# Clocks

Here We will study about general problems about clocks for mental ability.

Clocks are used to measure time. General clock has 12 numbers written on it,from 1 to 12, an hour hand and a minute hand.

Clock angle problems relate two different measurements: angles and time. The angle is typically measured in degrees from the mark of number 12 clockwise. The time is usually based on 12-hour clock.

A method to solve such problems is to consider the rate of change of the angle in degrees per minute. The hour hand of a normal 12-hour analogue clock turns 360° in 12 hours (720 minutes) or 0.5° per minute. The minute hand rotates through 360° in 60 minutes or 6° per minute.

#### Contents

## Hour hand

Hour hand completes its 1 rotation in 12 hours. It is denoted by a short length arm.

How many degrees does hour hand complete in 1 hour?

Hour hand completes 360 degree in 12 hours. So in 1 hour it covers

\[ \frac{360}{12} = 30 \]

And our final answer is 30 degrees \( _\square \)

## Minute hand

Minute hand completes one rotation in 60 minutes, making 6 degree per minute.

## Relative movement

As we have seen the hour hand completes 0.5 degree in 1 minute and minute hand completes 6 degree per minute, relative speed of minute hand is 5.5 degree per minute.

## Problem Type 1

What is the angle between the hands of clock at 1:00 pm?

The general formula is \(|30h-5.5m|\). Here \(h=1\) and \(m=0\). So angle=\(30^{\circ}\)

## Problem Type 2

At what tie between 1:00 and 2:00, will the hands of clock be together?

Here we have to do nothing, just in the above formula put angle as 0, \(0=|30-5.5m|\) or \(m=30/5.5=60/11\) min

So the time is \(1\) hr \(60/11\) minutes.

## Problem Type 3

Two clocks are set correctly at 9 am on Monday. Both the clocks gain 3 min and 5 min respectively in an hour. What time will the second clock register,if the first clock shows the time as 27 min past 6 pm on the same day?

In one hour the second clock gains \(2\) minutes more than first clock. Total time from 9 am to 6 pm on Monday=9 hours. So forst clock gains \(9×3=27\) minutes and shows the time as 6:27 pm. Also 2nd clock gains \(9×5=45\) minutes and show the time as 6:45 pm.

## Problem Type 4

The minute hand of a clock overtakes the hour hand at intervals of 63 minute of the correct time. How much does a clock gain or loose in a day?

In 60 minute, the minute hand gains 55 minutes space over the hour hand. To be together again, the minute hand must gain 60 minutes over the hour hand. 60 minute are gained in \((60/55×60)=720/11\) min

But they are together after 63 minute. Thus gain by the clock in 63 minute=\(720/11-63=27/11\) min

So gain by the clock in 24 hours=\((27/11×(60×24)/63))\) min=\(4320/77\) min gain.

## Try it yourself problems

What is the precise time that this happens? Round off to the nearest second; give the answer in the form \(\text{HMMSS}\). (For instance, if the answer were 2:12:41, type 21241.)

**Extra challenge**: How many times does a symmetric situation like this occur in a 12-hour period?

Liz has 2 clocks, after every 1 day, one clock will be 20 minutes faster, and the other clock will be 30 minutes slower. Liz adjusted the two clocks so that they show the same time and regularly comes back to check the clocks every 24 hours, how many days Liz has to wait until she can see the two clocks showing the same time again?

(Note: A.M. and P.M. doesn't matter, as long as the faces of the 2 clocks are the same)

This is one part of 1+1 is not = to 3.

When I left my home between 6:00 and 7:00, the hour and minute hands of my analog clock formed a \(110^{\circ}\) angle. When I returned home, the hour and minute hands of the clock also formed a \(110^{\circ}\) angle. If I left my home for more than five minutes, how many minutes was I away from my home?