Investment Banking puzzles, solved step by step

Puzzles: 100
Traced to a firm: 36
Topics: 12
Hard: 29

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031How many times do the hour hand and the minute hand of a clock overlap in 24 hours?Logic and brainteasersCoreConsulting style brainteasersSales and trading
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Pick before you count.
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22 times. The minute hand goes round 12 times in 12 hours and the hour hand once, so the minute hand gains 11 laps and passes the hour hand 11 times. The overlaps come every 12/11 hours, about 65 minutes 27 seconds apart, and the one you expect between 11 and 12 is the one at 12:00 itself. Two 12-hour cycles make 22, counting midnight once.
Why is it not once an hour?
Think of two runners on a track, one doing 12 laps while the other does 1. The faster runner passes the slower one only 11 times, because the slower runner's single lap is subtracted from the faster one's total. Overlaps count the laps the minute hand gains on the hour hand, and it gains 12 minus 1, which is 11, every 12 hours. The hour hand keeps creeping forward, so each catch-up takes a little longer than an hour.
Across 12 hours the hands meet 11 times, every 65 5/11 minutes, at 12:00, about 1:05, 2:11 and so on up to about 10:55; the next meeting is 12:00 again, so there is none between 11 and 12, and a full day holds 22.
When exactly do the hands meet?
The minute hand moves 6 degrees a minute and the hour hand half a degree, so the minute hand gains 5.5 degrees a minute. A full lap of 360 degrees takes 360 over 5.5, which is 65 5/11 minutes, so the meetings come at 12:00, about 1:05:27, about 2:10:55 and so on. The eleventh gap lands exactly back on 12:00, which is why there is no meeting between 11 and 12: the hands meet at about 10:54:33 and next at 12:00.
The relationship
$\text{gap} = \frac{360}{6 - 0.5} = \frac{720}{11}\text{ min} \approx 65.45\text{ min} \qquad \frac{24 \times 60}{720/11} = 22$
6 minute hand speed, degrees per minute
0.5 hour hand speed, degrees per minute
720/11 minutes between one overlap and the next
What it says in wordsDivide the minutes in a day by the minutes between meetings.
What do you say about the edges?
State how you count midnight. Counting from one midnight up to but not including the next gives 22; counting both midnights gives 23. That one sentence shows you saw the edge case rather than stumbling into it. The same lap method answers the follow-ups in seconds: the hands point in exactly opposite directions 22 times a day as well, and form a right angle 44 times, because a right angle happens twice per lap gained.
Where candidates lose it
Candidates say 24 because the hands seem to meet once an hour. The miss is forgetting that the hour hand moves, so each catch-up takes longer than 60 minutes and only 11 fit into 12 hours.
The second loss is producing 22 from memory with no reason behind it. Give the lap argument, 12 laps minus 1 lap, so the interviewer hears a method rather than a memorised number.
What the interviewer asks next
How many times a day do the hands point in exactly opposite directions?
At what exact time after 3:00 do the hands first overlap?
How many times a day do the hands form a right angle?
034Two fair dice are rolled and you are told that at least one of them shows a 4. What is the probability that the two dice sum to 7?ProbabilityCoreBulge bracket IBSales and trading
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Pick before you count.
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2/11, about 18.2%. Of the 36 equally likely outcomes, 11 contain at least one 4: six with the first die showing 4, six with the second, less the double 4 counted twice. Of those 11, only 4 and 3 and 3 and 4 sum to 7. The new information shrinks the world to 11 outcomes, and 2 of them qualify.
What does being told something actually change?
Imagine a teacher tells you that your friend is one of the 11 students who scored above 90. Your friend's chance of topping the class is now measured against those 11, not the whole class of 36. Conditional probability throws away every outcome the new information rules out and counts only inside what is left. Here the information rules out the 25 outcomes with no 4 anywhere, and 11 remain.
Of the 36 outcomes for two dice, 11 contain at least one 4 and only two of those, 4 and 3 and 3 and 4, sum to 7, so the probability is 2/11, about 18.2%, rather than 1/6.
Why is 1/6 tempting and wrong?
1/6 comes from fixing one die at 4 and asking for a 3 on the other, which is the right answer to 'the first die shows a 4'. 'At least one shows a 4' is a weaker statement: it lets the 4 sit on either die, which creates 11 outcomes rather than 6. If you counted the double 4 twice you would have 12 outcomes and get 2 in 12, exactly 1/6. Counted once, as it must be, there are 11, and the answer edges up to 2/11.
The relationship
$P(\text{sum }7 \mid \text{at least one }4) = \frac{2/36}{11/36} = \frac{2}{11}$
2/36 the chance of a 7 that contains a 4: (4,3) or (3,4)
11/36 the chance of at least one 4
What it says in wordsCount the outcomes that satisfy both conditions and divide by the outcomes that satisfy the information.
How do you check it quickly?
Use the complement for the denominator. The chance of no 4 on either die is 5/6 times 5/6, which is 25/36, so at least one 4 is 11/36. Two cells sum to 7 and contain a 4, so 2/36 over 11/36 is 2/11. Two routes to the same 11 is what the interviewer wants to hear before you commit to the answer.
Where candidates lose it
The trap is answering 1/6 by treating 'at least one die' as 'this die'. The two sound alike in a fast question and give different sets of outcomes.
The second loss is counting 12 outcomes with a 4 by forgetting that the double 4 sits in both lists. Picture the grid, or say the complement out loud, and the 11 is safe.
What the interviewer asks next
Now you are told the first die shows a 4. What is the probability of a 7?
Given at least one 6, what is the probability of a double?
Given that the sum is 7, what is the probability that at least one die shows a 4?
035One lender quotes 12% a year compounded monthly. Another quotes 12.5% a year compounded annually. Which loan is cheaper?Growth and compoundingCoreMiddle market IBPrivate equity
