Investment Banking puzzles, solved step by step
- Puzzles
- 100
- Traced to a firm
- 36
- Topics
- 12
- Hard
- 29
025Mental maths round: what is 17% of 340 plus 34% of 170?Bulge bracket IBMiddle market IB
Try it first
Your answer?
Show the worked solution
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.
Drawn to scale, a rectangle 340 wide and 17 tall has the same area as one 170 wide and 34 tall, so 17% of 340 and 34% of 170 are both 57.8 and together make 34% of 340, which is 115.6. The relationship17% x 340 the first term, 57.8 34% x 170 the second term, the same product with the factor of 2 moved across 34% x 340 the two equal terms combined What it says in wordsHalving the base and doubling the percentage leaves a percentage unchanged, so the two terms are equal and add to one simple product.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.
Where candidates lose it
The common loss is grinding out both products separately and dropping a decimal in one of them; 17 x 3.4 and 34 x 1.7 are easy to mangle under pressure, and the interviewer watches the hesitation.
The other loss is missing the point of the question. A mental maths round with suspiciously related numbers is inviting you to look for a shortcut; saying out loud that 34% of 170 is the same as 17% of 340 earns more credit than fast arithmetic.
What the interviewer asks next
- What is 8% of 25?
- What is 12.5% of 64 plus 25% of 32?
- What is 15% of 60 plus 30% of 30 plus 45% of 20?
045Twelve bankers meet at an offsite and every pair shakes hands exactly once. How many handshakes are there?Bulge bracket IBMiddle market IB
Try it first
Answer fast.
Show the worked solution
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.
Twelve people on a circle with one line for every pair make 66 lines: each person sits at the end of 11 lines, giving 132 line ends, and every line has two ends, so the count halves to 66. 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.
The relationship12 x 11 each person times the others they meet, counting every handshake from both sides 2 the two people in every handshake What it says in wordsPairs from a group of n are n times (n minus 1), halved.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.
Where candidates lose it
The fast wrong answer is 132, from 12 times 11 without halving. It treats one handshake between two people as two separate events.
The other slip is 144 or 78, from letting people shake their own hand. Say each person shakes hands with the other 11, and both errors disappear.
What the interviewer asks next
- How many people must be in the room for there to be 105 handshakes?
- At a dinner of six couples, everyone clinks glasses with everyone except their own partner. How many clinks?
- How many pairwise correlations does a 30-stock portfolio have?
077In your head, and out loud: what is 48 x 52, and what is 97 x 103?Bulge bracket IBMiddle market IB
Try it first
Pick the pair.
Show the worked solution
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.
A 50 by 50 square with a strip moved from its side to its base becomes 48 by 52 with a 2 by 2 corner missing, so 48 x 52 is 2,500 less 4, 2,496, and the same picture at 100 by 100 with a 3 by 3 corner missing gives 97 x 103 as 10,000 less 9, 9,991. 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.
The relationshipa the round number in the middle, 50 or 100 b the gap on each side, 2 or 3 What it says in wordsThe product of two numbers equally spaced around a middle is the middle squared minus the spacing squared.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.
Where candidates lose it
The common loss is adding the small square instead of subtracting it, giving 2,504 and 10,009. The missing corner picture settles the sign: the product is always a little less than the middle squared.
The second loss is reaching for long multiplication and taking twenty seconds for 48 x 52. Check the sum of the two numbers first; if its half is round, the whole thing is one subtraction.
What the interviewer asks next
- What is 995 x 1,005?
- What is 48 x 53, and how does the trick help even though the pair is not symmetric?
- Use the same idea to find 49 squared in your head.