Private Equity puzzles, solved step by step

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  1. 010In your head, and out loud: what are 48 x 52, 97 x 103 and 499 x 501?Mental mathsWarm upGeneral AtlanticNew York · 2026

    Try it first

    What is 97 x 103?

    Show the worked solution

    2,496, 9,991 and 249,999. Each pair sits evenly either side of a round number, so the product is that number squared less the gap squared. 48 x 52 is 50 squared less 2 squared, 2,500 less 4. 97 x 103 is 10,000 less 9. 499 x 501 is 250,000 less 1. Say the rule first, then the three answers come in seconds.

    What do the three pairs have in common?

    Look at where each pair sits. 48 and 52 are both 2 away from 50; 97 and 103 are both 3 away from 100; 499 and 501 are both 1 away from 500. Two numbers placed symmetrically around a round number multiply to that number squared, less the gap squared. In symbols, (a minus b) times (a plus b) is a squared minus b squared. The round number is easy to square, the small gap is easy to square, and one subtraction finishes the job.

    48 x 52 is a 50 x 50 square with a 2 x 2 corner taken away50 x 50= 2,5002 x 2 = 4 removedstrip 2 x 485048 x 52= 2,4964852strip moved48 x 52 = 2,500 - 4 = 2,49697 x 103 = 10,000 - 9 = 9,991499 x 501 = 250,000 - 1 = 249,999
    Cutting the 2 by 48 strip from a 50 by 50 square that is missing a 2 by 2 corner and laying it along the bottom gives a rectangle 48 wide and 52 tall, so 48 x 52 equals 2,500 less 4, which is 2,496.

    Why is the product always a little less than the square?

    Think of a square garden 50 metres on each side. If you make it 2 metres longer and 2 metres narrower, the strip you gain along one side is 2 by 48, but the strip you lose is 2 by 50. You lose slightly more than you gain, and the shortfall is exactly the small 2 by 2 corner, which is why the answer is always the square less the gap squared. That is the picture worth having in your head when the interviewer asks why the trick works.

    The relationship
    (a−b)(a+b)=a2−b2499×501=5002−12=249,999(a-b)(a+b) = a^2 - b^2 \qquad 499 \times 501 = 500^2 - 1^2 = 249{,}999
    athe round number in the middle, here 500
    bthe distance from it to each factor, here 1
    What it says in wordsSquare the middle, square the gap, subtract.

    Why would a growth equity interviewer bother with this?

    Because deal conversations are full of quick products: 48 months of a 52 lakh run rate, a multiple of 9.7x on 10.3 crore of EBITDA. The test is not whether you know the identity but whether you look for structure before you start grinding through digits. If the numbers are not symmetric, make them so: 47 x 55 is 51 squared less 4 squared, 2,601 less 16, which is 2,585. Say the method aloud so the interviewer hears the route, not just the result.

    Where candidates lose it

    The common loss is grinding through long multiplication out loud, which works but takes a minute and invites a slip in the carries. The interviewer is looking for the moment you spot the symmetry.

    The second trap is getting 10,009 for 97 x 103 by adding the gap squared. The product is always below the square, because the lost strip is bigger than the gained one.

    What the interviewer asks next

    • What is 51 x 49?
    • What is 46 x 54, and what is 46 x 55?
    • Square 105 in your head.

    Asked at General Atlantic, Generalist, New York, 2026 (Wall Street Oasis): The first round was behavioral with mental math at the end.

  2. 018A software company grows recurring revenue at 35% a year with an EBITDA margin of minus 10%. What is its Rule of 40 score? If growth slows to 25%, what margin does it need to stay at 40?Mental mathsCoreVista Equity PartnersAustin · 2021

    Try it first

    What margin keeps the score at 40 once growth is 25%?

    Show the worked solution

    The score is 25 today, and at 25% growth the margin must reach 15%. The Rule of 40 adds revenue growth and profit margin: 35 plus minus 10 is 25, well short of 40. If growth slows to 25%, the margin has to be 40 less 25, or 15%. That is a swing of 25 points of margin, from minus 10% to plus 15%, just to hold the bar.

    What is the Rule of 40 actually measuring?

    Think of a young restaurant that can either open new branches fast and lose money doing it, or open slowly and keep every branch profitable. Either can be healthy; losing money while also growing slowly is not. The Rule of 40 adds growth and margin so that a software company can trade one for the other, and treats a combined score of 40 or more as a sign of a healthy balance. Growth of 35% and a margin of minus 10% score 25, which says the company is spending more on growth than the growth is earning.

    The Rule of 40 is a trade-off line: slower growth must be bought back with margin-20%-10%0%10%20%30%40%0%10%20%30%40%50%Revenue growthEBITDA margin (vertical)growth + margin = 40today: 35 - 10 = 25slows, margin flat: 1525 + 15 = 40+25 points of marginScore now35 + (-10) = 25Margin at 25% growth40 - 25 = 15%
    A company growing 35% with a minus 10% margin scores 25, below the 40 line, and if growth slows to 25% it needs a 15% margin to reach 40, a climb of 25 points of margin.

    Why does slowing growth make the margin job so large?

    Because the score falls point for point. Losing 10 points of growth drops the score from 25 to 15 if nothing else moves, so the margin has to cover both the existing 15-point shortfall and the 10 points just lost. That is 25 points of margin. In practice it means cutting sales and marketing spend that was buying the growth, which is often exactly what happens when a buyout fund takes over a software business: growth slows by choice and margin is rebuilt.

