Financial Analysis puzzles, solved step by step

Puzzles: 100
Traced to a firm: 47
Topics: 13
Hard: 30

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012A supplier lists a part at Rs 100. It offers two schemes: 5% off every unit if you buy 1,000 or more, or 10% off only the units above 1,000. At what volume do the two schemes cost the same? And why might a buyer who needs 990 units order 1,000?Pricing, costing and unit economicsHardCorporate FP&ACost accounting
Try it first
At what order size do the two schemes cost exactly the same?
Show the worked solution
The schemes cost the same at 2,000 units, Rs 1,90,000 each. The all-units deal costs 95 times the quantity; the incremental deal costs Rs 1,00,000 for the first 1,000 and Rs 90 after that, which equals 95Q only at 2,000. Below that, all-units is cheaper. It also creates a cliff: 990 units cost Rs 99,000 but 1,000 cost Rs 95,000, so anyone ordering more than 950 should round up to 1,000.
How do you set the two schemes side by side?
Think of two mobile data plans, one with a flat lower price on everything once you cross a usage level and one that only discounts the extra data. They feel similar, but they reward different amounts of use. Write each scheme as total cost against quantity, then find where the two formulas are equal. All-units: 95Q once Q reaches 1,000. Incremental: 1,00,000 for the first 1,000 units, plus 90 for each unit beyond.
The relationship
$95Q = 100{,}000 + 90(Q - 1{,}000) \;\Rightarrow\; 5Q = 10{,}000 \;\Rightarrow\; Q = 2{,}000$
Q units ordered, at least 1,000
95 all-units price after 5% off
90 incremental price on units above 1,000
What it says in wordsThe incremental deal saves Rs 5 a unit more on every unit above 1,000, and needs 2,000 units in total to make up the Rs 5,000 head start the all-units deal gives at 1,000.
A quicker way to see it: at 1,000 units the all-units scheme is Rs 5,000 cheaper. Each extra unit costs Rs 95 under all-units and Rs 90 under incremental, so the incremental deal claws back Rs 5 a unit. It needs another 1,000 units to recover Rs 5,000, so the schemes meet at 2,000. Above that the incremental scheme wins: at 3,000 units it costs Rs 2,80,000 against Rs 2,85,000.
The all-units scheme drops the average price from Rs 100 to Rs 95 the moment an order reaches 1,000 units, while the incremental scheme lowers it gradually. The incremental average only reaches Rs 95 at 2,000 units, which is where the two schemes cost the same.
Why would anyone order more than they need?
Because the all-units scheme makes buying more cost less. 990 units at Rs 100 cost Rs 99,000, while 1,000 units at Rs 95 cost Rs 95,000, so ten extra parts arrive with Rs 4,000 back in your pocket. The same logic holds for any order above 950 units, because 1,000 units cost Rs 95,000 and anything from 951 to 999 costs more. A buyer with somewhere to store the spares should always round up.
Under the all-units scheme, 990 units cost Rs 99,000 but 1,000 units cost only Rs 95,000. Any order between 951 and 999 units costs more than rounding up to 1,000, which is the cliff an all-units discount creates.
Say the limit from both sides of the deal. Extra units carry storage, handling and the risk they are never used, so the saving is only real if the spares have a use. For the supplier, the cliff means giving away Rs 4,000 on a sale that would have happened anyway, which is why incremental schemes are common where buyers order close to a threshold.
Where candidates lose it
The common loss is answering 1,000, the point where the discounts start, instead of finding where the costs meet. Writing both costs as formulas takes ten seconds and makes 2,000 obvious.
The second loss is missing the cliff. Interviewers add the 990 question to see whether you notice that an all-units discount makes total cost fall as quantity rises, which is the opposite of what a cost curve normally does.
What the interviewer asks next
What is the smallest order at which rounding up to 1,000 saves money?
If storing each spare part costs Rs 3 a year, does the 990 buyer still round up?
Which scheme would you offer as the supplier, and why?
027A product sells at a 30% contribution margin. You raise the price by 5% and lose 8% of your volume. Is total contribution better or worse than before, and by how much?Pricing, costing and unit economicsCoreCorporate FP&ACost accounting
Try it first
Gut call: you gained 5% on price and lost 8% on volume. Where does total contribution land?
Show the worked solution