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Which loan is cheaper?
Show the worked solution
The 12.5% loan compounded annually is cheaper. 12% a year compounded monthly means 1% a month, and 1.01 to the twelfth is 1.1268, an effective rate of 12.68% a year. That is about 0.18 points more than 12.50%. On a Rs 100 crore loan the monthly loan costs about Rs 18 lakh a year more. Compare effective annual rates, never quoted rates with different compounding.
Why is 12% compounded monthly not really 12%?
Think of a savings account that credits interest every month. In February you earn interest on January's interest as well as on your deposit, so by December the year's earnings come to a little more than twelve single months. A rate compounded monthly is charged as one twelfth each month, and each month's interest then earns interest of its own, so the true yearly cost is higher than the quote. The true yearly cost is the effective annual rateThe rate that, charged once a year, costs the same as the quoted rate with its compounding. It is the only fair basis for comparing loans., and it is the only number worth comparing.
Loan A's 1% a month compounds to an effective 12.68% a year, while loan B charges 12.50% once, so loan B is cheaper by 0.18 points, about Rs 18 lakh a year on a Rs 100 crore loan.
How do you work out 1.01 to the twelfth in your head?
Expand it. Twelve months of 1% give 12% of simple interest. Every pair of months adds a little interest on interest, and there are 66 pairs, so add 66 times 0.01 squared, 0.66%. The first two terms give 12.66%, the next adds a sliver more, and the total is 12.68%, which beats 12.50% by 0.18 points. Saying the shortcut out loud shows you can reason about compounding without a calculator.
The relationship
$\text{EAR} = \left(1 + \frac{0.12}{12}\right)^{12} - 1 = 1.01^{12} - 1 \approx 12.68\%$
0.12/12 the monthly rate, one twelfth of the quoted 12%
12 the number of times interest is charged in a year
What it says in wordsCompound the monthly rate twelve times and subtract the one you started with to get the true yearly cost.
What else should you say before you finish?
Turn the comparison round as a check. The monthly quote that would match 12.5% effective is 12 times (1.125 to the power of one twelfth, minus 1), about 11.84%. Any lender quoting monthly compounding has to quote below 11.84% to beat the 12.5% annual loan. Then state the limit: fees, prepayment penalties and the timing of repayments can matter more than 0.18 points, so the effective rate is where a comparison starts, not where it ends.
Where candidates lose it
The fast answer is the 12% loan, because 12 is less than 12.5. It compares two numbers measured in different units: one is charged twelve times a year, the other once.
The second loss is knowing the rule but stalling on 1.01 to the twelfth. Have the shortcut ready, 12% plus 66 times 0.01 squared, and say 12.68% with confidence.
What the interviewer asks next
What is 12% compounded continuously, as an effective annual rate?
What monthly-compounded quote exactly matches 12.5% a year effective?
A lender charges 1% a month plus a 0.5% upfront fee on a one-year loan. What is the effective cost?
036You must staff a three-person deal team from eight analysts, but two of them refuse to work together. How many different teams are possible?Mental maths and countingCoreBulge bracket IBMiddle market IB
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Quick instinct: how many teams?
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50 teams. Choosing any 3 of 8 analysts gives 8 x 7 x 6 / 6 = 56 teams. The forbidden ones contain both analysts who refuse to work together, and the third seat can go to any of the other six, so there are 6 forbidden teams. 56 minus 6 leaves 50. Count everything, then subtract the cases the rule forbids.
Why count the teams you do not want?
Think of seating guests at a dinner when two of them have fallen out. Listing every acceptable table is slow; counting all possible tables and removing those that seat the two together is quick. When a rule forbids a small set of cases, count everything and subtract the forbidden ones, because the forbidden set is usually the easy one to count. Here a forbidden team is the two feuding analysts plus a third person, and there are only 6 choices for the third.
Of the 56 possible three-person teams from eight analysts, only the 6 that contain both A and B break the rule, so 50 remain: 20 with neither of the pair and 30 with exactly one.
How do you get 56 without a formula sheet?
Fill the team one seat at a time: 8 choices, then 7, then 6, which is 336 ordered picks. Each team of three turns up in 3 x 2 x 1 = 6 different orders, so divide by 6 to get 56 distinct teams. That is the combinationA selection where order does not matter. The number of ways to choose k from n is written n choose k. count, 8 choose 3. Subtract the 6 forbidden teams and 50 remain.
The relationship
$\binom{8}{3} - \binom{6}{1} = 56 - 6 = 50$
8 choose 3 every team of three from eight analysts
6 choose 1 teams holding both feuding analysts: one choice of third member from the other six
What it says in wordsAll teams, less the teams that contain the forbidden pair.
How do you check 50 a second way?
Count the allowed teams directly, split by how many of the feuding pair they include. Teams with neither of the two are 6 choose 3, which is 20; teams with exactly one are 2 times 6 choose 2, which is 30; together 50. Two methods agreeing is the answer the interviewer remembers, and the second method is the one you will need when a follow-up adds a second rule.
Where candidates lose it
The common wrong answer is 20: candidates take both feuding analysts out of the pool and count teams of three from the remaining six. That throws away the 30 perfectly good teams containing one of them.
The other slip is subtracting the feuding pair from 56 as if it were one team. The pair plus a third person can be formed 6 ways, so 6 teams go, not 1.
What the interviewer asks next
One senior analyst must be on every team, and the feud still applies. How many teams now?
What if the team is four people instead of three?
Two separate pairs refuse to work together. How many three-person teams are possible?

