To find when all three bells ring together again, we need to calculate the Least Common Multiple (LCM) of the given time intervals:

The first bell rings every 18 minutes. The second bell rings every 24 minutes. The third bell rings every 32 minutes. Step 1: Prime Factorization $18 = 2 \times 3^2$ $24 = 2^3 \times 3$ $32 = 2^5$

Step 2: Find LCM The LCM is found by taking the highest powers of each prime factor: The highest power of 2 is $2^5$ (from 32). The highest power of 3 is $3^2$ (from 18). Thus, LCM = $2^5 \times 3^2 = 32 \times 9 = 288$ minutes.

Step 3: Convert Minutes to Hours and Minutes 288 minutes = 4 hours 48 minutes.

Step 4: Add to Initial Time The bells first ring together at 8:00 AM. Adding 4 hours 48 minutes to 8:00 AM gives 12:48 PM.