Clocks: Formulas, Shortcuts and Solved Examples — MCQs with Answers (Set 1)

The hour hand starts at some position on the dial and returns to that same position after 12 hours. Returning to the same position means it has completed one full circle. One full circle is 360 degrees. So the hour hand traces 360 degrees in 12 hours.

If a watch shows a later time than the correct time, it has moved too far forward. A watch that reads ahead of the actual time is called fast. So a watch showing a time ahead of the correct time is too fast.

Between one 12 o’clock and the next, the hands coincide 11 times. This happens because the faster minute hand laps the hour hand 11 times in every 12 hours. A day of 24 hours is two such 12-hour spans, so the count would seem to be 11 x 2 = 22. It stays 22 rather than 24 because the coincidence exactly at 12 o’clock is shared between the two spans, not counted twice.

The clock face has 60 minute-spaces, matching the full 360-degree circle. Opposite directions means the hands form a straight line, which is 180 degrees. 180 degrees is half of 360 degrees, so half of 60 minute-spaces is 60 / 2 = 30 minute-spaces. So the hands are 30 minute-spaces apart when they point opposite ways.

The clock face has 60 minute-spaces, matching the full 360-degree circle. A right angle is 90 degrees, which is one quarter of 360 degrees. One quarter of 60 minute-spaces is 60 / 4 = 15 minute-spaces. So the hands are 15 minute-spaces apart at a right angle.

The minute hand completes a full 360-degree circle in 60 minutes. That is 60 minute-spaces, since the dial has 60 spaces. The hour hand moves only 30 degrees in 60 minutes. That is 5 minute-spaces, since each minute-space is 6 degrees. So the minute hand gains 60 – 5 = 55 minute-spaces on the hour hand every 60 minutes.

Let the clock start correct at 4:00 AM. It gains 9 minutes for every 24 hours of real time. After 1 day, real time elapsed is 1 x 24 x 60 = 1440 minutes. Over 1 day, the clock gains 9 x 1 = 9 minutes in total. So the clock’s own elapsed time is 1440 + 9 = 1449 minutes. Adding 1449 minutes to the start time 4:00 AM lands on 4:09 AM. Check: 1449 minutes is 24 hours and 9 minutes past the start.

Let the clock start correct at 6:00 AM. It loses 12 minutes for every 24 hours of real time. After 3 days, real time elapsed is 3 x 24 x 60 = 4320 minutes. Over 3 days, the clock loses 12 x 3 = 36 minutes in total. So the clock’s own elapsed time is 4320 – 36 = 4284 minutes. Adding 4284 minutes to the start time 6:00 AM lands on 5:24 AM. Check: 4284 minutes is 71 hours and 24 minutes past the start.

Let the clock start correct at 12:00 AM. It gains 10 minutes for every 24 hours of real time. After 1 day, real time elapsed is 1 x 24 x 60 = 1440 minutes. Over 1 day, the clock gains 10 x 1 = 10 minutes in total. So the clock’s own elapsed time is 1440 + 10 = 1450 minutes. Adding 1450 minutes to the start time 12:00 AM lands on 12:10 AM. Check: 1450 minutes is 24 hours and 10 minutes past the start.

Let the clock start correct at 7:00 AM. It loses 6 minutes for every 24 hours of real time. After 5 days, real time elapsed is 5 x 24 x 60 = 7200 minutes. Over 5 days, the clock loses 6 x 5 = 30 minutes in total. So the clock’s own elapsed time is 7200 – 30 = 7170 minutes. Adding 7170 minutes to the start time 7:00 AM lands on 6:30 AM. Check: 7170 minutes is 119 hours and 30 minutes past the start.

Let the clock start correct at 12:00 AM. It loses 3 minutes for every 24 hours of real time. After 3 days, real time elapsed is 3 x 24 x 60 = 4320 minutes. Over 3 days, the clock loses 3 x 3 = 9 minutes in total. So the clock’s own elapsed time is 4320 – 9 = 4311 minutes. Adding 4311 minutes to the start time 12:00 AM lands on 11:51 PM. Check: 4311 minutes is 71 hours and 51 minutes past the start.

Let the clock start correct at 11:00 AM. It gains 6 minutes for every 24 hours of real time. After 1 day, real time elapsed is 1 x 24 x 60 = 1440 minutes. Over 1 day, the clock gains 6 x 1 = 6 minutes in total. So the clock’s own elapsed time is 1440 + 6 = 1446 minutes. Adding 1446 minutes to the start time 11:00 AM lands on 11:06 AM. Check: 1446 minutes is 24 hours and 6 minutes past the start.