037Estimate how many cups of tea street vendors in Mumbai sell in a day.Estimation and market sizingCoreBulge bracket IBConsulting style brainteasers

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Which order of magnitude feels right?

Show the worked solution

Roughly 1.2 crore cups a day. Take about 2 crore people as a round assumption, 75% adults, 40% of them buying street tea on a given day and two cups each: 1.2 crore cups. Check it from the supply side: about one stall per 400 people gives 50,000 stalls, each selling about 252 cups a day, which is 1.26 crore. Two routes landing within 5% make the estimate credible.

How do you structure the estimate before any number?

Think of how you would guess the rotis a hostel mess makes in a day: count the students, how many eat in the mess, and how many rotis each one eats. A market sizing is a chain of multiplications, and the interviewer scores the chain more than the final figure. Say the chain first: people, then adults, then the share who buy street tea on a given day, then cups per buyer. Each link is one assumption you can defend or change, and round numbers keep the arithmetic in your head.

The demand route, 2 crore people to 1.5 crore adults to 60 lakh buyers at two cups, gives 1.20 crore cups; the supply route, 50,000 stalls at 252 cups each, gives 1.26 crore, within 5% of the first.

Why does a demand estimate need a supply check?

Each link in the demand chain is a guess, and errors multiply. A second route built from different assumptions, here stalls and cups per stall, tests the first: if both land close together, neither is likely to be badly wrong. One stall per 400 people gives 50,000 stalls. A busy stall open 14 hours and selling 18 cups an hour sells about 252 cups a day, so supply comes to about 1.26 crore cups, within 5% of demand.

