, and up 25% relative to where it started. The NPA ratio fell 1 percentage point, or 100 basis points, which is a 25% relative fall. Use points for a rate and per cent for an amount.
Why can one change honestly be called both 2 and 25?
Think of a student whose exam score goes from 40% to 50%. She scored 10 more marks out of 100, so she is up 10 points. Her score is also a quarter higher than before, up 25%. A percentage point measures the gap between two rates; a per cent measures that gap against the starting rate. Neither is wrong. The confusion comes from writing the same % sign for both, so the reader cannot tell which one you meant.
The margin moving from 8% to 10% is a rise of 2 percentage points, or 200 basis points, and 25% in relative terms. The NPA ratio falling from 4% to 3% is a fall of 1 percentage point, or 100 basis points, and a 25% relative fall.
Which label should lead, and when does the relative one matter?
Lead with points, because that is the convention in results notes, credit reports and board packs. A basis pointOne hundredth of a percentage point. 100 basis points equal 1 percentage point. is one hundredth of a point, which helps when moves are small: margin up 200 bps. The relative change matters when the rate drives an amount. On flat revenue of Rs 1,000 crore, EBITDA goes from Rs 80 crore to Rs 100 crore, which is a 25% rise in profit. So the 2 points and the 25% are describing two different things: the margin, and the profit it produces.
The relationship
$\Delta_{\text{pp}} = 10\% - 8\% = 2 \text{ pp} \qquad \Delta_{\text{rel}} = \frac{10\% - 8\%}{8\%} = 25\%$
pp percentage points, the plain gap between two rates
rel the relative change, the gap divided by the starting rate
What it says in wordsSubtract for the change in points; divide that by the start for the change in per cent.
For the NPA ratio, a falling ratio is good news, so say it plainly: gross NPA down 1 percentage point to 3%, or down 100 basis points. You can add that the ratio is a quarter lower. Say the limit too: a lower ratio can also come from loans growing faster than bad loans, so check the rupee amount of NPAs before calling it an improvement in asset quality.
Where candidates lose it
The loss is writing margin up 2% in a results note or saying it in an interview. A reader who takes you literally hears a move from 8% to 8.16%, and a credit reader will mark it as sloppy at once.
The second loss is choosing the bigger number to sound better: up 25% sounds stronger than up 2 points. Interviewers listen for whether you pick the label that fits, then offer the other as context.
What the interviewer asks next
Interest rates rise from 6.50% to 6.75%. How many basis points is that, and what is the relative change?
A bank's gross NPA ratio falls but its NPAs in rupees rise. How is that possible?
Margin is flat at 10% while revenue grows 20%. What happened to EBITDA?
011You have twelve coins that look identical. One is counterfeit and is either heavier or lighter than the rest; you do not know which. Using a balance scale only three times, find the counterfeit and say whether it is heavy or light.Logic and counting brainteasersWarm upConsulting-style caseBank credit
Try it first
How many coins go on each pan in the first weighing?
Show the worked solution
Weigh four against four with four left off, then use known good coins to mix the suspects. If the first weighing balances, the fake is among the four left off, and two more weighings find it. If it tips, the fake is one of four possibly heavy or four possibly light coins; weighing 1, 2 and 5 against 3, 6 and a good coin splits those eight into groups of three or fewer, and the third weighing names the coin and its direction.
How do you know three weighings can be enough?
Count the possible answers before touching a coin. Any of 12 coins could be the fake, and it could be heavy or light, so there are 24 answers. A balance has three outcomes, left down, right down or level, so three weighings give 3 x 3 x 3 = 27 outcomes. Twenty four answers fit inside twenty seven outcomes, but only just, so every weighing must split the remaining possibilities into three nearly equal groups. Think of the game of twenty questions: a question with three possible answers is only worth asking if all three answers are about equally likely.
Why four against four, and what happens next?
