Clocks: Formulas, Shortcuts and Solved Examples — MCQs with Answers (Set 3)

Let the hour be H = 9. At 9 o’clock, the hour hand sits 45 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 45-space gap, it needs (45 x 60) / 55 = 49 1/11 minutes. So the hands coincide at 49 1/11 minutes past 9 o’clock. Check: at that moment both hands point to the same mark on the dial.

Let the hour be H = 7. At 7 o’clock, the hour hand sits 35 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 35-space gap, it needs (35 x 60) / 55 = 38 2/11 minutes. So the hands coincide at 38 2/11 minutes past 7 o’clock. Check: at that moment both hands point to the same mark on the dial.

Let the time be 9:10, so the minutes past the hour are M = 10. For the angle formula, 12 o’clock counts as hour 0, so use H = 9 here. The minute hand moves 6 degrees per minute. So its angle measured from 12 is 6 x 10 = 60 degrees. The hour hand moves 0.5 degrees per minute, which is 30 degrees per hour plus a small drift from the minutes. So its angle measured from 12 is 30 x 9 + 0.5 x 10 = 275 degrees. The gap between the two angles is |275 – 60| = 215 degrees. Since 215 degrees is more than 180 degrees, the actual angle between the hands is 360 – 215 = 145 degrees.

Let the time be 4:50, so the minutes past the hour are M = 50. For the angle formula, 12 o’clock counts as hour 0, so use H = 4 here. The minute hand moves 6 degrees per minute. So its angle measured from 12 is 6 x 50 = 300 degrees. The hour hand moves 0.5 degrees per minute, which is 30 degrees per hour plus a small drift from the minutes. So its angle measured from 12 is 30 x 4 + 0.5 x 50 = 145 degrees. The gap between the two angles is |145 – 300| = 155 degrees. This is at most 180 degrees, so it is already the angle between the hands: 155 degrees.

Let the time be 8:50, so the minutes past the hour are M = 50. For the angle formula, 12 o’clock counts as hour 0, so use H = 8 here. The minute hand moves 6 degrees per minute. So its angle measured from 12 is 6 x 50 = 300 degrees. The hour hand moves 0.5 degrees per minute, which is 30 degrees per hour plus a small drift from the minutes. So its angle measured from 12 is 30 x 8 + 0.5 x 50 = 265 degrees. The gap between the two angles is |265 – 300| = 35 degrees. This is at most 180 degrees, so it is already the angle between the hands: 35 degrees.

Let the time be 1:00, so the minutes past the hour are M = 0. For the angle formula, 12 o’clock counts as hour 0, so use H = 1 here. The minute hand moves 6 degrees per minute. So its angle measured from 12 is 6 x 0 = 0 degrees. The hour hand moves 0.5 degrees per minute, which is 30 degrees per hour plus a small drift from the minutes. So its angle measured from 12 is 30 x 1 + 0.5 x 0 = 30 degrees. The gap between the two angles is |30 – 0| = 30 degrees. This is at most 180 degrees, so it is already the angle between the hands: 30 degrees.

Let the time be 10:10, so the minutes past the hour are M = 10. For the angle formula, 12 o’clock counts as hour 0, so use H = 10 here. The minute hand moves 6 degrees per minute. So its angle measured from 12 is 6 x 10 = 60 degrees. The hour hand moves 0.5 degrees per minute, which is 30 degrees per hour plus a small drift from the minutes. So its angle measured from 12 is 30 x 10 + 0.5 x 10 = 305 degrees. The gap between the two angles is |305 – 60| = 245 degrees. Since 245 degrees is more than 180 degrees, the actual angle between the hands is 360 – 245 = 115 degrees.

Let the time be 7:30, so the minutes past the hour are M = 30. For the angle formula, 12 o’clock counts as hour 0, so use H = 7 here. The minute hand moves 6 degrees per minute. So its angle measured from 12 is 6 x 30 = 180 degrees. The hour hand moves 0.5 degrees per minute, which is 30 degrees per hour plus a small drift from the minutes. So its angle measured from 12 is 30 x 7 + 0.5 x 30 = 225 degrees. The gap between the two angles is |225 – 180| = 45 degrees. This is at most 180 degrees, so it is already the angle between the hands: 45 degrees.

Let the time be 8:30, so the minutes past the hour are M = 30. For the angle formula, 12 o’clock counts as hour 0, so use H = 8 here. The minute hand moves 6 degrees per minute. So its angle measured from 12 is 6 x 30 = 180 degrees. The hour hand moves 0.5 degrees per minute, which is 30 degrees per hour plus a small drift from the minutes. So its angle measured from 12 is 30 x 8 + 0.5 x 30 = 255 degrees. The gap between the two angles is |255 – 180| = 75 degrees. This is at most 180 degrees, so it is already the angle between the hands: 75 degrees.

