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Clock Problems
Type 1: Finding Angle Between Minute And Hour Hands
This is the easiest type in clock problems. Two simple facts you should know to solve these problems are:
The hour hand finishes full rotation in 12 hours. That is, the hour hand traces 360 degrees in 12 hours.
The minute hand finishes full rotation in 60 minutes. That is, the minute hand traces 360 degrees in 60 minutes.
Now, lets us move to our example question…
Example Question 1: If the time is 5.30 am, what is the angle between minute and hour hands.
To solve this problem, you have to individually find the angles traced by hour hand and minute hand, and find the difference between the two.
Step 1. Angle for hour hand:
You know that the angle traced by hour hand in 12 hours is 360 degrees.
Now, as first step, you have to find the angle traced by hour hand at 5 hours 30 minutes or 5 1/2 hours or 11/2 hours.
Using direct proportion table below, you can easily find this angle. (You may know this simple method already)
Hours Angle
12 360
11/2 x
Since, hours and angle are in direct proportion (i.e angle increases as hour increases and vice versa), we can write…
12 / (11/2) = 360 / x
Or x = (360 / 12) x (11 / 2)
x = 165 degrees
Therefore, angle traced by hour hand at 5.30 am will be 165 degrees
Step 2. Angle for minute hand:
You know that the angle traced by minute hand in 60 minutes is 360 degrees
Now, you have to find the angle traced by the minute hand in 30 minutes (This is because, we are finding the angle for 5.30 am and minute part in 5.30 am is 30 minutes)
Now, you have to use the direct proportion table for minute hand
Minutes Angle
60 360
30 y
Since minutes and angle are in direct proportion, we can write,
60/30 = 360/y
Or y = 360 x 30 / 60 = 180 degrees
Therefore, angle traced by minute hand in 5.30 am will be 180 degrees
Step 3: Finding the difference
Therefore, angle between hour and minute hands in 5 hour 30 minutes will be the
difference between 165 and 180 degrees, which is 15 degrees.
Type 2: Finding Time If Angle Is Given
This type is the straight opposite of type 1 problems. In type 1, you have to find the angle if time is given. In this type, you will be given the angle and you will be finding the time.
Below 4 simple facts will help you to solve these types of problems:
If the hour and minute hands are in straight line and in same direction, the angle between them is 0 degrees or 360 degrees. And they are 0 minute spaces apart (they coincide).
If the hour and minute hands are in straight line and in opposite directions, the angle between them is 180 degrees. And they are 30 minute spaces apart.
If the angle between minute and hour hands is 90 degrees (right angle), then they will be 15 minute spaces apart.
Below data is based on measurement in clockwise direction:
Angle Between Minute And Hour Hand (in Degrees) Minute , Space Between Minute And Hour Hands (in Minutes)
30 , 5
60 , 10
90 ,15
120 , 20
150 , 25
180 , 30
210 , 35
240 , 40
270 , 45
300 , 50
330 , 55
360 , 60
Minute hand gains 55 minute spaces on hour hand in 60 minutes. This point confuses many youngsters. If you are not able to understand this, here is your explanation for this 4th point:
Whenever minute hand completes one full rotation, the hour hand will be moving by 5 minute spaces. For example, if the time is 2 o clock. Now minute hand will be at 12 and hour hand will be at 2.
When the minute hand completes one full rotation and reaches number 12 again, the hour hand will be pointing at 3.
In other words, whenever small hand (hour hand) moves 5 minute spaces, the long hand (minute hand) moves 60 minute spaces. That is, it will gain or be ahead of hour hand by 60 – 5 = 55 minutes.
You will understand type 2 after reading the below example:
Example Question 2: Assume that the time now is 5 pm. At what time from now, minute hand will be ahead of hour hand by 90 degrees?
At 5 pm, the minute and hour hands will be 25 minute spaces apart. If you do not understand this, please check the below diagram.
For minute hand to move ahead of the hour hand by 90 degrees, it should gain 15 minute spaces on hour hand. (If you want to know why 90 degrees corresponds to 15 minute spaces, please read again the facts we discussed at the start of this type 2)
But the minute hand is behind the hour hand by 25 minute spaces. Therefore, first the minute hand must coincide with hour hand by covering 25 minute spaces and then it has to move ahead by 15 minute spaces.
So, it has to gain a total of 25 + 15 = 40 minute spaces.
We already know, minute hand gains 55 minute spaces on hour hand in 60 minutes. (check the 4th fact if not clear)
Using direct proportion table, we can calculate the minutes required for minute hand to gain 40 minute spaces
Minute Space Gain Minutes
55 60
40 x
Here x is the number of minutes required for minute hand to move ahead of hour hand by 90 degrees.
55/40 = 60/x
Or x = 60 x 40 / 55
x = 2400/55 = 480/11 = 43 7/11 minutes.
Therefore, at 5 hour and 43 7/11 minutes, the minute hand will be ahead of hour hand by 90 degrees.
Note: In this question, we found the time when minute hand is ahead of hour hand by 90 degrees. But, if the question asks us to find the time when minute hand is behind hour hand by 90 degrees, we can repeat the same solution by calculating the time taken for minute hand to gain 25 – 15 = 10 minute spaces.
Type 3: Correct Time On Incorrect (Fast or Slow) Clocks
When some clocks are not perfect, they become either faster or slower than regular clocks. You may get problems based on incorrect clocks.
Below example will help you to solve these types of problems…
Example Question 3: Rohit buys a new clock and sets time to 5. pm (by seeing the correct time in a regular clock). But the new clock is faulty and it gains 20 minutes in 4 hours. After 3 days, Rohit sees that the faulty new clock is showing 9 pm. But, what will be the actual time in a regular correct clock?
Duration between 5 pm on day one to 9 pm on day 3 can be calculated as follows:
Time between 5 pm on day 1 to 5 pm on day 2 = 24 hours
Time between 5 pm on day 2 to 5 pm on day 3 = 24 hours
Time between 5 pm on day 3 to 9 pm on day 3 = 4 hours
Therefore, time between 5 pm on day 1 to 11 pm on day 3 = 24 + 24 + 4
= 52 hours or
52 x 60 = 3120 minutes
From the question, you know that hours and 20 minutes on faulty clock is same as 4 hours on regular clock.
To make calculations easier, we are converting above two times into minutes:
4 hours = 4 x 60 = 240 minutes and
4 hour 20 minutes = 4 x 60 + 20 = 260 minutes
Now you can calculate the minutes on a regular correct clock corresponding to 3120 minutes on the faulty clock using direct proportion table as shown below:
Minutes on faulty clock Minutes on regular clock
260 240
3120 x
260/3120 = 240/x
x = 240 x 3120 / 260 = 2880 minutes
Therefore, 3120 minutes on faulty clock will be equal to 2880 minutes on regular clock
Hence, difference in minutes between regular clock and faulty clock at 9 pm on third day, will be
3120 – 2880 = 240 minutes or 4 hours
Therefore, regular clock will be 4 hours behind faulty clock and hence the correct time will be 9 – 4 = 5 pm.
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