- Minute Spaces: The face or dial of the clock is a circle whose circumference is divided into 60 equal parts, named minute spaces.
- The hour hand and minute hand
A clock has two hands. The smaller hand is called the hour hand or shorthand and the larger one is called the minute hand or longhand. - In 60 minutes, the minute hand gains 55-minute spaces over the hour hand.
(In 60 minutes, the hour hand will move 5-minute spaces while the minute hand will move 60-minute spaces. In effect the space gain of minute hand with respect to hour hand will be 60 - 5 = 55 minutes.) - Both the hands of a clock coincide once every hour.
Example: Between 11 and 1'o clock, hands are together as shown in the adjacent figure.
- The hands of a clock are in the same straight line when they are coincident or opposite to each other.
- If two hands are in opposite direction. (180° apart), then they are 30 min spaces apart. This happens once in 1 h. In a period of 12 h both hands are in opposite direction 11 times and in a day both hands are in opposite direction 22 times.

- When the two hands of a clock are at right angles, they are 15-minute spaces apart.
This happens twice in 1 h. In a period of 12 h, the hands are at a right angle 22 times (2 common positions) and in a day both hands are at a right angle 44 times.

- When the hands of a clock are in opposite directions, they are 30-minute spaces apart.
- Angle traced by hour hand in 12 hrs = 360°
- Angle traced by minute hand in 60 min. = 360°.
Try yourself:An accurate clock shows 8 o'clock in the morning. Through how many degrees will the hour hand rotate when the clock shows 2 o'clock in the afternoon?
Explanation
Angle traced by the hour hand in 6 hours =

- If a watch or a clock indicates 9.15, when the correct time is 9, it is said to be 15 minutes too fast.
- If a watch or a clock indicates 8.45, when the correct time is 9, it is said to be 15 minutes too slow.
- The hands of a clock will be in a straight line but opposite in direction, 22 times in a day.
- The hands of a clock coincide 22 times in a day.
- The hands of a clock are straight 44 times in a day.
- The hands of a clock are at right angles 44 times a day.
- The two hands of a clock will be together between H and (H+1) o'clock at
(60 H/ 11) minutes past H o'clock. - The two hands of a clock will be in the same straight line but not together between H and (H+1) o'clock at

- The angle between hands of a clock
(i) When the minute hand is behind the hour hand, the angle between the two hands at M minutes past H 'o clock 
(ii) When the minute hand is ahead of the hour hand, the angle between the two hands at M minutes past H 'o clock
Try yourself:A clock starts at noon. By 10 minutes past 5, the hour hand has turned through:
Explanation
Angle traced by hour hand in 12 hrs = 360°.
Angle traced by hour hand in 5 hrs 10 min. i.e.

- The two hands of the clock will be at right angles between H and (H+1) o'clock at

If the minute hand of a clock overtakes the hour hand at intervals of M minutes of the correct time, the clock gains or loses in a day by 
Try yourself:An accurate clock shows 7 a.m. Through how many degrees will the hour hand rotate when the clock shows 1 p.m.?
Explanation
We know that angle traced by hour hand in 12 hrs. = 360°
From 7 to 1, there are 6 hours.
Angle traced by the hour hand in 6 hours = 6*(360/12) = 180°
Option B is the correct answer.
- Between H and (H+1) o'clock, the two hands of a clock are M minutes apart at

Solved Problems
Example 1: At what time between 4 and 5, the minute hand will be 2 minutes spaces ahead of the hour hand?
At 4 O'clock, the two hands are 20 min spaces apart.
In this case, the min hand will have to gain (20 + 2) i.e. 22 – minute spaces.
So, 22 – minute spaces will be gained in (60/55) × 22 = 24 min.
Example 2: The minute hand of a clock overtakes the hour hand at intervals of 65 minutes of correct time. How much does the clock lose or gain in 12 hours?
In a correct clock, the minute hand and hour hand should meet after every 65 5/11 min.
But we know that they are meeting after every 65 minutes.
So, the gain in 65 minutes is 5/11 minutes.
Gain in 12 hours = (120 × 60/65) × (5/11) = 720/143 = 5 5/143 min.
Try yourself:At what time between 5.30 and 6 will the hands of a clock be at right angles?
Explanation
Given: H = 5 and A = 90, since 5 and 6 lies in the first half, a positive sign is considered.
T = 2/11 [H*30±A]
T = 2/11 [5*30+90]
T = 2/11 [240] = 480/11= 43( 7/11)
Option C is the correct answer.
Example 3: A clock is set right at 10 am. The clock gains 5 minutes in 12 hours. What will be the true time when the clock indicates 3 pm on the next day?
Time from 10 am to 3 pm on the following day is 29 hrs.
Now, 12 hrs 5 min i.e. 145/12 hrs of this clock = 12 hours of correct clock.
So, 29 hours of this clock is (29 × 12 × 12)/145 = 144/5 = 28 hours 48 minutes.
So, the time is 12 minutes before 3 pm.
Example 4: The minute hand of a clock overtakes the hour band at interval of 60 mins. How much a day does the clock gain or lose?
In a correct clock, minute hand gains 55 minute spaces over the hour hand in 60 minutes.
Therefore, 60 min spaces are gained in (60 × 60)/55 = 720/11 minutes.
In other words, minute hand overtakes hour hand in every 720/11 minutes.
In the example given, minute hand overtakes hour band in 60 minutes.
Therefore, gain in 60 minute = (720/11) - 60 = 60/11 minutes.
Gain in 24 hours = (60/11) × 24 = 1440/11 min
gain/loss =
Since the sign is +ve, clock gains 1440/11 min in a day.
Try yourself:How many times in a day, the hands of a clock are straight?
Explanation
The hands of clocks make a straight line of 180° about 22 times in 24 hours. Also, the hands coincide 22 times in 24 hours, the coincidence of the hands also forms a straight line. Hence, the total straight lines are 22+22 = 44.
Option C is the correct answer.
Example 5: The minute hand of a clock overtakes the hour band at interval of 66 mins. How much a day does the clock gain or lose?
As we have seen previously, minute hand overtakes hour hand in every 720/11 minute.
In this example, minute hand overtakes hour band in 66 minutes
Therefore, loss in 66 minute = 66 - 720/11 = 6/11 minutes
Loss in 24 hours = 6/11 × 1/66 × 66 × 24 = 1440/121 min
gain/loss
Since the sign is -ve, the clock loses 1440/121 min a day.