Private Equity puzzles, solved step by step

Puzzles: 100
Traced to a firm: 22
Topics: 12
Hard: 30

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046In your head: you buy a business with EBITDA of 50 at 8x, funded 50% with debt. EBITDA grows to 75 over 5 years, cumulative free cash flow of 100 repays debt, and you exit at 8x. What are the money multiple and the IRR?Mental paper LBOsCoreNeuberger BermanNew York · 2022
Try it first
What is the money multiple on the equity?
Show the worked solution
2.5x and an IRR of about 20%. Entry EV is 50 x 8 = 400, half debt, so equity is 200. Exit EV is 75 x 8 = 600, and debt has fallen from 200 to 100, so exit equity is 500. 500 over 200 is 2.5x. Over five years that is about 20% a year, since 1.2 to the fifth is about 2.49.
What is the fastest way to do this in your head?
Buy a flat for Rs 40 lakh with a Rs 20 lakh loan. Five years later it is worth Rs 60 lakh and you have paid the loan down to Rs 10 lakh. Your stake went from Rs 20 lakh to Rs 50 lakh. Every paper LBO is two balance sheets, entry and exit, and the answer is exit equity over entry equity, where equity is EV less debt each time. Write the four numbers, subtract twice, divide once.
Entry EV of 400 is funded with 200 of debt and 200 of equity; at exit EV is 600 and debt 100, so equity is 500, a 2.5x return made up of 200 from EBITDA growth and 100 from debt paydown.
How do you get the IRR without a calculator?
Carry three anchors for a five-year hold. 2x in five years is about 15% a year, 2.5x is about 20%, and 3x is about 25%. 1.2 to the fifth is 1.2 x 1.2 = 1.44, then 1.73, 2.07 and 2.49, so 2.5x is a touch over 20%: 20.1%. Saying the anchor and then the check sounds far more assured than reaching for a formula.
The relationship
$MOIC = \frac{E_5 \, m - D_5}{E_0 \, m - D_0} = \frac{75 \times 8 - 100}{50 \times 8 - 200} = \frac{500}{200} = 2.5x$
E_0, E_5 EBITDA at entry and exit, 50 and 75
m the multiple at both ends, 8x
D_0, D_5 debt at entry and exit, 200 and 100
What it says in wordsThe money multiple is exit equity over entry equity, each one EV less debt.
Then split the gain, because the interviewer will ask where the return came from. EBITDA rose 25 at a constant 8x, adding 200 of value; cash flow repaid 100 of debt; the multiple did not move, so it added nothing. The limitation: the sums ignore fees at entry and any cash left on the balance sheet at exit, both of which a real model would carry.
Where candidates lose it
The common loss is computing the EV return, 600 over 400, and calling it 1.5x. The question is about the equity, and leverage plus paydown makes the equity grow much faster than the EV.
The second loss is forgetting the debt paydown and using 200 of debt at exit, which gives 400 of equity and 2.0x. Track the debt at both ends.
What the interviewer asks next
Exit at 7x instead of 8x. What are the multiple and IRR now?
What would the return have been with no debt at all?
How much of the return would you attribute to the sponsor's work, and how much to leverage?
Asked at Neuberger Berman, Private Equity, New York, 2022 (Wall Street Oasis): The most difficult was the more advanced industry-specific technicals and paperback LBOs
047Fund A compounds at 18% a year for 10 years. Fund B earns 25% a year for the first 5 years, then 10% a year for the next 5. Which ends with more money?Compounding and time valueCoreSecondaries and fund of funds
Try it first
Which fund ends ahead after ten years?
Show the worked solution
Fund A ends ahead: 5.23x against 4.91x. 1.18 to the tenth is about 5.23. For B, 1.25 to the fifth is about 3.05 and 1.1 to the fifth about 1.61, which multiply to 4.91. B leads at year five, 3.05x against 2.29x, but its slow second half lets A catch up around year 9. B's ten-year compound rate is 17.3%, below A's 18%.
Why does the fast starter lose?
A batsman who scores fast for 20 overs and then crawls can end below a partner who kept a steady rate all innings. Over a long period what matters is the compound rate across the whole span, and two halves at 25% and 10% compound to 17.3% a year, below a steady 18%. The early lead is real, but the second half compounds on a big base at a low rate.
Fund B's 25% start puts it at 3.05x after five years against A's 2.29x, but at 10% thereafter it is overtaken around year 9.1 and ends at 4.91x against A's 5.23x.
The relationship
$1.18^{10} \approx 5.23 \qquad 1.25^5 \times 1.10^5 \approx 3.05 \times 1.61 \approx 4.91$
1.18^10 Fund A's ten-year multiple
1.25^5 x 1.10^5 Fund B's two halves multiplied together
What it says in wordsMultiply the growth factors of each period; never average the rates.
Why is the average rate misleading here?
The simple average of 25% and 10% is 17.5%, but the compound rate is the geometric mean of the factors: the square root of 1.25 x 1.10, less one, which is 17.3%. The geometric mean is always below the simple average when rates differ, so a fund with uneven returns compounds more slowly than its average suggests. That gap widens the more the rates vary.
This matters to a secondaries or fund-of-funds buyer reading a track record. A fund that shows a strong early IRR may owe it to a few quick exits; what the LP takes home depends on the whole life. Say the limitation: real funds call and return capital over time, so their IRRs are not simple compound rates on one cheque.
