Five bells begin to toll together at intervals of 9s,6s,4s,10s and 8s,respectively. How many times will they toll together in the span of 1 h (excluding the toll at the start)?

Options

A. 5
B. 8
C. 10
D. Couldn't be determined

Answer

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

### Step 1: Identify the tolling intervals
The bells toll at the following intervals:
- Bell 1: 9 seconds
- Bell 2: 6 seconds
- Bell 3: 4 seconds
- Bell 4: 10 seconds
- Bell 5: 8 seconds

### Step 2: Find the Least Common Multiple (LCM)
To determine how often all five bells toll together, we need to calculate the LCM of the intervals: 9, 6, 4, 10, and 8.

**Finding the LCM:**
1. **Prime Factorization:**
- 9 = \(3^2\)
- 6 = \(2^1 \times 3^1\)
- 4 = \(2^2\)
- 10 = \(2^1 \times 5^1\)
- 8 = \(2^3\)

2. **Take the highest power of each prime:**
- For 2: \(2^3\) (from 8)
- For 3: \(3^2\) (from 9)
- For 5: \(5^1\) (from 10)

3. **Calculate the LCM:**
\[ \text{LCM} = 2^3 \times 3^2 \times 5^1 = 8 \times 9 \times 5 \]
\[ = 72 \times 5 = 360 \text{ seconds} \]

### Step 3: Convert the time span to seconds
We need to find out how many times they toll together in 1 hour.
1 hour = 60 minutes = 3600 seconds.

### Step 4: Calculate the number of tolls in 1 hour
To find how many times the bells toll together in 3600 seconds:
- The first toll occurs at 0 seconds (which we exclude).
- The subsequent tolls occur every 360 seconds.

To find the number of times they toll together in 3600 seconds:
\[ \text{Number of tolls} = \frac{3600 \text{ seconds}}{360 \text{ seconds}} = 10 \]
Since we exclude the first toll, the number of times they toll together is:
\[ 10 - 1 = 9 \]

### Final Answer
The bells will toll together **9 times** in one hour, excluding the initial toll.

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