Step	Demand route	Supply route
Start	2.0 crore people (assumed)	2.0 crore people (assumed)
First filter	75% adults: 1.5 crore	1 stall per 400 people: 50,000 stalls
Rate	40% buy today: 60 lakh buyers	14 hours x 18 cups: 252 a stall
Cups a day	1.20 crore	1.26 crore

Every input is an assumption stated out loud; the demand route gives 1.20 crore cups a day and the independent supply route 1.26 crore, so the answer is about 1.2 crore.

What do you say about the weakest link?

Name it before the interviewer does. The share of adults buying street tea on a given day drives the answer one for one: 30% instead of 40% cuts the demand estimate to 90 lakh cups. Office workers, drivers and shopkeepers buy several cups a day; many households brew at home and buy none. Then say which number you would go and observe, the stalls along one busy street, because that is the input you can check. The population figure is a round assumption to confirm against the latest census estimate, not a fact.

Where candidates lose it

Candidates jump to a final number with no chain, or build a ten-step chain and lose the room in the arithmetic. The interviewer wants four or five clear links, each stated as an assumption, with round numbers that multiply in your head.

The second loss is stopping at one route. A supply check costs thirty seconds and is the moment the estimate becomes believable.

What the interviewer asks next

What is the street tea market worth in rupees a year?
How would the estimate change in the monsoon months?
How many stalls would a new tea chain need to win 5% of this market?