Where candidates lose it
The common loss is backing B because it starts faster and its rates average almost the same. Averaging rates is the error: compounding multiplies factors, and the geometric mean sits below the arithmetic one.
The second loss is computing 1.25 to the fifth correctly and then adding the second half's growth instead of multiplying. Say each half as a factor, then multiply.
What the interviewer asks next
What rate in B's second half would make the two funds end level?
Why is a fund's early IRR often higher than its final IRR?
What is the geometric average of +50% and -50%, and why does that matter for volatile returns?
048Without a calculator, what is 1.1 to the power 7, to two decimal places?Mental mathsCoreLarge-cap buyout fund
Try it first
Which is right to two decimals?
Show the worked solution
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.
Seven 10% steps take 1 to 1.949, with each step adding a tenth of the bar before it, which lands just under the 2.0 that the rule of 72 predicts for seven years at 10%.
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.
The relationship
$(1+0.1)^7 = 1 + 7(0.1) + 21(0.1)^2 + 35(0.1)^3 + 35(0.1)^4 + \dots \approx 1.9487$
7, 21, 35, 35 the binomial coefficients for the seventh power
0.1 the 10% growth rate
What it says in wordsExpanding the bracket term by term gives 1.9487, with each term much smaller than the one before.
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.
Where candidates lose it
The common loss is adding seven lots of 10% to get 1.70, the simple-interest answer. The interviewer is checking whether compounding is automatic for you.
The second loss is rounding hard at every step, 1.2, 1.3, 1.5 and so on, which drifts well below the true value. Carry the decimals and round only once.
What the interviewer asks next
What is 1.1 to the 10th, roughly?
Use the same method for 1.08 to the 5th.
EBITDA of 100 grows 10% a year for 7 years. What is the exit EBITDA, and what if growth is 12%?
050You have two ropes and a lighter. Each rope takes exactly 60 minutes to burn, but they burn unevenly, so half a rope's length does not take 30 minutes. How do you measure exactly 45 minutes?Logic and brainteasersCoreLarge-cap buyout fund
Try it first
What is the key move?
Show the worked solution
Light rope A at both ends and rope B at one end, at the same moment. Rope A burns out after exactly 30 minutes, because two flames together consume 60 minutes of burning in half the time. At that moment rope B has 30 minutes of burning left, so light its other end. It burns out 15 minutes later, at 45 minutes in total.
Why does lighting both ends give exactly 30 minutes?
Two people painting a fence from opposite ends finish in half the time, even if some planks take longer than others, because they always work together on whatever is left. A rope holds 60 minutes of burning in total; two flames each burn their own share, and between them they use it up in 30 minutes, wherever they happen to meet. The uneven burn only changes where the flames meet, not when.
Rope A, lit at both ends, burns out at 30 minutes whatever its burn rate; rope B, lit at one end, then has exactly 30 minutes of burning left, and lighting its other end at that moment makes it last 15 more, so it goes out at 45.
Why does rope B have exactly 30 minutes left?
Because time, not length, is what the rope stores. Rope B has burned for 30 minutes at the moment rope A goes out, so whatever length remains holds exactly 30 minutes of burning, and two flames finish that in 15. You never need to know where on the rope the flame is. That is the general lesson: when a quantity is uneven in one dimension, measure it in the dimension that is reliable.
Say it in order, with the two events that trigger each action: at the start, light three ends; when A goes out, light B's last end; when B goes out, 45 minutes have passed. Interviewers mark you on the clean sequence as much as on the idea. The limitation of the puzzle is its assumption that both ends can be lit at exactly the same moment.
Where candidates lose it
The common loss is trying to use length: cutting a rope in half or marking three quarters of it. The question says the burn is uneven precisely to rule that out, and candidates who miss the word lose the room.
The second loss is lighting rope B's second end at the start as well, which gives 30 minutes, not 45. B must burn from one end for the first 30 minutes.
What the interviewer asks next
With the same two ropes, how would you measure 15 minutes?
What times between 0 and 120 minutes can you measure with two such ropes?
How many ropes would you need to measure 7.5 minutes?
052A company carries debt of 600 against EBITDA of 100, so it is levered 6.0x. It repays 100 of debt every year and EBITDA grows 10% a year. What is leverage after three years?Leverage and capital structureCoreMid-market buyout fundIndian mid-market PE
Try it first
Pick the closest figure for leverage at the end of year 3.
Show the worked solution
About 2.25x. Three repayments take debt from 600 to 300, and three years of 10% growth take EBITDA from 100 to 133.1. Leverage is 300 over 133.1, or 2.25x, down from 6.0x. Halving the debt alone would have left 3.0x; growth removes most of another turn, and that matters to a lender reading the covenant.
