Financial Analysis puzzles, solved step by step

Revenue rises 32%. Growth rates multiply, they do not add. Pair the price and volume factors first because they cancel neatly: 1.25 times 0.96 is exactly 1.20. Then 1.20 times 1.10 is 1.32. Adding 25, minus 4 and 10 gives 31%, which misses the cross terms, the growth earned on growth, worth one point here.

Why do the three changes multiply instead of add?

Think of a tea stall that sells 100 cups at Rs 10. Raise the price to Rs 12.50 and it now takes Rs 1,250 if it still sells 100 cups. If it then sells 4% fewer cups, those 4% are lost at the new, higher price, not the old one. Each change acts on revenue that the earlier changes have already moved, so the factors multiply. Revenue is price times volume times the exchange rate, and a product of three factors changes by the product of their changes.

How do you do it in your head without fumbling?

Choose the order. Multiplication does not care about order, but your head does. 0.96 is 24 over 25 and 1.25 is 5 over 4, so their product is 120 over 100, exactly 1.20, and the hard step disappears. Ten per cent of 1.20 is 0.12, so the last step is 1.32. Pair the factors the other way and you are stuck with 1.25 times 1.10, then 1.375 times 0.96, which is the same answer by a much longer road.

Why does the one-point gap matter on a real desk?

On a Rs 2,000 crore revenue line, one point is Rs 20 crore, which is a material miss in a budget bridge. The gap between adding and multiplying grows with the size of the changes, so it is small for 2% moves and large for 25% moves. When a board pack splits revenue growth into price, volume and currency, the cross terms have to sit somewhere, and an analyst should say where they put them rather than leave the bridge one point short.

531,441. Split the power: 9 to the 6th is 9 cubed, squared, and 9 cubed is 729. Square 729 as (730 minus 1) squared: 730 squared is 532,900, less twice 730, which is 1,460, plus 1, gives 531,441. Check it: an even power of 9 ends in 1, the digits add to 18, and 0.9 to the 6th is about 0.53, so the size is right.

How do you break a big power into steps you can hold in your head?

Think of carrying a heavy suitcase up six floors: you rest on the landings. Split the exponent into pieces you know, 6 = 3 x 2, so 9 to the 6th is 9 cubed, squared. Most people know 9 squared is 81 and 9 cubed is 729. That turns the question into one square, 729 times 729, which is still awkward until you move it next to a round number.

Why square 730 instead of 729?

Because 730 squared is easy: 73 squared is 5,329, so 730 squared is 532,900. The identity (a minus 1) squared equals a squared minus 2a plus 1 lets you square any number just below a round one with two easy corrections. Twice 730 is 1,460. So 532,900 less 1,460 is 531,440, and adding 1 gives 531,441. Each step is small enough to say out loud while you do it.

How do you check it before you say it?

Three checks take seconds. The last digit of an even power of 9 is always 1, the digit sum of any power of 9 is divisible by 9, and 9 to the 6th is 0.9 to the 6th times a million. Here the digits 5, 3, 1, 4, 4, 1 add to 18, and 0.9 to the 6th is about 0.53, so the answer must be about 530 thousand. That last check catches the common slip of being off by a factor of ten. A second route also helps: 81 squared is 6,561, and 6,561 times 81 is 524,880 plus 6,561, again 531,441.

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.

1/7 is 0.142857 repeating, 5/7 of 910 is 650, and 3/7 is 42.86%. 910 divides cleanly by 7 to give 130, and five of those is 650. For any seventh, the six digits 142857 repeat in the same order; only the starting digit changes. 3/7 is about 0.43, so it starts at the 4: 0.428571, or 42.86%.

Why should sevenths be memorable when they look messy?

Think of a clock face with six numbers on it instead of twelve. Whatever hour you start from, you read the same numbers in the same order. Sevenths work like that: every k/7 is the cycle 1, 4, 2, 8, 5, 7 read round the ring from a different starting digit. You only need to learn one string, 142857, and a way to pick the start: k/7 is roughly k x 0.14, so 2/7 starts at 2 (0.285714), 3/7 at 4 (0.428571) and 6/7 at 8 (0.857142).

When should you divide first and when should you use the decimal?

For 5/7 of 910, check whether 7 divides 910: 7 x 130 is 910, so divide first and multiply second, and the answer is a clean 650. When the number you are taking a fraction of is a multiple of the denominator, divide first; reach for the decimal only when it is not. For 5/7 of 1,000, there is no clean division, so use the cycle: 5/7 starts at the 7, 0.714285, and the answer is about 714.3. A check that works on both: 5/7 is a little over 70%, and 70% of 910 is 637, so 650 sits in the right place.

Interviewers use sevenths because the decimals look random and catch people who have only learned halves, quarters and eighths. The cycle gives you six decimal places in a second, which is more precision than you will ever need to say out loud.

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.

Root 50 is about 7.07, and 1.2% a day is about 19% a year. Root 49 is 7, and near a known square the curve is nearly straight, so add the step divided by twice the root: 7 + 1 / 14 = 7.0714 against a true 7.0711. Volatility scales with the square root of time, so multiply by root 250, about 15.81: 1.2 x 15.81 = 19.0%.

How do you get root 50 without a calculator?

Think of walking up a gentle hill: one more step forward lifts you by the slope at the point you are standing, and for a single step the hill may as well be straight. Near a square you know, the root curve is nearly straight, so the root of 49 plus 1 is 7 plus 1 divided by the slope's denominator, twice 7. That gives 7 + 1/14 = 7.0714; the true value is 7.0711, an error of 0.0004. The slope of the root curve at a point is one over twice the root, which is why the denominator is 14 and not 7.

Why does volatility scale with the square root of time?

Toss a coin 100 times and count heads minus tails. The lead is almost never 100; it is typically about 10, the root of 100, because the tosses partly cancel. Independent moves add in variance, not in standard deviation, so the spread after n days is the daily spread times root n. Daily variance is 1.2 squared, 1.44; over 250 days it is 360; the root of 360 is 19.0. Root 250 by the tangent at 256 is 16 - 6/32 = 15.812, close enough to the true 15.811.

When does the root-of-time rule mislead?

The rule assumes each day's move is independent of the last and drawn from the same distribution. Trending markets, where moves follow moves, make the true spread wider than root n suggests; mean-reverting ones make it narrower. Volatility also clusters, so a 1.2% daily figure measured in a calm month understates a stormy one. The day count is a convention: 250, 252 and 365 are all in use, and annualised volatilityThe standard deviation of returns restated to a one-year horizon, usually by multiplying a daily figure by the square root of the number of trading days. quoted on each will differ, so state the one you used. The limit to say: the rule converts a horizon, it does not forecast the year.

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.

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.