Financial Analysis puzzles, solved step by step

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All topicsAccounting flow riddles10Valuation and multiples riddles10Ratio and margin riddles8Cost of capital, leverage and rates8Compounding and time value8Mental maths8Probability and expected value9Working capital and cash riddles6Percentages and averages7Estimation and market sizing7Logic and counting brainteasers7Pricing, costing and unit economics6Data and statistics intuition6
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Showing 1–2 of 2 · filtered from 100Clear filters
  1. 033Convert to percentages in your head, quickly: 3/32, 5/16 and 7/64.Mental mathsWarm upTreasuryCorporate FP&A

    Try it first

    Which of these is 7/64 as a percentage?

    Show the worked solution

    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.

    Start from 1/8 = 12.5% and halve; every answer is built from these rungs1/812.5%halve1/166.25%halve1/323.125%halve1/641.5625%Each rung is half the one above it3/323 x 3.125or 1/8 less 1/329.375%5/165 x 6.25or 1/4 plus 1/1631.25%7/6412.5 less 1.56258/64 less 1/6410.9375%
    Halving 12.5% gives 6.25%, 3.125% and 1.5625% for sixteenths, thirty-seconds and sixty-fourths, so 3/32 is three rungs of 3.125% or 9.375%, 5/16 is five rungs of 6.25% or 31.25%, and 7/64 is 12.5% less one rung of 1.5625%, or 10.9375%.

    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.

    Where candidates lose it

    Candidates start long division, 3 divided by 32, and lose the decimal place or the tempo. The question is timed by feel: the interviewer wants an answer in a few seconds, and only the halving ladder gets you there.

    The second loss is rounding: 10.9% for 7/64 is fine as an estimate, but if the interviewer asked for the exact figure, it is 10.9375%, and the ladder gives it to you exactly.

    What the interviewer asks next

    • What is 11/64 as a percentage?
    • A bond is quoted at 98 and 5/32. What is that in decimal?
    • What is 1/128, and how do you get it from the ladder?
  2. 058Without paper and inside thirty seconds: what is 38 x 42, and then what is 67 x 73?Mental mathsWarm upPrivate equityConsulting-style case

    Try it first

    What is 38 x 42?

    Show the worked solution

    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.

    Numbers equally either side of a round number: square it, take off the gap squared40 x 3842 wide+2 x 38added-40 x 2 removed2 x 2 = 4short of 1,60038 x 42= (40 - 2) x (40 + 2)= 40 squared less 2 squared= 1,600 - 41,59667 x 73= (70 - 3) x (70 + 3)= 70 squared less 3 squared= 4,900 - 94,891Last-digit check: 8 x 2 ends in 6; 7 x 3 ends in 1
    Turning a 40 by 40 square into a 42 by 38 rectangle removes a strip of 80 and adds a strip of 76, so the rectangle is a 2 by 2 corner short of the square: 42 x 38 = 1,600 - 4 = 1,596, and the same move gives 67 x 73 = 4,900 - 9 = 4,891.

    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.

    The relationship
    (m−g)(m+g)=m2−g238×42=402−22=1,59667×73=702−32=4,891(m - g)(m + g) = m^2 - g^2 \qquad 38 \times 42 = 40^2 - 2^2 = 1{,}596 \qquad 67 \times 73 = 70^2 - 3^2 = 4{,}891
    mthe middle number, half the sum of the pair
    gthe gap from the middle to either number
    What it says in wordsThe product of a pair equally spaced around a middle number is the middle squared less the gap squared.

    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.

    Where candidates lose it

    Candidates round both numbers to 40 and say 1,600, treating the shortfall on one side as cancelling the excess on the other. In multiplication it does not: the gaps multiply into a 2 by 2 corner, and the product always lands below the square.

    The other loss is grinding through long multiplication aloud, 38 x 40 plus 38 x 2, and dropping a carry halfway. Spot the symmetry first, say the method in one line, and give the number.

    What the interviewer asks next

    • What is 46 x 54, and what is 99 x 101?
    • What is 38 x 44 in your head?
    • How would you square 65 quickly, and why does every number ending in 5 work the same way?