Why does leverage fall faster than the debt?
Think of a family loan measured against the family's income. Paying down the loan helps, and so does a pay rise, because the bank looks at how many years of income the loan represents. Leverage is debt divided by EBITDA, so it falls when the top shrinks and when the bottom grows, and here both are happening at once. Track both lines year by year rather than jumping to the end.
Debt falls from 600 to 300 while EBITDA rises from 100 to 133.1, so leverage drops from 6.0x to 2.25x; repayment alone would leave 3.0x and growth alone 4.51x.
Year Debt EBITDA Debt / EBITDA
0 600 100.0 6.00x
1 500 110.0 4.55x
2 400 121.0 3.31x
3 300 133.1 2.25x
Each year debt falls by a flat 100 and EBITDA rises by 10%, so the ratio drops from 6.00x to 4.55x, 3.31x and finally 2.25x.
How much of the drop came from growth?
Split the move multiplicatively, which is exact for a ratio. Halving the debt divides leverage by 2.0; growing EBITDA by 33.1% divides it by 1.331. Six divided by 2.0 and then by 1.331 is 2.25x. Measured that way, repayment does about 71% of the work and growth about 29%, so growth is worth roughly three quarters of a turn here. Say the limit too: the 100 a year of repayment is an assumption, and in a real deal the cash for it comes from the same EBITDA that is growing.
Where candidates lose it
The common slip is answering 3.0x: halving the debt and stopping. It treats leverage as a debt number when it is a ratio, and it misses the 33% rise in EBITDA that does a large share of the work.
The other slip is compounding the debt reduction, as if debt fell 100 then a percentage. The repayment is a flat 100 each year; only EBITDA compounds. Keep the two lines apart and the ratio falls out.
What the interviewer asks next
What leverage would a lender see at year 3 if EBITDA had fallen 10% a year instead?
The covenant steps down to 2.5x at year 3. Is there headroom?
Where does the 100 a year of repayment actually come from, and is it realistic at 6.0x?
053A commercial due diligence study costs Rs 2 crore. There is a 25% chance it uncovers a deal-breaker that would otherwise cost the fund Rs 30 crore. Is the study worth buying?Probability and expected value in dealsCoreMid-market buyout fund
Try it first
What is the most the fund should be willing to pay for this study?
Show the worked solution
Yes: the study is worth Rs 7.5 crore in expectation and costs Rs 2 crore, a net gain of Rs 5.5 crore. Without it, the fund carries a 25% chance of losing Rs 30 crore, an expected loss of Rs 7.5 crore. With it, the fund walks away in the bad case and pays only the fee. The study stays worth buying as long as the chance of a problem is above 2 over 30, about 6.7%.
What exactly is the fund buying?
Think of paying a mechanic to inspect a second-hand car before you buy it. The inspection is worth nothing if the car is sound and a great deal if the gearbox is about to fail. A diligence study is worth the loss it can prevent, multiplied by the chance that the loss is really there. It is not worth the loss itself, and it is not worth zero because the problem is unlikely. Draw both choices as branches and the answer is visible.
Proceeding blind carries a 25% chance of a Rs 30 crore loss, an expected loss of Rs 7.5 crore; buying the study caps the cost at its Rs 2 crore fee, so the study is worth Rs 5.5 crore more than it costs.
The relationship
$V = p \times L - c = 0.25 \times 30 - 2 = 5.5$
p the chance the deal-breaker exists and the study finds it, 25%
L the loss avoided by walking away, Rs 30 crore
c the cost of the study, Rs 2 crore
What it says in wordsThe net value of a test is the loss it prevents times the chance it prevents it, less what the test costs.
What assumptions is that answer resting on?
Two, and you should say both. First, the study is perfect: it always finds the problem when it is there and never raises a false alarm. If the study catches the problem only 60% of the time, its value falls to 0.25 x 0.6 x 30, Rs 4.5 crore, still above the fee. Second, walking away costs nothing beyond the fee. In a live auction, pausing for a study can lose the deal to a faster bidder, and that cost belongs on the yes branch.
Where candidates lose it
Candidates compare the Rs 2 crore fee with the Rs 30 crore loss and say yes because 30 is bigger, which would also say yes at a fee of Rs 20 crore. The comparison that decides it is the fee against the expected saving of Rs 7.5 crore.
The opposite slip is dismissing the study because the problem probably is not there. A 75% chance of a clean result is exactly why the study is cheap insurance, not why it is wasted.
What the interviewer asks next
What if the study also raises a false alarm on 10% of clean deals, and you would walk away from a good deal worth Rs 12 crore?
At what fee would you be indifferent?
How would you put a value on the time the study takes in a competitive auction?
054A company has an enterprise value of 500 and holds 300 of net cash, so its equity is worth 800. A buyer offers a 20% premium on the equity value. By what percentage does the implied enterprise value rise?Valuation riddlesCoreLarge-cap buyout fund
Try it first
