Question

In a zoo lions roar together at $10\text{am}$ . If the lions roar repeatedly at the definite interval of $10$ , $15$ and $12$ minutes respectively then at what time will they roar together again?
A. $11.30\text{ a}\text{.m}$
B. $11.45\text{ a}\text{.m}$
C. $12.00\text{ a}\text{.m}$
D. $11.00\text{ a}\text{.m}$

Answer 1

Hint: In this problem we need to calculate the time where the lion will roar together again. For this we need to calculate the LCM (Least Common Multiple) of the given three intervals. The LCM of the numbers is calculated from the factors of the numbers. So we will consider each number individually and calculate the factorization form of the numbers. After having the factorization form we can calculate the LCM of the given intervals. After that we can add the calculated value of LCM to the given initial time to get the required result.

Complete step by step solution:
Given that lions in a zoo will roar at $10\text{ a}\text{.m}$and the lions will roar repeatedly at the definite interval of $10$ , $15$ and $12$ minutes respectively.
Considering the number $10$. The factors of the number $10$ will be
$\begin{align}
  & 2\left| \!{\underline {\,
  10 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
From the above factors we can write the number $10$ as $10=2\times 5$ .
Considering the number $15$. The factors of the number $15$ will be
$\begin{align}
  & 3\left| \!{\underline {\,
  15 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5 \,}} \right. \\
 & 1 \\
\end{align}$
From the above factors we can write the number $15$ as $15=3\times 5$ .
Considering the number $12$. The factors of the number $12$ will be
$\begin{align}
  & 2\left| \!{\underline {\,
  12 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & 1 \\
\end{align}$
From the above factors we can write the number $12$ as $12=2\times 2\times 3$ .
From the factors of the numbers $10$ , $15$ and $12$ the lcm of the numbers $10$ , $15$ and $12$ will be
$\begin{align}
  & LCM=2\times 2\times 3\times 5 \\
 & \Rightarrow LCM=60 \\
\end{align}$
So the LCM of the given time intervals is $60$ minutes. We know that $60$ minutes are equal to $1$ hour. So the lions will roar together for one hour.
If the lions roar at $10\text{ a}\text{.m}$, then they will roar together after one hour, which means at $11\text{ a}\text{.m}$ .
Hence option – D is the correct answer.

Note: In this problem we have given all the time intervals in the same units that mean all the given intervals are in minutes. So we have directly calculated the LCM without any conversion. If they have given the time intervals in different units like in seconds or hours. We need to convert them into any one unit and then only we need to calculate the LCM of the intervals.