a man who went out between 5 & 6 o'clock & returned between 6 & 7 o'clock found that .....?

Question

the man found that the hands of the watch had exchanged places . when did he go out.?..following are the options:32*4/13 minutes past 5 o'clock , 27*2/13 minutes past 5 o'clock , 37*3/13 minutes past 5 o'clock , 31*5/13 minutes past 5 o'clock .

all the time is in p.m

Answer 1

Let's say he left at m minutes past 5 o'clock. Then the minute hand was on m and the hour hand was on 25 + m/12, since the hour hand was on 25 at exactly 5 o'clock and it moves 1/12 as fast as the minute hand.
When he returned, the hands had traded places, so the minute hand was now on 25 + m/12 and the hour hand was on m. But we know that it was between 6 and seven o'clock, so he returned at 25 + m/12 minutes past 6 o'clock. Since the hour hand was on 30 at exactly 6 o'clock, and it is still moving at 1/12 the speed of the minute hand, then its position m is the same as 30 + (25 + m/12)/12, that is,
m = 30 + (25 + m/12)/12. First multiply both sides by 12. Then 12m = 360 + 25 + m/12 = 385 + m/12. Multiply by 12 again.
144 m = 4620 + m, or 143m = 4620. Then
m = 4620/143 = 32 44/143 = 32 4/13.
So he left at 32 4/13 minutes past 5.

Answer 2

31-5/13 minutes past

Answer 3

a few minutes after 30 past five.

Answer 4

sorry to inform u honey that u forgot to mention am/pm