A red light flashes 3 times per minute and a green light flashes 5 times in two minutes at regular intervals. If both lights start flashing at the same time, how many times do they flash together in each hour ?

Options

A. 30
B. 29
C. 31
D. 32

Answer

To solve the problem of how many times the red and green lights flash together in one hour, we can follow these steps:

### Step 1: Determine the flashing frequency of each light.
- **Red light:** It flashes 3 times per minute.
- **Green light:** It flashes 5 times in 2 minutes, which means it flashes \( \frac{5}{2} = 2.5 \) times per minute.

### Step 2: Find the time interval for each light.
- **Red light:** The time interval between each flash is \( \frac{1 \text{ minute}}{3 \text{ flashes}} = \frac{1}{3} \text{ minutes} \) or 20 seconds.
- **Green light:** The time interval between each flash is \( \frac{2 \text{ minutes}}{5 \text{ flashes}} = \frac{2}{5} \text{ minutes} \) or 24 seconds.

### Step 3: Find the least common multiple (LCM) of the intervals.
- The red light flashes every 20 seconds and the green light flashes every 24 seconds. We need to find the LCM of 20 and 24.

- The prime factorization of 20 is \( 2^2 \times 5^1 \).
- The prime factorization of 24 is \( 2^3 \times 3^1 \).

- The LCM is found by taking the highest power of each prime:
- \( 2^3 \) (from 24)
- \( 3^1 \) (from 24)
- \( 5^1 \) (from 20)

- Therefore, \( \text{LCM} = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120 \).

### Step 4: Convert the LCM into minutes.
- The LCM of 20 seconds and 24 seconds is 120 seconds, which is equal to \( \frac{120}{60} = 2 \) minutes.

### Step 5: Calculate how many times they flash together in one hour.
- In one hour (60 minutes), the number of 2-minute intervals is \( \frac{60}{2} = 30 \).

### Conclusion:
Thus, the red and green lights flash together **30 times** in one hour.

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