Financial Analysis puzzles, solved step by step

About 5,100 people, as a range of roughly 3,400 to 5,100. Take a floor plate of 25,000 square feet. After lifts, stairs and services about 80% is usable, 20,000 square feet. At 100 square feet a seat that is 200 seats, and at 85% occupancy about 170 people a floor. Thirty floors gives about 5,100. Roomier offices at 150 square feet a seat bring it down to about 3,400.

Why start from one floor?

Think of estimating the people in a cinema: you count the seats in one row, then the rows. A tower is the same: thirty copies of one floor, so estimate one floor carefully and multiply, because every assumption is easier to defend on a single floor you can picture. Define the answer too: people who work in the building, meaning assigned staff, not visitors and not the count present on any one day.

What goes into one floor?

Four assumptions, each said out loud. A floor plate of about 25,000 square feet, typical for a large commercial tower. About 20% of that is the coreThe central block of a floor holding lifts, stairs, toilets, shafts and service rooms, which cannot be used for desks., leaving 20,000 usable. At 100 square feet a seat, including aisles and meeting rooms, the floor holds 200 seats, and with 85% of seats assigned that is about 170 people. The rest are vacant desks, new hires not yet in, or space held for growth.

Which assumption would you flex, and how would you check it?

Seat density moves the answer most. A dense back office runs near 100 square feet a seat; a bank's front office with larger desks and more meeting rooms nearer 150. At 150 square feet a seat the tower holds about 3,400 people, so give the answer as about 3,400 to 5,100. Then trim for floors that hold no desks: if the lobby and a plant floor take two of the thirty, the dense case falls to about 4,760. A check from another side: assume a dozen lifts, each carrying about 15 people a trip on a round trip of about 3 minutes. That moves about 12 x 15 x 20, or 3,600 people an hour, which fits a building of a few thousand people arriving over a morning, not tens of thousands.

Fill the 3-litre bottle, pour it into the 4, fill the 3 again and pour until the 4 is full: 2 litres are left in the 3-litre bottle. The 4 already holds 3 litres, so it takes only 1 more. That is four moves. A longer route fills the 4 first and ends with 2 litres in the 4-litre bottle after six moves.

How do you avoid getting lost halfway through?

Think of following a recipe that says pour half the milk into the other pan: if you do not write down how much is in each pan, two steps later you have no idea. Track the pair of levels after every single move, written as (3-litre, 4-litre), and the puzzle becomes a short walk through a few states instead of a juggling act. There are only three kinds of move: fill a bottle from the tap, empty a bottle, or pour one into the other until the source is empty or the target is full.

Say the states out loud as you go: (0, 0), (3, 0), (0, 3), (3, 3), (2, 4). The interviewer can check each one in a second, and if you slip, you can see where.

Which amounts can these two bottles measure at all?

Every pour moves water in steps of 3 and 4, so the amounts you can isolate are the combinations of 3 and 4 with whole numbers in front: 4 minus 3 is 1, 3 times 2 minus 4 is 2, and so on. Because 3 and 4 share no common factor bigger than 1, these bottles can measure every whole number of litres from 1 to 4, and up to 7 if you count both bottles together. With a 4-litre and a 6-litre bottle you could never get an odd number, since every amount would be a multiple of 2. Saying that rule turns a party trick into reasoning.

The 12.5% annual loan is cheaper. Twelve per cent compounded monthly is an effective 12.68% a year, against 12.50%. A 12% nominal rate charged monthly means 1% a month, and interest earns interest eleven more times within the year: 1.01 to the power 12, less 1, is 12.68%. Convert every quote to an effective annual rate before comparing.

Why is 12% a year compounded monthly more than 12%?

Leave a credit card balance unpaid and the interest charged in January is itself charged interest in February. The bank's 'monthly rate' grows faster than the same rate charged once a year. A nominal rate tells you how interest is quoted; the effective annual rate tells you what it costs, and only effective rates can be compared. Here 12% a year becomes 1% a month, and after twelve months of compounding Rs 100 has grown to Rs 112.68.

