Four bells starts ringing together and then ring at an interval of 24 sec, 36 sec, 48 sec and 64 sec respectively. How many times will they ring together is 7 hrs 12min ?

Options

A. 45
B. 46
C. 48
D. 44

Answer

To solve the problem of how many times the four bells will ring together in 7 hours and 12 minutes, we will follow these steps:

### Step 1: Find the Least Common Multiple (LCM)
We need to find the LCM of the ringing intervals of the four bells: 24 seconds, 36 seconds, 48 seconds, and 64 seconds.

**Finding the Prime Factorization:**
- **24** = 2^3 × 3^1
- **36** = 2^2 × 3^2
- **48** = 2^4 × 3^1
- **64** = 2^6

**Finding the LCM:**
To find the LCM, take the highest power of each prime factor:
- For 2: the highest power is 2^6 (from 64)
- For 3: the highest power is 3^2 (from 36)

Thus, the LCM = 2^6 × 3^2 = 64 × 9 = 576 seconds.

### Step 2: Convert the Time Period into Seconds
Now, we need to convert 7 hours and 12 minutes into seconds.

**Calculating Seconds:**
- 7 hours = 7 × 60 minutes = 420 minutes
- 12 minutes = 12 minutes
- Total minutes = 420 + 12 = 432 minutes

Now convert minutes to seconds:
- Total seconds = 432 minutes × 60 seconds/minute = 25920 seconds.

### Step 3: Calculate How Many Times the Bells Ring Together
To find how many times the bells ring together in the total time period, divide the total seconds by the LCM.

**Calculation:**
Number of times = Total seconds / LCM = 25920 seconds / 576 seconds = 45.

### Step 4: Include the Initial Ring
Since the bells ring together at the start, we need to add 1 to the count of times they ring together.

**Final Count:**
Total times they ring together = 45 + 1 = 46.

### Conclusion
The bells will ring together **46 times** in 7 hours and 12 minutes.

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