GMATinsight wrote:
At what time between 4 and 5 o’clock in a 12-hour clock, the min hand and hour hand of the clock will be at right angle for the second time?
A. 4:38
B. 4:38 2/11
C. 4:37
D. 4:37 2/11
E. 4:39
The first time between 4 and 5 o’clock that the minute hand and the hour hand form a right angle is around 4:05. We say “around 4:05” since it’s not exactly at 4:05. That is because the hour hand is moving slightly away from the number 4 on the clock when the minute hand is exactly pointing at the number 1 on the clock. However, we don’t need to know the exact time when the minute and hour hands form a right angle the first time since what we really need to determine is the exact time the second they form a right angle.
We can test the answer choices (keep in mind that there are 60 (minute) intervals on a circular clock, so each interval is 360/60 = 6 degrees and there are 12 numbers on the clock, so each pair of consecutive numbers is separated by 360/12 = 30 degrees).
A. 4:38
At 4:38, the minute hand sweeps 6 x 38 = 228 degrees (counting from the number 12 on the clock) while the hour hand sweeps 30 x 4 + 30 x 38/60 = 120 + 19 = 139 degrees (again, counting from the number 12 on the clock). Therefore, at 4:38, the two hands of the clock form an angle of 228 - 139 = 89 degrees.
Since 90 > 89 and the minute hand moves faster than the hour hand, the exact time when they form a right angle should be greater than 4:38, so the correct answer is either B or E. Let’s test choice E first (since it’s easier and if it’s not the correct answer, then choice B must be the correct answer).
E. 4:39
At 4:39, the minute hand sweeps 6 x 39 = 234 degrees (counting from the number 12 on the clock) while the hour hand sweeps 30 x 4 + 30 x 39/60 = 120 + 19.5 = 139.5 degrees (again, counting from the number 12 on the clock). Therefore, at 4:39, the two hands of the clock form an angle of 234 - 139.5 = 94.5 degrees.
We see that this is more than 90 degrees, so the correct time must be between 4:38 and 4:39 and it will be closer to 4:38 than it is to 4:39 since 89 degrees is closer to 90 degrees than 94.5 degrees. So choice B, or 4:38 2/11, must be the correct answer.
Answer: B _________________
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