Calculating Simultaneous Clock Rings

The problem requires finding the next time all four table clocks ring together. They ring at intervals of 10 minutes, 15 minutes, 20 minutes, and 25 minutes. To find when they will ring together again, we need to calculate the Least Common Multiple (LCM) of these intervals.

Finding the LCM of Ringing Intervals

First, find the prime factorization of each interval:

• $10 = 2 \times 5$
• $15 = 3 \times 5$
• $20 = 2^2 \times 5$
• $25 = 5^2$

The LCM is found by taking the highest power of each prime factor present in any factorization:

$ \text{LCM}(10, 15, 20, 25) = 2^2 \times 3^1 \times 5^2 $

$ \text{LCM} = 4 \times 3 \times 25 = 12 \times 25 = 300 \text{ minutes} $

Determining the Next Ring Time

The clocks will ring together every 300 minutes.

Convert 300 minutes into hours:

$ \frac{300 \text{ minutes}}{60 \text{ minutes/hour}} = 5 \text{ hours} $

They all rang together at 10 a.m. They will ring together again after 5 hours.

Adding 5 hours to 10 a.m.:

$10:00 \text{ a.m.} + 5 \text{ hours} = 15:00$

In standard time, 15:00 is 3:00 p.m.

Final Answer

The clocks will ring together again at 3:00 p.m.