Private Equity puzzles, solved step by step

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

Showing 41–50 of 100

041Estimate the annual power bill of a data centre with 20 MW of IT capacity, running at 70% utilisation, with a power usage effectiveness (PUE) of 1.4 and a tariff of Rs 7 per kWh.Market sizing and estimationHardInfrastructure fund
Try it first
Which is closest to the yearly bill?
Show the worked solution
About Rs 120 crore a year. 20 MW at 70% use is 14 MW of IT load. A PUE of 1.4 means the site draws 1.4 times that for cooling and losses, 19.6 MW. Running all year, 19.6 MW times 8,760 hours is about 171.7 million kWh. At Rs 7 a kWh that is about Rs 1,202 million, or Rs 120 crore. The tariff is an assumption to check.
What does PUE add to the bill?
A home air conditioner does not just cool the room; it also uses power to run its own compressor and fan. Power usage effectivenessTotal power drawn by the whole facility divided by the power used by the IT equipment alone. 1.0 would mean no overhead at all. is total site power divided by IT power, so a PUE of 1.4 means every 1 MW of servers needs another 0.4 MW for cooling, power conversion and lighting. Here that is 5.6 MW on top of 14 MW of IT load, and the bill pays for all 19.6.
20 MW of IT capacity at 70% use draws 19.6 MW once a PUE of 1.4 adds cooling and losses, which over 8,760 hours is 171.7 million kWh and about Rs 120 crore a year at an assumed Rs 7 per kWh.
How do you keep the units straight out loud?
Convert once, early, and say it. One megawatt running for one hour is 1,000 kWh, so 19.6 MW for a year is 19.6 x 8,760 x 1,000, about 171.7 million kWh. Then price it: Rs 7 times 171.7 million is about Rs 1,202 million. A crore is ten million, so divide by ten: Rs 120 crore. Most wrong answers to this question are right in method and a factor of ten off in units.
The relationship
$\text{Bill} = C \times u \times PUE \times 8760 \times 1000 \times p = 20 \times 0.7 \times 1.4 \times 8760 \times 1000 \times 7$
C IT capacity, 20 MW
u utilisation, 70%
PUE total power over IT power, 1.4
p tariff, Rs 7 per kWh, an assumption
What it says in wordsMultiply the power actually drawn by the hours in a year, convert to kWh, and price it.
Now the investor's view. Power is usually the largest running cost of a data centre, so PUE is an economic number, not a technical one: improving it from 1.4 to 1.3 saves about Rs 8.6 crore a year here. Many colocation contracts pass power through to tenants, which shifts who bears the tariff risk. Ask how the contracts treat power before you treat the bill as the owner's cost.
Where candidates lose it
The common loss is a factor of ten in the units: candidates convert MW to kWh correctly and then slip between million and crore, landing at Rs 12 crore or Rs 1,200 crore. Say each conversion out loud.
The second loss is pricing only the IT load and forgetting the PUE, which understates the bill by 40% and misses the point of the question.
What the interviewer asks next
What tariff would make the bill Rs 150 crore?
If utilisation rises to 90%, what happens to the bill and to the PUE?
In a colocation contract, who usually pays for power, and why does that matter to the investor?
042Which makes more money on the same cheque: a 25% IRR for 3 years, or a 20% IRR for 5 years?Returns mathsWarm upMid-market buyout fund
Try it first
Which ends with more money?
Show the worked solution
The 20% for 5 years makes more money: 2.49x against 1.95x. 1.25 cubed is about 1.95 and 1.2 to the fifth is about 2.49. On a cheque of 100 that is a profit of 149 against 95. The 25% deal is faster, but the money is back after three years, and it only catches up if it can be reinvested at about 13% for the remaining two.
Why does the higher IRR make less money?
A car doing 100 km an hour for three hours covers 300 km; one doing 80 km an hour for five hours covers 400. Speed and distance are different questions. IRR measures how fast money grows, the multiple measures how much money you end with, and a longer hold at a lower speed can end further ahead. Here 1.25 cubed is 1.95 and 1.2 to the fifth is 2.49.
Compounding at 25% for three years ends at 1.95x, while 20% for five years ends at 2.49x, so the higher IRR makes less money unless its proceeds can be reinvested at about 12.9% for the two years it is not running.
So which would an LP prefer?
It depends on what happens to the money after year three. If the LP can redeploy A's proceeds at more than about 12.9% a year, A ends ahead; below that, B wins. Reinvested at 20%, A reaches 2.81x by year five; parked at 8%, only 2.28x. That is why LPs read IRR and the money multiple together, and why GPs who sell early to protect a high IRR are sometimes accused of leaving money on the table.
The relationship
$1.25^3 = 1.95 \qquad 1.20^5 = 2.49 \qquad (1+r)^2 = \frac{2.49}{1.95} \Rightarrow r \approx 12.9\%$
1.25^3 A's money multiple after three years
1.20^5 B's money multiple after five years
r the reinvestment rate at which A catches B by year five