Let the time be 6:40, so the minutes past the hour are M = 40. For the angle formula, 12 o’clock counts as hour 0, so use H = 6 here. The minute hand moves 6 degrees per minute. So its angle measured from 12 is 6 x 40 = 240 degrees. The hour hand moves 0.5 degrees per minute, which is 30 degrees per hour plus a small drift from the minutes. So its angle measured from 12 is 30 x 6 + 0.5 x 40 = 200 degrees. The gap between the two angles is |200 – 240| = 40 degrees. This is at most 180 degrees, so it is already the angle between the hands: 40 degrees.

Let the time be 3:10, so the minutes past the hour are M = 10. For the angle formula, 12 o’clock counts as hour 0, so use H = 3 here. The minute hand moves 6 degrees per minute. So its angle measured from 12 is 6 x 10 = 60 degrees. The hour hand moves 0.5 degrees per minute, which is 30 degrees per hour plus a small drift from the minutes. So its angle measured from 12 is 30 x 3 + 0.5 x 10 = 95 degrees. The gap between the two angles is |95 – 60| = 35 degrees. This is at most 180 degrees, so it is already the angle between the hands: 35 degrees.

Let the time be 9:00, so the minutes past the hour are M = 0. For the angle formula, 12 o’clock counts as hour 0, so use H = 9 here. The minute hand moves 6 degrees per minute. So its angle measured from 12 is 6 x 0 = 0 degrees. The hour hand moves 0.5 degrees per minute, which is 30 degrees per hour plus a small drift from the minutes. So its angle measured from 12 is 30 x 9 + 0.5 x 0 = 270 degrees. The gap between the two angles is |270 – 0| = 270 degrees. Since 270 degrees is more than 180 degrees, the actual angle between the hands is 360 – 270 = 90 degrees.

Let the time be 2:10, so the minutes past the hour are M = 10. For the angle formula, 12 o’clock counts as hour 0, so use H = 2 here. The minute hand moves 6 degrees per minute. So its angle measured from 12 is 6 x 10 = 60 degrees. The hour hand moves 0.5 degrees per minute, which is 30 degrees per hour plus a small drift from the minutes. So its angle measured from 12 is 30 x 2 + 0.5 x 10 = 65 degrees. The gap between the two angles is |65 – 60| = 5 degrees. This is at most 180 degrees, so it is already the angle between the hands: 5 degrees.

Let the time be 4:00, so the minutes past the hour are M = 0. For the angle formula, 12 o’clock counts as hour 0, so use H = 4 here. The minute hand moves 6 degrees per minute. So its angle measured from 12 is 6 x 0 = 0 degrees. The hour hand moves 0.5 degrees per minute, which is 30 degrees per hour plus a small drift from the minutes. So its angle measured from 12 is 30 x 4 + 0.5 x 0 = 120 degrees. The gap between the two angles is |120 – 0| = 120 degrees. This is at most 180 degrees, so it is already the angle between the hands: 120 degrees.

Let the time be 12:30, so the minutes past the hour are M = 30. For the angle formula, 12 o’clock counts as hour 0, so use H = 0 here. The minute hand moves 6 degrees per minute. So its angle measured from 12 is 6 x 30 = 180 degrees. The hour hand moves 0.5 degrees per minute, which is 30 degrees per hour plus a small drift from the minutes. So its angle measured from 12 is 30 x 0 + 0.5 x 30 = 15 degrees. The gap between the two angles is |15 – 180| = 165 degrees. This is at most 180 degrees, so it is already the angle between the hands: 165 degrees.

Let the time be 4:10, so the minutes past the hour are M = 10. For the angle formula, 12 o’clock counts as hour 0, so use H = 4 here. The minute hand moves 6 degrees per minute. So its angle measured from 12 is 6 x 10 = 60 degrees. The hour hand moves 0.5 degrees per minute, which is 30 degrees per hour plus a small drift from the minutes. So its angle measured from 12 is 30 x 4 + 0.5 x 10 = 125 degrees. The gap between the two angles is |125 – 60| = 65 degrees. This is at most 180 degrees, so it is already the angle between the hands: 65 degrees.

Let the time be 12:10, so the minutes past the hour are M = 10. For the angle formula, 12 o’clock counts as hour 0, so use H = 0 here. The minute hand moves 6 degrees per minute. So its angle measured from 12 is 6 x 10 = 60 degrees. The hour hand moves 0.5 degrees per minute, which is 30 degrees per hour plus a small drift from the minutes. So its angle measured from 12 is 30 x 0 + 0.5 x 10 = 5 degrees. The gap between the two angles is |5 – 60| = 55 degrees. This is at most 180 degrees, so it is already the angle between the hands: 55 degrees.

Let the time be 3:20, so the minutes past the hour are M = 20. For the angle formula, 12 o’clock counts as hour 0, so use H = 3 here. The minute hand moves 6 degrees per minute. So its angle measured from 12 is 6 x 20 = 120 degrees. The hour hand moves 0.5 degrees per minute, which is 30 degrees per hour plus a small drift from the minutes. So its angle measured from 12 is 30 x 3 + 0.5 x 20 = 100 degrees. The gap between the two angles is |100 – 120| = 20 degrees. This is at most 180 degrees, so it is already the angle between the hands: 20 degrees.