Venture Capital puzzles, solved step by step

Puzzles: 100
Traced to a firm: 7
Topics: 12
Hard: 30

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011In your head, no calculator: a SaaS company has ARR of Rs 42 crore growing 65% a year, net burn of Rs 3.1 crore a month, net new ARR last quarter of Rs 6.2 crore and Rs 55 crore of cash. What are next year's ARR, the quarterly burn multiple and the runway?Mental maths and speed testsCoreVista Equity PartnersAustin · 2021
Try it first
What is the quarterly burn multiple?
Show the worked solution
About Rs 69.3 crore of ARR, a burn multiple of 1.5 and about 18 months of runway. 42 x 1.65 is 42 plus 21 plus 6.3. A quarter's burn is 9.3, and 9.3 over 6.2 is 1.5 because both are multiples of 3.1. Cash of 55 over 3.1 a month is a shade under 55 over 3, so about 17.7 months.
How do you multiply by 1.65 without a calculator?
Break the awkward multiplier into pieces you already know. A shopkeeper adding 65% to a Rs 42 cost does not multiply by 0.65; he adds half, then a bit more. 65% is 50% plus 15%, so 42 x 1.65 is 42 plus 21 plus 6.3, which is Rs 69.3 crore. Fifteen percent is itself 10% plus half of that again: 4.2 plus 2.1. Every step is a halving or a shift of the decimal point, which is fast and hard to get wrong out loud.
Why does the burn multiple need care before any arithmetic?
Because the two inputs come in different periods. Burn is quoted per month and net new ARR per quarter. A burn multiple divides the cash burned by the new annual recurring revenue added over the same stretch of time, so three months of burn, Rs 9.3 crore, goes over the quarter's Rs 6.2 crore. Then notice that 9.3 is 3 x 3.1 and 6.2 is 2 x 3.1, so the ratio is exactly 3 over 2, or 1.5. Interviewers choose numbers like these on purpose; spotting the common factor is part of the test.
Splitting 1.65 into 1 plus a half plus 15% gives Rs 69.3 crore of ARR, matching periods turns the burn multiple into 9.3 over 6.2, which is 1.5, and rounding 3.1 down to 3 then shaving 3% gives about 17.7 months of runway.
The relationship
$\text{BM} = \frac{3 \times 3.1}{6.2} = 1.5 \qquad \text{Runway} = \frac{55}{3.1} \approx 17.7 \text{ months}$
3 x 3.1 net burn over one quarter, Rs crore
6.2 net new ARR added in the quarter, Rs crore
55 cash in the bank, Rs crore
What it says in wordsPut burn and new revenue on the same period before dividing, and divide cash by monthly burn for runway.
What do you say once the three numbers are out?
Check them against each other, because that is what a growth investor does next. Growing ARR by Rs 27.3 crore next year needs about Rs 6.8 crore of net new ARR a quarter, a little above the Rs 6.2 crore just achieved, so the 65% plan is plausible but not banked. At a 1.5 burn multiple that growth costs about Rs 41 crore of burn in the year, and with Rs 55 crore of cash and about 18 months of runway, the company will want to raise again within roughly a year, before the runway gets short. That one sentence turns arithmetic into a view on the company.
Where candidates lose it
The costly slip is dividing the monthly burn by the quarterly net new ARR and announcing a burn multiple of 0.5. It makes the company look three times more efficient than it is, and the interviewer chose mixed periods to see whether you notice.
The second loss is going silent while you calculate. Say the shortcut as you use it: half, then fifteen percent; three parts over two parts; fifty five over three, then shave a little.
What the interviewer asks next
If net burn rises 20% next year while ARR grows 65%, what happens to the burn multiple if net new ARR grows in step with ARR?
How much cash should the company raise to have 24 months of runway at the current burn?
Is a burn multiple of 1.5 good for a company of this size, and what would change your view?
Asked at Vista Equity Partners, Private Equity, Austin, 2021 (Wall Street Oasis): Mental math and tech/SaaS-specific sector insights
030Estimate 1.07 to the power 10 and 0.93 to the power 10 in your head, and say why the two answers are not reciprocals.Mental maths and speed testsCoreGrowth equityMulti-stage VC
Try it first
Which pair is closest to the two answers?
Show the worked solution
About 1.97 and about 0.48. For the first, the rule of 72 says 7% doubles money in about 10.3 years, so ten years gives just under 2. For the second, ln 0.93 is about minus 0.0725, ten years of it is minus 0.725, and e to that is about 0.48. They are not reciprocals because 0.93 is not 1 over 1.07: undoing a 7% rise takes only a 6.5% fall.
How do you get each number without a calculator?
