Financial Analysis puzzles, solved step by step

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

1,596 and 4,891. Both pairs sit equally either side of a round number, so each product is that round number squared less the gap squared. 38 x 42 = 40 squared less 2 squared = 1,600 - 4 = 1,596. 67 x 73 = 70 squared less 3 squared = 4,900 - 9 = 4,891. Two easy steps replace four partial products.

Why does a pair either side of a round number multiply so neatly?

Picture a square courtyard 40 tiles by 40. Rebuild it 42 tiles wide but only 38 deep. You lose a row of 40 tiles along the bottom and gain a column of 38 down the side. The column is two tiles shorter than the row, and the missing piece is a 2 by 2 corner. Any two numbers equally either side of a middle number multiply to the middle squared less the gap squared, because (a - b)(a + b) = a squared less b squared.

So the work is two squares you already know. For 67 x 73 the middle is 70 and the gap is 3: 4,900 less 9 is 4,891. A quick check catches slips: 7 x 3 ends in 1, and so does 4,891; 8 x 2 ends in 6, and so does 1,596. Both products sit just under the square, as they must.

What do you do when the pair is not centred on a round number?

Find the middle anyway, or nudge one number. 38 x 44 has a middle of 41 and a gap of 3: 1,681 less 9 is 1,672. Or reuse what you have: 38 x 42 is 1,596, add two more 38s and get 1,672. When the sum is odd, as in 37 x 42, the middle is 39.5, which is messy; adjust instead: 38 x 42 less one 42 is 1,554. The skill is choosing the anchor you can square instantly, then correcting by a small, easy amount.

Why does a finance interviewer test this?

Interviewers use mental maths to see whether you can keep a number straight under pressure, which is what a live model review or a client call demands. They watch the method as much as the answer: a candidate who says 'that is 40 squared less 4' sounds in control. The limit is honest too: the trick only helps when a pair straddles a square you know. For 38 x 57, round and correct, or split 57 into 50 and 7, rather than forcing the pattern.

Rs 180. The 18% is charged on the pre-tax price, so the MRP is the base times 1.18. The base is 1,180 / 1.18 = Rs 1,000 and the tax is Rs 180. As a share of the inclusive price, tax is 18 / 118, about 15.25%, not 18%. Taking 18% of Rs 1,180 charges tax on the tax and gives a wrong Rs 212.40.

Why is the tax not 18% of the price you pay?

A restaurant bill of Rs 1,100 that includes a 10% service charge contains Rs 100 of service, not Rs 110: the 10% was added to the Rs 1,000 of food, not to the final total. A rate applied to a base and then included in the price is rate over one plus rate of the inclusive price, never the rate itself. For 18% that share is 18 / 118, about 15.25%. The base is the starting point, and the inclusive price is the base grown by 18%.

How do you strip tax out of any inclusive price?

The rates in the table are for illustration. GST rates vary by product and change over time, so confirm the current rate for the item before you rely on a figure.

Where else does the same slip turn up?

Everywhere a percentage of one base gets quoted against another. A 25% markup on cost is only a 20% margin on price, because 25 / 125 = 20%. A fund that charges 2% on assets is not taking 2% of the return. An analyst who multiplies an inclusive revenue line by the GST rate overstates the tax the company passes on to the government. Before applying any percentage, ask what it is a percentage of, and convert if the base you hold is a different one. The test is quick: 1,000 plus 18% of 1,000 must give back the MRP.