Classic Investment Banking Brain Teasers

Classic: 'What's the angle between the hour and minute hands at 3:15?' The minute hand at :15 points at 90° (the 3). The hour hand is 15/60 = 1/4 of the way past 3, and each hour is 30°, so it's at 90° + 7.5° = 97.5°. The angle between them is 7.5°, not 0° (the common trap). Another: 'How many times a day do the hands overlap?' 22 times (not 24 — the hands overlap once roughly every 65.5 minutes, giving 11 overlaps per 12 hours). The pattern: don't trust the 'obvious' answer; work the mechanics.

Banks test fast arithmetic because you'll do it live on calls. Practice: percentages (what's 13% of 250? 10% = 25, 3% = 7.5, so 32.5), squares, and reciprocals for quick P/E ↔ earnings-yield conversions (a 20x P/E = a 5% earnings yield — directly useful for accretion/dilution intuition). Trick for ×11: 'multiply by 10 and add the number' (36 × 11 = 360 + 36 = 396). For division, estimate the bracket first, then refine. Speak the steps so the interviewer follows your method even if you misstep.

Common: 'I flip a fair coin until I get heads. What's the expected number of flips?' Answer: 2 (geometric distribution, E = 1/p = 1/0.5). 'What's the probability of rolling at least one 6 in four rolls of a die?' = 1 − (5/6)^4 ≈ 51.8% — compute the complement, a recurring trick. 'Two cards drawn from a deck, probability both are aces?' = 4/52 × 3/51 ≈ 0.45%. The meta-lesson: for 'at least one' problems, compute 1 minus the probability of none; for expectations, look for the geometric or linearity-of-expectation shortcut.

'How many golf balls fit in a 747?' or 'How many gas stations are in the US?' — these test structured estimation, not a memorized number. Build a transparent chain: state a starting population, apply ratios, and round to clean numbers. Gas stations: ~330M people / ~4 per household ≈ 80M households; assume ~1 station per ~2,500 people → ~130,000 stations (the real figure is ~115,000–150,000, so a defensible chain wins). Always state assumptions explicitly, keep the arithmetic clean, and sanity-check the final magnitude.