Six bells commence tolling together and toll at interval of 2, 4 , 6, 8 , 10 , 12 minutes respectively . In 30 hours , how many times do they toll together ?

Answer

To solve the problem of how many times the six bells toll together in 30 hours, we can follow these steps:

### Step 1: Identify the tolling intervals
The bells toll at the following intervals:
- Bell 1: 2 minutes
- Bell 2: 4 minutes
- Bell 3: 6 minutes
- Bell 4: 8 minutes
- Bell 5: 10 minutes
- Bell 6: 12 minutes

### Step 2: Find the Least Common Multiple (LCM)
To determine when all the bells toll together, we need to find the LCM of their intervals.

1. **Prime factorization of each interval:**
- 2 = 2
- 4 = 2²
- 6 = 2 × 3
- 8 = 2³
- 10 = 2 × 5
- 12 = 2² × 3

2. **Take the highest power of each prime:**
- For 2: the highest power is 2³ (from 8)
- For 3: the highest power is 3¹ (from 6 and 12)
- For 5: the highest power is 5¹ (from 10)

3. **Calculate the LCM:**
\[ \text{LCM} = 2³ \times 3¹ \times 5¹ = 8 \times 3 \times 5 = 120 \text{ minutes} \]

### Step 3: Convert the LCM to hours
120 minutes is equal to:
\[ \frac{120}{60} = 2 \text{ hours} \]

### Step 4: Determine how many times they toll together in 30 hours
Now, we need to find out how many 2-hour intervals fit into 30 hours:
\[ \text{Number of intervals} = \frac{30}{2} = 15 \]

### Step 5: Count the total tolls
Since they toll together at the start (time = 0), we add 1 to the number of intervals:
\[ \text{Total tolls} = 15 + 1 = 16 \]

### Final Answer
The six bells toll together **16 times** in 30 hours.

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