Start with the one you know. The rule of 72A shortcut for compounding: money growing at r per cent a year doubles in roughly 72 divided by r years. says money growing at r% doubles in about 72 over r years, so at 7% it doubles in about 10.3 years and ten years leaves you just short of 2. For the fall, work in natural logs, which turn compounding into adding. ln(1 minus x) is about minus x minus half of x squared, so ln 0.93 is about minus 0.07 minus 0.00245, which is minus 0.0725. Ten years gives minus 0.725. e to minus 0.693 is exactly one half, and minus 0.725 is a little further down, so the answer is a shade under a half: about 0.48.
The relationship
$\ln(1 \pm x) \approx \pm x - \tfrac{x^2}{2} \qquad 10\ln 1.07 \approx 0.676 \Rightarrow 1.97 \qquad 10\ln 0.93 \approx -0.725 \Rightarrow 0.48$
x the yearly rate, 0.07
x squared over 2 the compounding correction, 0.00245, which has the same sign whichever way the rate goes
ln the natural log, which turns repeated multiplying into adding
What it says in wordsTen years of compounding is ten times the log of one year's factor, and the squared term always pulls the result down.
Rising 7% a year for ten years turns 1.00 into 1.97 and falling 7% a year turns it into 0.484, below the 0.508 a true mirror would give, so the two paths together leave 0.952, not 1.00.
Why are the two answers not mirror images?
Everyday version first: a shirt marked up 7% and then marked down 7% ends below its starting price, because the markdown is taken on the higher price. The x squared term in the log has the same sign whichever way the rate goes, so it drags both paths down: the up path gains a little less than 7% a year in log terms and the down path loses a little more. The true mirror of a 7% rise is a 6.5% fall. Put the two paths together and 1.07 x 0.93 is 0.9951 a year, so ten years of each leaves 0.952: a round trip that still loses about 5%.
Then say why a growth investor cares. Swings cost compound growth even when the average yearly change is zero. A company whose revenue alternates between up 7% and down 7% averages zero change but ends smaller, losing roughly half the square of the swing every year. The limit is that this is a small effect at 7%; it becomes large at the 30% and 50% swings early-stage revenue can show.
Where candidates lose it
The quick wrong answer to the second number is 0.51, one over 1.97, because the candidate assumes a 7% fall reverses a 7% rise. It does not, and the last clause of the question is there to test exactly that.
The other loss is answering 1.70 and 0.30, as if the rate added up in a straight line. Ten years at 7% nearly doubles money; simple interest would add only 70%.
What the interviewer asks next
Estimate 1.12 to the power 6 using the rule of 72.
A portfolio company's revenue rises 50% and then falls 50%. Where does it end, and what does that say about volatile growth?
What annual rate turns Rs 100 into Rs 300 over ten years?
042A mental maths set from a first-round interview: work out 49 x 51, 998 x 1,002 and 12.5% of Rs 3,680 crore, each inside ten seconds.Mental maths and speed testsWarm upGeneral AtlanticNew York · 2026
Try it first
What is 998 x 1,002?
Show the worked solution
2,499, then 9,99,996, then Rs 460 crore. The first two use the same shortcut: numbers equally spaced either side of a round number multiply to its square minus the gap squared, so 49 x 51 is 2,500 minus 1, and 998 x 1,002 is 10,00,000 minus 4. The third is a fraction in disguise: 12.5% is one eighth, so halve Rs 3,680 crore three times, to 1,840, 920 and 460.
Why does 49 x 51 come out one short of 2,500?
Picture a square garden 50 metres a side. Take a one-metre strip off the bottom, 50 square metres, and lay it along the right-hand side. It only fits for 49 metres, so one square metre is left over. A rectangle 49 by 51 is a 50 by 50 square with one corner square missing, so the product is 2,500 minus 1, or 2,499. The same picture works for any pair spread evenly around a round number: the product is the middle number squared minus the gap squared.
The relationship
$(a - b)(a + b) = a^2 - b^2 \qquad 49 \times 51 = 50^2 - 1^2 = 2{,}499 \qquad 998 \times 1{,}002 = 1{,}000^2 - 2^2 = 9{,}99{,}996$
a the round number in the middle, 50 or 1,000
b how far each factor sits from it, 1 or 2
a squared minus b squared the product, always slightly below the square
What it says in wordsTwo numbers spread evenly either side of a round number multiply to that number squared, less the gap squared.
49 x 51 is a 50 by 50 square with one unit square missing, so it equals 2,499, and 12.5% of Rs 3,680 crore is one of eight equal blocks, Rs 460 crore, found by halving three times.
How do you take 12.5% of anything in a few seconds?
Recognise the fraction first. 12.5% is one eighth, and dividing by eight is the same as halving three times, which is easier to do aloud than any long division. Rs 3,680 crore halves to 1,840, then 920, then 460. Keep a short list of these anchors ready: 12.5% is an eighth, 37.5% three eighths, 16.7% a sixth, 6.25% a sixteenth. Interviewers who give percentage questions usually pick one of them, because the speed of recognising the fraction is what they are testing.
