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. 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.