Clocks and Calendar Problems — MCQs with Answers (Set 2)

A mirror image time follows this rule: mirror time equals 11:60 minus the actual time. At 8:20, subtract the hours first. 11 minus 8 equals 3. Then subtract the minutes: 60 minus 20 equals 40. So the mirror image reads 3:40. This rule works because a mirror reverses the direction the hands appear to move in.

The hands form a right angle 22 times in every 12-hour block, not 24 times. The gap between the hands sweeps a full circle every 12/11 hours. A right angle occurs twice in each such sweep, giving 22 occurrences in 12 hours. A full 24-hour day contains exactly 2 such 12-hour blocks. Total right angles equal 22 times 2, which is 44.

The hands coincide 11 times in every 12-hour block, not 12 times. The minute hand only laps the hour hand 11 times in 12 hours. The last coincidence of one block lands exactly at the block’s end, which is shared with the next block’s start, so it is not counted twice. Monday noon to Wednesday noon spans exactly 4 blocks of 12 hours. Total coincidences equal 11 times 4, which is 44.

A right angle means the hands are exactly 90 degrees apart. This means 6 x M minus (30 x H plus 0.5 x M) equals 90. So 5.5 x M equals 90 plus 30 x H, and M equals (180 plus 60 x H) divided by 11. Between 1:00 and 2:00, H equals 1. So M equals 240 divided by 11, which is 21 and 9/11 minutes. The first right angle after 1:00 falls at 1:21 9/11.

A right angle means the hands are exactly 90 degrees apart. This means 6 x M minus (30 x H plus 0.5 x M) equals 90. So 5.5 x M equals 90 plus 30 x H, and M equals (180 plus 60 x H) divided by 11. Between 4:00 and 5:00, H equals 4. So M equals 420 divided by 11, which is 38 and 2/11 minutes. The first right angle after 4:00 falls at 4:38 2/11.

The hands are opposite when they are 180 degrees apart. This means 6 x M minus (30 x H plus 0.5 x M) equals 180. So 5.5 x M equals 180 plus 30 x H, and M equals (360 plus 60 x H) divided by 11. Between 2:00 and 3:00, H equals 2. So M equals 480 divided by 11, which is 43 and 7/11 minutes. The hands are opposite at exactly 2:43 7/11.

The hands coincide when the hour hand’s angle equals the minute hand’s angle. This means 30 x H plus 0.5 x M equals 6 x M. So 30 x H equals 5.5 x M, and M equals 60 x H divided by 11. Between 3:00 and 4:00, H equals 3. So M equals 180 divided by 11, which is 16 and 4/11 minutes. The hands coincide at exactly 3:16 4/11.

The clock loses 15 minutes for every 24-hour day that passes. From Monday noon to the following Monday noon is exactly 7 days. Total loss equals 15 minutes times 7, which is 105 minutes. That equals 1 hour and 45 minutes. A losing clock always falls behind the real time, so subtract this from the correct time. Noon minus 1 hour 45 minutes gives 10:15 AM. The clock shows 10:15 AM at real noon the following Monday.

The clock gains 5 minutes for every 2 real hours, which is 2.5 minutes per hour. From 9:00 AM to 3:00 PM, exactly 6 real hours pass. Total gain equals 2.5 minutes times 6, which is 15 minutes. A gaining clock always runs ahead of the real time. So add this gain to the correct time. 3:00 PM plus 15 minutes gives 3:15 PM. The faulty clock shows 3:15 PM at real 3:00 PM.

The watch loses 3 minutes for every real hour that passes. From 8:00 AM to 2:00 PM, exactly 6 real hours pass. Total loss equals 3 minutes times 6, which is 18 minutes. A losing watch always falls behind the real time. So subtract this loss from the correct time. 2:00 PM minus 18 minutes gives 1:42 PM. The faulty watch shows 1:42 PM at real 2:00 PM.

The clock gains 2 minutes for every real hour that passes. From 6:00 AM to 12:00 noon, exactly 6 real hours pass. Total gain equals 2 minutes times 6, which is 12 minutes. A gaining clock always runs ahead of the real time. So add this gain to the correct time. 12:00 noon plus 12 minutes gives 12:12 PM. The faulty clock shows 12:12 PM at real noon.

The angle between the hands equals |30 x H minus 5.5 x M| degrees. At 11:50, H equals 11 and M equals 50. So the angle equals |30 x 11 minus 5.5 x 50|, which is |330 minus 275|, or 55 degrees. This value is already under 180 degrees, so it needs no further adjustment. The angle between the hands at 11:50 is 55 degrees.

The angle between the hands equals |30 x H minus 5.5 x M| degrees. At 9:40, H equals 9 and M equals 40. So the angle equals |30 x 9 minus 5.5 x 40|, which is |270 minus 220|, or 50 degrees. This value is under 180 degrees, so no further adjustment is needed. The angle between the hands at 9:40 is 50 degrees.

The angle between the hands equals |30 x H minus 5.5 x M| degrees. At 6:20, H equals 6 and M equals 20. So the angle equals |30 x 6 minus 5.5 x 20|, which is |180 minus 110|, or 70 degrees. This value is already under 180 degrees, so it is the final answer. The angle between the hands at 6:20 is 70 degrees.

The angle between the hands equals |30 x H minus 5.5 x M| degrees. At 4:20, H equals 4 and M equals 20. So the angle equals |30 x 4 minus 5.5 x 20|, which is |120 minus 110|, or 10 degrees. The minute hand has moved 120 degrees from 12, while the hour hand has moved to 120 degrees too, since it drifts as the minutes pass. The gap between the two hands at 4:20 is 10 degrees.

The angle between the hands equals |30 x H minus 5.5 x M| degrees. At 2:30, H equals 2 and M equals 30. So the angle equals |30 x 2 minus 5.5 x 30|, which is |60 minus 165|, or 105 degrees. This value is already less than 180 degrees, so no further adjustment is needed. The angle between the hands at 2:30 is 105 degrees.

The angle between the hands equals |30 x H minus 5.5 x M| degrees. At 7:30, H equals 7 and M equals 30. So the angle equals |30 x 7 minus 5.5 x 30|, which is |210 minus 165|, or 45 degrees. A common mistake is to treat the hour hand as still sitting exactly on the 7. But the hour hand also drifts by 0.5 degrees for every minute that passes. The correct angle between the hands at 7:30 is 45 degrees.

The angle between the hands equals |30 x H minus 5.5 x M| degrees. At 9:00, H equals 9 and M equals 0. So the angle equals |30 x 9 minus 5.5 x 0|, which is |270 minus 0|, or 270 degrees. Since 270 is more than 180, subtract it from 360 instead. That gives 360 minus 270, which is 90 degrees. The angle between the hands at 9:00 is 90 degrees.

The angle between the hands equals |30 x H minus 5.5 x M| degrees, where H is the hour and M is the minutes. At 4:00, H equals 4 and M equals 0. So the angle equals |30 x 4 minus 5.5 x 0|, which is |120 minus 0|. That works out to 120 degrees. Since 120 is less than 180, this is already the smaller angle between the hands. The angle between the hands at 4:00 is 120 degrees.