# Clocks

In this wiki we will study common problems about clocks for mental ability. Clocks are used to measure time. A clock in general 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

The hour hand, the shorter of the two hands, completes its 1 rotation in 12 hours.

How many degrees does the hour hand of a clock turn in 1 hour?

The hour hand turns 360 degrees in 12 hours. So in 1 hour it covers

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

degrees. \( _\square \)

## Minute Hand

The minute hand completes one rotation in 60 minutes, rotating 6 degree per minute.

## Relative Movement

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

## Problem Type 1

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

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

## Problem Type 2

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

Here we have to do nothing. Just equate the above formula with zero: \(0=|30-5.5m| \implies m=\frac{30}{5.5}=\frac{60}{11}.\)

So the time is \(1\) hour \(\frac{60}{11}\) minutes. \(_\square\)

## Problem Type 3

Two clocks are set correctly at 9 am on Monday. 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 minutes past 6 pm on the same day?

The time that elapses from 9 am to 6 pm on the same day is 9 hours. So the first clock gains \(9×3=27\) minutes and shows the time as 6:27 pm, while the second clock gains \(9×5=45\) minutes and shows the time as 6:45 pm. \(_\square\)

## Problem Type 4

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

In 60 minutes, 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 minutes are gained in \(\frac{60}{55}\times 60=\frac{720}{11}\) minutes. But they are given to be together after 63 minute. Thus the gain by the clock in 63 minutes is \(\frac{720}{11}-63=\frac{27}{11}\) minutes.

Therefore, the gain by the clock in 24 hours is \[\frac{27}{11}\times \frac{60\times 24}{63} =\frac{4320}{77}\] minutes. \(_\square\)

## Try-it-yourself Problems

At some time close to a quarter past two, the hour and minute hands of the clock are mirror images of each other, with respect to the horizontal line through 3 o'clock.

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. Everyday, one clock is 20 minutes faster while the other clock is 30 minutes slower. Liz adjusts the two clocks so that they show the same time and comes back to check the clocks every 24 hours. How many days does Liz have to wait until she can see the two clocks showing the same time again?

**Note:** AM and PM don'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 and returned home before 7:00, how many minutes was I away from my home?

I have three analog clocks. Clock 1 is correct, Clock 2 gains one minute every hour, and Clock 3 loses one minute every hour. I set them all to the correct time at \(12\) AM. At \(8 \! : \! 20\) PM, the angle between the hour and minute hands on Clock 1 is \(A^{\circ}\), the angle between the hour and minute hands on Clock 2 is \(B^{\circ}\), and the angle between the hour and minute hands on Clock 3 is \(C^{\circ}\). Assume \(0 \leq A, B, C \leq 180\). Find the value of \(\left \lfloor A + B + C \right \rfloor\).