Financial Analysis puzzles, solved step by step

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 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.

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.