038A distressed company's assets will be worth 160 or 40 next year with equal probability, and it owes 100 of debt due then. With zero interest rates and risk-neutral pricing, what are the equity and the debt worth today, and why is the equity not worthless?Expected value and gamesHardBulge bracket IBConsulting style brainteasers
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What is the equity worth today?
Show the worked solution
The equity is worth 30 and the debt 70. If assets reach 160, lenders get 100 and shareholders 60; if assets fall to 40, lenders take all 40 and shareholders get nothing, but never less than nothing. Half of 60 is 30 for equity; half of 100 plus half of 40 is 70 for debt. Together they equal the 100 the assets are worth. Limited liability makes equity a call option on the assets.
Why is the equity worth anything when assets only cover the debt?
Think of a student who borrows to start a food stall and can walk away from the loan if the stall fails. In a good year the student keeps everything above the loan; in a bad year the lender keeps the stall and the student loses nothing more. Because shareholders can lose at most what they put in but keep every rupee above the debt, equity is a call optionThe right, but not the obligation, to buy an asset at a fixed price. It pays the amount by which the asset ends above that price, or nothing. on the assets with a strike price equal to the debt. An option has value even when it sits exactly at the money, which is where this company stands today.
Debt pays the asset value up to 100 and equity pays everything above 100, so at assets of 40 debt gets 40 and equity 0, at 160 debt gets 100 and equity 60, and today equity is worth 30 and debt 70.
How do you price the two claims?
With zero interest rates and risk-neutral pricing, each claim is worth its average payoff. Equity pays 60 or 0 and is worth 30; debt pays 100 or 40 and is worth 70; the two add back to the asset value of 100. The debt trades at 70 for a promise of 100, a yield of about 42.9%, which is how a market prices distress.
The relationship
$E = \tfrac12\max(160-100,0) + \tfrac12\max(40-100,0) = 30 \qquad D = 100 - 30 = 70$
max(V - 100, 0) what shareholders get: assets above the debt, never below zero
100 today's asset value, the average of 160 and 40
What it says in wordsEquity is the average of its floored payoffs; debt is whatever is left of the assets.
What happens if the company takes more risk?
Widen the outcomes to 190 or 10, with the same average of 100. Equity now pays 90 or 0 and is worth 45; debt pays 100 or 10 and is worth 55: extra risk moves 15 of value from lenders to shareholders without the company being worth a rupee more. That is why lenders to weak companies write covenants against new risky projects, and why restructuring bankers ask who gains from each option the board is weighing.
Where candidates lose it
Candidates subtract the debt from today's assets, get zero, and call the equity worthless. That treats the equity as if it had to settle today and ignores that shareholders keep the upside while being protected from the downside.
The second slip is pricing the debt at its face value of 100. Lenders carry the bad state, so their claim is worth 70, and the market shows that as a high yield.
What the interviewer asks next
The outcomes become 190 or 10. What are equity and debt worth now?
Why might shareholders of this company vote for a risky project with a negative expected value?
How does a positive interest rate change the answer?
039A company earns Rs 10 a share, has a 10% cost of equity, does not grow and pays out everything. It decides to retain half its earnings and reinvest them at a 10% return. Does the share price change?Valuation riddlesCoreElite boutique IBBulge bracket IB
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Does the price move?
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No, the price stays at Rs 100. Paying out Rs 10 forever at a 10% cost of equity is worth 10 over 10%, Rs 100. Retaining half and reinvesting at 10% gives growth of 50% times 10%, which is 5%, on a Rs 5 dividend: 5 over (10% minus 5%) is also Rs 100. Reinvesting at exactly the cost of capital swaps cash today for cash later at a fair rate, so value does not change.
Why does growth not add value here?
Imagine lending your bonus to a friend at exactly the rate your bank pays. You will have more money later, but you are no richer today, because the bank would have paid you the same. Retained earnings create value only if the company reinvests them at more than shareholders could earn elsewhere at the same risk, which is the cost of equity. At exactly 10% the company is just the bank: the growth is real, but it is paid for rupee for rupee by the dividend given up.
Rs 10 a year forever and Rs 5 a year growing at 5% are both worth Rs 100 at a 10% cost of equity, even though the growing stream only overtakes after about 14 years; reinvesting at 15% would lift the price to Rs 200, at 6% it would cut it to Rs 71.
How do the two dividend streams compare?
The flat stream pays Rs 10 every year. The growing one starts at Rs 5 and rises 5% a year, so it overtakes Rs 10 only after about 14 years. Discounted at 10%, the early shortfall and the later surplus cancel exactly, and both streams are worth Rs 100. The tool is the Gordon growth modelA valuation of a stream that grows at a constant rate forever: the next payment divided by the discount rate minus the growth rate.: price equals next year's dividend over the cost of equity minus growth, and growth equals the share retained times the return on the money reinvested.
The relationship
$P = \frac{D_1}{r - g} = \frac{5}{0.10 - 0.5 \times 0.10} = \frac{5}{0.05} = 100$
D1 next year's dividend, half of Rs 10
r the cost of equity, 10%
g growth: half retained, times a 10% return on it
What it says in wordsA smaller dividend that grows is worth exactly the same as the full dividend when the growth is bought at the cost of capital.
When would the decision change the price?
Change the return on the reinvested money. At 15%, growth is 7.5% and the price doubles to Rs 200; at 6%, growth is 3% and the price falls to about Rs 71. The same retention policy creates or destroys value depending only on whether the return beats 10%. That is the sentence the interviewer is waiting for: growth is not good in itself, profitable growth is.
Where candidates lose it
The usual answer is that the price rises because the company now grows. Growth sounds good, but it is bought with the dividend given up, and at a 10% return it costs exactly what it is worth.
The opposite slip is saying the price halves because the dividend halves. That ignores the growth the retained cash buys. Run both legs through the same formula and they cancel.
What the interviewer asks next
What if the retained earnings are reinvested at 15%?
Why might the market still cheer a company that announces growth at its cost of capital?
How does this connect to return on invested capital in a DCF?
040You have eight gold bars that look identical, and one is lighter than the rest. Using a balance scale, what is the fewest weighings that guarantees you find the light bar?Logic and brainteasersCoreConsulting style brainteasersSales and trading
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What is the minimum?
Show the worked solution
Two weighings. Put three bars on each side and leave two off. If the scale balances, the light bar is one of the two left off, and weighing them against each other finds it. If one side is lighter, the light bar is among those three: weigh one against one, and if they balance it is the third. A weighing has three outcomes, so two weighings separate up to 3 x 3 = 9 bars.
Why is halving the wrong instinct?
Think of guessing a number with questions that can be answered higher, lower or spot on. Each answer splits the possibilities three ways, not two. A balance scale gives three outcomes, left lighter, right lighter or level, so each weighing should split the suspects into three groups, not two. Halving, four against four, uses only two of those outcomes and needs 3 weighings for eight bars.
