A, B and C run an a circular track whose radius is 21 m. Both A and B run in the opposite direction of C. If the speed of A, B and C is 10, 15 and 35 m/ minutes respectively. Then find after how much time, they will collectively meet first time and also find after how much time they will reach at the starting point together.

Answer

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

1. The time after which A, B, and C will collectively meet for the first time.
2. The time after which they will all reach the starting point together.

### Step 1: Calculate the circumference of the circular track

The formula for the circumference \( C \) of a circle is given by:
\[ C = 2 \pi r \]
Where \( r \) is the radius of the circle. Given that the radius \( r = 21 \) m, we can substitute this value into the formula.

\[ C = 2 \times \frac{22}{7} \times 21 \]
Calculating this gives:
\[ C = 2 \times 22 \times 3 = 132 \text{ m} \]

### Step 2: Determine the speeds of A, B, and C

The speeds are given as follows:
- