About Clock Puzzles
A clock puzzle asks about the positions of the two hands, and the whole topic rests on two rates: the minute hand sweeps 6 degrees a minute and the hour hand half a degree, so the angle between them changes by 5.5 degrees a minute. Because 5.5 is 11/2, the times that answer these questions come out in elevenths of a minute, which is why they read like 16 4/11 minutes past 3.
What you need to understand
- The minute hand sweeps 6 degrees a minute and the hour hand 0.5 degrees a minute.
- So the gap between the hands opens or closes at 5.5 degrees a minute, which is 11/2.
- Times therefore come out in elevenths of a minute, not whole minutes. An answer of 16 4/11 minutes past 3 is exact rather than approximate.
- The hour hand does not stand still. At half past three it is halfway between the 3 and the 4.
- At H o'clock the gap between the hands is 30H degrees, which is the starting point for most of these questions.
- The angle between the hands is the smaller of the gap and 360 minus the gap.
- Hands in opposite directions are 180 degrees apart; hands at right angles are 90 degrees apart.
How to work through these questions
- Work out the gap at the hour named, which is 30 degrees for each hour past twelve.
- Decide whether the minute hand has to close that gap or open it.
- Divide the change needed by 5.5 to get the time, or double it and divide by 11 to keep the arithmetic exact.
- Check the answer lies inside the hour the question asks about.
- If the question says for the first time, take the earlier of the two solutions.
- Convert the minutes into a mixed number to match the options.
Mistakes that cost marks
- Treating the hour hand as though it stays on the hour, which ignores its half degree a minute.
- Giving the reflex angle, 360 minus the angle, rather than the angle between the hands.
- Answering with a decimal when the exact answer is a mixed number in elevenths.
- Taking the later crossing when the question asks for the first.
- Reporting a time outside the hour, such as one falling exactly on the next hour.
- Dividing the gap by 6 instead of 5.5, which assumes the hour hand is still.
Worked example
At what time between 3 o'clock and 4 o'clock will the hands of a clock coincide?
- At 3 o'clock the hour hand is on the 3 and the minute hand on the 12, so the gap between them is 90 degrees.
- The minute hand gains 6 - 0.5 = 5.5 degrees a minute on the hour hand.
- Closing 90 degrees at 5.5 degrees a minute takes 90 / 5.5 = 180/11 = 16 4/11 minutes.
- So the hands coincide at 16 4/11 minutes past 3.
Answer: 16 4/11 minutes past 3
Practice questions with answers
A few Clock Puzzles questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
What is the angle between the hour hand and the minute hand of a clock at 4:44?
Answer: Option D — with explanation
At 4:44 the hour hand has moved 4 x 30 + 44 x 0.5 = 142 degrees from 12, and the minute hand has moved 44 x 6 = 264 degrees.
The gap is 122 degrees, and the angle between the hands is the smaller of that and its complement, which is 122 degrees.
Common mistakes
- treated the hour hand as though it stayed on the hour: 92
- added 30 degrees for the hour instead of subtracting it: 152
- gave the reflex angle rather than the angle between the hands: 238
Question 2
At what time between 8 o'clock and 9 o'clock will the hour hand and the minute hand of a clock coincide?
-
A
20 9/11 minutes past 8
-
B
43 7/11 minutes past 8
-
C
21 9/11 minutes past 8
-
D
22 9/11 minutes past 8
Answer: Option B — with explanation
At 8 o'clock the minute hand is 240 degrees behind the hour hand. The minute hand gains 6 - 0.5 = 5.5 degrees a minute, so it gains 240 degrees in 240 / 5.5 = 43 7/11 minutes.
That is 43 7/11 minutes past 8.
Common mistakes
- forgot that the hour hand moves as the minute hand closes the gap: 22 9/11 minutes past 8
- divided the gap by 6, as though the hour hand were standing still: 21 9/11 minutes past 8
- allowed for the hour hand moving too far: 20 9/11 minutes past 8
Question 3
How many degrees does the hour hand of a clock turn through in 80 minutes?
Answer: Option A — with explanation
The hour hand covers 360 degrees in 12 hours, which is 720 minutes, so it turns through 360 / 720 = 0.5 degrees a minute.
In 80 minutes it turns through 80 x 0.5 = 40 degrees.
Common mistakes
- multiplied by the number of hours in a half day: 480
- used the minute hand's 6 degrees a minute: 80
- halved the half degree a minute: 20
Question 4
How many degrees does the minute hand of a clock turn through in 2 minutes?
Answer: Option D — with explanation
The minute hand completes a full circle of 360 degrees in 60 minutes, so it turns through 6 degrees a minute.
In 2 minutes it turns through 2 x 6 = 12 degrees.
Common mistakes
- gave the angle left rather than the angle turned: 348
- halved the six degrees a minute: 6
- used the hour hand's half degree a minute: 1
Question 5
What is the angle between the hour hand and the minute hand of a clock at 7:16?
Answer: Option D — with explanation
At 7:16 the hour hand has moved 7 x 30 + 16 x 0.5 = 218 degrees from 12, and the minute hand has moved 16 x 6 = 96 degrees.
The gap is 122 degrees, and the angle between the hands is the smaller of that and its complement, which is 122 degrees.
Common mistakes
- gave the reflex angle rather than the angle between the hands: 238
- added 30 degrees for the hour instead of subtracting it: 152
- treated the hour hand as though it stayed on the hour: 92
Frequently asked questions
Why are the answers in elevenths of a minute?
Because the gap changes at 5.5 degrees a minute, which is 11/2. Dividing a whole number of degrees by 11/2 leaves a denominator of 11, and it never cancels.
Does the hour hand really move during the hour?
Yes. It moves half a degree a minute, so at half past three it stands halfway between the 3 and the 4. Ignoring that movement is the commonest mistake in the topic.
How do I find both times the hands are at right angles?
Solve for a gap of 90 degrees and again for a gap of minus 90 degrees. Both give a time inside the hour, and the earlier of the two is the first.
What is the angle between the hands at half past three?
The hour hand stands at 105 degrees and the minute hand at 180, so the gap is 75 degrees rather than 90.
Should I answer in degrees or in time?
Whichever the question asks for. Some ask for the angle at a given time and others for the time at which a given angle occurs, so read the last line carefully.