Better, by about 7.3%. Take a price of 100 with variable cost 70, so each unit contributes 30. After the rise the price is 105 and each unit contributes 35. Selling 92 units instead of 100 gives 92 x 35 = 3,220 against 3,000 before. Volume could fall as far as 14.3% before the price rise stopped paying.
Why does 5% on price beat 8% on volume?
Picture a tea stall that sells a cup for 10 rupees with 7 rupees of milk, leaves and sugar in it. Raise the price to 10.50 and the stall's profit per cup goes from 3 to 3.50, a sixth more. A price rise falls straight to contribution, so its effect is measured against the margin, not against the price. The volume loss, by contrast, removes whole cups and their contribution at the same rate as the lost units, 8%.
Raising price from 100 to 105 lifts contribution per unit from 30 to 35, a 16.7% gain, so 92 units at 35 earn 3,220 against 3,000 for 100 units at 30, and volume would need to fall 14.3% to erase the gain.
How much volume could you lose before the price rise hurts?
Set new contribution equal to old: units times 35 must equal 100 times 30, so units can fall to 3,000 over 35, which is 85.7. The breakeven volume loss for a price rise is the price change divided by the new margin: 5 over 35, or 14.3%. That is why thin-margin businesses gain so much from price and lose so much from discounts: at a 30% margin, a 5% discount needs volume to rise by 5 over 25, a full 20%, just to stand still.
The relationship
$\Delta V_{\text{breakeven}} = -\frac{\Delta p}{m + \Delta p} = -\frac{5\%}{30\% + 5\%} = -14.3\%$
Delta p the price change as a share of the old price, 5%
m the contribution margin before the change, 30%
What it says in wordsVolume can shrink by the price change divided by the new margin before the rise stops paying.
Say the limit too. The sum treats variable cost per unit as fixed and ignores any fixed costs that could be cut when volume falls, and it says nothing about customers who leave and do not come back next year.
Where candidates lose it
The usual loss is adding the two percentages: plus 5 and minus 8 is minus 3, so the move looks bad. That treats the price change as if it acted on revenue, when it lands entirely on the contribution slice.
The second loss is working in revenue instead of contribution. Revenue does fall slightly, 105 x 92 is 9,660 against 10,000, and a candidate who stops there gets the answer backwards.
What the interviewer asks next
At the same 30% margin, how much must volume rise to justify a 10% price cut?
How does the answer change if the margin is 70%, as in software?
Which fixed costs would you look at if volume did fall 8%?
045A snack pack shrinks from 100 g to 90 g while its price stays at Rs 50. What is the effective price rise per gram?Pricing, costing and unit economicsWarm upCorporate FP&ACost accounting
Try it first
Pick the effective price rise.
Show the worked solution
About 11.1%. Before, Rs 50 buys 100 g, Rs 0.500 a gram. After, Rs 50 buys 90 g, Rs 0.556 a gram. The rise is 100 over 90 less one, which is 10 over 90, or 11.1%. In general a size cut of x at the same price is a price rise of x divided by (1 minus x), always a little more than the cut.
Why is a 10% smaller pack more than a 10% price rise?
Think of sharing the same Rs 50 bill among 10 friends instead of 11. Each pays more, and the increase is measured against the new, smaller headcount. Price per gram is price divided by grams, so cutting the grams by 10% divides the price by 0.9, which raises it by 1/0.9 less one, 11.1%. The percentage fall in grams is measured on 100; the percentage rise in price per gram is measured on what is left, 90.
Rs 50 for 100 g is Rs 0.500 a gram and Rs 50 for 90 g is Rs 0.556 a gram, an 11.1% rise, and the gap between the cut and the rise grows with the cut: 20% smaller is 25% dearer and 25% smaller is 33.3% dearer.
The relationship
$\text{Price rise} = \frac{1}{1-x} - 1 = \frac{x}{1-x} = \frac{0.10}{0.90} = 11.1\%$
x the cut in pack size, as a decimal
What it says in wordsA size cut of x at a fixed price is a price rise of x over one minus x.
How would a finance team use this?
Inflation in raw materials pushes consumer companies to protect a price point that shoppers remember, such as Rs 10 or Rs 50, and adjust grams instead. To compare price changes across products, convert every pack-size change into a price per unit, or a volume decline will be misread as stable pricing. In a revenue bridge, the same rupee sales on fewer grams show up as a price gain, not as flat sales. Say the limitation: price per gram is the right comparison only if quality and the mix of the pack stay the same.
Where candidates lose it
