Investment Banking puzzles, solved step by step

About 11.1%. A price of 100 falls 10% to 90. Getting back to 100 needs a rise of 10 on a base of 90, which is 10 divided by 90, or 11.1%. The general rule: to recover a loss of L, you need L divided by (1 minus L), which grows much faster than the loss itself.

Why is the recovery bigger than the fall?

A 10% discount on a 100 rupee shirt takes it to 90. If the shop then marks it up by 10%, the tag reads 99, not 100, because the markup is taken on 90. Percentages are always measured against a base, and after a fall the base is smaller, so the same rupee gain is a bigger percentage.

What do you add to sound like a banker rather than a calculator?

Give the number, then the pattern. Large drawdowns are hard to climb out of: a fund that loses 50% has to double just to break even. That one sentence shows you see why lenders and investors care about downside, which is the real reason the question is asked.

115.6. Notice that 34% of 170 is the same as 17% of 340: halving one number and doubling the other leaves a product unchanged. So the sum is 17% of 340 twice, which is 34% of 340. That is 30% of 340, which is 102, plus 4% of 340, which is 13.6, giving 115.6.

What is the trick hiding in the numbers?

Two plots of land, one 34 metres by 17 and one 17 metres by 34, have the same area; turning a rectangle on its side does not change it. Percentages behave the same way, because x% of y is x times y divided by 100. The percentage and the base can trade places, so 34% of 170 equals 17% of 340: each is 17 x 340 / 100, which is 57.8. Once you see that, the question is one product, not two.

How do you do 34% of 340 in your head?

Split it into easy pieces: 30% of 340 is 102 and 4% of 340 is 13.6, so the total is 115.6. Or notice that 34% of 340 is 34 x 3.4, and 34 x 34 is 1,156, so the answer is 115.6. Spotting a structure first and calculating second is the habit the interviewer is checking, and it usually turns two awkward products into one easy one.

How do you check it before you say it?

Estimate first: 17% is a little more than a sixth, and a sixth of 340 is about 57, so each term is near 57 and the sum near 115. Then check each term directly: 10% of 340 is 34 and 7% is 23.8, so 17% is 57.8; 30% of 170 is 51 and 4% is 6.8, so 34% of 170 is 57.8. The two terms match, which confirms the swap.

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.

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.

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.

2,496 and 9,991. Both pairs sit the same distance either side of a round number, so use the difference of squares: (a minus b)(a plus b) is a squared minus b squared. 48 x 52 is 50 squared less 2 squared, 2,500 minus 4, 2,496. 97 x 103 is 100 squared less 3 squared, 10,000 minus 9, 9,991. Say the middle number and the gap out loud, then subtract.

Why is the product always a little less than the middle number squared?

Take a square courtyard 50 paces on each side, 2,500 square paces. Cut a strip 2 paces wide off one side and lay it along the bottom. The courtyard is now 48 by 52, but the strip you moved was only 48 long, so a 2 by 2 corner is left bare. Two numbers either side of a round one multiply to the round number squared minus the gap squared, because the strip you move never quite fills the corner. 48 x 52 is 2,500 less 4, which is 2,496; the bare corner is the 4.

How do you spot when the trick applies?

Add the two numbers and halve: if the result is round, the trick is on. 48 plus 52 is 100, half is 50, gap 2. 97 plus 103 is 200, half is 100, gap 3. Any pair that straddles a round number is a difference of squares: 73 x 67 is 70 squared less 3 squared, 4,900 minus 9, 4,891. If the pair does not straddle a round number, fall back on splitting one factor: 48 x 53 is 48 x 52 plus 48, 2,496 plus 48, 2,544. The reflex to look for the structure first is the skill the question tests.

Why does a banking interview bother with this?

Because the same reflex speeds up every number you say in a meeting. A 4% discount on a Rs 1,040 crore valuation, or a share price of Rs 97 times 103 crore shares, is a difference of squares in disguise, and getting it in two seconds without a calculator buys you credibility. Say your working out loud as you go: middle, gap, square, subtract. The interviewer is listening for the method as much as the answer, and a candidate who mumbles the right number silently earns less than one who narrates the route.

Division C grew fastest, at 25%, but Division A contributed most, adding Rs 42 crore of the Rs 111 crore increase. Growth rates are 42/420 = 10%, 39/260 = 15% and 30/120 = 25%. Rupee increases are 42, 39 and 30. A's slower rate on a much bigger base adds more. Total revenue grew 13.9%, and A supplied 37.8% of that growth.

Why are these two different questions?

A child's pocket money rising from 100 to 150 is 50% growth; a parent's salary rising from 1 lakh to 1.1 lakh is 10%. The household budget still cares far more about the salary. A growth rate measures change relative to each part's own size; contribution measures change in rupees, which is what adds up to the total. Online tests ask both in one question because candidates who rush answer the same division twice.

How do you do it fast under a timer?

Compute the increases first, because they are subtractions: 42, 39, 30. Then compute rates only where needed, using easy fractions: 42 on 420 is a tenth, 30 on 120 is a quarter, 39 on 260 sits between them at 15%. Write the two added columns next to the table rather than holding the numbers in your head, because the next question in the set often reuses them.

That weighting is the third trap a test can set. The simple average of 10%, 15% and 25% is 16.7%, which is wrong because it gives the small division the same say as the large one. Total revenue went from 800 to 911, a rise of 13.9%. If an option shows 16.7%, it is there to catch exactly that shortcut.