In a school, the bell for the middle school rings every 30 minutes and the bell for the senior school rings every 40 minutes. The two bells start to ring together at 8.00 a.m. At what time will they ring together again?

Answer

To solve the problem step by step, we need to find the time when both bells will ring together again after starting at 8:00 a.m.

### Step 1: Identify the ringing intervals
- The middle school bell rings every 30 minutes.
- The senior school bell rings every 40 minutes.

### Step 2: Find the Least Common Multiple (LCM)
To find out when both bells will ring together again, we need to calculate the LCM of 30 and 40.

#### Step 2.1: Prime Factorization
- 30 can be factored into prime numbers:
- \(30 = 2 \times 3 \times 5\)
- 40 can be factored into prime numbers:
- \(40 = 2^3 \times 5\)

#### Step 2.2: Determine the LCM
To find the LCM, we take the highest power of each prime number that appears in the factorizations:
- For 2: the highest power is \(2^3\) (from 40)
- For 3: the highest power is \(3^1\) (from 30)
- For 5: the highest power is \(5^1\) (common in both)

Now, we multiply these together:
\[ LCM = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 \]

Calculating this step by step:
- \(8 \times 3 = 24\)
- \(24 \times 5 = 120\)

So, the LCM of 30 and 40 is 120 minutes.

### Step 3: Convert LCM to hours and minutes
120 minutes is equivalent to:
\[ 120 \div 60 = 2 \text{ hours} \]

### Step 4: Calculate the time when both bells will ring together again
Since both bells start ringing together at 8:00 a.m., we add 2 hours to this time:
\[ 8:00 \text{ a.m.} + 2 \text{ hours} = 10:00 \text{ a.m.} \]

### Final Answer
The two bells will ring together again at **10:00 a.m.**

---