One bell rings at intervals of 40 minutes and another at intervals of 30 minutes. If they ring at 10 am together, when will they ring together again for the next time?
The problem requires us to find the least common multiple (LCM) of 40 minutes and 30 minutes. The LCM represents the shortest time interval at which both bells will ring together again.
Here's how to solve it:
1. **Find the prime factors of each interval:** * 40 = 2 x 2 x 2 x 5 (2^3 x 5) * 30 = 2 x 3 x 5
2. **Determine the LCM:** The LCM is found by taking the highest power of each prime factor present in either number. * LCM(40, 30) = 2^3 x 3 x 5 = 8 x 3 x 5 = 120 minutes
3. **Convert minutes to hours:** 120 minutes is equal to 2 hours.
4. **Calculate the next time:** Since they rang together at 10:00 am, and they will ring together again after 2 hours, the next time will be 12:00 noon.