Six against six can never balance, which throws away one of the three outcomes. Four against four with four left off splits the 24 answers into 8, 8 and 8. If it balances, the eight coins on the scale are proven good, and you can use them as reference weights in the next weighing. Weigh three of the suspects, 9, 10 and 11, against three good coins. Level means coin 12 is fake and one last weighing against a good coin says heavy or light. A tip tells you both that the fake is among 9 to 11 and which way it is off, so weighing 9 against 10 finishes it.
Weighing four coins against four splits the 24 possible answers into three groups of 8. Each branch then uses a second weighing that mixes suspects with known good coins, so every one of the 24 answers ends at a single coin and a heavy or light verdict after three weighings.
If the first weighing tips with the left side heavy, the fake is one of 1 to 4 and heavy, or one of 5 to 8 and light. Now move coins between pans. Weigh 1, 2 and 5 against 3, 6 and good coin 9: each outcome leaves at most three suspects, each with a known direction. Level leaves 4 heavy, 7 light or 8 light, and weighing 7 against 8 settles it. Left heavy leaves 1 heavy, 2 heavy or 6 light; weigh 1 against 2. Left light leaves 3 heavy or 5 light; weigh 3 against a good coin.
The relationship
$2N \le 3^{n} - 3 \quad\Rightarrow\quad N \le \frac{3^{3}-3}{2} = 12$
N number of coins
n number of weighings, here 3
2N possible answers: each coin, heavy or light
What it says in wordsThree weighings can handle at most twelve coins when you must also say heavy or light, so twelve is the hardest version that still works.
Where candidates lose it
The usual loss is opening with six against six, or splitting into halves out of habit. The scale tips, you have twelve candidates and two weighings left, and nine outcomes cannot cover them.
The second loss is keeping the suspects in fixed groups after the first weighing. The solution only works because you move coins between pans and bring in coins already proven good. Say that idea out loud even if the details of a branch take you a moment.
What the interviewer asks next
What if you only had to find the fake, not say whether it is heavy or light?
With four weighings, how many coins could you handle?
If you are told the fake is heavier, how many coins can three weighings handle?
019A plant's output grows 10% every month. Roughly how many months until output doubles, and how long until it is eight times today's level? Use the rule of 72, then check it against the exact answer.Compounding and time valueWarm upConsulting-style caseCorporate FP&A
Try it first
How long until output is eight times today's level?
Show the worked solution
Output doubles in about 7.2 months and reaches eight times in about 21.6 months, by the rule of 72. Divide 72 by the growth rate in per cent: 72 over 10 is 7.2 months per doubling, and eight times is three doublings. The exact answer, the log of 2 over the log of 1.1, is 7.27 months, or 21.8 months for eight times, so the rule is within a fifth of a month.
Why does eight times take three doublings and not 70 months?
Think of a rumour that doubles the number of people who know it every week. One becomes two, then four, then eight in three weeks, not eight weeks. Compounding growth multiplies, so the question to ask is how many doublings fit in, not how many 10% steps add up to 700%. Two times two times two is eight, so eight times today's output is three doublings away. Dividing 700% by 10% a month treats each month's growth as if it were earned only on the starting level.
The relationship
$t_{2x} = \frac{\ln 2}{\ln 1.10} = \frac{0.693}{0.0953} = 7.27 \qquad \text{rule of 72: } \frac{72}{10} = 7.2$
ln 2 the natural log of 2, about 0.693
ln 1.10 the natural log of one month's growth factor, about 0.0953
72 a convenient number close to 100 times ln 2, with many divisors
What it says in wordsDoubling time is the log of two divided by the log of one period's growth factor; the rule of 72 approximates that with one division.
At 10% a month, output reaches twice today's level after 7.27 months, four times after 14.5 and eight times after 21.8. The rule of 72 marks at 7.2, 14.4 and 21.6 months sit just short of each, so the shortcut is close enough to say out loud.
Where does the rule of 72 work, and where does it slip?
The exact doubling time uses the log of 1 plus the rate, which is a little less than the rate itself, and 72 is chosen to compensate for typical rates. The rule is within about 1% for rates from 5% to 10% a period and drifts at the extremes: at 2% it overstates, and at 50% it understates. At 10% the gap is only 0.07 months. A check in whole months helps too: 1.1 to the 7th is 1.95 and to the 8th is 2.14, so output passes double during month 8.