What would the monthly lender need to quote to match?

Run the conversion backwards: the monthly rate that compounds to 12.5% is 1.125 to the power 1/12, less 1, about 0.98% a month, or a nominal 11.84% a year. Any monthly quote above 11.84% is dearer than 12.5% annual. For a quick mental check, the extra from monthly compounding at these rates is roughly half the rate squared: 0.5 x 0.12 x 0.12 is 0.72 points, close to the true 0.68.

What else decides which loan is really cheaper?

The effective rate prices the money, not the whole deal. Processing fees, prepayment penalties and insurance bundled into the loan all add to the cost and must be folded in. Watch for flat-rate quotes too: a 7% flat rate on a three-year loan charges interest on the original amount even as it is repaid, which works out to an effective rate of about 13.6% a year. Confirm how any lender computes its rate before comparing; the limit of the EAR is that it compares like with like only once every cost is in it.

Net income rises Rs 7.5 crore, cash falls Rs 2.5 crore, and net fixed assets are Rs 10 crore higher. Income statement: depreciation down 10, pre-tax profit up 10, tax up 2.5, net income up 7.5. Cash flow: net income up 7.5 but the add-back down 10, so cash from operations down 2.5. Balance sheet: cash down 2.5, fixed assets up 10, assets up 7.5, matched by retained earnings up 7.5.

Why does profit rise while cash falls?

A shopkeeper who decides his delivery van will last ten years instead of five writes off half as much each year. The van is the same van, the fuel bill is the same, and no customer paid him more. Changing a useful life changes when the cost of an asset is recognised, not whether it is paid, so the extra profit is an accounting rearrangement with one real consequence: a higher tax bill. Depreciation falls 10, pre-tax profit rises 10, tax at 25% rises 2.5, and net income rises 7.5.

On the cash flow statement, start from net income, up 7.5, and add back depreciation, which is now 10 lower than before. Cash from operations is 7.5 - 10 = -2.5. Nothing changes in investing or financing, so cash falls 2.5. On the balance sheet, cash is down 2.5 and net fixed assets are up 10, because less accumulated depreciation has been charged against them: assets up 7.5. Retained earnings carry the 7.5 of extra net income, so liabilities and equity are also up 7.5.

Why is the higher profit not good news?

Because nothing about the business improved. The same machines wear out on the same schedule; the company will replace them on the same date for the same money; and it has handed Rs 2.5 crore to the tax authority earlier than it needed to. A profit increase that comes with a cash decrease and no operating change is cosmetic, and the choice to make it is itself a signal: management may be reaching for a target. It also breaks comparability with peers who kept the shorter lives, and the total charge over the asset's life is unchanged, so the profit that appears now is borrowed from later years.

What if tax depreciation did not follow book?

In practice the tax authority sets its own depreciation rates, and in India book depreciation under the Companies Act and tax depreciation under the Income Tax Act are computed separately; confirm the current rules before relying on this. Then cash tax does not change: net income still rises 7.5, but a deferred tax liabilityTax that the accounts recognise as owed on profit already reported, but that the tax return has not yet charged, so it will be paid in a later period. of 2.5 is booked instead of extra cash tax. On the cash flow statement, net income up 7.5, add-back down 10, deferred tax up 2.5: cash from operations is 0.0. The limit: in that version the change is purely cosmetic, cash untouched, which is why analysts read the depreciation policy note before they read the profit line.

112, and Rs 560 crore. For 17.5%, build it from blocks: 10% of 640 is 64, 5% is half of that, 32, and 2.5% is half again, 16; 64 + 32 + 16 = 112. For 12.5%, spot the fraction: it is one eighth, so halve 4,480 three times, to 2,240, 1,120 and 560. Check each with a second route before you say it.

