Two persons A and B are running on a circular track at a speed of 15 m/s and 25 m/s respectively. The circumference of a circular track is 1200 m. If they start at the same place, at the same time and also in the same direction then find after how much time will they meet first? And also find after how much time will they reach at the starting point together.

Answer

To solve the problem, we need to find two things:

1. The time after which persons A and B will meet for the first time.
2. The time after which both A and B will reach the starting point together.

### Step 1: Determine the relative speed
- Speed of A = 15 m/s
- Speed of B = 25 m/s

Since both are running in the same direction, we find the relative speed:
\[ \text{Relative Speed} = \text{Speed of B} - \text{Speed of A} = 25 \, \text{m/s} - 15 \, \text{m/s} = 10 \, \text{m/s} \]

### Step 2: Calculate the time to meet for the first time
To find out when they will meet, we need to determine how long it takes for B to gain a full lap (1200 m) on A:
\[ \text{Time to meet} = \frac{\text{Distance}}{\text{Relative Speed}} = \frac{1200 \, \text{m}}{10 \, \text{m/s}} = 120 \, \text{seconds} \]

### Step 3: Determine the time to reach the starting point together
To find the time when they both reach the starting point together, we need to find the least common multiple (LCM) of the times taken by A and B to complete one lap.

- Time taken by A to complete one lap:
\[ \text{Time}_{A} = \frac{\text{Circumference}}{\text{Speed of A}} = \frac{1200 \, \text{m}}{15 \, \text{m/s}} = 80 \, \text{seconds} \]

- Time taken by B to complete one lap:
\[ \text{Time}_{B} = \frac{\text{Circumference}}{\text{Speed of B}} = \frac{1200 \, \text{m}}{25 \, \text{m/s}} = 48 \, \text{seconds} \]

Now, we find the LCM of 80 seconds and 48 seconds.

### Step 4: Calculate the LCM
- The prime factorization of 80 is \( 2^4 \times 5 \).
- The prime factorization of 48 is \( 2^4 \times 3 \).

The LCM is calculated by taking the highest power of each prime:
\[ \text{LCM} = 2^4 \times 3^1 \times 5^1 = 16 \times 3 \times 5 = 240 \, \text{seconds} \]

### Final Answers
1. They will meet for the first time after **120 seconds**.
2. They will reach the starting point together after **240 seconds**.