Growth per period Rule of 72 Exact Gap
2% 36.00 35.00 +2.8%
5% 14.40 14.21 +1.4%
10% 7.20 7.27 -1.0%
25% 2.88 3.11 -7.3%
50% 1.44 1.71 -15.8%
Periods to double, with the rule's error as a share of the exact answer. It is within about 1% at 5% and 10% a period, and drifts further at very low and very high rates.
Say the limit in context. Output growing 10% a month for two years is a startup or a ramp-up, not a steady state; a plant hits capacity, demand or cash constraints long before it has grown nearly tenfold. The interviewer wants the arithmetic, then one sentence of business sense.
Where candidates lose it
The common loss is answering about 70 months for eight times, treating the growth as simple rather than compound. It is the same instinct that underestimates how quickly compounding builds.
The second loss is reaching for a calculator-style log in your head and stalling. Give the rule of 72 answer first, then say the exact number is a little longer, about 7.3 months.
What the interviewer asks next
Using the same idea, how long does it take for prices to halve in value at 6% inflation?
Why do some people use 69 or 70 instead of 72?
Output grows 10% a month for a year. What is the annual growth rate?
022Estimate how many people work in a 30-storey office tower in Mumbai.Estimation and market sizingWarm upHSBCCentral · 2026
Try it first
Which order of magnitude feels right?
Show the worked solution
About 5,100 people, as a range of roughly 3,400 to 5,100. Take a floor plate of 25,000 square feet. After lifts, stairs and services about 80% is usable, 20,000 square feet. At 100 square feet a seat that is 200 seats, and at 85% occupancy about 170 people a floor. Thirty floors gives about 5,100. Roomier offices at 150 square feet a seat bring it down to about 3,400.
Why start from one floor?
Think of estimating the people in a cinema: you count the seats in one row, then the rows. A tower is the same: thirty copies of one floor, so estimate one floor carefully and multiply, because every assumption is easier to defend on a single floor you can picture. Define the answer too: people who work in the building, meaning assigned staff, not visitors and not the count present on any one day.
What goes into one floor?
Four assumptions, each said out loud. A floor plate of about 25,000 square feet, typical for a large commercial tower. About 20% of that is the coreThe central block of a floor holding lifts, stairs, toilets, shafts and service rooms, which cannot be used for desks., leaving 20,000 usable. At 100 square feet a seat, including aisles and meeting rooms, the floor holds 200 seats, and with 85% of seats assigned that is about 170 people. The rest are vacant desks, new hires not yet in, or space held for growth.
The relationship
$N = \frac{25{,}000 \times 0.80}{100} \times 0.85 \times 30 = 200 \times 0.85 \times 30 \approx 5,100$
25,000 floor plate, square feet
0.80 usable share after the core
100 square feet per seat, aisles and meeting rooms included
0.85 share of seats occupied
30 floors
What it says in wordsSeats on one floor, times the share filled, times the number of floors.
One 25,000 square foot floor, 80% usable at 100 square feet a seat, holds 200 seats, of which 170 are occupied. Thirty such floors put about 5,100 people in the tower.
Which assumption would you flex, and how would you check it?
Seat density moves the answer most. A dense back office runs near 100 square feet a seat; a bank's front office with larger desks and more meeting rooms nearer 150. At 150 square feet a seat the tower holds about 3,400 people, so give the answer as about 3,400 to 5,100. Then trim for floors that hold no desks: if the lobby and a plant floor take two of the thirty, the dense case falls to about 4,760. A check from another side: assume a dozen lifts, each carrying about 15 people a trip on a round trip of about 3 minutes. That moves about 12 x 15 x 20, or 3,600 people an hour, which fits a building of a few thousand people arriving over a morning, not tens of thousands.
Where candidates lose it
The common loss is guessing the whole building in one leap, which gives answers anywhere from 1,000 to 50,000 with nothing to defend. One floor, four assumptions, then multiply.
The second loss is forgetting the core and the empty desks, which overstates the answer by a third. Saying both out loud is what the interviewer marks.
What the interviewer asks next
How many lifts does the tower need to get everyone in between 8.30 and 9.30?
With hybrid work, how many people are in the building on a typical Friday?
What would you charge a coffee chain for the ground floor, and how would you justify it?
Asked at HSBC, Sales and Trading, Central, 2026 (Wall Street Oasis): How many employees in London hsbc building How many taxis are in HK central
023A customer who owes your company Rs 50 lakh goes bankrupt and will pay nothing. Walk the write-off through the three statements at a 25% tax rate, first when no provision was held against the debt, then when a full provision was booked last year.Accounting flow riddlesWarm upBig FourCorporate FP&A
Try it first
A full provision for this debt was booked last year. How much does this year's net income fall when the debt is written off?
Show the worked solution
Without a provision, net income falls Rs 37.5 lakh; with a full provision already booked, it does not move. With no provision, the Rs 50 lakh is an expense, tax falls Rs 12.5 lakh, and net income falls 37.5. Cash rises 12.5 from lower tax, receivables fall 50 and retained earnings fall 37.5. With a provision booked last year, the write-off only removes the debt and the provision together, so profit and cash are unchanged.
What happens when nothing was set aside?
Think of a shopkeeper with a customer who has run up a tab and then left town. The money was counted as owed; now it never will be. Writing off a debt with no provision turns an asset into an expense in one step: receivables fall Rs 50 lakh and a bad debt expense of Rs 50 lakh hits profit. Assuming the write-off is tax deductible, tax falls Rs 12.5 lakh, so net income falls Rs 37.5 lakh. On the cash flow statement, add back the Rs 50 lakh because no cash moved; cash from operations rises Rs 12.5 lakh, the tax saved. The balance sheet balances: assets fall 37.5 (receivables down 50, cash up 12.5) and retained earnings fall 37.5.
With no provision held, writing off Rs 50 lakh cuts net income by Rs 37.5 lakh after tax and raises cash by Rs 12.5 lakh of tax saved. With a full provision booked last year, the write-off removes the debt and the provision together, so neither profit nor cash moves this year.
Why does the provision case show nothing this year?
Because the loss was already recognised. A provision for doubtful debtsAn amount set aside against receivables the company expects not to collect. It reduces net receivables on the balance sheet and is charged to profit when it is created. is booked when a loss becomes likely, and that is when profit takes the hit. Accounting recognises a loss when it becomes probable, not when the customer finally fails, so the write-off later is a tidy-up between two balance sheet lines. Gross receivables fall Rs 50 lakh, the provision against them falls Rs 50 lakh, and net receivables are unchanged. Under Ind AS 109 and IFRS 9 companies provide for expected credit losses in advance, which is this logic applied to a whole loan book.
The relationship
$\Delta NI = -50 \times (1 - 0.25) = -37.5 \qquad \Delta\text{Cash} = +50 \times 0.25 = +12.5$
50 the debt written off, Rs lakh
0.25 tax rate
ΔNI change in net income with no provision held
What it says in wordsWithout a provision, profit falls by the after-tax loss and cash rises by the tax saved.
What about tax in the provision case?
This is the follow-up that separates good answers. In many tax systems a general provision is not deductible until the debt is actually written off, so the tax saving arrives now even though the expense was booked last year. The books handle that with a deferred tax asset created last year and reversed this year: cash tax falls Rs 12.5 lakh now, with no effect on this year's profit. Treat the rule as something to confirm for the jurisdiction where the company files tax, rather than a fixed fact.
Where candidates lose it
The common loss is putting the Rs 50 lakh through profit again in the provision case, which counts the same loss twice: once when the provision was made and again on write-off.
The second loss is in the no-provision case: saying cash falls by 50. No cash moved when the customer failed; the cash was never received. The only cash effect is the tax saved.
What the interviewer asks next
The provision booked last year was only Rs 30 lakh. Walk the write-off through now.
The customer later pays Rs 10 lakh after all. What happens in each statement?
How would a rising provision show up in a company's cash flow statement?
033Convert to percentages in your head, quickly: 3/32, 5/16 and 7/64.Mental mathsWarm upTreasuryCorporate FP&A
Try it first
Which of these is 7/64 as a percentage?
