Three bells chime at an interval of 18, 24 and 32 minutes respectively. At a certain time they begin to chime together. What length of time will elapse before they chime together again?

Options

A. 2 hours 24 minutes
B. 4 hours 48 minutes
C. 1 hour 36 minutes
D. 5 hours

Answer

To find the time that will elapse before the three bells chime together again, we need to determine the least common multiple (LCM) of the intervals at which they chime. The intervals are 18 minutes, 24 minutes, and 32 minutes.

### Step 1: Find the prime factorization of each interval.
- **18**:
- \(18 = 2 \times 3^2\)
- **24**:
- \(24 = 2^3 \times 3\)
- **32**:
- \(32 = 2^5\)

### Step 2: Identify the highest power of each prime factor.
- For the prime number **2**: The highest power is \(2^5\) (from 32).
- For the prime number **3**: The highest power is \(3^2\) (from 18).

### Step 3: Calculate the LCM using the highest powers.
The LCM is calculated by multiplying the highest powers of all prime factors:
\[ \text{LCM} = 2^5 \times 3^2 \]
Calculating this:
\[ 2^5 = 32 \quad \text{and} \quad 3^2 = 9 \]
Now multiply these results:
\[ \text{LCM} = 32 \times 9 = 288 \]

### Step 4: Convert the LCM into hours and minutes.
Since the LCM is 288 minutes, we convert this into hours and minutes:
- There are 60 minutes in an hour.
- To find the number of hours, divide 288 by 60:
\[ 288 \div 60 = 4 \quad \text{(whole hours)} \]
- To find the remaining minutes, calculate:
\[ 288 - (4 \times 60) = 288 - 240 = 48 \quad \text{(remaining minutes)} \]

### Final Answer:
The time that will elapse before the three bells chime together again is **4 hours and 48 minutes**.

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