Two bells ring at intervals of 18 minutes and 24 minutes respectively. If they ring together at 9:00 a.m., at what time will they next ring together?

Step 1: Understand the Problem

We are given two bells that ring at regular intervals. One bell rings every 18 minutes, and the other rings every 24 minutes. We need to find when they will ring together again after their initial simultaneous ringing.

Step 2: Find the Least Common Multiple (LCM)

To find when the bells will ring together again, we need to find the least common multiple (LCM) of their ringing intervals. The LCM will be the smallest number of minutes after which both bells will have completed a whole number of their respective cycles.

Step 3: Prime Factorization of 18 and 24

First, we find the prime factorization of each number. For 18, we have $2 \times 3 \times 3 = 2 \times 3^2$. For 24, we have $2 \times 2 \times 2 \times 3 = 2^3 \times 3$.

Step 4: Calculate the LCM

To find the LCM, we take the highest power of each prime factor present in the factorizations. The highest power of 2 is $2^3$ (from 24), and the highest power of 3 is $3^2$ (from 18). Multiplying these gives us $8 \times 9 = 72$. So, the bells will ring together again after 72 minutes.

Step 5: Convert Minutes to Hours and Minutes

Since there are 60 minutes in an hour, 72 minutes can be converted to hours and minutes. 72 minutes is equal to 1 hour and 12 minutes ($72 - 60 = 12$).

Step 6: Determine the Next Ringing Time

The bells rang together at 9:00 a.m. Adding 1 hour and 12 minutes to this time gives us 10:12 a.m. This is when they will next ring together.

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