Show the worked solution
3/32 is 9.375%, 5/16 is 31.25% and 7/64 is 10.9375%. Anchor on 1/8 = 12.5% and halve: 1/16 is 6.25%, 1/32 is 3.125% and 1/64 is 1.5625%. Then multiply the rung by the numerator, or step from a nearby eighth: 7/64 is 1/8 less 1/64, which is 12.5 minus 1.5625.
Why should powers of two be easy?
Think of cutting a pizza: halve it, halve the halves, halve again, and you have eighths without measuring anything. Fractions with 2, 4, 8, 16, 32 or 64 at the bottom work the same way. Every power-of-two fraction is 12.5% halved some number of times, so you only need one anchor and the habit of halving. 1/8 is 12.5%, 1/16 is 6.25%, 1/32 is 3.125% and 1/64 is 1.5625%. The digits 125, 625, 3125, 15625 repeat in every problem of this kind.
Halving 12.5% gives 6.25%, 3.125% and 1.5625% for sixteenths, thirty-seconds and sixty-fourths, so 3/32 is three rungs of 3.125% or 9.375%, 5/16 is five rungs of 6.25% or 31.25%, and 7/64 is 12.5% less one rung of 1.5625%, or 10.9375%.
When should you multiply up, and when should you step down from an eighth?
Small numerators multiply cleanly: 3 x 3.125 is 9.375 and 5 x 6.25 is 31.25. When the numerator sits one below a multiple of the anchor, subtract instead: 7/64 is 8/64 less 1/64. That turns a seven-times table of 1.5625 into one subtraction, 12.5 less 1.5625, which is 10.9375%. Checking 3/32 the same way, 4/32 is 12.5%, less 3.125% is 9.375%, which matches. Two routes agreeing is the habit that catches a slipped digit.
Where does this show up on a desk? Some bond prices are quoted in thirty-seconds of a point, so 3/32 of a point is 0.09375. Ownership splits and waterfall percentages often come in halves and quarters of halves. And an interviewer who asks this is really watching how fast you find a structure instead of reaching for longhand division.
Where candidates lose it
Candidates start long division, 3 divided by 32, and lose the decimal place or the tempo. The question is timed by feel: the interviewer wants an answer in a few seconds, and only the halving ladder gets you there.
The second loss is rounding: 10.9% for 7/64 is fine as an estimate, but if the interviewer asked for the exact figure, it is 10.9375%, and the ladder gives it to you exactly.
What the interviewer asks next
What is 11/64 as a percentage?
A bond is quoted at 98 and 5/32. What is that in decimal?
What is 1/128, and how do you get it from the ladder?
038You have a 3-litre bottle, a 4-litre bottle and a tap. Neither bottle has markings. How do you measure exactly 2 litres?Logic and counting brainteasersWarm upNomuraNew York · 2026
Try it first
What is the fewest number of fills and pours that leaves exactly 2 litres?
Show the worked solution
Fill the 3-litre bottle, pour it into the 4, fill the 3 again and pour until the 4 is full: 2 litres are left in the 3-litre bottle. The 4 already holds 3 litres, so it takes only 1 more. That is four moves. A longer route fills the 4 first and ends with 2 litres in the 4-litre bottle after six moves.
How do you avoid getting lost halfway through?
Think of following a recipe that says pour half the milk into the other pan: if you do not write down how much is in each pan, two steps later you have no idea. Track the pair of levels after every single move, written as (3-litre, 4-litre), and the puzzle becomes a short walk through a few states instead of a juggling act. There are only three kinds of move: fill a bottle from the tap, empty a bottle, or pour one into the other until the source is empty or the target is full.
Filling the 3-litre bottle, pouring it into the 4, filling the 3 again and topping up the 4 leaves exactly 2 litres in the 3-litre bottle after four moves, because the 4-litre bottle had room for only 1 more litre.
Say the states out loud as you go: (0, 0), (3, 0), (0, 3), (3, 3), (2, 4). The interviewer can check each one in a second, and if you slip, you can see where.
Which amounts can these two bottles measure at all?
Every pour moves water in steps of 3 and 4, so the amounts you can isolate are the combinations of 3 and 4 with whole numbers in front: 4 minus 3 is 1, 3 times 2 minus 4 is 2, and so on. Because 3 and 4 share no common factor bigger than 1, these bottles can measure every whole number of litres from 1 to 4, and up to 7 if you count both bottles together. With a 4-litre and a 6-litre bottle you could never get an odd number, since every amount would be a multiple of 2. Saying that rule turns a party trick into reasoning.
Where candidates lose it
