Financial Analysis puzzles, solved step by step

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.

About 32 lakh litres a day, inside a plausible range of roughly 18 to 47 lakh. Assume about 1.3 crore people in four-person homes, 32.5 lakh households. If 80% buy packaged milk at about a litre a day, that is 26 lakh litres. Tea stalls, restaurants and sweet shops add perhaps a quarter more. Each input is an assumption to state aloud and test.

What is the natural unit for a daily staple?

Think about your own home: someone buys one or two packets every morning, and the number depends on how many people share the kitchen, not on who drinks what. For something bought every day by almost every home, the household is the unit, because the buying decision and the quantity are set per kitchen. Start from people, convert to homes, decide what share buys the product, and multiply by the daily quantity. Then add the buyers who are not homes.

Every number in the chain is an assumption, not a fact, and should be said that way. The population depends on where you draw the boundary: the city proper and the wider metropolitan region differ by a large factor, so confirm the latest census or municipal estimate and say which one you are using. The 80% packaged share leaves room for loose milk from local dairies, which is still common in parts of the city.

How do you cross-check it without reusing the same assumptions?

Use a check that rests on different inputs. Per head, 32.5 lakh litres across 1.3 crore people is about 250 ml a day: two or three cups of tea at about 50 ml of milk each plus a glass for a child. That is believable. From the supply side, if a road tanker carries about 15,000 litres, which is an assumption to state, the city needs about 217 tankers a day. A good estimate survives a check built from a different direction; if the per-head number had come out at two litres, you would know a step was wrong.

What would sharpen the estimate most?

The packaged share is the biggest swing, so it is the first thing to research, followed by the trade uplift, which is hard to observe because tea stalls and sweet shops often buy loose milk in bulk. Seasonality matters too: festival weeks lift demand for sweets and milk. The limit to say: this is a demand estimate on an average day, and a dairy planning supply would also need the peak day and the share each brand holds.

The 12.5% annual loan is cheaper. Twelve per cent compounded monthly is an effective 12.68% a year, against 12.50%. A 12% nominal rate charged monthly means 1% a month, and interest earns interest eleven more times within the year: 1.01 to the power 12, less 1, is 12.68%. Convert every quote to an effective annual rate before comparing.

Why is 12% a year compounded monthly more than 12%?

Leave a credit card balance unpaid and the interest charged in January is itself charged interest in February. The bank's 'monthly rate' grows faster than the same rate charged once a year. A nominal rate tells you how interest is quoted; the effective annual rate tells you what it costs, and only effective rates can be compared. Here 12% a year becomes 1% a month, and after twelve months of compounding Rs 100 has grown to Rs 112.68.

What would the monthly lender need to quote to match?

Run the conversion backwards: the monthly rate that compounds to 12.5% is 1.125 to the power 1/12, less 1, about 0.98% a month, or a nominal 11.84% a year. Any monthly quote above 11.84% is dearer than 12.5% annual. For a quick mental check, the extra from monthly compounding at these rates is roughly half the rate squared: 0.5 x 0.12 x 0.12 is 0.72 points, close to the true 0.68.

What else decides which loan is really cheaper?

The effective rate prices the money, not the whole deal. Processing fees, prepayment penalties and insurance bundled into the loan all add to the cost and must be folded in. Watch for flat-rate quotes too: a 7% flat rate on a three-year loan charges interest on the original amount even as it is repaid, which works out to an effective rate of about 13.6% a year. Confirm how any lender computes its rate before comparing; the limit of the EAR is that it compares like with like only once every cost is in it.

Net income rises Rs 7.5 crore, cash falls Rs 2.5 crore, and net fixed assets are Rs 10 crore higher. Income statement: depreciation down 10, pre-tax profit up 10, tax up 2.5, net income up 7.5. Cash flow: net income up 7.5 but the add-back down 10, so cash from operations down 2.5. Balance sheet: cash down 2.5, fixed assets up 10, assets up 7.5, matched by retained earnings up 7.5.