    The relationship
    Score=g+m35+(−10)=25m=40−25=15%\text{Score} = g + m \qquad 35 + (-10) = 25 \qquad m = 40 - 25 = 15\%
    grevenue growth, usually recurring revenue, in per cent
    mprofit margin in per cent, often EBITDA or free cash flow margin
    What it says in wordsGrowth and margin trade one for one, so the margin needed is 40 less the growth rate.

    What should you say about the rule's limits?

    That it is a heuristic, not a law. Different investors use different margin definitions, EBITDA, operating or free cash flow, and the same company can pass on one and fail on another. The rule also treats a point of growth and a point of margin as equal, which is untrue for a small company where growth compounds into a much larger future business. Use it to frame the trade-off, then look at what is driving each side: retention, sales efficiency and gross margin.

    Where candidates lose it

    The common slip is with the sign: adding 35 and 10 to get 45 and declaring the company healthy. A negative margin subtracts.

    The second loss is giving 15% without noticing the size of the move. The interviewer wants to hear that going from minus 10% to plus 15% is a 25-point swing and what kind of cost cutting it implies.

    What the interviewer asks next

    • What growth rate would the company need to reach 40 at its current margin?
    • Which margin definition would you use, and why?
    • Why might a buyout fund accept a lower score at entry?

    Asked at Vista Equity Partners, Private Equity, Austin, 2021 (Wall Street Oasis): They asked mental math and asked about tech/saas-specific sector insights

  3. 059Quick-fire round, about ten seconds each: 17 x 23, 1.08 cubed, 7/8 as a percentage, and 45% of 360. Give each answer and the trick that gets you there.Mental mathsWarm upOaktree Capital ManagementLos Angeles · 2022

    Try it first

    Which trick turns 17 x 23 into a one-step sum?

    Show the worked solution

    391, about 1.26, 87.5% and 162. 17 x 23 is (20 - 3)(20 + 3), so 400 - 9. 1.08 cubed is about 1 + 3 x 0.08 + 3 x 0.0064, which is 1.259, close to the exact 1.2597. Seven eighths is one less one eighth, 100% - 12.5%. And 45% of 360 is half of 360 less a twentieth of it, 180 - 18.

    Why name the trick rather than just give the number?

    A shopkeeper who adds a bill in his head is not calculating faster than you; he has a handful of shortcuts he has used ten thousand times. Speed in a quick-fire round comes from recognising which shortcut a question is built for, so say the shortcut as you give the answer. It shows the interviewer the answer is reliable, and it protects you when a number comes out slightly off: the method is audible even if the last digit slips.

    Each answer comes from a named trick, not from faster arithmeticDifference of squares17 x 23(20 - 3)(20 + 3) = 400 - 9391Binomial: 1 + 3x + 3x squared1.08 cubed1 + 3(0.08) + 3(0.0064) = 1.2592about 1.26Known fraction7/8 as a %1 - 1/8 = 100% - 12.5%87.5%Split the percentage45% of 36050% - 5% = 180 - 1816220 sq-3 sq10.240.0197 of 8 = 87.5%50% less 5%
    Each of the four answers comes from one named trick: 17 x 23 is 400 less 9, 1.08 cubed is about 1 plus 0.24 plus 0.019, seven eighths is 100% less 12.5%, and 45% of 360 is 180 less 18.

    How do the two harder ones work?

    For 1.08 cubed, expand (1 + x) cubed as 1 + 3x + 3x squared + x cubed with x = 0.08. When x is small, the first two terms carry almost everything and the third is a small correction: 1 + 0.24 + 0.0192 is 1.2592, within 0.001 of the exact 1.2597. That is three years of 8% growth, about 26%, which is why the trick earns its place on a returns desk. For 17 x 23, check the trick fits: it only works when the two numbers sit the same distance either side of a round number.

    The relationship
    (a−b)(a+b)=a2−b2(1+x)3≈1+3x+3x2(a-b)(a+b) = a^2 - b^2 \qquad (1+x)^3 \approx 1 + 3x + 3x^2
    athe round number in the middle, here 20
    bthe distance either side, here 3
    xthe growth rate, here 0.08
    What it says in wordsA product of two numbers equally spaced around a round number is that number squared less the gap squared; a small rate cubed is about one plus three times the rate.

    Fractions and percentages are best memorised in eighths: 12.5%, 25%, 37.5% and so on up to 87.5%. Percentages of awkward numbers split into easy pieces: 10%, 5%, 50%. With those four habits, most of what turns up in a quick-fire round is one step.

    Where candidates lose it

    The trap is going silent and grinding long multiplication for 17 x 23 or 1.08 cubed. Ten seconds is not enough, and the interviewer hears nothing to rescue. Reach for the shortcut out loud.

    The second slip is on 1.08 cubed: answering 1.24 by tripling 8% and forgetting the compounding term. Three years at 8% is about 26%, not 24%; the extra two points are the growth on the growth.

    What the interviewer asks next

    • What is 1.1 to the power 5, to two decimals?
    • What is 48 x 52?
    • If a date falls on a Monday this year, what day is it next year?

    Asked at Oaktree Capital Management, Generalist, Los Angeles, 2022 (Wall Street Oasis): Quick mental math questions are unexpected. Was asked around 6 of them.