Five bells begin to toll together and toll respectively at intervals of 6, 5, 7, 10 and 12 seconds. How many times will they toll together in one hour excluding the one at the start?

Options

A. 7
B. 8
C. 9
D. 10

Explanation

To solve the problem of how many times five bells toll together in one hour (excluding the one at the start), we will follow these steps:

### Step 1: Identify the tolling intervals of the bells
The bells toll at the following intervals:
- Bell 1: 6 seconds
- Bell 2: 5 seconds
- Bell 3: 7 seconds
- Bell 4: 10 seconds
- Bell 5: 12 seconds

### Step 2: Find the Least Common Multiple (LCM)
To find out when all the bells toll together, we need to calculate the LCM of the intervals (6, 5, 7, 10, and 12).

**Finding LCM:**
1. **Prime Factorization:**
- 6 = 2 × 3
- 5 = 5
- 7 = 7
- 10 = 2 × 5
- 12 = 2² × 3

2. **Take the highest power of each prime:**
- 2² (from 12)
- 3¹ (from 6 or 12)
- 5¹ (from 5 or 10)
- 7¹ (from 7)

3. **Calculate LCM:**
LCM = 2² × 3¹ × 5¹ × 7¹ = 4 × 3 × 5 × 7

Now, calculate:
- 4 × 3 = 12
- 12 × 5 = 60
- 60 × 7 = 420

So, the LCM is 420 seconds.

### Step 3: Convert LCM to minutes
Since there are 60 seconds in a minute, we convert the LCM from seconds to minutes:
- 420 seconds ÷ 60 seconds/minute = 7 minutes

### Step 4: Determine how many times they toll together in one hour
Now, we need to find out how many times the bells toll together in one hour (60 minutes):
- Total time in one hour = 60 minutes
- Time interval for tolling together = 7 minutes

To find the number of times they toll together:
- Number of times = 60 minutes ÷ 7 minutes = approximately 8.57

Since we are excluding the initial toll at the start, we take the whole number part:
- Therefore, they toll together **8 times** in one hour.

### Final Answer
The bells toll together **8 times** in one hour, excluding the initial toll.

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