The GATE Grind

GATE 2023 ME – Question 4

General Aptitude · Quantitative Aptitude: Arithmetic and Number Computation · 1 mark · Multiple choice

The minute-hand and second-hand of a clock cross each other _______ times between 09:15:00 AM and 09:45:00 AM on a day.

  1. 30
  2. 15
  3. 29
  4. 31

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Correct answer: (A) 30

Explanation

The second hand turns at $360°$ per minute and the minute hand at $6°$ per minute, so the second hand gains $354°$ per minute. At 9:15:00 the minute hand is at $90°$ and the second hand is at $0°$, so the second hand has to gain $90°$ for the first crossing, and a further $360°$ for each later one. In 30 minutes it gains $354 \times 30 = 10620°$. The crossings happen at gains of $90 + 360k$ with $90 + 360k \le 10620$, that is $k \le 29.25$, so $k = 0, 1, \ldots, 29$, which is 30 crossings.