Why does profit rise while cash falls?

A shopkeeper who decides his delivery van will last ten years instead of five writes off half as much each year. The van is the same van, the fuel bill is the same, and no customer paid him more. Changing a useful life changes when the cost of an asset is recognised, not whether it is paid, so the extra profit is an accounting rearrangement with one real consequence: a higher tax bill. Depreciation falls 10, pre-tax profit rises 10, tax at 25% rises 2.5, and net income rises 7.5.

On the cash flow statement, start from net income, up 7.5, and add back depreciation, which is now 10 lower than before. Cash from operations is 7.5 - 10 = -2.5. Nothing changes in investing or financing, so cash falls 2.5. On the balance sheet, cash is down 2.5 and net fixed assets are up 10, because less accumulated depreciation has been charged against them: assets up 7.5. Retained earnings carry the 7.5 of extra net income, so liabilities and equity are also up 7.5.

Why is the higher profit not good news?

Because nothing about the business improved. The same machines wear out on the same schedule; the company will replace them on the same date for the same money; and it has handed Rs 2.5 crore to the tax authority earlier than it needed to. A profit increase that comes with a cash decrease and no operating change is cosmetic, and the choice to make it is itself a signal: management may be reaching for a target. It also breaks comparability with peers who kept the shorter lives, and the total charge over the asset's life is unchanged, so the profit that appears now is borrowed from later years.

What if tax depreciation did not follow book?

In practice the tax authority sets its own depreciation rates, and in India book depreciation under the Companies Act and tax depreciation under the Income Tax Act are computed separately; confirm the current rules before relying on this. Then cash tax does not change: net income still rises 7.5, but a deferred tax liabilityTax that the accounts recognise as owed on profit already reported, but that the tax return has not yet charged, so it will be paid in a later period. of 2.5 is booked instead of extra cash tax. On the cash flow statement, net income up 7.5, add-back down 10, deferred tax up 2.5: cash from operations is 0.0. The limit: in that version the change is purely cosmetic, cash untouched, which is why analysts read the depreciation policy note before they read the profit line.

The real raise is 3.77%, so the 4% shortcut overstates it by about a quarter of a point. Real growth divides growth factors: 1.10 / 1.06 = 1.0377. The shortcut subtracts the rates, which ignores that the raise is itself paid in money worth 6% less. At low rates the gap is small; at a 30% raise with 20% inflation the shortcut says 10% against a true 8.33%.

Why divide rather than subtract?

Measure your pay in lunches instead of rupees. Last year a thali cost Rs 100 and you earned Rs 100 a day: one thali. This year the thali costs Rs 106 and you earn Rs 110, so you can buy 110 / 106 = 1.0377 thalis. Your real pay rose 3.77%, because real growth is the ratio of two growth factors, not the difference between two rates. The relation is usually credited to the economist Irving Fisher.

The second form shows exactly where the shortcut goes wrong. The 4% gap is right in the numerator; it just has to be shrunk by dividing by 1.06. The error is small when inflation is small and grows as inflation rises.

When does the shortcut become a real error?

Three places. When rates are high, as the chart shows. When the rate compounds: over ten years, 4% a year grows purchasing power by 48.0% but the true 3.77% grows it by 44.8%, a gap of about three points. And when a budget is being judged: an FP&A analyst splitting revenue growth into price and volume uses exactly this division, and subtracting instead misstates the volume growth the business really delivered. For quick talk in a meeting, 4% is fine; in a model, divide.

Light rope A at both ends and rope B at one end, together. When A burns out, 30 minutes have passed; light B's other end, and B finishes 15 minutes later, at 45 minutes. Lighting both ends halves any rope's remaining burn time whatever its speed profile, because two flames use the time up together. B has 30 minutes of burn left at the half hour, and two flames clear that in 15.

Why does lighting both ends give exactly half, when the rope burns unevenly?