What it says in wordsCompare the two multiples, then ask what rate A's money must earn in the gap years to catch up.
Say the limitation. Both deals here are a single cheque in and out. A fund's IRR also depends on when capital is called and returned, and a high IRR on a small, quick deal can flatter a fund that made little money overall.
Where candidates lose it
The common loss is picking the higher IRR on reflex. The interviewer is checking whether you know that IRR is a rate and says nothing on its own about how much money comes back.
The second loss is giving the right answer without the reinvestment point. The full answer is that B makes more money, and A wins only if its proceeds can be redeployed at about 13% or better.
What the interviewer asks next
What IRR over 3 years would match 2.49x?
Why might a GP sell a winner early even though holding would make more money?
How does the timing of capital calls affect a fund's IRR but not its multiple?
043An asset earns 9% a year unlevered and debt costs 12%. What is the return on equity at 0%, 40% and 70% debt funding? When does adding leverage lower the equity return?Leverage and capital structureHardLarge-cap buyout fundPrivate credit
Try it first
At 70% debt, roughly what does equity earn?
Show the worked solution
9%, 7% and 2%: here every turn of debt lowers the equity return, because debt costs more than the asset earns. On 100 of assets earning 9, debt of 40 costs 4.8 and leaves 4.2 on equity of 60, which is 7%. Debt of 70 costs 8.4 and leaves 0.6 on equity of 30, which is 2%. Leverage helps only when the asset earns more than the debt costs; when it earns less, leverage works in reverse.
How does leverage change the equity return?
Borrow at 12% to buy a flat that earns 9% in rent and you lose 3% a year on every borrowed rupee, and the more you borrow, the bigger the loss as a share of your own money. Equity keeps whatever the asset earns after the debt is paid, so each unit of debt adds the gap between the asset return and the debt cost, times the debt-to-equity ratio. With a 9% asset and 12% debt that gap is minus 3 points.
The relationship
$r_E = r_A + (r_A - r_D)\frac{D}{E} = 9\% + (9\% - 12\%) \times \frac{70}{30} = 2\%$
r_A the unlevered asset return, 9%
r_D the cost of debt, 12%
D/E debt over equity, 70/30 at 70% debt
What it says in wordsEquity earns the asset return plus the asset-debt gap, scaled up by how much debt sits on each unit of equity.
With an asset earning 9%, debt at 12% drags the equity return down to 7% at 40% debt and 2% at 70%, while debt at 6% lifts it to 11% and 16%, because leverage magnifies whatever the gap between the two rates is.
Why does a buyout investor worry about this?
A business bought at a high multiple has a low unlevered return: at 12x EBITDA, pre-tax cash yield on the purchase price is in single digits. If borrowing costs rise above that yield, the leverage that used to lift returns now pulls them down, and the deal depends entirely on growth and multiple expansion. This is called negative leverage, and it is why higher rates squeeze what sponsors can pay.
Debt share Equity return, debt at 12% Equity return, debt at 6%
0% 9% 9%
40% 7% 11%
70% 2% 16%
The same three capital structures give falling returns when debt costs more than the asset earns and rising returns when it costs less.
Say the limitations. The sums ignore tax: interest is usually deductible, which lowers the after-tax cost of debt and moves the break-even. And they treat the asset return as fixed. A growing business can out-earn expensive debt later even if it does not today, which is the bet some sponsors knowingly make.
Where candidates lose it
The common loss is the reflex that leverage always increases equity returns. It magnifies the spread, and when the spread is negative it magnifies losses of return. Interviewers ask this exactly to catch that reflex.
The second loss is working in percentages of the asset instead of the equity. Compute earnings, subtract interest, and divide by the equity cheque: 0.6 on 30 is 2%, not 0.6%.
What the interviewer asks next
At what cost of debt does leverage stop mattering for this asset?
How does the interest tax shield change the 70% debt answer at a 25% tax rate?
If the asset return grows 1 point a year, how long before 70% debt at 12% starts to help?
044Work backwards: EBITDA grows from 100 to 140 over 5 years and the business exits at 9x. Entry debt is 5x EBITDA and 200 of it is repaid by exit. What is the highest entry multiple that still returns 3x the equity?Mental paper LBOsHardNeuberger BermanNew York · 2022
Try it first
Where does the maximum entry multiple land?
Show the worked solution
About 8.2x. Exit EV is 140 x 9 = 1,260. Debt falls from 500 to 300, so exit equity is 960. A 3x return allows entry equity of 960 / 3 = 320. Add the 500 of entry debt and the most you can pay is an EV of 820, which is 8.2x entry EBITDA. Pay more and the 3x target is missed; fees and costs would lower the ceiling further.
Why work backwards instead of guessing an entry price?