How do you take an awkward percentage of anything in your head?

When a restaurant bill is Rs 640 and you want to tip 17.5%, nobody multiplies by 0.175. You find 10%, which is just moving the decimal point, and then cut pieces off it. Any percentage can be built from 10%, 5% (half of 10%), 1% and their halves, and each block is one easy operation on the last. For 17.5% of 640: 10% is 64, 5% is 32, 2.5% is 16, and the three add to 112.

Check with a second route: 20% is a fifth, 128, and 17.5% is 2.5 points less, so 128 minus 16 is 112. Two routes agreeing is what lets you say a number aloud with confidence in an interview where speed is being watched.

Which percentages are fractions in disguise?

Some percentages are clean fractions, and dividing is faster than building. 12.5% is one eighth, so 12.5% of Rs 4,480 crore is three halvings: 2,240, 1,120, 560. The same idea gives 37.5% as three eighths (560 x 3 = 1,680) and 87.5% as the whole less one eighth (4,480 minus 560 = 3,920). Desk interviewers like these because they turn up constantly in fee splits, stake sizes and tax at round rates.

What makes a mental answer trustworthy under time pressure?

Two habits. First, estimate before computing: 17.5% is a little under a fifth, so the answer must be a little under 128. If your arithmetic produces 1,120 or 11.2, the estimate catches the slipped decimal at once. Second, keep the units: the second answer is Rs 560 crore, not 560. An interviewer marks a right number with units and a sanity check far above a right number mumbled after a long silence. Speak the route as you go, so a slip is visible and recoverable.

134,547. Split 301 into 300 and 1. Three hundred times 447 is 447 x 3 with two zeros: 1,341, then 134,100. Add one more 447: 134,547. Check before you say it: 300 x 450 is 135,000, so the answer should sit a little below that; the last digit must be 1 x 7, a 7; and the digit sums agree, since 301 reduces to 4, 447 to 6, 4 x 6 = 24 reduces to 6, and 1+3+4+5+4+7 = 24 also reduces to 6.

Why split 301 rather than attack both numbers?

If a shop sells 301 items at Rs 447 each, nobody in the shop multiplies 301 by 447 digit by digit. They price 300 items and add one more. Break the awkward factor into a round number and a small remainder, multiply the round part first, and the remainder is one easy addition. 447 x 3 is 1,200 plus 141, which is 1,341; two zeros make it 134,100; plus 447 is 134,547. Say each intermediate number aloud as you reach it, so the interviewer can follow and a slip is caught before it compounds.

The split can go the other way too. 447 is 450 less 3, so 301 x 447 = 301 x 450 minus 301 x 3. The first is 300 x 450 plus 450, 135,450; the second is 903; the difference is 134,547. Having two routes is what turns a hopeful answer into a checked one.

How do you check an answer you cannot write down?

Three checks, each a few seconds. The estimate: 300 x 450 is 135,000, and you used one factor slightly above and one slightly below, so the answer should be close to it, which 134,547 is. The last digit: 1 x 7 ends in 7, and so does the answer. The digit sums, an old bookkeeper's test: reduce each number to a single digit by adding its digits, 301 gives 4 and 447 gives 15 then 6; the product 24 reduces to 6, and the answer's digits 1+3+4+5+4+7 = 24 also reduce to 6. A wrong answer rarely survives all three checks, and running them aloud shows the interviewer a habit, not a lucky number. The digit-sum test has a blind spot: it misses errors that are exact multiples of 9, including two digits swapped, which is why the estimate sits beside it.

What is the interviewer marking?

Pace, an audible route and a check, in that order. A right answer after a long silence scores below a right answer reached in three spoken steps, because the desk wants to hear how you handle a number, not whether you can. The limitation is that this trick suits factors near a round number. For something like 347 x 683, say that you would estimate, 350 x 700 less a bit, give about 237,000, and label it an estimate; pretending to exact arithmetic you cannot check is the real failure.