Anil sets an alarm in two different clocks The first alarm rings after every 3 hours and the second alarm rings after every 2 hours and 30 minutes. If both the alarms rang together at 2 p.m., what will be the time when they wtll next nng together?

Options

A. 2 a.m
B. 4 a.m
C. 5 a.m
D. 3 a m.

Answer

To find out when both alarms will next ring together, we need to determine the least common multiple (LCM) of the two time intervals: 3 hours and 2 hours 30 minutes.

### Step-by-step Solution:

1. **Convert the time intervals into the same unit**:
- The first alarm rings every 3 hours.
- The second alarm rings every 2 hours and 30 minutes. We can convert this into hours:
\[ 2 \text{ hours} + 30 \text{ minutes} = 2.5 \text{ hours} \]

2. **Identify the time intervals**:
- First alarm: 3 hours
- Second alarm: 2.5 hours

3. **Convert the hours into a fraction**:
- 3 hours can be written as \( \frac{3}{1} \).
- 2.5 hours can be written as \( \frac{5}{2} \).

4. **Find the LCM of the two fractions**:
- To find the LCM of two fractions \( \frac{a}{b} \) and \( \frac{c}{d} \), we use the formula:
\[ \text{LCM}\left(\frac{a}{b}, \frac{c}{d}\right) = \frac{\text{LCM}(a, c)}{\text{GCD}(b, d)} \]
- Here, \( a = 3 \), \( b = 1 \), \( c = 5 \), and \( d = 2 \).

5. **Calculate LCM of the numerators**:
- LCM(3, 5) = 15 (since 3 and 5 are co-prime).

6. **Calculate GCD of the denominators**:
- GCD(1, 2) = 1 (since 1 is a factor of every integer).

7. **Calculate the LCM of the two time intervals**:
\[ \text{LCM}\left(3, 2.5\right) = \frac{15}{1} = 15 \text{ hours} \]

8. **Determine the next time both alarms will ring together**:
- Since both alarms rang together at 2 p.m., we add 15 hours to this time:
\[ 2 \text{ p.m.} + 15 \text{ hours} = 5 \text{ a.m. (next day)} \]

### Final Answer:
The next time both alarms will ring together is at **5 a.m. the next day**.