If you want to arrive at a wedding by 7 and the drive takes two hours with half an hour of traffic, you leave at 4:30. You fix the end point and walk back. A target multiple fixes the exit equity you need relative to the cheque, so the maximum price is found by running the LBO in reverse: exit value, less exit debt, divided by the target, plus entry debt. Every number you need is in the question.
Exit EV of 1,260 less 300 of remaining debt leaves exit equity of 960, which supports entry equity of 320 at a 3x target; adding 500 of entry debt sets the maximum entry EV at 820, or 8.2x EBITDA of 100.
The relationship
$EV_{0}^{max} = \frac{E_5 \times m_{exit} - D_5}{\text{target}} + D_0 = \frac{1260 - 300}{3} + 500 = 820$
E_5 x m_exit exit EV, 140 x 9
D_5 debt left at exit, 300
target the money multiple required, 3x
D_0 entry debt, 5 x 100 = 500
What it says in wordsThe most you can pay is the exit equity divided by the target multiple, plus the debt you borrow at entry.
What does 8.2x tell the investment committee?
It is a ceiling, not an offer. Entering at 8.2x and exiting at 9x means the plan relies on a little multiple expansion as well as 40% EBITDA growth and 200 of debt paydown, so the committee will test each of those three. 3x over five years is an IRR of about 25%: 3 to the power one fifth is about 1.246.
Say the limitations. Transaction fees and financing costs come out of the equity at entry, so the true ceiling on the headline price is lower. A real model would also hold some cash at exit and might pay interest in kind. And if the lenders will not provide 5x at an 8.2x price, the equity cheque grows and the ceiling falls.
Where candidates lose it
The common loss is forgetting to add the entry debt back: candidates divide exit equity by 3, get 320, and say 3.2x. That is the equity cheque over EBITDA, not the enterprise multiple.
The second loss is using entry debt instead of exit debt when computing exit equity, which ignores the 200 repaid and makes the ceiling look lower than it is. Track debt at both ends.
What the interviewer asks next
If the exit multiple is 8x instead of 9x, what is the new ceiling?
How much does the ceiling fall if 20 of fees are paid out of the equity at entry?
What IRR does 3x in five years imply, and what multiple would 25% need?
Asked at Neuberger Berman, Private Equity, New York, 2022 (Wall Street Oasis): Interviews 4-5 were very technical again and also included multiple paperback LBOs and other, more advanced technicals.
045A seller wants 100 upfront plus an earnout of 50 if year-2 EBITDA hits a target. You think the chance of hitting it is 40%. At a 12% discount rate, what is the earnout worth to you today, and what is the total price in value terms?Probability and expected value in dealsHardMid-market buyout fund
Try it first
What is the earnout worth to you today?
Show the worked solution
The earnout is worth about 15.9 to you, so the price in value terms is about 115.9. A 40% chance of 50 is 20 on average, payable in two years, and 20 / 1.12^2 is 15.9. The headline price is 150, but you are paying about 116 in value. A seller who thinks the chance is 80% values the same earnout at 31.9, which is why earnouts close price gaps.
How do you value a payment that may not happen?
A shopkeeper who offers a Rs 500 voucher to anyone who spends Rs 5,000 next month books its cost by how many customers she expects to claim it, not by the face value. An earnout is a contingent payment, so you value it as chance times amount, discounted for the wait: 0.4 x 50 / 1.12^2 is about 15.9. That number, not 50, is what the earnout adds to your price.
The relationship
$V = \frac{p \times EO}{(1+r)^t} = \frac{0.4 \times 50}{1.12^2} \approx 15.9$
p your estimate of the chance the target is hit, 40%
EO the earnout payment, 50
r the discount rate, 12%
t years until payment, 2
What it says in wordsAn earnout is worth the chance of paying it, times the amount, discounted to today.
The same 50 earnout is worth 15.9 today to a buyer who puts the chance at 40% and 31.9 to a seller who puts it at 80%, so the buyer feels it is paying 115.9 while the seller feels it is receiving 131.9.
Why does an earnout make a deal possible?
Because both sides can be right in their own terms. The seller, confident in the plan, values the package at about 132; the buyer, more cautious, values it at about 116; and the contract lets each side bet on its own view. If the target is hit, the buyer pays more but has a business that performed. If not, the buyer has paid 100 for a business that did not.
Say the risks. Earnouts are a common source of disputes: how EBITDA is defined, which costs the buyer loads onto the business after closing, and whether the seller's management stays to run it. A buyer controls the business that decides the payout, so the contract needs a tight EBITDA definition and agreed accounting policies.
Where candidates lose it
The common loss is counting the earnout at its face value of 50, which makes the price look like 150 and either overpays or kills the deal. The other is applying the probability but forgetting two years of waiting.