The equity premium is 20%. What is the premium on the enterprise value?
Show the worked solution
The implied enterprise value rises 32%, from 500 to 660. A 20% premium on equity of 800 is 160, taking the offer to 960. The 300 of cash is worth 300 to everyone, so the whole 160 is a premium on the operating business. With net cash, the premium on EV is always larger than the headline premium on equity; with net debt, it is smaller.
Why can the premium not land on the cash?
Picture buying a shop that has Rs 3 lakh sitting in its till. You may pay extra for the shop's location and customers, but nobody pays Rs 3.6 lakh for Rs 3 lakh of cash: the cash is worth its face value to every buyer. The equity price is cash plus the business, and since the cash is fixed at 300, the full 160 of premium has to sit on the business. That turns a 20% premium into a 32% premium on the part the buyer is really valuing.
Equity rises 20% from 800 to 960, but the 300 of cash does not move, so the enterprise value rises from 500 to 660, a 32% premium; with no cash the same offer would be 20% on both.
The relationship
$\Delta EV\% = \frac{p \times E}{E - C} = \frac{0.20 \times 800}{800 - 300} = 32\%$
p the premium on equity, 20%
E equity value before the offer, 800
C net cash, 300
What it says in wordsThe premium on the business is the rupee premium divided by the business value, which is equity less cash.
Why would a buyout investor care which premium you quote?
Because the sponsor underwrites the business, not the cash. If the business earned EBITDA of 50, the market price would be 10x and the offer is 13.2x. A 20% headline premium sounds modest, but on a cash-rich target it can mean paying a third more for the operations. The limit runs the other way too: on a company with large net debt, a 20% equity premium is a much smaller premium on EV, so headline premiums are never comparable across capital structures without this step.
Where candidates lose it
The common answer is 20%: candidates assume the premium spreads evenly across equity and enterprise value. It only does when there is no cash and no debt.
The second slip is subtracting the cash from the wrong side and getting a smaller number, as though cash dilutes the premium. Cash is the one part of the company with no premium on it, which is why the rest carries more.
What the interviewer asks next
Now the company has 300 of net debt instead of cash. What is the premium on EV?
If the business earns EBITDA of 50, what multiple does the buyer pay? (13.2x)
Why might a target with large cash still demand a premium on the cash, and how would you respond?
055A sponsor buys a business with EBITDA of 80 at 10x, funded 60% with debt. EBITDA stays flat for four years and the exit is also at 10x. How much debt must be repaid for the sponsor to make 2x its money?Mental paper LBOsCoreMid-market buyout fundIndian mid-market PE
Try it first
How much debt has to be repaid over the four years?
Show the worked solution
The company must repay 320, which is the whole equity cheque. The business costs 800, funded by 480 of debt and 320 of equity. With flat EBITDA and the same multiple it is still worth 800 at exit, so equity of 640 needs debt to fall to 160. That is 80 a year for four years, equal to all of EBITDA, so in practice this deal cannot reach 2x without growth.
Why does the repayment equal the equity cheque?
Think of a house bought for Rs 80 lakh with a Rs 48 lakh loan and Rs 32 lakh of your own money. If the house is still worth Rs 80 lakh years later, the only way your share doubles to Rs 64 lakh is for the loan to shrink to Rs 16 lakh. When the enterprise value does not move, equity gains exactly what the debt loses, so doubling the equity means repaying an amount equal to the original equity. That holds at any leverage, which makes it a fast check.
With enterprise value stuck at 800, equity doubles from 320 to 640 only if debt falls from 480 to 160; that repayment of 320 is 80 a year, equal to all of EBITDA before year 1 interest of about 38.
Is that repayment realistic?
Now test the answer against the business. Repaying 80 a year means every rupee of EBITDA goes to the lenders as principal. Interest on 480 of debt at 8% is about 38 in year one alone, before any tax or capex, so the cash to repay 80 a year simply does not exist. The honest conclusion is the useful one: a flat business bought at 10x with 6x debt cannot reach 2x from paydown. It needs EBITDA growth, a higher exit multiple, or a lower entry price.
The relationship
$\text{Repayment} = (2 - 1) \times E_0 = 320 \qquad \text{IRR} = 2^{1/4} - 1 \approx 18.9\%$
E_0 the sponsor's equity at entry, 320
2 the target money multiple
1/4 four years of hold
What it says in wordsWith no change in enterprise value, the debt repaid must equal the gain you want on the equity.
Where candidates lose it
Candidates get 320 and stop, which wins half the point. The interviewer is waiting for the second sentence: 80 a year is all of EBITDA, so the plan does not work. Answering the arithmetic without testing it against the cash flow misses why the question was asked.
The other slip is answering 160, the debt left at exit, instead of the 320 repaid. Say both numbers so there is no doubt which you mean.
What the interviewer asks next
How much EBITDA growth gets the same deal to 2x with no debt repaid at all?