What is the interviewer actually checking?
Speed sets are less about arithmetic than about whether you look for structure before you calculate. Saying the shortcut out loud, square minus one, or halve three times, shows a method that will also work on the next question. Two habits help. Check the size of your answer against an anchor: 49 x 51 must sit just under 2,500. And say the number back with its units, Rs 460 crore, not just 460. The limit of these tricks is that they only work on numbers chosen to suit them; for awkward numbers, round and say so.
Where candidates lose it
The common loss on the products is grinding through long multiplication and running out of time, or writing 10,00,004 because the sign of the correction is guessed. The product of numbers either side of a round number is always below its square.
On the percentage, candidates multiply 3,680 by 0.125 digit by digit and fumble. Name the fraction, one eighth, and halve three times; the clock is part of the question.
What the interviewer asks next
Work out 97 x 103 and 4.95 x 5.05 the same way.
What is 37.5% of Rs 2,400 crore?
Estimate 1,999 squared 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.
053A company's valuation rises 150% at its Series B, falls 60% at its Series C and rises 50% at its Series D, one round a year. What is the net change from the Series A valuation, and what steady annual rate over the three years gives the same result?Mental maths and speed testsCoreSeries A to C VCGrowth equity
Try it first
Net change from Series A to Series D?
Show the worked solution
Up 50% in all, or about 14.5% a year. Turn each round into a multiplier: up 150% is x 2.5, down 60% is x 0.4, up 50% is x 1.5. The shortcut is that 2.5 x 0.4 is exactly 1, so the Series C fall wiped out the Series B gain, and the net is just Series D's x 1.5. The annual rate is the cube root of 1.5, which sits between 14% and 15%.
Why can the three percentages not simply be added?
A shopkeeper who marks a shirt up 150% and then runs a 60% off sale is not ahead by 90%. The markup took Rs 100 to Rs 250, and the sale took 60% of Rs 250, landing back at Rs 100. Each percentage is measured against whatever the value was just before it, so moves in a chain multiply rather than add. Adding gives +140%, a number that describes nothing in this company's history. Multiplying gives 2.5 x 0.4 x 1.5, which is 1.5: a valuation of 100 went to 250, back to 100, then to 150.
From an index of 100 the valuation rises to 250 at Series B, falls back to 100 at Series C and ends at 150 after Series D, so the three rounds multiply to 1.5x, about 14.5% a year, not the +140% that adding the percentages suggests.
How do you find the annual rate without a calculator?
You need the number that, cubed, gives 1.5. Bracket it. 1.14 cubed is about 1.48 and 1.15 cubed is about 1.52, so the rate sits just below 14.5%, and saying 'about 14.5% a year' is the right precision for the room. The exact figure is 14.47%. Check it the other way: half of 50% is 25%, far too high, which is the arithmetic average and the error the question is fishing for.
The relationship
$(1+r)^3 = 2.5 \times 0.4 \times 1.5 = 1.5 \quad\Rightarrow\quad r = 1.5^{1/3} - 1 = 14.5\%$
2.5, 0.4, 1.5 the three rounds written as multipliers
r the steady annual rate with the same end result
What it says in wordsMultiply the rounds to get the total, then take the root for the number of years to get the steady rate.
One sentence of judgement helps. A company that fell 60% in one round and still ended up 50% ahead over three years has had a volatile path, and the steady 14.5% hides that volatility completely. Investors who entered at the Series B price are down 40% at Series D, which is why the entry round matters as much as the company's overall path.
Where candidates lose it
The fast wrong answer is +140%, from adding the three percentages. The second is averaging them to about 47% a round. Both treat percentages as if they shared a base, which is the exact mistake the question is built to catch.
The slower loss is spotting 2.5 x 0.4 = 1 but then stumbling on the cube root. Bracket it between 1.14 and 1.15 out loud; an interviewer wants to see the method, not four decimal places.
What the interviewer asks next
An investor came in at Series B. What is their multiple at Series D?
What single fall at Series C would have left the company flat over the three rounds?
Why can a flat overall path still leave some investors well below their entry price?