Candidates start pouring in their head without writing states and lose track after three moves, then restart. It looks like panic even when it is not. Say the pair of numbers after each move and the interviewer follows you.
The second loss is stopping at a long route without checking for a short one. Mentioning that you found 2 litres in four moves, and that a six-move route also works, shows you looked for the efficient answer.
What the interviewer asks next
With the same bottles, how do you measure exactly 1 litre?
With a 6-litre and a 9-litre bottle, can you measure 4 litres?
How would you prove that a given amount cannot be measured?
Asked at Nomura, Equity Capital Markets, New York, 2026 (Wall Street Oasis): How much water can you fill using 1 3liter and 1 4liter bottle using each other?
040Receivables are Rs 300 crore at quarter end and revenue for the quarter was Rs 450 crore. A junior analyst reports days sales outstanding of 243 days. What went wrong, and what is the right figure?Working capital and cash riddlesWarm upCorporate FP&ABig Four
Try it first
What is the right DSO?
Show the worked solution
The right DSO is about 61 days; the junior set a quarter's revenue against a full year of days. DSO is receivables over revenue times the days in the same period. Either annualise revenue, 450 x 4 = 1,800, giving 300 / 1,800 x 365 = 60.8 days, or keep the quarter and multiply by its 91.25 days. The 243 is exactly four times too large.
What does DSO actually measure?
Think of a tailor who sells Rs 1,000 of clothes a day and is owed Rs 30,000 by customers. Customers are, on average, 30 days behind. DSO is receivables divided by revenue per day, so the revenue and the day count must come from the same period. Rs 450 crore over a quarter is about Rs 4.93 crore a day, and Rs 300 crore of receivables is about 61 days of that.
Dividing receivables of Rs 300 crore by one quarter's revenue of Rs 450 crore and multiplying by 365 days gives 243 days, exactly four times the right answer of about 61 days, which comes from annualising revenue or using the 91 days in the quarter.
The relationship
$\text{DSO} = \frac{\text{Receivables}}{\text{Revenue in period}} \times \text{days in period} = \frac{300}{450} \times 91.25 = 60.8$
Revenue in period sales over the same span of time as the day count
days in period 365 for a year, about 91 for a quarter
What it says in wordsMatch the revenue period to the day count, and DSO is the number of days of sales still unpaid.
How would you have caught 243 days before it went out?
Run a plausibility check before any number leaves your desk. 243 days says the average customer pays eight months after the sale, which almost no ordinary business tolerates, so the number should have triggered a second look on sight. A second check: compare with last quarter's DSO computed the same way. A sudden fourfold jump is almost always a formula or period error, not a collections crisis. Say the limitation of the corrected figure too: quarter-end receivables against average daily sales can swing with seasonality, so analysts often use average receivables over the period, and a strong last month of the quarter will push DSO up even when customers pay on time.
Where candidates lose it
The error itself is the trap: quarterly revenue with 365 days. It happens most when a template built for annual numbers is fed quarterly data. Spot the factor of four and say so.
The second loss is fixing it silently. The interviewer wants you to name the rule, match the period of revenue to the days you multiply by, and to say how you would catch it next time.
What the interviewer asks next
If the last month of the quarter carried half the quarter's sales, how would that distort DSO?
How would you compute days payable outstanding from quarterly data?
Why might an analyst prefer average receivables to the quarter-end figure?
041A distributor prices at cost plus 25%. A competitor says it earns a 25% gross margin. Who keeps more on every Rs 100 of sales, and what markup would give the distributor a 25% margin?Ratio and margin riddlesWarm upCorporate FP&ACost accounting
Try it first
What markup on cost delivers a 25% gross margin?
Show the worked solution
The competitor keeps more: Rs 25 against Rs 20 on every Rs 100 of sales. Cost plus 25% means a price of 1.25 times cost, so on Rs 100 of sales cost is Rs 80 and the margin is 20%. A 25% gross margin needs cost of Rs 75 on a price of Rs 100, which is a markup of 25 over 75, or 33.3%.
Why does the same 25% mean different money?