Weighing bars 1, 2 and 3 against 4, 5 and 6 sends the search down one of three branches, and a single second weighing inside each branch names the light bar, so two weighings cover all eight bars.
Why can it not be done in one weighing?
One weighing has only three possible results, and any of the eight bars could be the light one. Three outcomes cannot point to eight different answers, so one weighing is never enough, and two weighings, with 3 x 3 = 9 outcome paths, are the fewest that can cover eight bars. That counting argument is the proof the interviewer wants: it shows two is a floor, not just a method that happened to work.
The relationship
$3^{w} \ge 8 \;\Rightarrow\; w \ge \log_3 8 \approx 1.89 \;\Rightarrow\; w = 2$
w the number of weighings
3^w the number of different outcome paths that many weighings can produce
What it says in wordsYou need enough weighings for the outcome paths to outnumber the bars.
What does the puzzle teach beyond the scale?
The same counting sits behind any search. The fewest questions you need is set by how many answers each question can give, not by how clever the questions are. With nine bars, two weighings still suffice; with ten, you need a third. Saying where the method breaks shows the interviewer you understand why it works, which is the point of asking it.
Where candidates lose it
Most candidates halve: four against four, then two against two, then one against one, and answer three. The method finds the bar but misses that a level scale is an outcome too, and it carries information.
The second loss is giving two with no lower-bound argument. Say why one weighing cannot do it: three outcomes cannot separate eight bars.
What the interviewer asks next
What is the largest number of bars that two weighings can handle?
Now the odd bar could be lighter or heavier, and you do not know which. How many weighings for 12 bars?
What changes if two of the eight bars are light?
042A company has revenue of Rs 730 crore and collects from customers in 45 days. Next year revenue grows 20% and collection slows to 60 days. How much cash does the receivables line absorb?Accounting riddlesCoreBulge bracket IBMiddle market IB
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How much cash does the receivables line absorb next year?
Show the worked solution
About Rs 54 crore. Receivables today are Rs 730 crore times 45 over 365, which is Rs 90 crore. Next year revenue is Rs 876 crore and collection takes 60 days, so receivables are Rs 144 crore. The increase of Rs 54 crore is cash the business has earned but not collected. Rs 18 crore of it comes from growth and Rs 36 crore from slower collection, twice as much.
Why does growth eat cash at all?
Think of a tailor who lets regular customers pay a month and a half after collecting their clothes. If the business grows, more money is out with customers at any moment, even though nobody is paying late. Receivables are revenue earned but not yet collected, so when revenue grows at the same collection period the balance grows with it, and the increase is cash the business has to fund. Here 20% more revenue at 45 days lifts receivables from Rs 90 crore to Rs 108 crore.
Receivables rise from Rs 90 crore to Rs 144 crore: Rs 18 crore of the increase comes from 20% growth at the old 45 days and Rs 36 crore from collecting in 60 days instead, so slower collection absorbs twice as much cash as growth.
How do you split the Rs 54 crore?
Take the two changes one at a time. Growth first: Rs 876 crore at the old 45 days is Rs 108 crore, so growth absorbs Rs 18 crore. Then slower collection: going from 45 to 60 days on Rs 876 crore adds 15 days of sales, Rs 36 crore. Fifteen extra days of collection on the larger business costs twice as much cash as 20% growth. The line to watch is days sales outstandingReceivables divided by revenue, times 365: the average number of days customers take to pay..
The relationship
$\Delta AR = \left(876 \times \tfrac{45}{365} - 90\right) + 876 \times \tfrac{60-45}{365} = 18 + 36 = 54$
876 x 45/365 next year's receivables if collection had not slowed
90 today's receivables, 730 x 45/365
876 x 15/365 the extra 15 days of sales tied up by slower collection
What it says in wordsThe change in receivables is the growth effect at the old collection period plus the slower collection on the new revenue.
Does the order of the split matter?
A little. Slower collection first, on today's revenue, costs Rs 30 crore, and growth then adds Rs 24 crore; the Rs 6 crore difference is growth and slower collection compounding each other. The total is Rs 54 crore whichever way you split it, and slower collection is the bigger cause either way. Say which order you used. Then say what it means: a business growing fast while collecting slowly can report rising profit and falling cash at the same time, which is the pattern a lender looks for first.
Where candidates lose it
Candidates answer Rs 144 crore, the new balance, rather than Rs 54 crore, the change. Cash is absorbed by the increase in receivables, not by their level.
The second loss is blaming it all on growth. Splitting the change shows that slower collection costs twice as much, and that is the insight the question is built to draw out.
What the interviewer asks next
Collection improves to 30 days instead. How much cash does the receivables line release?
Payables also stretch from 30 to 45 days of costs. How does that change the answer?
Why might a company's collection period lengthen as it grows?
045Twelve bankers meet at an offsite and every pair shakes hands exactly once. How many handshakes are there?Mental maths and countingWarm upBulge bracket IBMiddle market IB
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Answer fast.
Show the worked solution
66 handshakes. Each of the 12 bankers shakes hands with the other 11, which gives 12 x 11 = 132, but every handshake has two people in it and so has been counted twice. Half of 132 is 66. The same number falls out of adding 11 + 10 + 9 and so on down to 1, or from 12 choose 2.
Why do you halve the count?
Think of counting the phone calls in a group where everyone rang everyone else once. If each person reports the calls they took part in, every call shows up in two reports. Each handshake involves two people, so counting from each person's side counts every handshake exactly twice, and the true number is half. Twelve people times eleven others is 132 ends of handshakes, which is 66 handshakes.
Twelve people on a circle with one line for every pair make 66 lines: each person sits at the end of 11 lines, giving 132 line ends, and every line has two ends, so the count halves to 66.
How do you check 66 another way?
Let people arrive one at a time. The second person shakes 1 hand, the third 2, and so on to the twelfth, who shakes 11, and adding 1 through 11 gives 66. Pair the terms to add them quickly: 1 and 11, 2 and 10, and so on make five pairs of 12 plus a 6 in the middle, 66. Two routes to the same number is the check interviewers like to hear.
The relationship
$\binom{12}{2} = \frac{12 \times 11}{2} = 66 = 1 + 2 + \cdots + 11$
12 x 11 each person times the others they meet, counting every handshake from both sides
2 the two people in every handshake
What it says in wordsPairs from a group of n are n times (n minus 1), halved.
Where does this count show up on a desk?
Anywhere pairs matter. The number of pairs grows roughly with the square of the group: double the group to 24 and the handshakes rise to 276, more than four times as many. That is why a 12-stock portfolio has 66 pairwise correlationsMeasures of how closely two things move together, from minus 1 to plus 1. to estimate, and why a deal with many parties needs far more conversations than its headcount suggests.
Where candidates lose it
The fast wrong answer is 132, from 12 times 11 without halving. It treats one handshake between two people as two separate events.
The other slip is 144 or 78, from letting people shake their own hand. Say each person shakes hands with the other 11, and both errors disappear.
What the interviewer asks next
How many people must be in the room for there to be 105 handshakes?
At a dinner of six couples, everyone clinks glasses with everyone except their own partner. How many clinks?
How many pairwise correlations does a 30-stock portfolio have?

