Find Next Simultaneous Ring Time

This problem involves finding the Least Common Multiple (LCM) of the ringing intervals to determine when the phones will ring together again.

1. Identify Ringing Intervals

• Phone 1 rings every 20 seconds.
• Phone 2 rings every 24 seconds.
• Phone 3 rings every 30 seconds.

2. Calculate the LCM

To find when they will ring together, we need the LCM of 20, 24, and 30.

• Prime factorization of 20: $2^2 \times 5$
• Prime factorization of 24: $2^3 \times 3$
• Prime factorization of 30: $2 \times 3 \times 5$
• The LCM is found by taking the highest power of each prime factor present: $2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120$.

The LCM is $120$ seconds.

3. Convert LCM to Minutes

Convert the interval back into minutes:

$ 120 \text{ seconds} = \frac{120}{60} \text{ minutes} = 2 \text{ minutes} $

4. Determine Next Ring Time

Add this interval to the time they last rang together:

Last ring time: 11:25 a.m.

Interval: 2 minutes

Next ring time: $ 11:25 \text{ a.m.} + 2 \text{ minutes} = 11:27 \text{ a.m.} $

Therefore, the phones will next ring together at 11:27 a.m.