# Clocks

### Clocks Important Formulas

- A clock is a complete circle having 360 degrees. It is divided into 12 equal parts i.e. each part:

$\frac{360}{12}=30$ - As the minute hand takes a complete round in one hour it covers 360 degrees in 60 min.

Minute Hand covers $\frac{360}{60}=6\frac{degree}{minute}$ - Also, as the hour hand covers just one part out of the given 12 parts in one hour, this implies

Hour Hand covers 300 in 60 min. i.e. 1/2 degree per minute. - Therefore, the relative speed of the minute hand is $6\text{\u2013}\frac{1}{2}=5\frac{1}{2}degrees$
- Every hour, both the hands coincide once. In 12 hours, they will coincide 11 times.

It happens due to only one such incident between 12 and 1’o clock. - The hands are in the same straight line when they are coincident or opposite to each other.
- When the two hands are at a right angle, they are 15-minute spaces apart.
- In one hour, they will form two right angles and in 12 hours there are only 22 right angles. It happens due to right angles formed by the minute and hour hand at 3’o clock and 9’o clock.
- When the hands are in opposite directions, they are 30-minute spaces apart.
- If a clock indicates 9.15, when the correct time is 9, it is said to be 15 minutes too fast. On the other hand, if it indicates 8.45, when the correct time is 9, it is said to be 15 minutes too slow.
- If both the hour hand and minute hand move at their normal speeds, then both the hands meet after $65\frac{5}{11}minutes$.
- 22 times in a day, the hands of a clock will be in a straight line but opposite in direction.
- 44 times in a day, the hands of a clock will be straight.
- 44 times in a day, the hands of a clock are at right angles.
- 22 times in a day, the hands of a clock coincide.

## Problems on angles of Clocks Tricks

- Speed of the hour hand = 0.5 degrees per minute (dpm) {The hour hand completes a full circle or 360 degrees in 12 hours or 720 minutes}
- Speed of the minute hand = 6 dpm {The minute hand completes a full circle in 60 minutes}
- At ‘n’ o’ clock, the angle of the hour hand from the vertical is 30n

**Example 1:** What is the angle between the hands of the clock at 7:20

Answer: At 7 o’ clock, the hour hand is at 210 degrees from the vertical.

In 20 minutes, Hour hand = 210 + 20*(0.5) = 210 + 10 = 220 {The hour hand moves at 0.5 dpm}

Minute hand = 20*(6) = 120 {The minute hand moves at 6 dpm}. Difference or angle between the hands = 220 – 120 = **100 degrees**

## Problems on incorrect clocks

Such sort of problems arise when a clock runs faster or slower than expected pace. When solving these problems it is best to keep track of the correct clock.

**Example 2:** A watch gains 5 seconds in 3 minutes and was set right at 8 AM. What time will it show at 10 PM on the same day?

Answer: The watch gains 5 seconds in 3 minutes => 100 seconds in 1 hour. From 8 AM to 10 PM on the same day, time passed is 14 hours. In 14 hours, the watch would have gained 1400 seconds or 23 minutes 20 seconds. So, when the correct time is 10 PM, the watch would show 10 : 23 : 20 PM