A watch showed a time of fourteen minutes past nine (9 hrs and 14 mins). The positions of the hour-hand and the minute-hand of the watch are exactly interchanged. The new time shown by the watch is closest to which one of the following?

Use standard clock-angle relations: the hour hand moves 30° per hour (0.5° per minute) and the minute hand 6° per minute. At 9:14 the hour-hand angle = 30·9 + 0.5·14 = 277° and the minute-hand angle = 6·14 = 84°. After exact interchange the new minute-angle = 277°, so the minutes = 277/6 = 46.1667 minutes (46 minutes 10 seconds). The new hour-angle must be 84°, which corresponds to roughly 2 hours plus those minutes, so the new time ≈ 2:46:10 — i.e. about 14 minutes to 3. Hence option (3) Fourteen minutes to three is closest.