Financial Analysis puzzles, solved step by step

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.

1,596 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.

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.