Question

At what time between 4 and 5, will the hands of a clock coincide?
A) 15.81 min
B) 21.81 min
C) 23.81 min
D) 33.48 min

Answer 1

Hint:
We can find the change in the angle of the minute hand per minute by dividing the whole angle by 60. Then, we can find the change in the angle of the hour hand per minute by dividing the whole angle by 12 and 60. Then we can assume the hands will coincide after x minutes past 4. Then we can take the change in angle in x minutes after 4 and equate them. Then we can solve for x to obtain the required answer.

Complete step by step solution:
We know that the minute hand of a clock completes 1 round in 60 minutes. Then the angle subtended in one minute by the minute hand is given by,
 $ \Rightarrow 1\min = \dfrac{{360^\circ }}{{60}}$
On division we get,
 $ \Rightarrow 1\min = 6^\circ $
Let the hands of the clock coincide after x minutes.
So, the angle subtended by the minute hand in x minutes is given by x times the angle subtended in one minute. It is given by,
 $ \Rightarrow x\min = 6x^\circ $
The angle subtended in x minute by the minute hand $ = 6x^\circ $ …. (1)
Now consider the hour hand. It completes one round in 12 hours. So, angle subtended by the hour hand in one hour is,
 $ \Rightarrow 1hour = \dfrac{{360^\circ }}{{12}}$
On division we get,
 $ \Rightarrow 1hour = 30^\circ $
We know that there are 60 minutes in one hour. So, angle subtended by the hour hand in one minute is given by,
 $ \Rightarrow 1\min = \dfrac{{30^\circ }}{{60}}$
On division we get,
 \[ \Rightarrow 1\min = 0.5^\circ \]
So, the angle subtended by the hour hand in x minutes is given by x times the angle subtended in one minute. It is given by,
 \[ \Rightarrow x\min = 0.5x^\circ \]
We need to find the time between 4 and 5. So we can take the angle subtended by the hour hand x minutes past 4. So, we can write the angle subtended as
The angle subtended by the hour hand x minutes past 4 \[ = 0.5\left( {4 \times 60 + x} \right)^\circ \] … (1)
When the two hands coincide, the angle subtended by them will be equal. So, we can equate equations (1) and (2). So, we can write,
 \[ \Rightarrow 6x^\circ = 0.5\left( {4 \times 60 + x} \right)^\circ \]
On simplification, we get,
 \[ \Rightarrow 12x^\circ = \left( {240 + x} \right)^\circ \]
On taking like terms at on side we get,
 \[ \Rightarrow 12x - x = 240\]
On adding like terms, we get,
 \[ \Rightarrow 11x = 240\]
On dividing the equation by 11 we get,
 \[ \Rightarrow x = \dfrac{{240}}{{11}}\]
On division we get,
 \[ \Rightarrow \] x=21.81
So, the hands of a clock coincide $21.81$ minutes past 4.

Therefore, the correct answer is option B, 21.81 min.

Note:
We must take the angle subtended by each hand in one minute. We must note that the hands must coincide between 4 and 5. We are equating the angle subtended from 12. So, we must make sure, we add the angle up to 4 or the minutes till 4 before equating the angles. Most of the time students forget to consider that angle. We can verify our answer by checking whether the minutes comes between 20 and 25 which are the corresponding minutes for 4 and 5.