The trap answer is 10%, the size cut itself. It measures the change against the old pack, while the price per gram has to be measured against the new one.
A second wrong answer is 9.1%, which comes from dividing 10 by 110. That is the rule for the opposite case, a pack that grows 10% at the same price, which is a 9.1% price cut.
What the interviewer asks next
A pack grows from 100 g to 120 g at the same price. What is the price change per gram?
The pack shrinks to 90 g and the price falls to Rs 48. What is the net price change per gram?
How would you show this in a price-volume bridge?
060A product sells at a price that gives a 40% contribution margin. If you cut the price by 10%, by how much must volume rise to keep total contribution unchanged?Pricing, costing and unit economicsCoreCorporate FP&ACost accounting
Try it first
By how much must volume rise?
Show the worked solution
Volume must rise by a third, 33.3%, just to stand still. On a Rs 100 price, contribution is Rs 40 a unit. Cut the price 10% and variable cost stays at Rs 60, so contribution falls to Rs 30, a quarter lower. Selling 40 / 30 = 1.333 times as many units keeps total contribution flat. The general rule is the cut divided by the margin less the cut: 10 / (40 - 10).
Why does a 10% price cut need far more than 10% more volume?
A samosa seller charges Rs 20 and spends Rs 12 on ingredients, so each samosa leaves Rs 8. Cutting the price by Rs 2 does not cut the ingredient bill; it cuts the Rs 8 to Rs 6. A price cut comes entirely out of contribution, because variable cost does not fall with the price, so the volume needed to make it back is set by the margin, not the price. In the puzzle the 10% cut takes a quarter of the contribution, so volume must rise by 33.3%.
Total contribution is the area of each rectangle: 100 units at Rs 40 before the cut, and Rs 30 a unit after it, so the rectangle must widen to 133.3 units, a rise of 33.3%, to keep the area at 4,000.
What is the general rule, and how fast does it bite?
The relationship
$\text{volume rise needed} = \frac{\text{cut}}{\text{margin} - \text{cut}} = \frac{10\%}{40\% - 10\%} = 33.3\%$
cut the price reduction as a share of the old price
margin contribution margin, contribution per unit over price, before the cut
What it says in wordsThe volume needed rises sharply as the cut approaches the margin, and a cut as large as the margin can never be made back.
Contribution margin Cut 5% Cut 10% Cut 20%
20% 33.3% 100.0% not possible
40% 14.3% 33.3% 100.0%
60% 9.1% 20.0% 50.0%
The volume rise needed to hold contribution flat grows quickly as the price cut approaches the margin; at a 20% margin a 20% cut leaves nothing per unit, so no volume is enough.
The table shows why low-margin businesses fear discounting. At a 20% margin a 10% cut needs volume to double; at 60% it needs only 20% more. The rule also runs the other way: a 10% price rise lifts contribution to Rs 50 a unit, so volume can fall by up to 20% before total contribution drops. Price rises are usually safer than they feel and price cuts riskier than they look.
What would you check before backing the cut?
Three things. First, whether demand will really grow by a third, which depends on how sensitive buyers are to price and whether rivals match the cut. Second, capacity: a third more units may need overtime or a new shift, which turns fixed costs into step costs. Third, the existing customers who would have paid Rs 100 anyway now pay Rs 90, which is where most of the loss sits. The limit of the rule is that it holds contribution flat; it says nothing about cash tied up in the extra stock and receivables.
Where candidates lose it
The instinct is to say 10%, or 11.1% if the candidate remembers that a fall needs a bigger rise to recover. Both hold revenue flat, not contribution, and miss that the cut lands entirely on the Rs 40 margin.
The other slip is applying the 10% cut to variable cost as well, as if costs fell with the price. They do not, which is the whole reason the answer is a third rather than a tenth.
What the interviewer asks next
At a 25% contribution margin, what volume rise does a 10% cut need?
By how much can volume fall after a 10% price rise before contribution drops?
Fixed costs are Rs 20 a unit at today's volume. Does the answer change, and when would it?
076A subscription product earns Rs 500 a month per customer at a 70% gross margin. Monthly churn is 3%, and it costs Rs 6,000 to acquire a customer. What is the customer lifetime value, the LTV to CAC ratio, and the payback period in months?Pricing, costing and unit economicsCoreCorporate FP&ABusiness finance
