Question

The traffic light at three different signal points change after every 45 seconds, 75 seconds and 90 seconds respectively. If all change simultaneously at 7:20:15 hours, then they will change again simultaneously at
A. 7:27:50 hours
B. 7:27:45 hours
C. 7:28:00 hours
D. 7:27:30 hours

Answer 1

Hint: To solve this question, we only need the basic concept of the least common multiple. We know that for any two or more than two numbers, the least common multiple is the smallest number which is divisible by all the numbers of which we are taking the LCM.

Complete step-by-step answer:

In this question, we have to find the time after 7:20:15 hours, at which all the three signal points change. We have been given the question that after every 45 seconds one of the signal changes. Another signal changes after 75 seconds and the third signal changes after 90 seconds. Now, we are asked to find that, at what time all the three signals will change simultaneously after 7:20:15 hours.
To find this, we should know what the least common multiple is. The least common multiple is nothing but the smallest natural number which is divisible by all the numbers of which we are taking the LCM.
So, we have to find the least common multiple of 45, 75 and 90.
We know that the factors of $45=3\times 3\times 5$.
The factors of $75=3\times 5\times 5$.
The factors of $90=2\times 3\times 3\times 5$.
So, we get the LCM of 45, 75 and 90 as $2\times 3\times 9\times 5\times 5\Rightarrow 450$.
We can see that the LCM of 45, 75 and 90 is 450. Therefore, three different signal points will all change simultaneously after 450 seconds or we can write, 450 seconds = $\left( 7\times 60+30 \right)$ seconds.
  $\begin{align}
  & \Rightarrow 7\min +30\sec \\
 & \Rightarrow 7\min 30\sec \\
\end{align}$
After 7:20:15 hours, we can say that all signal points will change after 7 min 30 sec or we can say that 7:20:15 hours + 7 min 30 sec which is 7:27:45 hours.
Hence, option (B) is the correct answer.

Note: The possible mistake that the students can make while solving the question in hurry is not changing the unit of 450 sec. Instead it must be written as 7 min 30 sec.