Clocks: Formulas, Shortcuts and Solved Examples — MCQs with Answers (Set 2)

Let the correct time be 3:00 and the watch’s time be 3:16. The watch already shows 16 minutes past the hour, while the correct time is exactly on the hour. So the watch is ahead of the correct time by 16 – 0 = 16 minutes. A watch that reads ahead of the correct time is running fast. So the watch is 16 minutes too fast.

Let the correct time be 4:00 and the watch’s time be 4:08. The watch already shows 8 minutes past the hour, while the correct time is exactly on the hour. So the watch is ahead of the correct time by 8 – 0 = 8 minutes. A watch that reads ahead of the correct time is running fast. So the watch is 8 minutes too fast.

Let the correct time be 9:00 and the watch’s time be 9:39. The watch’s minute hand still needs 60 – 39 = 21 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 21 minutes, meaning it is 21 minutes too slow.

Let the hour be H = 10. At 10 o’clock, the hour hand is 50 minute-spaces ahead of the minute hand. Opposite directions (180 degrees apart) means the hands are 30 minute-spaces apart, since 60 spaces span the full 360-degree dial. The minute hand must gain enough on the hour hand to shrink the gap from 50 spaces down to 30 spaces. So it must gain 50 – 30 = 20 minute-spaces. The minute hand gains 55 minute-spaces on the hour hand every 60 minutes. So the time needed is (20 x 60) / 55 = 21 9/11 minutes. So the hands point in opposite directions at 21 9/11 minutes past 10 o’clock. Check: at that moment the hands are 30 minute-spaces (180 degrees) apart, on opposite sides of the dial.

Let the hour be H = 9. At 9 o’clock, the hour hand is 45 minute-spaces ahead of the minute hand. Opposite directions (180 degrees apart) means the hands are 30 minute-spaces apart, since 60 spaces span the full 360-degree dial. The minute hand must gain enough on the hour hand to shrink the gap from 45 spaces down to 30 spaces. So it must gain 45 – 30 = 15 minute-spaces. The minute hand gains 55 minute-spaces on the hour hand every 60 minutes. So the time needed is (15 x 60) / 55 = 16 4/11 minutes. So the hands point in opposite directions at 16 4/11 minutes past 9 o’clock. Check: at that moment the hands are 30 minute-spaces (180 degrees) apart, on opposite sides of the dial.

Let the hour be H = 7. At 7 o’clock, the hour hand is 35 minute-spaces ahead of the minute hand. Opposite directions (180 degrees apart) means the hands are 30 minute-spaces apart, since 60 spaces span the full 360-degree dial. The minute hand must gain enough on the hour hand to shrink the gap from 35 spaces down to 30 spaces. So it must gain 35 – 30 = 5 minute-spaces. The minute hand gains 55 minute-spaces on the hour hand every 60 minutes. So the time needed is (5 x 60) / 55 = 5 5/11 minutes. So the hands point in opposite directions at 5 5/11 minutes past 7 o’clock. Check: at that moment the hands are 30 minute-spaces (180 degrees) apart, on opposite sides of the dial.

Let the hour be H = 8. At 8 o’clock, the hour hand is 40 minute-spaces ahead of the minute hand. Opposite directions (180 degrees apart) means the hands are 30 minute-spaces apart, since 60 spaces span the full 360-degree dial. The minute hand must gain enough on the hour hand to shrink the gap from 40 spaces down to 30 spaces. So it must gain 40 – 30 = 10 minute-spaces. The minute hand gains 55 minute-spaces on the hour hand every 60 minutes. So the time needed is (10 x 60) / 55 = 10 10/11 minutes. So the hands point in opposite directions at 10 10/11 minutes past 8 o’clock. Check: at that moment the hands are 30 minute-spaces (180 degrees) apart, on opposite sides of the dial.

Let the hour be H = 5. At 5 o’clock, the hour hand is 25 minute-spaces ahead of the minute hand. A right angle (90 degrees) means the hands are 15 minute-spaces apart, since 60 spaces span the full 360-degree dial. The minute hand should end up 15 spaces behind the hour hand. So it only needs to close the 25-space gap down to 15 spaces. So the minute hand must gain 25 – 15 = 10 minute-spaces on the hour hand. The minute hand gains 55 minute-spaces on the hour hand every 60 minutes. So the time needed is (10 x 60) / 55 = 10 10/11 minutes. So the hands are at right angles, with the minute hand behind the hour hand, at 10 10/11 minutes past 5 o’clock.

Let the hour be H = 6. At 6 o’clock, the hour hand is 30 minute-spaces ahead of the minute hand. A right angle (90 degrees) means the hands are 15 minute-spaces apart, since 60 spaces span the full 360-degree dial. The minute hand should end up 15 spaces behind the hour hand. So it only needs to close the 30-space gap down to 15 spaces. So the minute hand must gain 30 – 15 = 15 minute-spaces on the hour hand. The minute hand gains 55 minute-spaces on the hour hand every 60 minutes. So the time needed is (15 x 60) / 55 = 16 4/11 minutes. So the hands are at right angles, with the minute hand behind the hour hand, at 16 4/11 minutes past 6 o’clock.