If the exit multiple rises to 11x, how much repayment do you still need?
2x in four years is about 18.9% a year. Would a fund accept that on this risk?
056Two companies each have EBITDA of 100 and an enterprise value of 900. One spends 10 a year on capex, the other 50. If you could value a business on one metric only, levered free cash flow, EBIT or EBITDA, which would you choose, and which of these two is cheaper?Valuation riddlesCoreWarburg PincusSan Francisco · 2014
Try it first
Both trade at 9.0x EBITDA. What are they on EBITDA less capex?
Show the worked solution
Choose levered free cash flow, and Company A is the cheaper one. Both cost 9.0x EBITDA, but A keeps 90 after capex and B only 50, so the same 900 buys cash at 10.0x in A and 18.0x in B. After 25% tax the gap holds at 13.3x against 24.0x. Levered free cash flow is the one number that has already paid for capex, working capital, interest and tax: it is what an owner can actually take out.
What does EBITDA leave out that matters here?
Picture two taxis, each bringing in Rs 1 lakh a month after fuel and the driver. One is new and needs little upkeep; the other is old and eats half its takings in repairs. Before repairs they look identical, and nobody would pay the same for both. EBITDA is earnings before the business pays for the assets it uses up, so two companies with equal EBITDA can leave their owners very different amounts of cash. Company B spends half its EBITDA just to keep running.
Both companies cost 900 for EBITDA of 100, 9.0x, but after capex Company A keeps 90 and Company B keeps 50, so the same price is 10.0x A's cash and 18.0x B's; after tax the gap is 13.3x against 24.0x.
Why levered free cash flow rather than EBIT?
EBIT does subtract depreciation, which here happens to equal capex, so EBIT would rank the two correctly. But depreciation is an accounting estimate of past spending, not the cash going out this year, and it ignores working capital, interest and tax. Levered free cash flow is the only one of the three that is cash in the owner's hand after every claim ahead of the owner has been paid. Assume no debt and depreciation equal to capex: A's free cash flow is 100 less tax of 22.5 less capex of 10, which is 67.5; B's is 37.5.
What is the limit of your answer?
Say two things. First, one year of free cash flow can be lumpy: a big capex year or a working capital swing can make a good business look expensive. With a single period of data, levered free cash flow is the most honest number, but it is also the noisiest. Second, levered cash flow belongs to the equity, so you compare it with equity value, not enterprise value. In this example there is no debt, so the two are the same; with debt you would divide by equity value instead.
Where candidates lose it
Candidates pick EBITDA because buyout people talk in EBITDA multiples all day. The question is built to see whether you know what EBITDA leaves out: on these two companies it gives the same answer for a business that keeps 90 and one that keeps 50.
The second loss is choosing levered free cash flow and then dividing it by enterprise value. Levered cash is the equity's cash; set it against the equity value, or say clearly that with no debt the two coincide.
What the interviewer asks next
Company B's capex is growth capex for a new plant. Does that change your view?
Which of the three metrics would you pick for a bank, and why?
How would you adjust one year of free cash flow that includes a large working capital release?
Asked at Warburg Pincus, Private Equity, San Francisco, 2014 (Wall Street Oasis): If you knew nothing about a company or industry and were able to use one metric as a means of valuation, what would you choose?
057Without a calculator: what is 1,000 divided by 7, to three decimal places? And what is 22 divided by 0.35?Mental mathsCoreSoftware buyout
Try it first
Which first step makes 22 / 0.35 easy?
Show the worked solution
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.
Dividing 1,000 by 7 leaves remainders 6, 4, 5, 1, 3 and 2 before 6 returns, so the digits 857142 repeat and the answer is 142.857; 22 / 0.35 rescales to 440 / 7, which is 62 and six sevenths, about 62.86.
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.
The relationship
$\frac{22}{0.35} = \frac{2200}{35} = \frac{440}{7} = 62\tfrac{6}{7} \approx 62.857$
x100 multiplying top and bottom by the same number leaves the value unchanged
440/7 after dividing 2,200 and 35 by 5
6/7 the leftover, read off the 142857 cycle as 0.857
What it says in wordsRescale the divisor to a whole number, cancel what you can, then divide.
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.
Where candidates lose it
The usual slip on 22 / 0.35 is moving the decimal the wrong way and answering 6.29 or 628.6. Rescale out loud, 2,200 over 35, and the size of the answer is obvious before you start.
On 1,000 / 7, candidates stop at 142.8 or round to 142.86 when asked for three decimals. Keep going until a remainder repeats: the cycle gives you every digit you need and shows you know why.
What the interviewer asks next
What is 5/7 as a decimal, without dividing?
Divide 1 by 13 to six decimals. What is the repeating cycle?
A company earns 38 on capital of 0.45 crore units. Quickly, what is the ratio?

