Multiple choice

Mr. Alaram designed a new clock, which has 12 divisions as usual, represented by the numbers 1 to 12. But the speed of the minute hand is directly proportional to the digit, which it is going to approach, i.e. if the minute hand is between 1 and 2, its speed is proportional to 2 and if it is between 11 and 12, its speed is proportional to 12. The speed of the minute hand, when it is between 12 and 1 is equal to that of an ordinary clock and both the clocks were set at 9 : 00 am. What will be the time in the ordinary clock, if the new clock shows 9 : 30 am?

  1. 9 : 12 : 15 am

  2. 9 : 20 : 15 am

  3. 9 : 24 : 15 am

  4. 9 : 30 : 15 am

  5. 9 : 15 : 15 am
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A Correct answer
Explanation

The minute hand speed is proportional to the digit it is approaching. Between 12 and 1, speed is 1x (normal). Between 1 and 2, speed is 2x, and so on. To reach 30 minutes, the hand covers the distance from 12 to 1 (5 mins at 1x speed = 5 mins) and 1 to 6 (25 mins distance). However, the speed varies: 1-2 (speed 2), 2-3 (speed 3), 3-4 (speed 4), 4-5 (speed 5), 5-6 (speed 6). The time taken is 5/1 + 5/2 + 5/3 + 5/4 + 5/5 + 5/6 = 5 + 2.5 + 1.66 + 1.25 + 1 + 0.83 = 12.25 minutes. Thus, 9:30 on the new clock corresponds to 9:12:15 on the ordinary clock.

AI explanation

The speed of the minute hand in the ordinary clock is 6 degrees per minute, meaning it takes exactly 5 minutes to cover one division. In the new clock, the speed is proportional to the digit it approaches, so for the first division (approaching 1), its speed equals the ordinary clock at 6 degrees per minute, taking 5 minutes to reach 1. For the second division (approaching 2), its speed doubles to 12 degrees per minute, covering the 30-degree division in 2.5 minutes; applying this varying rate across the divisions yields a total real time of 9:12:15 am for the new clock to display 9:30 am.

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