Let the hour be H = 4. At 4 o’clock, the hour hand is 20 minute-spaces ahead of the minute hand. A right angle (90 degrees) means the hands are 15 minute-spaces apart, since 60 spaces span the full 360-degree dial. The minute hand should end up 15 spaces ahead of the hour hand. So it must close the 20-space gap completely, then gain 15 more. So the minute hand must gain 20 + 15 = 35 minute-spaces on the hour hand. The minute hand gains 55 minute-spaces on the hour hand every 60 minutes. So the time needed is (35 x 60) / 55 = 38 2/11 minutes. So the hands are at right angles, with the minute hand ahead of the hour hand, at 38 2/11 minutes past 4 o’clock.

Let the hour be H = 4. At 4 o’clock, the hour hand is 20 minute-spaces ahead of the minute hand. A right angle (90 degrees) means the hands are 15 minute-spaces apart, since 60 spaces span the full 360-degree dial. The minute hand should end up 15 spaces behind the hour hand. So it only needs to close the 20-space gap down to 15 spaces. So the minute hand must gain 20 – 15 = 5 minute-spaces on the hour hand. The minute hand gains 55 minute-spaces on the hour hand every 60 minutes. So the time needed is (5 x 60) / 55 = 5 5/11 minutes. So the hands are at right angles, with the minute hand behind the hour hand, at 5 5/11 minutes past 4 o’clock.

Let the hour be H = 7. At 7 o’clock, the hour hand is 35 minute-spaces ahead of the minute hand. A right angle (90 degrees) means the hands are 15 minute-spaces apart, since 60 spaces span the full 360-degree dial. The minute hand should end up 15 spaces ahead of the hour hand. So it must close the 35-space gap completely, then gain 15 more. So the minute hand must gain 35 + 15 = 50 minute-spaces on the hour hand. The minute hand gains 55 minute-spaces on the hour hand every 60 minutes. So the time needed is (50 x 60) / 55 = 54 6/11 minutes. So the hands are at right angles, with the minute hand ahead of the hour hand, at 54 6/11 minutes past 7 o’clock.

Let the hour be H = 3. At 3 o’clock, the hour hand is 15 minute-spaces ahead of the minute hand. A right angle (90 degrees) means the hands are 15 minute-spaces apart, since 60 spaces span the full 360-degree dial. The minute hand should end up 15 spaces ahead of the hour hand. So it must close the 15-space gap completely, then gain 15 more. So the minute hand must gain 15 + 15 = 30 minute-spaces on the hour hand. The minute hand gains 55 minute-spaces on the hour hand every 60 minutes. So the time needed is (30 x 60) / 55 = 32 8/11 minutes. So the hands are at right angles, with the minute hand ahead of the hour hand, at 32 8/11 minutes past 3 o’clock.

Let the hour be H = 6. At 6 o’clock, the hour hand is 30 minute-spaces ahead of the minute hand. A right angle (90 degrees) means the hands are 15 minute-spaces apart, since 60 spaces span the full 360-degree dial. The minute hand should end up 15 spaces ahead of the hour hand. So it must close the 30-space gap completely, then gain 15 more. So the minute hand must gain 30 + 15 = 45 minute-spaces on the hour hand. The minute hand gains 55 minute-spaces on the hour hand every 60 minutes. So the time needed is (45 x 60) / 55 = 49 1/11 minutes. So the hands are at right angles, with the minute hand ahead of the hour hand, at 49 1/11 minutes past 6 o’clock.

Let the hour be H = 5. At 5 o’clock, the hour hand is 25 minute-spaces ahead of the minute hand. A right angle (90 degrees) means the hands are 15 minute-spaces apart, since 60 spaces span the full 360-degree dial. The minute hand should end up 15 spaces ahead of the hour hand. So it must close the 25-space gap completely, then gain 15 more. So the minute hand must gain 25 + 15 = 40 minute-spaces on the hour hand. The minute hand gains 55 minute-spaces on the hour hand every 60 minutes. So the time needed is (40 x 60) / 55 = 43 7/11 minutes. So the hands are at right angles, with the minute hand ahead of the hour hand, at 43 7/11 minutes past 5 o’clock.

Let the hour be H = 7. At 7 o’clock, the hour hand is 35 minute-spaces ahead of the minute hand. A right angle (90 degrees) means the hands are 15 minute-spaces apart, since 60 spaces span the full 360-degree dial. The minute hand should end up 15 spaces behind the hour hand. So it only needs to close the 35-space gap down to 15 spaces. So the minute hand must gain 35 – 15 = 20 minute-spaces on the hour hand. The minute hand gains 55 minute-spaces on the hour hand every 60 minutes. So the time needed is (20 x 60) / 55 = 21 9/11 minutes. So the hands are at right angles, with the minute hand behind the hour hand, at 21 9/11 minutes past 7 o’clock.

