Private Equity puzzles, solved step by step

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.

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.

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.

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.

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.

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.

The square root of 2.5 is about 1.58, so 2.5x over two years is about 58% a year. 1.5 squared is 2.25 and 1.6 squared is 2.56, so the root lies between them and much nearer 1.6: 2.5 is 25 of the 31 hundredths along, giving about 1.581. One Newton step from 1.6 gives 1.5813, against a true 1.5811.

How do you find a square root you have not memorised?

A train leaves one station at 6:00 and reaches the next at 9:00. A signal four fifths of the way along is passed at about 8:24: you slide between the two times you know. Find the two squares you know that bracket the number, then slide between them in proportion: the root sits about as far along as the number sits between the squares. 1.5 squared is 2.25 and 1.6 squared is 2.56. The target 2.5 is 0.25 above the lower square out of a gap of 0.31, about 0.81 of the way, so the root is about 1.5 plus 0.081, or 1.581.

How do you sharpen the estimate and check it?

Use one Newton step: take a guess, divide the number by it, and average the two. From 1.6, 2.5 divided by 1.6 is 1.5625, and the average of 1.6 and 1.5625 is 1.58125. One averaging step from a good guess gets you to three decimals, close enough for any interview. Check by squaring: 1.58 squared is 2.4964, just under 2.5, so the root is a hair above 1.58.

Now the investing use. A growth deal returning 2.5x in two years earns the square root of 2.5, less one, each year: about 58%. The fast answer, 150% over two years so 75% a year, ignores compounding: 1.75 squared is 3.06, not 2.5. Say the limitation: short holds produce eye-catching IRRs on modest money, so investors read the multiple alongside it.

About 1.95. Multiply up one step at a time: 1.21, 1.331, 1.4641, 1.6105, 1.7716, then 1.7716 plus a tenth of itself is 1.9487. Each step is the last number plus a tenth of it, which is easy to do in your head. Check with the rule of 72: at 10%, money doubles in about 7.2 years, so after 7 years it should be just under 2.

What is the cleanest way to do it in your head?

Adding 10% is the friendliest step there is: move the decimal one place and add. A bill of 1,771 plus 10% is 1,771 plus 177.1, which is 1,948.1. Multiplying by 1.1 is the same as adding a tenth of the number to itself, so seven careful additions beat any half-remembered formula. Keep four decimals until the last step, then round.

How do you check it a second way?

Two checks. First, the rule of 72: 72 divided by 10 is 7.2 years to double, so seven years should give a little under 2. Second, expand the bracket. (1 + 0.1) to the 7th is 1 + 0.7 + 0.21 + 0.035 + 0.0035 + smaller terms, which sums to about 1.9487. The terms shrink fast, so the first five get you there.

Why a buyout interviewer asks this: a business growing 10% a year for seven years almost doubles. That is the sort of number you need to sanity-check a model's exit EBITDA in two seconds. The limitation of the rule of 72 is that it drifts at high rates; at 10% it is close, at 50% it is not.

1,000 / 7 is 142.857 and 22 / 0.35 is about 62.86. Seven goes into 1,000 142 times with 6 left, and six sevenths is 0.857142, repeating. For the second, multiply both numbers by 100 to get 2,200 / 35, divide both by 5 to get 440 / 7, and that is 62 with 6 left: 62 and six sevenths, about 62.86. Both answers end in the same sevenths.

Why is one seventh worth memorising?

Think of a clock face with six numbers on it instead of twelve. Every seventh, whatever it is, lands on the same circle of six digits, 1, 4, 2, 8, 5, 7, and only the starting point changes. Once you know 1/7 = 0.142857, every other seventh is a rotation of the same six digits: 2/7 starts at the 2, 3/7 at the 4, and 6/7 at the 8. That turns both of these divisions into one fact you already hold.

How do you show the long division out loud?

Seven into 1,000 is 142, because 7 x 142 is 994, leaving 6. Bring down a zero: 60, seven goes 8 times, 4 left. 40 gives 5, with 5 left. 50 gives 7, with 1 left. The moment a remainder you have seen before comes back, the digits start repeating, so you can stop dividing and write the cycle. Three decimals is 142.857; the next digit is a 1, so no rounding is needed.

Interviewers ask these at desks that live in spreadsheets because quick division is how you sanity check a multiple or a margin in a meeting. A 22 crore profit on a 0.35 crore unit is about 63 units. Saying the rescaling step out loud matters more than speed: it shows a method you would trust on any number.

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.

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.

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.

Down 1%. Start from 100. A 20% rise takes it to 120, a 25% fall takes away 30 to leave 90, and a 10% rise adds 9 to reach 99. In one line, 1.2 x 0.75 x 1.1 = 0.99. Percentage changes multiply; adding them gives plus 5%, which never happened.

Why can you not just add the percentages?

A shop marks a Rs 100 kurta up 20% to Rs 120, then runs a 25% sale, then nudges the price up 10%. The sale takes 25% of 120, which is 30, not 25. Every percentage is measured against the level it starts from, so each change has a different base and the changes multiply rather than add. Adding them treats every change as if it were applied to the original 100.

How do you do 1.2 x 0.75 x 1.1 fast?

Pick the pair that cancels nicely. 1.2 times 0.75 is exactly 0.9, because three quarters of 1.2 is 0.9, and 0.9 times 1.1 is 0.99. Saying the intermediate 0.9 out loud lets the interviewer follow you and catch a slip early. The order does not matter: multiplication gives the same answer whichever change comes first.

In private equity this is the same arithmetic as a multiple that expands 20% at one point and contracts later, or a business that grows, shrinks and recovers. Revenue up 20% and back down 20% does not return to the start: it ends 4% lower.

994. Treat 12.5% as one eighth and halve 1,368 three times: 684, 342, 171. For a third of 2,469, split it into 2,400 and 69: a third of each is 800 and 23, so 823. Then 171 plus 823 is 994. Common percentages are fractions in disguise, and fractions are faster in your head.

Why turn 12.5% into a fraction?

Multiplying by 0.125 in your head means three digits of decimals to keep track of. Halving is something you have done since school, like splitting a restaurant bill between two, then four, then eight friends. 12.5% is exactly one eighth, so three halvings give the answer with no decimals at all. 1,368 halves to 684, then 342, then 171.

How do you divide by three cleanly?

Split the number into a part that divides easily and a small remainder. 2,469 is 2,400 plus 69, and a third of each is 800 and 23, so a third of the whole is 823 with no long division. Check it: 823 times 3 is 2,469. The same trick works for any divisor: pick the nearest round multiple, then handle the leftover.

Buyout interviews use questions like this as a warm-up, often before a paper LBO, to see whether you will cope with arithmetic out loud. Saying each step lets the interviewer follow you and makes a slip easy to catch.