Think of a vegetable seller who buys tomatoes for Rs 80 a kilo and adds Rs 20. To him, that is 25% on what he paid. To his customer, Rs 20 out of the Rs 100 she paid is 20%. Same Rs 20, two percentages. Markup divides profit by cost; margin divides it by price, and price is always the bigger number, so a margin is always smaller than the markup that produced it. The distributor's 25% markup is a 20% margin; the competitor's 25% margin is a 33.3% markup.
On Rs 100 of sales, pricing at cost plus 25% means cost of Rs 80 and Rs 20 kept, while a 25% gross margin means cost of Rs 75 and Rs 25 kept, a markup of 33.3% on cost.
Markup on cost Margin on price Kept per Rs 100 of sales
10% 9.1% Rs 9.1
20% 16.7% Rs 16.7
25% 20.0% Rs 20.0
50% 33.3% Rs 33.3
100% 50.0% Rs 50.0
Margin equals markup divided by one plus markup, so the gap widens as the markup grows: a 100% markup is only a 50% margin.
Where does this mix-up cost real money?
In pricing conversations, sales teams often quote markups because the numbers sound larger, while finance teams report margins. If a target says 25% margin and the price list is built at cost plus 25%, every sale falls 5 rupees short per Rs 100, a fifth of the profit the plan assumed. Discounts make it worse: a 10% discount off a cost-plus-25% price leaves a price of Rs 90 on cost of Rs 80, an 11.1% margin, so half the profit per unit is gone. Say the conversion formula out loud whenever someone gives you a percentage of profit, and ask which base it is on.
Where candidates lose it
The trap is answering that both keep the same, because both say 25%. The question is built to see whether you ask: 25% of what?
The second loss is getting the conversion backwards and saying a 25% margin needs a 20% markup. Check with Rs 100 of price: cost Rs 75, profit Rs 25, and 25 over 75 is a third.
What the interviewer asks next
What margin does a 40% markup give?
A retailer offers 20% off a product priced at cost plus 50%. What margin is left?
Why do sales teams often prefer to talk in markups?
045A snack pack shrinks from 100 g to 90 g while its price stays at Rs 50. What is the effective price rise per gram?Pricing, costing and unit economicsWarm upCorporate FP&ACost accounting
Try it first
Pick the effective price rise.
Show the worked solution
About 11.1%. Before, Rs 50 buys 100 g, Rs 0.500 a gram. After, Rs 50 buys 90 g, Rs 0.556 a gram. The rise is 100 over 90 less one, which is 10 over 90, or 11.1%. In general a size cut of x at the same price is a price rise of x divided by (1 minus x), always a little more than the cut.
Why is a 10% smaller pack more than a 10% price rise?
Think of sharing the same Rs 50 bill among 10 friends instead of 11. Each pays more, and the increase is measured against the new, smaller headcount. Price per gram is price divided by grams, so cutting the grams by 10% divides the price by 0.9, which raises it by 1/0.9 less one, 11.1%. The percentage fall in grams is measured on 100; the percentage rise in price per gram is measured on what is left, 90.
Rs 50 for 100 g is Rs 0.500 a gram and Rs 50 for 90 g is Rs 0.556 a gram, an 11.1% rise, and the gap between the cut and the rise grows with the cut: 20% smaller is 25% dearer and 25% smaller is 33.3% dearer.
The relationship
$\text{Price rise} = \frac{1}{1-x} - 1 = \frac{x}{1-x} = \frac{0.10}{0.90} = 11.1\%$
x the cut in pack size, as a decimal
What it says in wordsA size cut of x at a fixed price is a price rise of x over one minus x.
How would a finance team use this?
Inflation in raw materials pushes consumer companies to protect a price point that shoppers remember, such as Rs 10 or Rs 50, and adjust grams instead. To compare price changes across products, convert every pack-size change into a price per unit, or a volume decline will be misread as stable pricing. In a revenue bridge, the same rupee sales on fewer grams show up as a price gain, not as flat sales. Say the limitation: price per gram is the right comparison only if quality and the mix of the pack stay the same.
Where candidates lose it
The trap answer is 10%, the size cut itself. It measures the change against the old pack, while the price per gram has to be measured against the new one.
A second wrong answer is 9.1%, which comes from dividing 10 by 110. That is the rule for the opposite case, a pack that grows 10% at the same price, which is a 9.1% price cut.
What the interviewer asks next
A pack grows from 100 g to 120 g at the same price. What is the price change per gram?
The pack shrinks to 90 g and the price falls to Rs 48. What is the net price change per gram?
How would you show this in a price-volume bridge?

