To find the next time when all three bells (P, Q, and R) will ring together, we need to find the least common multiple (LCM) of their ringing intervals.
The ringing intervals are:
P every 20 minutes,
Q every 30 minutes, and
R every 50 minutes.
The LCM of 20, 30, and 50 is the smallest positive integer that is divisible by each of these numbers. Factorizing each number into its prime factors, we get:
\begin{align*}
20 & = 2^2 \times 5, \\
30 & = 2 \times 3 \times 5, \\
50 & = 2 \times 5^2.
\end{align*}
Now, take the highest power of each prime factor that appears in the factorization of these numbers:
\begin{align*}
\text{The highest power of 2 is} & \ 2^2, \\
\text{The highest power of 3 is} & \ 3 \ (\text{since} \ 3 \ \text{appears in the factorization of} \ 30), \\
\text{The highest power of 5 is} & \ 5^2.
\end{align*}
Multiply these highest powers to find the LCM:
\[ LCM(20, 30, 50) = 2^2 \times 3 \times 5^2 = 4 \times 3 \times 25 = 300. \]
So, the LCM of the ringing intervals is 300 minutes.
Now, to find when the three bells will ring together again after 12:00 PM, we add 300 minutes to the initial time:
\[ 12:00 PM + 300 \ \text{minutes} = 5:00 PM. \]
Therefore, the correct answer is (A) 5:00 PM.