065An online aptitude test has 50 questions in 12 minutes, five options each, and no penalty for wrong answers. You can answer about 30 questions carefully in the time, at 85% accuracy. Should you guess the other 20, and what is your expected score?Mental maths and speed testsWarm upVista Equity PartnersAustin · 2022
Try it first
What is your expected score if you guess the last 20?
Show the worked solution
Yes, guess every remaining question; the expected score rises from 25.5 to 29.5. Your 30 careful answers at 85% are worth 25.5 points. A blind guess on a five-option question is right one time in five, and a wrong answer costs nothing, so 20 guesses add 20 x 0.2, or 4 expected points. A blank is a sure zero. The only real decision is leaving yourself the last 30 seconds to fill them in.
Why is a blind guess worth anything at all?
A free raffle ticket is worth taking even if the odds are poor, because it costs nothing. When wrong answers carry no penalty, every guess has a positive expected value, one point times the chance of being right, and a blank has an expected value of exactly zero. With five options that chance is 1 in 5, so each guess is worth 0.2 points and twenty are worth 4. On a 50-question test, 4 points is often the gap between two scoring bands.
Thirty careful answers at 85% give 25.5 expected points, and filling the remaining 20 with blind guesses adds 20 x 0.2 = 4 more, lifting the expected score to 29.5 at no cost because wrong answers are not penalised.
The relationship
$E[\text{score}] = 30 \times 0.85 + 20 \times \tfrac{1}{5} = 25.5 + 4 = 29.5$
30 x 0.85 careful answers times your accuracy
20 x 1/5 blind guesses times the chance a random pick is right
What it says in wordsAdd what the careful answers are worth to what the guesses are worth; blanks add nothing.
What changes if wrong answers are penalised, and how do you manage the clock?
Suppose a wrong answer cost a quarter of a point. A blind guess would then be worth 0.2 minus 0.8 x 0.25, which is exactly zero. The rule is to guess whenever the chance of being right times the reward beats the chance of being wrong times the penalty, and eliminating even one option tips a penalised guess back into positive value. On the clock: 12 minutes for 50 questions is 14.4 seconds each. Skip any question that will take more than about 30 seconds, come back if time allows, and stop answering carefully with half a minute left to fill every blank.
Two honest notes. The 4 points are an average: 20 guesses have a standard deviation of about 1.8 points, and there is roughly a 1% chance none of them land. And the no-penalty rule is the premise of this question; tests differ, so check the instructions of the one you are sitting before deciding.
Where candidates lose it
The trap is leaving blanks out of a sense that guessing is unserious. On a no-penalty test that throws away free expected points, and the people who score well on these tests always fill every answer.
The second loss is spending too long on hard questions early. At 14.4 seconds a question, one stubborn item can cost you three easy ones; skip, mark and return.
What the interviewer asks next
If a wrong answer cost a quarter point, would you still guess?
You can eliminate one option on each of the 20. What is the guess value now?
Would you rather answer 35 at 75% accuracy or 30 at 85%?
Asked at Vista Equity Partners, Technology, Media and Telecom, Austin, 2022 (Wall Street Oasis): Application requires a CCAT, 50 questions in 12 minutes.
084Without a calculator, what is the IRR of an investment that returns 2.5x the money in 4 years? Use anchors you already know, such as 2x in 3 years and 3x in 5 years.Mental maths and speed testsCoreGrowth equityFund of funds and LPs
Try it first
Pick the closest before you work it.
Show the worked solution
About 26%; the exact figure is 25.7%. Two anchors bracket it: 2x in 3 years is 26.0% and 3x in 5 years is 24.6%. For a sharper figure, take ln 2.5, about 0.92, divide by 4 to get 0.23, and add half its square: about 25.6%. Check by squaring twice: 1.26 squared is 1.59, and 1.59 squared is about 2.5.
Which anchors should you carry into the room?
You judge a distance on a road by the milestones, not by pacing it out. Carry a few multiple and years pairs and interpolate between them rather than calculating from scratch. 2x in 3 years is 26.0%, 3x in 5 years is 24.6%, 2x in 4 years is 18.9% and 3x in 4 years is 31.6%. The target, 2.5x in 4 years, lies between the last two and is bracketed by the first two.
IRRs for 2x, 2.5x and 3x over 3, 4 and 5 years show that the anchors 2x in 3 years and 3x in 5 years both sit near 25%, and that 2.5x in 4 years is 25.7%, well below the 37.5% a simple average gives.
How do you land on 26% out loud?