pp	percentage points, the plain gap between two rates
rel	the relative change, the gap divided by the starting rate

N	number of coins
n	number of weighings, here 3
2N	possible answers: each coin, heavy or light

ln 2	the natural log of 2, about 0.693
ln 1.10	the natural log of one month's growth factor, about 0.0953
72	a convenient number close to 100 times ln 2, with many divisors

Growth per period	Rule of 72	Exact	Gap
2%	36.00	35.00	+2.8%
5%	14.40	14.21	+1.4%
10%	7.20	7.27	-1.0%
25%	2.88	3.11	-7.3%
50%	1.44	1.71	-15.8%

Try it first

Why can one change honestly be called both 2 and 25?

Which label should lead, and when does the relative one matter?

Where candidates lose it

What the interviewer asks next

Try it first

How do you know three weighings can be enough?

Why four against four, and what happens next?

Where candidates lose it

What the interviewer asks next

Try it first

Why does eight times take three doublings and not 70 months?

Where does the rule of 72 work, and where does it slip?

Where candidates lose it

What the interviewer asks next

Try it first

Why start from one floor?

What goes into one floor?

Which assumption would you flex, and how would you check it?

Where candidates lose it

What the interviewer asks next

Try it first

What happens when nothing was set aside?

Why does the provision case show nothing this year?

What about tax in the provision case?

Where candidates lose it

What the interviewer asks next

Try it first

Why should powers of two be easy?

When should you multiply up, and when should you step down from an eighth?

Where candidates lose it

What the interviewer asks next

Try it first

How do you avoid getting lost halfway through?

Which amounts can these two bottles measure at all?

Where candidates lose it

What the interviewer asks next

Try it first

What does DSO actually measure?

How would you have caught 243 days before it went out?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the same 25% mean different money?

Where does this mix-up cost real money?

Where candidates lose it

What the interviewer asks next

Try it first

Why is a 10% smaller pack more than a 10% price rise?

How would a finance team use this?

Where candidates lose it

What the interviewer asks next