Financial Analysis puzzles, solved step by step

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  1. 032A machine turns a Rs 10 note into a Rs 20 note, but each use takes exactly one year, costs Rs 2 of electricity, and the machine breaks after 10 uses. At a 10% interest rate, what would you pay for it today?Compounding and time valueHardGoldman SachsDallas · 2026

    Try it first

    Before you discount anything: what is the machine's real net gain per use, measured at the end of the year?

    Show the worked solution

    About Rs 43. Each use ties up a Rs 10 note for a year, which at 10% costs you Rs 11 by year end, plus Rs 2 of electricity, and pays Rs 20. That is Rs 7 a year for ten years, and Rs 7 times the ten-year annuity factor of 6.1446 is Rs 43.01. If the electricity is paid at the start of each year the net is Rs 6.80 and the value Rs 41.78.

    What is one use of the machine actually worth?

    Think of lending a friend Rs 10 for a year and getting Rs 20 back. You doubled your note, but you also went a year without it, and that year had a price: whatever the money would have earned in the bank. The machine does not create Rs 10 a year; it creates Rs 20 at year end in exchange for Rs 10 today, and Rs 10 today is worth Rs 11 at year end at 10%. Take off Rs 2 of electricity and each use leaves Rs 7, received one year after you feed it the note.

    Ten uses, Rs 7 net each, discounted back to todayOne use, valued at year end:+20 out - 10 x 1.1 note in - 2 power = 7Sum of ten present valuesRs 43.01 (power paid up front: 41.78)6.36Yr 15.79Yr 25.26Yr 34.78Yr 44.35Yr 53.95Yr 63.59Yr 73.27Yr 82.97Yr 92.70Yr 10Dashed outline: Rs 7 received at the end of each year. Filled: what it is worth today at 10%.Forget the Rs 1 a year the note could have earned and you get Rs 49.16, too much.
    Each of the ten uses leaves Rs 7 at the end of its year once the Rs 11 cost of the note and the Rs 2 of electricity are taken off the Rs 20, and the ten present values fall from Rs 6.36 to Rs 2.70 and sum to Rs 43.01.
    The relationship
    PV=(20−10(1.1)−2)×1−1.1−100.10=7×6.1446=43.01PV = \big(20 - 10(1.1) - 2\big)\times\frac{1-1.1^{-10}}{0.10} = 7 \times 6.1446 = 43.01
    10(1.1)the Rs 10 note's value at year end had you kept it at 10%
    6.1446the present value of Rs 1 a year for ten years at 10%
    What it says in wordsFind the net gain of one use at the end of its year, then value ten of them as an annuity.

    Why does the timing of the electricity change the answer?

    The question does not say when you pay for power. If you pay at the end of the year, the net is the Rs 7 above. If you pay at the start, alongside the note, the Rs 2 also costs you a year of interest, Rs 2.20 by year end, and the net drops to Rs 6.80, worth Rs 41.78. State the timing assumption out loud, because an interviewer who wrote this puzzle is listening for whether you notice that cash flows need a date. The other common answer, Rs 49.16, discounts Rs 8 a year and is wrong for a reason, not a rounding: it treats the Rs 10 note as free to borrow.

    One check makes the answer believable. You could reproduce the machine with a bank loan: borrow Rs 10 at 10%, run the machine, repay Rs 11 and the Rs 2 of power, keep Rs 7. A buyer will pay up to the present value of that stream, and no more, because the bank can supply the money at 10% anyway.

    Where candidates lose it

    The usual answer is Rs 49.16: Rs 8 a year, discounted. It misses that the note fed in each year is capital with a cost. Candidates who think of each use as a project, with an outflow today and an inflow in a year, do not make this mistake.

    The second loss is giving one number with no assumption. Say when you assume the electricity is paid, give both values if asked, and the interviewer hears someone who dates every cash flow.

    What the interviewer asks next

    • What would you pay if the machine could be used once a year forever?
    • At what interest rate is the machine worth nothing?
    • If the machine could run two notes at once, what would it be worth?

    Asked at Goldman Sachs, Summer Analyst Interview, Dallas, 2026 (Wall Street Oasis): A mad scientist invents a machine that turns a standard $10 bill into a $20 bill

  2. 083A phone sells for Rs 57,000 cash today, or on a 'no-cost EMI' of Rs 10,000 a month for six months, the first instalment due a month from now. What interest rate is hidden in the EMI?Compounding and time valueHardBank creditCorporate FP&A

    Try it first

    Roughly what annual rate is hidden in the EMI?