Try it first
Before you work it: how long does the average customer stay?
Show the worked solution
Lifetime value is about Rs 11,667, LTV to CAC is about 1.9x, and payback is about 17 months. Each customer brings Rs 350 of gross profit a month, 70% of Rs 500. At 3% churn the average customer stays 1 / 0.03 = 33.3 months, so LTV is Rs 350 x 33.3. The Rs 6,000 acquisition cost divided by Rs 350 a month is paid back in 17.1 months.
Why is the average customer life one over churn?
Think of a hostel where 3 residents in every 100 move out each month and are replaced. A resident faces a 3% chance of leaving every month, so on average the wait before leaving is 1 / 0.03 months. With a constant monthly churn rate, the expected customer life is one divided by that rate: at 3%, 33.3 months. Lifetime value is the gross profit the customer brings each month times that life.
Use gross profit, not revenue. The Rs 150 a month it costs to serve the customer is spent whether or not the customer was worth acquiring, so it cannot pay back the acquisition cost. Rs 350 x 33.33 months is Rs 11,667, and against a CACCustomer acquisition cost: marketing, sales and onboarding spend divided by the number of customers it won. of Rs 6,000 that is 1.94x.
The relationship
$\text{LTV} = \frac{m \times \text{ARPU}}{c} = \frac{0.70 \times 500}{0.03} = 11{,}667$
m gross margin, 70%
ARPU revenue per customer per month, Rs 500
c monthly churn, 3%, so 1/c is the average life in months
What it says in wordsLifetime value is monthly gross profit divided by the monthly churn rate.
A customer who stays earns back the Rs 6,000 acquisition cost in month 17.1, but the average customer, allowing for 3% monthly churn, crosses it only in month 23.7, and the cohort curve flattens toward a lifetime value of Rs 11,667.
Why are there two payback answers?
Payback is the month in which cumulative gross profit covers the Rs 6,000 spent on day zero. For a customer who stays, that is 6,000 / 350 = 17.1 months, the standard answer. Across a whole cohort payback is slower, because some customers leave before they have repaid their share of the acquisition cost. Weight each month by the chance the customer is still there and the average customer crosses Rs 6,000 in month 23.7. Give 17 months and name the cohort figure in one sentence.
Is 1.9x good enough?
Many subscription investors use a rule of thumb of 3x or better with payback inside about a year; treat those as conventions, not laws. At 1.9x this business recovers its acquisition cost with a thin cushion, and the ratio flatters it, because LTV here is undiscounted and ignores any cost of retaining customers. Churn is the strongest lever: cutting it from 3% to 2% lifts LTV by half to Rs 17,500, while a 10% price rise at the same churn only reaches Rs 12,833.
Where candidates lose it
Candidates compute lifetime value on revenue: Rs 500 x 33.3 = Rs 16,667, and report 2.8x. The Rs 150 a month it costs to serve each customer is real money, and a customer who covers only that cost has repaid nothing of the acquisition spend. Lifetime value is a profit measure and runs on gross margin.
The second loss is stating 17 months as if it were the whole truth. It assumes the customer stays. One sentence on the cohort version, and on the fact that none of this is discounted, turns a correct calculation into an analyst's answer.
What the interviewer asks next
Discount the lifetime value at 1% a month. What does it become? (Roughly Rs 350 / (0.03 + 0.01) = Rs 8,750.)
Which lifts LTV to CAC more: halving churn or raising the price 20%?
How would you estimate churn for a product that launched only eight months ago?
094A cafe sells coffee at Rs 120 a cup, though each cup costs Rs 140 to serve, to bring people in. Each food item earns Rs 60 of contribution. What share of coffee buyers must also buy a food item for the coffee to pay its way?Pricing, costing and unit economicsCoreCorporate FP&ACost accounting
Try it first
What attach rate makes the coffee break even?
Show the worked solution
One coffee buyer in three, 33.3%, must also buy a food item. Each cup loses Rs 20: Rs 140 to serve against Rs 120 of price. Each food item adds Rs 60 of contribution. So one food sale covers the loss on three coffees, and the break-even attach rate is 20 / 60. Two checks matter: only food bought because of the coffee counts, and Rs 140 must be the true extra cost of a cup, not a share of rent.
How do you price a product that loses money on purpose?
