Certainly! Let's solve this problem step-by-step.
### Step 1: Understand the Problem
Three bells ring at different intervals:
- Bell 1 rings every 36 minutes.
- Bell 2 rings every 45 minutes.
- Bell 3 rings every 60 minutes.
They all ring together at 8:00 AM. We need to find the next time they will ring together again.
### Step 2: Find the Least Common Multiple (LCM)
To determine when the bells will ring together again, we need to find the Least Common Multiple (LCM) of the three intervals (36, 45, and 60 minutes). This is because the LCM is the smallest number that is a multiple of all the intervals.
1. Find the prime factorizations:
- 36 = 2² × 3²
- 45 = 3² × 5
- 60 = 2² × 3 × 5
2. LCM Calculation:
- The LCM will include the highest powers of all prime factors present in the factorizations.
- LCM = 2² × 3² × 5
3. Compute the LCM:
- [tex]\( 2² = 4 \)[/tex]
- [tex]\( 3² = 9 \)[/tex]
- [tex]\( 5 = 5 \)[/tex]
Multiply these together:
[tex]\[
LCM = 4 × 9 × 5 = 180
\][/tex]
### Step 3: Determine the Next Time
Now that we know the bells will ring together again in 180 minutes, we need to calculate the exact time this will occur.
Since they initially ring together at 8:00 AM, we add 180 minutes to this time.
### Step 4: Convert Minutes to Hours and Minutes
1. Convert 180 minutes to hours and minutes:
- 180 minutes is equivalent to 3 hours (because [tex]\( 180 ÷ 60 = 3 \)[/tex] hours).
2. Add these 3 hours to the initial time (8:00 AM):
- 8:00 AM + 3 hours = 11:00 AM
### Conclusion
The bells will ring together again at 11:00 AM.
