A and B start running at the same time and from the same point around a circle. If A can complete one round in 40 seconds and B in 50 seconds, how many seconds will they take to reach the starting point simultaneously?

Options

A. 10
B. 200
C. 90
D. 2000

Answer

To solve the problem of how many seconds A and B will take to reach the starting point simultaneously, we need to find the least common multiple (LCM) of the times taken by A and B to complete one round.

### Step-by-Step Solution:

1. **Identify the time taken by A and B to complete one round:**
- A takes 40 seconds to complete one round.
- B takes 50 seconds to complete one round.

2. **Find the LCM of the two times:**
- The LCM of two numbers is the smallest number that is a multiple of both.
- To find the LCM of 40 and 50, we can list the multiples or use the prime factorization method.

3. **Prime Factorization:**
- The prime factorization of 40 is:
- \( 40 = 2^3 \times 5^1 \)
- The prime factorization of 50 is:
- \( 50 = 2^1 \times 5^2 \)

4. **Determine the LCM using the highest powers of each prime factor:**
- For the prime number 2, the highest power is \( 2^3 \) (from 40).
- For the prime number 5, the highest power is \( 5^2 \) (from 50).
- Therefore, the LCM is:
\[ LCM = 2^3 \times 5^2 = 8 \times 25 = 200 \]

5. **Conclusion:**
- A and B will take 200 seconds to reach the starting point simultaneously.

### Final Answer:
A and B will take **200 seconds** to reach the starting point simultaneously.

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