Let the hour be H = 3. At 3 o’clock, the hour hand is 15 minute-spaces ahead of the minute hand. A right angle (90 degrees) means the hands are 15 minute-spaces apart, since 60 spaces span the full 360-degree dial. The minute hand should end up 15 spaces behind the hour hand. So it only needs to close the 15-space gap down to 15 spaces. So the minute hand must gain 15 – 15 = 0 minute-spaces on the hour hand. The minute hand gains 55 minute-spaces on the hour hand every 60 minutes. So the time needed is (0 x 60) / 55 = 0 minutes. So the hands are at right angles, with the minute hand behind the hour hand, at 0 minutes past 3 o’clock.

Let the hour be H = 4. At 4 o’clock, the hour hand sits 20 minute-spaces ahead of the minute hand. The minute hand moves 60 minute-spaces every 60 minutes. The hour hand moves only 5 minute-spaces every 60 minutes. So the minute hand gains 60 – 5 = 55 minute-spaces on the hour hand every 60 minutes. To close the 20-space gap, it needs (20 x 60) / 55 = 21 9/11 minutes. So the hands coincide at 21 9/11 minutes past 4 o’clock. Check: at that moment both hands point to the same mark on the dial.

Let the hour be H = 1. At 1 o’clock, the hour hand sits 5 minute-spaces ahead of the minute hand. The minute hand moves 60 minute-spaces every 60 minutes. The hour hand moves only 5 minute-spaces every 60 minutes. So the minute hand gains 60 – 5 = 55 minute-spaces on the hour hand every 60 minutes. To close the 5-space gap, it needs (5 x 60) / 55 = 5 5/11 minutes. So the hands coincide at 5 5/11 minutes past 1 o’clock. Check: at that moment both hands point to the same mark on the dial.

Let the hour be H = 5. At 5 o’clock, the hour hand sits 25 minute-spaces ahead of the minute hand. The minute hand moves 60 minute-spaces every 60 minutes. The hour hand moves only 5 minute-spaces every 60 minutes. So the minute hand gains 60 – 5 = 55 minute-spaces on the hour hand every 60 minutes. To close the 25-space gap, it needs (25 x 60) / 55 = 27 3/11 minutes. So the hands coincide at 27 3/11 minutes past 5 o’clock. Check: at that moment both hands point to the same mark on the dial.

Let the hour be H = 2. At 2 o’clock, the hour hand sits 10 minute-spaces ahead of the minute hand. The minute hand moves 60 minute-spaces every 60 minutes. The hour hand moves only 5 minute-spaces every 60 minutes. So the minute hand gains 60 – 5 = 55 minute-spaces on the hour hand every 60 minutes. To close the 10-space gap, it needs (10 x 60) / 55 = 10 10/11 minutes. So the hands coincide at 10 10/11 minutes past 2 o’clock. Check: at that moment both hands point to the same mark on the dial.

Let the hour be H = 6. At 6 o’clock, the hour hand sits 30 minute-spaces ahead of the minute hand. The minute hand moves 60 minute-spaces every 60 minutes. The hour hand moves only 5 minute-spaces every 60 minutes. So the minute hand gains 60 – 5 = 55 minute-spaces on the hour hand every 60 minutes. To close the 30-space gap, it needs (30 x 60) / 55 = 32 8/11 minutes. So the hands coincide at 32 8/11 minutes past 6 o’clock. Check: at that moment both hands point to the same mark on the dial.

Let the hour be H = 10. At 10 o’clock, the hour hand sits 50 minute-spaces ahead of the minute hand. The minute hand moves 60 minute-spaces every 60 minutes. The hour hand moves only 5 minute-spaces every 60 minutes. So the minute hand gains 60 – 5 = 55 minute-spaces on the hour hand every 60 minutes. To close the 50-space gap, it needs (50 x 60) / 55 = 54 6/11 minutes. So the hands coincide at 54 6/11 minutes past 10 o’clock. Check: at that moment both hands point to the same mark on the dial.

Let the hour be H = 8. At 8 o’clock, the hour hand sits 40 minute-spaces ahead of the minute hand. The minute hand moves 60 minute-spaces every 60 minutes. The hour hand moves only 5 minute-spaces every 60 minutes. So the minute hand gains 60 – 5 = 55 minute-spaces on the hour hand every 60 minutes. To close the 40-space gap, it needs (40 x 60) / 55 = 43 7/11 minutes. So the hands coincide at 43 7/11 minutes past 8 o’clock. Check: at that moment both hands point to the same mark on the dial.

Let the hour be H = 3. At 3 o’clock, the hour hand sits 15 minute-spaces ahead of the minute hand. The minute hand moves 60 minute-spaces every 60 minutes. The hour hand moves only 5 minute-spaces every 60 minutes. So the minute hand gains 60 – 5 = 55 minute-spaces on the hour hand every 60 minutes. To close the 15-space gap, it needs (15 x 60) / 55 = 16 4/11 minutes. So the hands coincide at 16 4/11 minutes past 3 o’clock. Check: at that moment both hands point to the same mark on the dial.