6	minute hand speed, degrees per minute
0.5	hour hand speed, degrees per minute
720/11	minutes between one overlap and the next

2/36	the chance of a 7 that contains a 4: (4,3) or (3,4)
11/36	the chance of at least one 4

0.12/12	the monthly rate, one twelfth of the quoted 12%
12	the number of times interest is charged in a year

8 choose 3	every team of three from eight analysts
6 choose 1	teams holding both feuding analysts: one choice of third member from the other six

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Why is it not once an hour?

When exactly do the hands meet?

What do you say about the edges?

Where candidates lose it

What the interviewer asks next

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What does being told something actually change?

Why is 1/6 tempting and wrong?

How do you check it quickly?

Where candidates lose it

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Why is 12% compounded monthly not really 12%?

How do you work out 1.01 to the twelfth in your head?

What else should you say before you finish?

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What the interviewer asks next

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Why count the teams you do not want?

How do you get 56 without a formula sheet?

How do you check 50 a second way?

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How do you structure the estimate before any number?

Why does a demand estimate need a supply check?

What do you say about the weakest link?

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Why is the equity worth anything when assets only cover the debt?

How do you price the two claims?

What happens if the company takes more risk?

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Why does growth not add value here?

How do the two dividend streams compare?

When would the decision change the price?

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Why is halving the wrong instinct?

Why can it not be done in one weighing?

What does the puzzle teach beyond the scale?

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Why does growth eat cash at all?

How do you split the Rs 54 crore?

Does the order of the split matter?

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Why do you halve the count?

How do you check 66 another way?

Where does this count show up on a desk?

Where candidates lose it

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