Let the clock start correct at 12:00 AM. It loses 10 minutes for every 24 hours of real time. After 1 day, real time elapsed is 1 x 24 x 60 = 1440 minutes. Over 1 day, the clock loses 10 x 1 = 10 minutes in total. So the clock’s own elapsed time is 1440 – 10 = 1430 minutes. Adding 1430 minutes to the start time 12:00 AM lands on 11:50 PM. Check: 1430 minutes is 23 hours and 50 minutes past the start.

Let the clock start correct at 9:00 AM. It gains 2 minutes for every 24 hours of real time. After 1 day, real time elapsed is 1 x 24 x 60 = 1440 minutes. Over 1 day, the clock gains 2 x 1 = 2 minutes in total. So the clock’s own elapsed time is 1440 + 2 = 1442 minutes. Adding 1442 minutes to the start time 9:00 AM lands on 9:02 AM. Check: 1442 minutes is 24 hours and 2 minutes past the start.

Let the clock start correct at 12:00 AM. It loses 5 minutes for every 24 hours of real time. After 5 days, real time elapsed is 5 x 24 x 60 = 7200 minutes. Over 5 days, the clock loses 5 x 5 = 25 minutes in total. So the clock’s own elapsed time is 7200 – 25 = 7175 minutes. Adding 7175 minutes to the start time 12:00 AM lands on 11:35 PM. Check: 7175 minutes is 119 hours and 35 minutes past the start.

Let the clock start correct at 9:00 AM. It loses 9 minutes for every 24 hours of real time. After 3 days, real time elapsed is 3 x 24 x 60 = 4320 minutes. Over 3 days, the clock loses 9 x 3 = 27 minutes in total. So the clock’s own elapsed time is 4320 – 27 = 4293 minutes. Adding 4293 minutes to the start time 9:00 AM lands on 8:33 AM. Check: 4293 minutes is 71 hours and 33 minutes past the start.

Let the clock start correct at 9:00 AM. It gains 15 minutes for every 24 hours of real time. After 2 days, real time elapsed is 2 x 24 x 60 = 2880 minutes. Over 2 days, the clock gains 15 x 2 = 30 minutes in total. So the clock’s own elapsed time is 2880 + 30 = 2910 minutes. Adding 2910 minutes to the start time 9:00 AM lands on 9:30 AM. Check: 2910 minutes is 48 hours and 30 minutes past the start.

Let the clock start correct at 9:00 AM. It loses 19 minutes for every 24 hours of real time. After 5 days, real time elapsed is 5 x 24 x 60 = 7200 minutes. Over 5 days, the clock loses 19 x 5 = 95 minutes in total. So the clock’s own elapsed time is 7200 – 95 = 7105 minutes. Adding 7105 minutes to the start time 9:00 AM lands on 7:25 AM. Check: 7105 minutes is 118 hours and 25 minutes past the start.

Let the correct time be 9:00 and the watch’s time be 9:44. The watch’s minute hand still needs 60 – 44 = 16 minutes to reach the next hour mark. A watch that has not yet caught up to the hour is running behind the correct time. So the watch is behind by 16 minutes, meaning it is 16 minutes too slow.

Let the correct time be 6:00 and the watch’s time be 6:40. The watch’s minute hand still needs 60 – 40 = 20 minutes to reach the next hour mark. A watch that has not yet caught up to the hour is running behind the correct time. So the watch is behind by 20 minutes, meaning it is 20 minutes too slow.

Let the correct time be 1:00 and the watch’s time be 1:14. The watch already shows 14 minutes past the hour, while the correct time is exactly on the hour. So the watch is ahead of the correct time by 14 – 0 = 14 minutes. A watch that reads ahead of the correct time is running fast. So the watch is 14 minutes too fast.

Let the correct time be 1:00 and the watch’s time be 1:54. The watch’s minute hand still needs 60 – 54 = 6 minutes to reach the next hour mark. A watch that has not yet caught up to the hour is running behind the correct time. So the watch is behind by 6 minutes, meaning it is 6 minutes too slow.

Let the correct time be 7:00 and the watch’s time be 7:56. The watch’s minute hand still needs 60 – 56 = 4 minutes to reach the next hour mark. A watch that has not yet caught up to the hour is running behind the correct time. So the watch is behind by 4 minutes, meaning it is 4 minutes too slow.

Let the correct time be 9:00 and the watch’s time be 9:36. The watch already shows 36 minutes past the hour, while the correct time is exactly on the hour. So the watch is ahead of the correct time by 36 – 0 = 36 minutes. A watch that reads ahead of the correct time is running fast. So the watch is 36 minutes too fast.

Let the correct time be 3:00 and the watch’s time be 3:21. The watch already shows 21 minutes past the hour, while the correct time is exactly on the hour. So the watch is ahead of the correct time by 21 – 0 = 21 minutes. A watch that reads ahead of the correct time is running fast. So the watch is 21 minutes too fast.