The second loss is treating the 40% as a fact. It is your estimate; the seller's is higher. Say that the gap between the two estimates is exactly what the earnout is for.
What the interviewer asks next
What chance of hitting the target makes the earnout worth 25 to you?
Would you rather the earnout be all-or-nothing or paid in proportion to EBITDA achieved, and why?
How would you protect the seller from the buyer managing EBITDA down after closing?
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%?
049EBITDA grows from 100 to 250 over 6 years. What is the compound annual growth rate?Compounding and time valueWarm upMid-market buyout fundIndian mid-market PE
Try it first
Pick the CAGR.
Show the worked solution
About 16.5% a year. CAGR is the ratio of end to start, 2.5, raised to one over the number of years, less one. 1.15 to the sixth is about 2.31 and 1.17 to the sixth about 2.57, so the rate sits about three quarters of the way between them, near 16.5%. Dividing the 150% gain by 6 gives 25%, which would compound to 381.
Why is 25% wrong?
A savings account that pays interest on interest grows faster each year, so it needs a lower rate than you might think to reach a target. CAGR is the single rate that, compounded every year, turns the start into the end, so it is the sixth root of 2.5, not 150% divided by six. 25% compounded for six years would reach 381, far past 250.
Compounding at 16.5% a year takes EBITDA from 100 to 250 in six years, while the 25% from dividing 150% by six would compound to 381; bracketing between 1.15 and 1.17 to the sixth finds the rate.
The relationship
$CAGR = \left(\frac{250}{100}\right)^{1/6} - 1 \approx 16.5\%$
250 / 100 the ratio of end value to start value, 2.5
1/6 one over the number of years
What it says in wordsThe compound growth rate is the ratio of end to start, rooted by the number of years, less one.
How do you find a sixth root in your head?
Bracket it with rates whose sixth powers you can build. 1.15 squared is 1.3225, cubed that is about 2.31. 1.17 squared is 1.3689, cubed that is about 2.57. 2.5 sits about 0.74 of the way from 2.31 to 2.57, so the rate is about 15% plus 0.74 of 2 points, near 16.5%. A cross-check with the rule of 72: at 16.5% money doubles in about 4.4 years, and 2.5x in 6 years is a little more than one doubling, which fits.
Say what the number hides. A CAGR smooths the path: a business could have been flat for four years and then jumped, and the CAGR would be the same. A buyout investor asks for the yearly figures before trusting the rate, and checks whether the 250 includes acquisitions.
Where candidates lose it
The common loss is dividing the total growth by the years and saying 25%. That is the average simple growth, and it overstates the compound rate badly over six years.
The second loss is freezing on the sixth root. You do not need logarithms; bracket the rate between two you can compute and slide.
What the interviewer asks next
What CAGR turns 100 into 300 over 5 years?
If EBITDA grew 40% in year one and was flat after, what is the CAGR over six years?
How would you strip acquired EBITDA out of the growth rate?
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?

C	IT capacity, 20 MW
u	utilisation, 70%
PUE	total power over IT power, 1.4
p	tariff, Rs 7 per kWh, an assumption

1.25^3	A's money multiple after three years
1.20^5	B's money multiple after five years
r	the reinvestment rate at which A catches B by year five

r_A	the unlevered asset return, 9%
r_D	the cost of debt, 12%
D/E	debt over equity, 70/30 at 70% debt

Debt share	Equity return, debt at 12%	Equity return, debt at 6%
0%	9%	9%
40%	7%	11%
70%	2%	16%

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What does PUE add to the bill?

How do you keep the units straight out loud?

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Why does the higher IRR make less money?

So which would an LP prefer?

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How does leverage change the equity return?

Why does a buyout investor worry about this?

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Why work backwards instead of guessing an entry price?

What does 8.2x tell the investment committee?

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How do you value a payment that may not happen?

Why does an earnout make a deal possible?

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

How do you get the IRR without a calculator?

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Why does the fast starter lose?

Why is the average rate misleading here?

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

How do you check it a second way?

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Why is 25% wrong?

How do you find a sixth root in your head?

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Why does lighting both ends give exactly 30 minutes?

Why does rope B have exactly 30 minutes left?

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