E_0, E_5	EBITDA at entry and exit, 50 and 75
m	the multiple at both ends, 8x
D_0, D_5	debt at entry and exit, 200 and 100

1.18^10	Fund A's ten-year multiple
1.25^5 x 1.10^5	Fund B's two halves multiplied together

7, 21, 35, 35	the binomial coefficients for the seventh power
0.1	the 10% growth rate

Year	Debt	EBITDA	Debt / EBITDA
0	600	100.0	6.00x
1	500	110.0	4.55x
2	400	121.0	3.31x
3	300	133.1	2.25x

Try it first

What is the fastest way to do this in your head?

How do you get the IRR without a calculator?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the fast starter lose?

Why is the average rate misleading here?

Where candidates lose it

What the interviewer asks next

Try it first

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

How do you check it a second way?

Where candidates lose it

What the interviewer asks next

Try it first

Why does lighting both ends give exactly 30 minutes?

Why does rope B have exactly 30 minutes left?

Where candidates lose it

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Why does leverage fall faster than the debt?

How much of the drop came from growth?

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What exactly is the fund buying?

What assumptions is that answer resting on?

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Why can the premium not land on the cash?

Why would a buyout investor care which premium you quote?

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Why does the repayment equal the equity cheque?

Is that repayment realistic?

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What does EBITDA leave out that matters here?

Why levered free cash flow rather than EBIT?

What is the limit of your answer?

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Why is one seventh worth memorising?

How do you show the long division out loud?

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