A shopkeeper who sells milk below cost does it because people who come in for milk also pick up bread and biscuits. The milk is a door, and its price is justified by what walks through it. A loss leader pays its way when the profit on what customers also buy covers the loss on the leader, so the number to manage is the attach rate. Here, each coffee loses Rs 20 and each food item earns Rs 60.
The relationship
$a^{*} = \frac{\text{loss per leader}}{\text{contribution per add-on}} = \frac{140 - 120}{60} = \frac{1}{3}$
a* the break-even attach rate: share of leader buyers who also buy the add-on
contribution price less the variable cost of the add-on
What it says in wordsThe break-even attach rate is the loss on the leader divided by the contribution on what it brings in.
Three coffees at a Rs 20 loss each balance one food item at Rs 60 of contribution, so the coffee pays its way once one coffee buyer in three also buys food.
What makes the one-in-three figure misleading?
Two things. First, only incremental food counts. If half the people buying food would have come in for lunch anyway, their food was not brought in by the cheap coffee, and the coffee cannot claim it. The attach rate that matters is the share of coffee buyers who buy food because they came for coffee, which is lower than the attach rate on the till receipts. Measuring it needs a comparison: a price test in a few outlets, or the behaviour of customers before and after the coffee price changed.
Second, ask what is inside the Rs 140. If it includes a share of rent, staff and the coffee machine, those costs are there whether or not one more cup is sold. Suppose the extra cost of a cup, beans, milk and the cup itself, is Rs 70. Then each coffee contributes Rs 50 at the margin and needs no food to pay its way; it only looks like a loss after fixed costs are spread across it. Decisions about one more sale use contribution; fully loaded cost answers a different question, whether the whole cafe covers its overheads.
What would you track after launch?
Attach rate by time of day, average food contribution per coffee buyer, and the number of new customers the coffee brings in. A cafe that hits 40% attach in the morning and 15% in the afternoon may want the cheap coffee only before noon. The limitation is that habit takes time: a promotion judged on its first two weeks can look worse than it will in month three.
Where candidates lose it
The common slip is setting the attach rate at the point where food covers the full price of the coffee, or dividing the wrong way and getting three food items per coffee. Set the loss per coffee against the gain per food item and it falls out: 20 / 60.
The bigger loss is stopping at a third. The interviewer is waiting for the two questions behind it: is the food sale caused by the coffee, and is Rs 140 really the extra cost of a cup? Either can reverse the conclusion.
What the interviewer asks next
The cafe raises coffee to Rs 130. What attach rate does it now need?
Half of the food buyers would have come anyway. What measured attach rate do you need on the till receipts?
How would you test whether the cheap coffee is bringing in new customers at all?

Q	units ordered, at least 1,000
95	all-units price after 5% off
90	incremental price on units above 1,000

Delta p	the price change as a share of the old price, 5%
m	the contribution margin before the change, 30%

cut	the price reduction as a share of the old price
margin	contribution margin, contribution per unit over price, before the cut

Contribution margin	Cut 5%	Cut 10%	Cut 20%
20%	33.3%	100.0%	not possible
40%	14.3%	33.3%	100.0%
60%	9.1%	20.0%	50.0%

Try it first

How do you set the two schemes side by side?

Why would anyone order more than they need?

Where candidates lose it

What the interviewer asks next

Try it first

Why does 5% on price beat 8% on volume?

How much volume could you lose before the price rise hurts?

Where candidates lose it

What the interviewer asks next

Try it first

Why is a 10% smaller pack more than a 10% price rise?

How would a finance team use this?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a 10% price cut need far more than 10% more volume?

What is the general rule, and how fast does it bite?

What would you check before backing the cut?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the average customer life one over churn?

Why are there two payback answers?

Is 1.9x good enough?

Where candidates lose it

What the interviewer asks next

Try it first

How do you price a product that loses money on purpose?

What makes the one-in-three figure misleading?

What would you track after launch?

Where candidates lose it

What the interviewer asks next