Picture two friends eating one plate of biryani from opposite sides. One side is all rice and goes fast, the other is full of bones and goes slowly. They will meet somewhere off centre, but if the plate takes one person 60 minutes, two people eating at the same moment finish it in 30, because between them they are always doing two minutes of eating every minute. The rope's total burn time is fixed at 60 minutes; two flames consume it twice as fast, so it lasts 30 minutes regardless of where the slow sections are.

That is the whole trick. Measuring by length fails, which is why cutting a rope in half or marking its midpoint gets you nothing: the midpoint of the string is not the midpoint of the time. Measuring by burn time is exact.

Where do the last 15 minutes come from?

Rope B was lit at one end at the same moment as A. When A dies at minute 30, B has burned for 30 minutes, so it holds exactly 30 minutes of burn time, although nobody knows how long the remaining piece is. Lighting B's unlit end at that moment halves its remaining 30 minutes to 15, so B goes out at 30 plus 15, or 45 minutes. Both ropes are doing the same job: A is a 30 minute timer, and B is a 60 minute timer that you shorten halfway through.

Say the steps in order and say the start condition: all three ends must be lit at the same instant. An interviewer listening to a phone screen is checking that the sequence is clean and that you explain why it works, not just that you have heard the riddle before.

What else can two ropes measure?

With one rope you can time 60 or 30 minutes. With two you can add 15, 45, 75 and 90: 90 by burning one rope at both ends and then the other at one end, 75 by burning A at both ends, then B at one end for 30 minutes and lighting its second end. Each rope gives you only halving moves on whatever time it has left, so the times you can build are sums of 60, 30 and 15. You cannot get 20 minutes with these ropes, and saying why is a strong close.

112, and Rs 560 crore. For 17.5%, build it from blocks: 10% of 640 is 64, 5% is half of that, 32, and 2.5% is half again, 16; 64 + 32 + 16 = 112. For 12.5%, spot the fraction: it is one eighth, so halve 4,480 three times, to 2,240, 1,120 and 560. Check each with a second route before you say it.

How do you take an awkward percentage of anything in your head?

When a restaurant bill is Rs 640 and you want to tip 17.5%, nobody multiplies by 0.175. You find 10%, which is just moving the decimal point, and then cut pieces off it. Any percentage can be built from 10%, 5% (half of 10%), 1% and their halves, and each block is one easy operation on the last. For 17.5% of 640: 10% is 64, 5% is 32, 2.5% is 16, and the three add to 112.

Check with a second route: 20% is a fifth, 128, and 17.5% is 2.5 points less, so 128 minus 16 is 112. Two routes agreeing is what lets you say a number aloud with confidence in an interview where speed is being watched.

Which percentages are fractions in disguise?

Some percentages are clean fractions, and dividing is faster than building. 12.5% is one eighth, so 12.5% of Rs 4,480 crore is three halvings: 2,240, 1,120, 560. The same idea gives 37.5% as three eighths (560 x 3 = 1,680) and 87.5% as the whole less one eighth (4,480 minus 560 = 3,920). Desk interviewers like these because they turn up constantly in fee splits, stake sizes and tax at round rates.

What makes a mental answer trustworthy under time pressure?

Two habits. First, estimate before computing: 17.5% is a little under a fifth, so the answer must be a little under 128. If your arithmetic produces 1,120 or 11.2, the estimate catches the slipped decimal at once. Second, keep the units: the second answer is Rs 560 crore, not 560. An interviewer marks a right number with units and a sanity check far above a right number mumbled after a long silence. Speak the route as you go, so a slip is visible and recoverable.

No: 20% then 10% off is 28% off, not 30%; buy two get one free is a 33.3% discount per unit. Successive discounts multiply, because the second applies to the already reduced price: 0.80 x 0.90 = 0.72, so you pay 72% of the original. Buy two get one free means paying for two units and taking three, so each unit costs two thirds of its price, a third off, but only if you wanted three.