Route one: interpolate on a log scale. 2.5 sits about 55% of the way from 2 to 3 in log terms, so the IRR sits about 55% of the way from 18.9% to 31.6%, near 26%. Route two uses the log directly. The continuous growth rate is the log of the multiple divided by the years, and adding half its square converts it to an annual rate: 0.23 plus 0.026 is about 25.6%.
The relationship
$\text{IRR} \approx \frac{\ln M}{n} + \frac{1}{2}\left(\frac{\ln M}{n}\right)^2 = 0.229 + 0.026 \approx 25.6\%$
M money multiple, 2.5
n years, 4
ln M natural log of the multiple, about 0.92
What it says in wordsSpread the log of the multiple evenly over the years, then nudge it up slightly for annual compounding.
Then check. Squaring twice is the fastest test of a four year rate: 1.26 squared is 1.59, and 1.59 squared is 2.52, close enough to 2.5. The check matters more than the method, because it catches a slip in either route in five seconds.
Where candidates lose it
The expensive slip is the simple average: a 150% gain over 4 years is 37.5% a year. It ignores compounding, and it overstates the IRR by more than ten points. Anyone who has done this before hears it immediately.
The second loss is getting to 25.7% in silence. The interviewer wants the anchors and the check said aloud, because that is how you would sanity check a fund's reported return on a call.
What the interviewer asks next
What IRR is 3x in 7 years?
A fund makes 2.5x in 4 years, but half its money was only invested for the last 2 years. Is its IRR above or below 26%?
Why do investors in a fund ask for both the multiple and the IRR?
096A timed cognitive test item: if 4 associates screen 60 pitch decks in 3 hours, how many decks do 6 associates screen in 5 hours, working at the same rate?Mental maths and speed testsWarm upSeed and early-stage VCMulti-stage VC
Try it first
Answer in under thirty seconds.
Show the worked solution
150 decks. Reduce everything to one unit first: 4 associates working 3 hours is 12 associate-hours, and 60 decks over 12 is 5 decks per associate-hour. Six associates for five hours is 30 associate-hours, so 30 x 5 = 150. As a check, scale the original: 60 x 6/4 x 5/3 is also 150. This assumes everyone works at the same steady rate.
Why reduce to a rate per associate-hour?
If 2 cooks make 40 rotis in an hour, one cook makes 20 an hour, and any kitchen size or shift length follows from that one number. Work problems become simple once you find the output of one worker in one unit of time, because the total is then that rate times workers times hours. Here 60 decks came from 4 x 3 = 12 associate-hours, so the rate is 5 decks per associate-hour. Everything else is multiplication.
Four associates working three hours make 12 associate-hours, and 60 decks over 12 cells is 5 decks per associate-hour; six associates working five hours make 30 such cells, so they screen 30 x 5 = 150 decks.
The relationship
$D = r \times a \times h \qquad r = \frac{60}{4 \times 3} = 5 \qquad D = 5 \times 6 \times 5 = 150$
D decks screened
r decks per associate-hour
a number of associates
h hours worked
What it says in wordsOutput is the rate of one person for one hour, times the people, times the hours.
How do you avoid the slip under time pressure?
Ask which way each change pushes the answer before you multiply. More associates and more hours both raise output, so both ratios must be greater than one: 6/4 and 5/3, never 4/6 or 3/5. The fast wrong answer, 90, applies the 6/4 and stops. On a timed test the danger is not hard arithmetic but a missed step, so a five-second check that the answer moved in the right direction by roughly the right amount is worth it: one and a half times the people for two thirds more time should give about two and a half times the output, and 60 x 2.5 is 150.
Where would this break down in a real deal team?
The model assumes every associate screens at the same steady pace and that output scales with hours. Real screening slows late in a long day, and a bigger team spends time coordinating and avoiding duplicate work. Saying so is unnecessary on the test itself, but it is the right instinct if the same question comes up in an interview: the arithmetic answer is the ceiling, and the real number is usually lower.
Where candidates lose it
The common slip is to scale for only one of the two changes, giving 90 from the associates alone or 100 from the hours alone. Under a clock, candidates grab the first ratio they see and move on.
The other loss is inverting a ratio, as if more associates meant fewer decks. Find the unit rate first and the direction takes care of itself.
What the interviewer asks next
How many associates are needed to screen 300 decks in 4 hours?
If two of the six associates work at half speed, how many decks are screened in 5 hours?
A partner reviews 20% of screened decks at 3 decks an hour. How many partner-hours does the 5-hour session create?

