Financial Analysis puzzles, solved step by step

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  1. 021You invest Rs 10,000 in a fund when its NAV is Rs 100, and another Rs 10,000 when the NAV is Rs 50. What is your average cost per unit, and why is it not Rs 75?Percentages and averagesCoreCorporate FP&ABusiness finance

    Try it first

    What is your average cost per unit?

    Show the worked solution

    About Rs 66.67 a unit, not Rs 75. The first Rs 10,000 buys 100 units at Rs 100 and the second buys 200 units at Rs 50, so Rs 20,000 buys 300 units. Rs 75 is the average of the two prices, which would be your cost only if you had bought equal numbers of units. Equal rupee amounts buy more units when the price is low, so the average cost is the harmonic mean of the prices.

    Why does the cheap purchase count for more?

    Think of filling your scooter with Rs 500 of petrol every week. In a week when petrol is cheap, Rs 500 buys more litres; when it is dear, fewer. At the end of the month the litres you own lean towards the cheap weeks. A fixed rupee amount buys more units when the price is low, so the low price carries more weight in your average cost than the high one. Here Rs 10,000 buys 100 units at Rs 100 but 200 units at Rs 50, so the Rs 50 price carries two thirds of the weight.

    The relationship
    average cost=total spenttotal units=20,000100+200=66.67=21/100+1/50\text{average cost} = \frac{\text{total spent}}{\text{total units}} = \frac{20{,}000}{100 + 200} = 66.67 = \frac{2}{1/100 + 1/50}
    total spentRs 20,000 across both purchases
    total units100 units at NAV 100 plus 200 at NAV 50
    2 / (1/100 + 1/50)the harmonic mean of the two prices
    What it says in wordsAverage cost is money spent over units bought, which for equal rupee amounts is the harmonic mean of the prices.
    Fixed rupees buy more units when the price is low100 unitsRs 10,000at NAV 100200 unitsRs 10,000at NAV 50Units bought with each Rs 10,000300 unitsfor Rs 20,000= Rs 66.67 a unitThree averages of the prices 100 and 505010066.7Your average cost70.7Geometric mean75.0Simple average priceAverage cost = total spent / total units = harmonic mean
    Rs 10,000 buys 100 units at a NAV of 100 and 200 units at a NAV of 50, so Rs 20,000 buys 300 units at an average cost of Rs 66.67. That harmonic mean sits below both the geometric mean of 70.7 and the simple average price of 75.

    When is Rs 75 the right answer?

    When you buy equal numbers of units rather than equal rupee amounts. Buy 100 units at Rs 100 and 100 units at Rs 50 and you spend Rs 15,000 on 200 units, which is exactly Rs 75 a unit. So the right average depends on what you held fixed: units fixed gives the simple average of prices, money fixed gives the harmonic mean. This is the arithmetic behind the idea of rupee cost averagingInvesting the same rupee amount at regular intervals, so that more units are bought when prices are low and fewer when they are high..

    What you hold fixedUnits boughtSpent, RsAverage cost, Rs
    Rs 10,000 each time30020,00066.67
    100 units each time20015,00075.00
    The same two prices give two different average costs, depending on whether the rupee amount or the number of units was held constant.

    Does the lower average cost mean you made money?

    No, and saying so is the mark of a careful answer. A lower average cost is arithmetic about what you paid, not a return. If the NAV stays at Rs 50, your 300 units are worth Rs 15,000 against Rs 20,000 invested, a 25% loss. Your break-even NAV is Rs 66.67, not Rs 75, which is the useful fact. The effect also needs prices to vary: at a constant price every average is the same.

    Where candidates lose it

    The common loss is answering Rs 75 by averaging the two prices. It ignores that the purchases bought different numbers of units, and it is the same slip people make when they average two speeds for a journey.

    The second loss is going too far the other way and describing the lower average cost as a gain. Give Rs 66.67, say break-even is at that NAV, and stop short of claiming anything about returns.

    What the interviewer asks next

    • You invest Rs 10,000 at NAV 100, then 50, then 100 again. What is your average cost?
    • What NAV do you need to break even on the two purchases?
    • Why is the harmonic mean of two different positive numbers always below their simple average?
  2. 036A business has a 10% operating margin. Next year revenue grows 20% and costs grow 15%. What is the new margin?Percentages and averagesCoreCorporate FP&ABusiness finance

    Try it first

    Quick call: where does the margin land?

    Show the worked solution

    The new margin is 13.75%. Take revenue of 100, so cost is 90 and profit 10. Revenue grows to 120 and cost to 90 x 1.15 = 103.5, leaving profit of 16.5. The margin is 16.5 over 120, which is 13.75%. Profit itself grows 65%, because a 5 point gap between growth rates is large next to a 10% margin.

    Why is the answer not simply 15%?

    Think of a tea stall that takes in Rs 100 a day and spends Rs 90. If takings rise 20% and spending rises 15%, the extra takings are Rs 20 but the extra spending is Rs 13.50, because 15% is taken on 90, not on 100. Growth rates act on their own bases, so a margin moves by the gap in rupees, not by the gap in percentage points. Work in rupees on a base of 100 and the trap disappears: profit goes from 10 to 16.5 on revenue of 120.