Why do discounts not simply add?

A Rs 1,000 kurta at 20% off costs Rs 800. The cashier then takes 10% off what you are about to pay, which is Rs 800, so the second discount is worth Rs 80, not Rs 100. You pay Rs 720. Each successive discount is taken off a price that the earlier discounts have already reduced, so discounts combine by multiplying what is left: 0.80 x 0.90 = 0.72, or 28% off. The order does not matter, 10% then 20% also gives 0.72, but the base does.

The shortfall from simple addition is the product of the two discounts: 20% x 10% = 2 points, the discount on the discount. It is small here and grows with the size of the cuts: 50% then 50% off is 75% off, not 100%.

What is buy two, get one free really worth?

You pay for two units and walk out with three, so each unit costs two thirds of the list price. Buy two get one free is a 33.3% discount per unit, deeper than the flat 30%, but only on the condition that you buy three. Buy one, get the second at half price is milder: one and a half prices for two units, 25% off each. If you needed only one, the effective discount of either offer is zero.

Where does this show up at work?

Everywhere rates are applied one after another. A distributor margin taken on a price that already carries a retailer margin, a trade discount followed by a cash discount, a fee charged after another fee: all compound. An FP&A analyst who adds layered discounts to estimate net price will overstate the discount and understate revenue per unit. The same arithmetic runs in reverse for mark-ups: a 20% price rise followed by a 20% cut leaves you 4% below where you started, because 1.2 x 0.8 = 0.96.

134,547. Split 301 into 300 and 1. Three hundred times 447 is 447 x 3 with two zeros: 1,341, then 134,100. Add one more 447: 134,547. Check before you say it: 300 x 450 is 135,000, so the answer should sit a little below that; the last digit must be 1 x 7, a 7; and the digit sums agree, since 301 reduces to 4, 447 to 6, 4 x 6 = 24 reduces to 6, and 1+3+4+5+4+7 = 24 also reduces to 6.

Why split 301 rather than attack both numbers?

If a shop sells 301 items at Rs 447 each, nobody in the shop multiplies 301 by 447 digit by digit. They price 300 items and add one more. Break the awkward factor into a round number and a small remainder, multiply the round part first, and the remainder is one easy addition. 447 x 3 is 1,200 plus 141, which is 1,341; two zeros make it 134,100; plus 447 is 134,547. Say each intermediate number aloud as you reach it, so the interviewer can follow and a slip is caught before it compounds.

The split can go the other way too. 447 is 450 less 3, so 301 x 447 = 301 x 450 minus 301 x 3. The first is 300 x 450 plus 450, 135,450; the second is 903; the difference is 134,547. Having two routes is what turns a hopeful answer into a checked one.

How do you check an answer you cannot write down?

Three checks, each a few seconds. The estimate: 300 x 450 is 135,000, and you used one factor slightly above and one slightly below, so the answer should be close to it, which 134,547 is. The last digit: 1 x 7 ends in 7, and so does the answer. The digit sums, an old bookkeeper's test: reduce each number to a single digit by adding its digits, 301 gives 4 and 447 gives 15 then 6; the product 24 reduces to 6, and the answer's digits 1+3+4+5+4+7 = 24 also reduce to 6. A wrong answer rarely survives all three checks, and running them aloud shows the interviewer a habit, not a lucky number. The digit-sum test has a blind spot: it misses errors that are exact multiples of 9, including two digits swapped, which is why the estimate sits beside it.

What is the interviewer marking?

Pace, an audible route and a check, in that order. A right answer after a long silence scores below a right answer reached in three spoken steps, because the desk wants to hear how you handle a number, not whether you can. The limitation is that this trick suits factors near a round number. For something like 347 x 683, say that you would estimate, 350 x 700 less a bit, give about 237,000, and label it an estimate; pretending to exact arithmetic you cannot check is the real failure.