Correct Option (B)
To determine when the three bells will ring together again, it is necessary to find the Least Common Multiple (LCM) of their individual ringing intervals.
- First bell rings every 18 minutes.
- Second bell rings every 24 minutes.
- Third bell rings every 32 minutes.
The prime factorization of the intervals is as follows:
- 18 = 2 × 3²
- 24 = 2³ × 3
- 32 = 2⁵
The LCM(18, 24, 32) is calculated by taking the highest power of all prime factors present:
LCM = 2⁵ × 3² = 32 × 9 = 288 minutes.
Convert 288 minutes into hours and minutes:
288 minutes ÷ 60 minutes/hour = 4 hours and 48 minutes.
Given that all three bells rang together at 8:00 in the morning, the next time they will ring together is determined by adding this duration:
8:00 + 4 hours 48 minutes = 12:48.
Incorrect Options:
Options A (12:40hrs), C (12:56hrs), and D (13:04hrs) are incorrect because they do not correspond to the calculated Least Common Multiple of the ringing intervals (18, 24, and 32 minutes). The bells will only ring together at precise intervals that are exact multiples of their LCM. Any other time would imply that at least one bell would not complete its cycle precisely, or their ringing would not coincide simultaneously.