3 x 3.1	net burn over one quarter, Rs crore
6.2	net new ARR added in the quarter, Rs crore
55	cash in the bank, Rs crore

x	the yearly rate, 0.07
x squared over 2	the compounding correction, 0.00245, which has the same sign whichever way the rate goes
ln	the natural log, which turns repeated multiplying into adding

a	the round number in the middle, 50 or 1,000
b	how far each factor sits from it, 1 or 2
a squared minus b squared	the product, always slightly below the square

2.5, 0.4, 1.5	the three rounds written as multipliers
r	the steady annual rate with the same end result

Try it first

How do you multiply by 1.65 without a calculator?

Why does the burn multiple need care before any arithmetic?

What do you say once the three numbers are out?

Where candidates lose it

What the interviewer asks next

Try it first

How do you get each number without a calculator?

Why are the two answers not mirror images?

Where candidates lose it

What the interviewer asks next

Try it first

Why does 49 x 51 come out one short of 2,500?

How do you take 12.5% of anything in a few seconds?

What is the interviewer actually checking?

Where candidates lose it

What the interviewer asks next

Try it first

Why can the three percentages not simply be added?

How do you find the annual rate without a calculator?

Where candidates lose it

What the interviewer asks next

Try it first

Why is a blind guess worth anything at all?

What changes if wrong answers are penalised, and how do you manage the clock?

Where candidates lose it

What the interviewer asks next

Try it first

Which anchors should you carry into the room?

How do you land on 26% out loud?

Where candidates lose it

What the interviewer asks next

Try it first

Why reduce to a rate per associate-hour?

How do you avoid the slip under time pressure?

Where would this break down in a real deal team?

Where candidates lose it

What the interviewer asks next