In 60 minutes, how many minutes does the minute hand gain on the hour hand?

55 minutes. The minute hand completes a full 360-degree circle in 60 minutes. That is 60 minute-spaces, since the dial has 60 spaces. The hour hand moves only 30 degrees in 60 minutes. That is 5 minute-spaces, since each minute-space is 6 degrees. So the minute hand gains 60 - 5 = 55 minute-spaces on the hour hand every 60 minutes.

How many minute-spaces apart are the hands of a clock when they are at right angles to each other?

15. The clock face has 60 minute-spaces, matching the full 360-degree circle. A right angle is 90 degrees, which is one quarter of 360 degrees. One quarter of 60 minute-spaces is 60 / 4 = 15 minute-spaces. So the hands are 15 minute-spaces apart at a right angle.

How many minute-spaces apart are the hands of a clock when they point in exactly opposite directions?

30. The clock face has 60 minute-spaces, matching the full 360-degree circle. Opposite directions means the hands form a straight line, which is 180 degrees. 180 degrees is half of 360 degrees, so half of 60 minute-spaces is 60 / 2 = 30 minute-spaces. So the hands are 30 minute-spaces apart when they point opposite ways.

In a period of 24 hours, how many times do the hour hand and minute hand of a clock coincide?

22 times. Between one 12 o'clock and the next, the hands coincide 11 times. This happens because the faster minute hand laps the hour hand 11 times in every 12 hours. A day of 24 hours is two such 12-hour spans, so the count would seem to be 11 x 2 = 22. It stays 22 rather than 24 because the coincidence exactly at 12 o'clock is shared between the two spans, not counted twice.

At what time between 9 and 10 o'clock will the hands of a clock be in the same straight line but not together (i.e. pointing in opposite directions)?

16 4/11 minutes past 9. 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.

At what time between 10 and 11 o'clock will the hands of a clock be in the same straight line but not together (i.e. pointing in opposite directions)?

21 9/11 minutes past 10. 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.

Clocks: Formulas, Shortcuts and Solved Examples

Clock problems ask for angles, overlaps and faulty clocks. They all depend on how fast each hand moves.

This lesson gives each rule with the reason it works. It then solves four examples step by step and lists the traps that cost marks.

The idea in brief

Minute hand: 6° a minute.
Hour hand: 0.5° a minute.
Gain: 55 minute spaces an hour.
Overlaps: 11 in 12 hours, 22 in a day.
Trap: the hour hand moves between numbers.

How a Clock Moves

What do clock questions ask?

They ask for the angle between the hands. Or the time when the hands overlap, make a right angle, or point opposite ways. Or how much a faulty clock gains or loses.

All of them use the speeds of the two hands.

How fast does each hand move?

The minute hand goes round 360° in 60 minutes. That is 6° per minute.

The hour hand goes round 360° in 12 hours. That is 30° per hour, or 0.5° per minute.

So the minute hand gains 5.5° every minute on the hour hand.

The minute hand moves 6° a minute, the hour hand 0.5° a minute. The gap grows by 5.5° a minute.

Check 1

The hour hand moves 6° every minute.

Core Rules and Why They Work

How do you find the angle at a given time?

Angle = |30H − 5.5M|, where H is the hour and M the minutes. If it is over 180°, subtract it from 360°.

Why it works: the hour hand sits at 30H + 0.5M degrees. The minute hand sits at 6M degrees. Their gap is 5.5M − 30H.

A clock at 3:40. The hour hand is at 110° and the minute hand at 240°, so the angle between them is 130°.

What are minute spaces?

A clock face has 60 minute spaces. Hands at right angles are 15 spaces apart. Opposite hands are 30 spaces apart. Hands together are 0 apart.

Why it works: 15 spaces is a quarter of 60. And 30 spaces is half of it.

How much does the minute hand gain in an hour?

It gains 55 minute spaces in 60 minutes.

Why it works: the minute hand moves 60 spaces an hour. The hour hand moves 5 spaces an hour. The difference is 55.

How often do the hands overlap?

They overlap 11 times in 12 hours, and 22 times in 24 hours.

Why it works: the hands start together at 12. The minute hand laps the hour hand once every 12/11 hours. In 12 hours that is 11 laps, not 12.

How do you find when the hands reach a given gap?

Find the gap needed at H o’clock, then divide it by 5.5 to get the minutes after H.

Why it works: the gap grows by 5.5° every minute. So gap ÷ 5.5 is the time taken.

Check 2

The hands of a clock overlap 12 times in 12 hours.

Four Solved Examples

Easy, step by step: what is the angle between the hands at 3:40?

Step 1: hour hand = 3 × 30 + 40 × 0.5 = 110°.

Step 2: minute hand = 40 × 6 = 240°.

Step 3: gap = 240 − 110 = 130°.

Medium, step by step: at what time between 4 and 5 do the hands overlap?

Step 1: at 4:00 the hour hand is 120° ahead of the minute hand.

Step 2: the minute hand gains 5.5° every minute.

Step 3: time = 120 ÷ 5.5 = 240/11 = 21 9/11 minutes.

Answer: 21 9/11 minutes past 4.

Exam level, step by step: at what time between 9 and 10 are the hands in a straight line but not together?

Step 1: at 9:00 the minute hand is 270° behind the hour hand, or 90° ahead going the other way.

Step 2: for a straight line, the gap must be 180°. The minute hand must gain 270 − 180 = 90°.

Step 3: time = 90 ÷ 5.5 = 180/11 = 16 4/11 minutes.

Answer: 16 4/11 minutes past 9.

Exam level, step by step: at what time between 10 and 11 are the hands in a straight line but not together?

Step 1: at 10:00 the hour hand is 300° ahead of the minute hand.

Step 2: the minute hand must gain 300 − 180 = 120°.

Step 3: time = 120 ÷ 5.5 = 21 9/11 minutes.

Answer: 21 9/11 minutes past 10.

Check 3

In 60 minutes the minute hand gains 55 spaces on the hour hand.

Common Traps and Shortcuts

Which traps cost marks?

Forgetting that the hour hand moves too. At 3:40 it is not on the 3.

Giving the large angle. Always give the smaller one, 180° or less.

Using 6° for both hands. The hour hand moves 0.5°.

Mixing up “together” and “straight line”. Together means a gap of 0°. Straight line means 180°.

Which shortcuts save time?

Use 30H − 5.5M. Take the positive value.

For overlaps, the time after H o’clock is 60H/11 minutes.

For a right angle or a straight line, add or subtract 15 or 30 spaces, then use the gain of 55 spaces an hour.

Draw a small clock. It stops sign errors.

What patterns come up in exams?

Four patterns repeat. The angle at a time, the overlap time, the straight-line time, and a fast or slow clock.

For a faulty clock, find how much it gains or loses in one true hour. Then scale up.

Count how many times the hands overlap in a day. The answer is 22.

Check 4

The hands of a clock are at right angles when they are 15 minute spaces apart.

0/0 understanding checks correct

Was this helpful?

Take a Quantitative Aptitude QuizRandom practice question from this subject →

Practice 68 MCQs on this topic

Set 1 Set 2 Set 3