    Revenue adds 20 on a base of 100; cost takes back only 13.5 on a base of 9010Old profit10% of 100+20Revenue+20% x 100-13.5Cost+15% x 9016.5New profiton 120Revenue100 to 120+20%Cost90 to 103.5+15%Profit10 to 16.5+65%Margin16.5 / 12013.75%Not 10% + (20% - 15%) = 15%the two growth rates act on different bases
    On revenue of 100, profit of 10 gains 20 from revenue growth and loses 13.5 to cost growth, ending at 16.5 on revenue of 120, a margin of 13.75% rather than the 15% you get by adding the growth gap to the margin.
    The relationship
    m1=1−(1−m0)1+gc1+gr=1−0.90×1.151.20=13.75%m_1 = 1 - (1-m_0)\frac{1+g_c}{1+g_r} = 1 - 0.90 \times \frac{1.15}{1.20} = 13.75\%
    m0, m1the old and new margins
    g_r, g_crevenue growth and cost growth
    1 - m0cost as a share of revenue before the change
    What it says in wordsThe new cost share is the old cost share scaled by cost growth over revenue growth, and the margin is what is left.

    Why does a thin margin make the move so large?

    Profit is the small gap between two big numbers, so a modest difference in their growth rates is a big change to the gap. The thinner the starting margin, the more operating leverage a growth gap carries: here profit grows 65% on 20% revenue growth. Run the same 20% and 15% from a 30% margin and the new margin is 32.9%, a smaller relative move, because cost is a smaller share of revenue. Say the limit too: the formula assumes all cost grows at 15%. In practice fixed costs grow slowly and variable costs track volume, so you would split the cost line before trusting the answer.

    Where candidates lose it

    The fast wrong answer is 15%: 10% plus the 5 point gap between the growth rates. It treats both rates as if they applied to the same base, when cost growth applies to 90 and revenue growth to 100.

    The second loss is stopping at the margin. The interviewer often wants the profit growth too, and 65% on 20% revenue growth is the sentence that shows you understand operating leverage.

    What the interviewer asks next

    • What cost growth would keep the margin at exactly 10%?
    • If half the cost base is fixed and does not grow, what is the new margin?
    • Revenue falls 10% and costs fall 5%. What happens to a 10% margin?
  3. 049A table shows segment revenue for two years: A Rs 400 crore to Rs 460 crore, B Rs 250 crore to Rs 240 crore, C Rs 350 crore to Rs 420 crore. Which segment's share of total revenue rose the most, and by how many percentage points?Percentages and averagesCoreBarclaysNew York · 2026

    Try it first

    Which segment gained the most share?

    Show the worked solution

    Segment C, up 2.5 percentage points, from 35.0% to 37.5%. Total revenue rises from Rs 1,000 crore to Rs 1,120 crore, 12% growth. C grew 20%, and 420 of 1,120 is 37.5%. A grew 15% but its 460 of 1,120 is 41.1%, only 1.1 points above 40.0%. B shrank 4% and lost 3.6 points, falling to 21.4%.

    Why is growth not the same as gaining share?

    Think of a household where every earner got a raise this year. Whose share of the family income went up most? Not the one with the biggest raise in rupees, and not necessarily the one with a big raise at all, but the one whose raise beat the household's overall rise by the widest margin. A segment gains share only by growing faster than the total, so the first number to compute is total growth, here 12%. A grew 15%, three points faster than the total; C grew 20%, eight points faster; B shrank.

    Shares are computed on the new total; growth faster than 12% is what gains shareSegmentYear 1ShareYear 2ShareGrowthChangeSegment A40040.0%46041.1%+15%+1.1 ptsSegment B25025.0%24021.4%-4%-3.6 ptsSegment C35035.0%42037.5%+20%+2.5 ptsTotal1,0001,120+12%Rs crore. Shares of each year's own total.A grew 15%but gained only1.1 points:the total grew12%, so A beatit by only 3C grew 20%,8 points faster,and gained 2.5
    Total revenue grows 12% from Rs 1,000 crore to Rs 1,120 crore, so segment C, growing 20%, lifts its share from 35.0% to 37.5%, while A grows 15% but gains only 1.1 points and B loses 3.6 points.
    The relationship
    Δs=s0×g−G1+GC:0.35×0.20−0.121.12=2.5 pts\Delta s = s_0 \times \frac{g - G}{1 + G} \qquad C: 0.35 \times \frac{0.20 - 0.12}{1.12} = 2.5\text{ pts}
    s0the segment's share in year one
    gthe segment's own growth
    Ggrowth of the total, 12%
    What it says in wordsThe change in share is the old share times how far the segment outgrew the total, scaled down by the total's growth.

    How do you answer this fast in a timed test?

    Sum each column first: 1,000 and 1,120. Compare every segment's growth with 12%, and only A and C can have gained. Then use the formula on those two: C is 0.35 x 8/112, which is 2.5 points; A is 0.40 x 3/112, about 1.07. Shares always add to 100%, so the gains and losses must net to zero, which gives a free check: 2.50 + 1.07 - 3.57 = 0. Watch the units in the answer options: C's share rose 2.5 percentage points, which is a 7.1% rise in its share. Timed numerical tests often offer both, and only one matches the question's wording.

    Where candidates lose it

    The trap is picking A because it added Rs 60 crore on the largest base, or because 15% sounds strong. Share depends on growth relative to the total, and A beat the total by only three points.

    The second loss is mixing percentage points and per cent. A share moving from 35.0% to 37.5% is up 2.5 points or about 7.1%. Read which one the question asks for before you choose.

    What the interviewer asks next

    • What growth would segment B have needed to hold its 25% share?
    • If total revenue had grown 20%, which segments would have gained share?
    • Why might a segment that loses share still be the most valuable one?

    Asked at Barclays, Investment Banking, New York, 2026 (Wall Street Oasis): The numerical section involved interpreting tables and charts quickly