john123 wrote:At what time after 3:10 am, the acute angle made by the minute and the hour-hand is double to that of at 3:10 am?
At 10 minutes past the hour, the minute hand is on the 2, that is 1/6 of the way away from the 12, or 60 degrees. The hour hand, for its part, should be 1/6 of the way between the 3 and the 4. Given that there are 30 degrees for every five minutes, that will mean 5 degrees below the 3, whereas the minute hand is 30 degrees above the 3.
Therefore, the two hands are 35 degrees apart. What we need is to find the moment afterwards when they move to 70 degrees apart.
Given the constant movement of both hands, it is easier to do this by trial-and-error than to attempt to find any sort of algorithm to relate the movement of the two except to say that when the hour hand moves one degree, the minute hand moves 12.
First, let's try 3:30. At this time, the minute hand is on the 6 and the hour hand will never go below the 4 until the beginning of the next hour, so that gives us an automatic 60 degrees. At 6:30 the hour hand will be halfway between the 3 and the 4, so 15 degrees above the 4. Thus, we will be at 75 degrees apart. Too far.
What about 3:29? This moves the minute hand back 6 degrees and the second hand, 1/2 degree. So now we are at 69.5 degrees apart. The trouble is, if we move the minute hand forward by 1/2 a degree (representing 1/24 of a minute), the hour hand will gain on it by 1/24 a degree. Remember: when the minute hand moves forward by 12, the hour hand moves forward by 1, so for every 12 degrees of movement of the minute hand the hour and the minute hand grow further apart by a total of 11 degrees. So if we want our total movement to be 0.5, we take:
12/11 = x/0.5
6/11 = x
So, the minute hand needs to move 6/11 of a degree forward from 3:29 to give us that 70 degree space we want. Since there are 6 degrees per minute of movement, then it follows that the minute hand will need to move 1/11 of a minute. So, we are at 3:29:05 and 5/11 of a second.
In all honesty, though, I think the real GMAT would ask for something like 75 degrees (exactly 3:30), or else specify that the hour hand stays directly on the 3, because this is just ludicrous the way I've interpreted it...
