IMAT Logical Reasoning · Spatial and Sequence Reasoning

Clock Hands, Elapsed Time and Days of the Week

Clock hands turn at fixed rates, and days repeat in cycles of 7, so time puzzles reduce to a rate times a time, or a remainder.

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Why this matters

A 2025 ministry question gave a time on an analogue clock and asked what it shows after the minute hand completes a number of turns. A 2013 Cambridge question asked for the time shown by a clock with an unusual face.

Concept 1 of 3: Clock hands: rotations, degrees and elapsed time

The minute hand makes one full turn every hour, so its turns are hours in disguise. The hour hand makes one full turn every 12 hours, so it moves 30° per hour. Convert the turn or the angle into minutes, then add the minutes to the starting time.

Definition

Minute hand: 1 full turn = 60 minutes; it moves $360° / 60 = 6°$ per minute.
Hour hand: 1 full turn = 12 hours; it moves $360° / 12 = 30°$ per hour, which is $0.5°$ per minute.
A quarter turn of the minute hand is 15 minutes; 0.75 of a turn is 45 minutes.
Past 12:00 the time continues on the other half of the day: on a 24-hour clock, add 12 to afternoon hours.

Converting hand movement to time

\text{minutes} = 60 \times (\text{minute-hand turns}) = 2 \times (\text{hour-hand degrees})

$6°$ minute hand, per minute
$0.5°$ hour hand, per minute

Worked example

A clock shows 10:10 in the morning. The minute hand then completes

2\tfrac{1}{3}

turns. What time is it now, and through how many degrees has the hour hand turned?

Practice this conceptself-check · 3 quick reps

Degrees of the hour hand are not minutes

The hour hand moves only 0.5° per minute, so 75° of the hour hand is 150 minutes, not 75. Always say which hand has moved before converting.

Concept 2 of 3: The angle between the hour hand and the minute hand

Measure both hands from 12 o'clock. The minute hand is at 6° per minute. The hour hand is at 30° per hour plus the extra 0.5° per minute it has crept since the hour began. The angle between them is the difference, and you take the smaller way round.

Definition

At $H$ hours and $M$ minutes, measured clockwise from 12:

Minute hand: $6M$ degrees.
Hour hand: $30H + 0.5M$ degrees (use $H$ from 0 to 11).
The angle between them is the difference; if it is over 180°, subtract it from 360°.
The hands overlap when the difference is 0. This happens 11 times in 12 hours, about every 65.5 minutes.

Angle between the hands

\theta = \lvert 30H - 5.5M \rvert \qquad (\text{if } \theta > 180°, \text{ use } 360° - \theta)

$H$ hour shown, 0 to 11
$M$ minutes past the hour

Worked example

What is the angle between the hands of a clock at 4:20?

Practice this conceptself-check · 4 quick reps

The hour hand moves between the hours

At 7:30 the hour hand is not on the 7: it has moved

30 \times 0.5° = 15°

towards the 8. Leaving it on the hour mark gives an angle that is off by

0.5M

degrees.

Concept 3 of 3: Days of the week and times across midnight

The days of the week repeat every 7 days, so only the remainder after dividing by 7 matters. In the same way the hours repeat every 24. The one real danger is counting: "day 1" is the start day itself, so day 100 is only 99 days later.

Definition

Day of the week: $N$ days after a given day, move forward by the remainder of $N \div 7$ .
Inclusive counting: if the start is day 1, then day $N$ is $N - 1$ days after it.
A normal year has 365 days $= 52 \times 7 + 1$ , so the same date falls one weekday later next year (two later after 29 February).
24-hour time: add the hours and subtract 24 if the result passes midnight; the date moves on by one day.

Weekday after N days

\text{shift} = N \bmod 7

$N \bmod 7$ the remainder when $N$ is divided by 7

Worked example

Today is a Tuesday. What day of the week will it be in 45 days?

Practice this conceptself-check · 3 quick reps

Day 1 is the starting day

If a course starts on day 1, day 100 is 99 days later, not 100. Counting both ends (or neither) moves the answer one weekday, and that neighbouring day is always an option.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Clock hands: rotations, degrees and elapsed time
Converting hand movement to time
$\text{minutes} = 60 \times (\text{minute-hand turns}) = 2 \times (\text{hour-hand degrees})$
The angle between the hour hand and the minute hand
Angle between the hands
$\theta = \lvert 30H - 5.5M \rvert \qquad (\text{if } \theta > 180°, \text{ use } 360° - \theta)$
Days of the week and times across midnight
Weekday after N days
$\text{shift} = N \bmod 7$

Watch out for (3)

Degrees of the hour hand are not minutes
→ Clock hands: rotations, degrees and elapsed time
The hour hand moves between the hours
→ The angle between the hour hand and the minute hand
Day 1 is the starting day
→ Days of the week and times across midnight

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