Question

Three bells ring once every 12 minutes, 20 minutes and 36 minutes respectively. If all of them ring together at 6:30 am, then at what time will they all ring together for the next time on the same day morning.
(a) 8:30
(b) 9:30
(c) 10:30
(d) 11:30

Answer 1

Hint: To solve this question, we will use the concept of LCM (Least common multiple). We will find the LCM of the three given bells ringing time that will give us the exact tile on which all of them will ring again.

Complete step-by-step answer:
We are given that the first bell rings every 12 minutes, the second bell rings every 20 minutes and lastly, the third bell rings every 36 minutes. We are also given that they ring together all at 6:30 am. So now, let us define the LCM of some numbers. LCM or the least common multiple is defined as the smallest positive number that is a multiple of two or more numbers. Using this, let us find the LCM of our given three numbers. Our given numbers are 12, 20 and 36. The LCM of 12, 20 and 36 will give us the exact time duration after which all the three bells will ring again.
So, the LCM of 12, 20 and 36 are given as below
\[\begin{align}
  & 2\left| \!{\underline {\,
  12,20,36 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6,10,18 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3,5,9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  1,5,3 \,}} \right. \\
 & 5\left| \!{\underline {\,
  1,5,1 \,}} \right. \\
 & \text{ 1,1,1} \\
\end{align}\]
Hence, we have obtained the LCM of 12, 20 and 36 as
\[LCM=2\times 2\times 3\times 3\times 5=180\min \]
Hence, they will ring again after 180 minutes. Now we have 180 minutes = 3 hours.
So, they will ring again after 3 hours, i.e. 6:30 + 3 hours = 9:30 am. Hence, all three bells will ring together again at 9:30 am.
Therefore, option (b) is the right answer.

So, the correct answer is “Option (b)”.

Note: To convert minutes into hours, we have used the conversion formula as 1 hour = 60 minutes.
\[\Rightarrow 60\min =1hour\]
\[\Rightarrow 1\min =\dfrac{1}{60}hour\]
\[\Rightarrow 180\min =\dfrac{1}{60}\times 180=3hours\]