    Show the worked solution

    About 1.49% a month, or 19.4% a year effective. The real price is the Rs 57,000 cash price, so choosing the EMI means borrowing Rs 57,000 and repaying Rs 60,000 in six monthly instalments. The rate that makes six payments of Rs 10,000 worth exactly Rs 57,000 today is 1.49% a month. The forgone Rs 3,000 discount is the interest, charged on a balance that shrinks every month.

    Where is the interest, if the EMI says no-cost?

    A friend offers to sell you his bike for Rs 57,000 today, or for Rs 60,000 paid over six months. Nobody calls the second option interest-free; it is a loan of Rs 57,000 with Rs 3,000 of interest. A shop doing the same thing is no different. When a cash discount is available, the discount you give up by paying in instalments is the interest on the loan. The 'no-cost' label means the instalments add up to the sticker price, not that money has no time value.

    So the question becomes: at what monthly rate is a stream of six Rs 10,000 payments, starting in one month, worth Rs 57,000 today? That is the internal rate of return of the loan.

    The relationship
    57,000=∑k=1610,000(1+r)k=10,000×1−(1+r)−6r  ⇒  r≈1.49%57{,}000 = \sum_{k=1}^{6} \frac{10{,}000}{(1+r)^k} = 10{,}000 \times \frac{1-(1+r)^{-6}}{r} \;\Rightarrow\; r \approx 1.49\%
    rthe monthly interest rate hidden in the instalments
    kthe month each Rs 10,000 instalment is paid
    What it says in wordsThe hidden rate is the one that discounts the six instalments back to the cash price.
    The forgone Rs 3,000 discount is the interest on a no-cost EMICash flows+Rs 57,000: the phone, today-Rs 10,000 at the end of months 1 to 6month 0123456Six EMIs, face value10,00010,00010,00010,00010,00010,000Discounted at 1.49% a month9,8549,7099,5679,4279,2899,153Cash price today57,000Rs 3,000 of interestdiscounted EMIs sum to exactly 57,000Hidden rate: 1.49% a month = 19.4% a year effective (17.8% simple annual)
    Six Rs 10,000 instalments add to Rs 60,000 at face value, but discounted at 1.49% a month they are worth exactly the Rs 57,000 cash price, so the Rs 3,000 discount given up is interest at 19.4% a year effective.

    How do you solve it without a spreadsheet?

    Guess and check. The annuity factor must be 57,000 / 10,000 = 5.70. At 1% a month, six payments are worth 5.80 times one payment; at 2% they are worth 5.60. Halfway gives about 1.5%, and 1.5% gives 5.697, a whisker under 5.70, so the answer is just below: 1.49%. Then compound, do not multiply: 1.0149 to the power 12 is 1.194, so the effective annual rate is 19.4%, against 17.8% if you simply multiply by 12.

    A second check uses the average balance. The loan starts at Rs 57,000 and is paid down to zero over six months; the balance at the start of each month averages about Rs 33,700. Rs 3,000 of interest on about Rs 33,700 over six months is 8.9%, or about 17.8% a year simple, which matches 1.49% x 12.

    What changes the answer in real life?

    Timing matters a great deal. If the first instalment is due today rather than in a month, you really borrow only Rs 47,000 and repay Rs 50,000 over five months, and the hidden rate jumps to 2.10% a month, about 28% a year. Any processing fee, or tax charged on the interest component, raises the true cost further, so always ask what you pay on day one. Check the specific charges and the current tax treatment with the lender. And if there was never a cash discount to forgo, the EMI really is close to free for the buyer, because the retailer, not the buyer, is paying the lender.

    Where candidates lose it

    The quick wrong answer is about 5%: Rs 3,000 on Rs 57,000. Some candidates double it to 10.5% for a year and feel careful. Both ignore that the loan is repaid monthly, so the average balance is only a little over half of Rs 57,000, and the rate on what is actually owed is far higher than the headline.

    The second loss is multiplying the monthly rate by 12 and calling it the annual rate. 17.8% is a nominal rate; money compounding monthly at 1.49% grows 19.4% in a year. Give both and name which is which.

    What the interviewer asks next

    • If the first instalment is paid at purchase, what is the hidden rate? (About 2.1% a month.)
    • The cash discount is only Rs 1,000. What is the hidden rate now, roughly?
    • Who pays the lender when the retailer offers no cash discount at all, and why would the retailer do that?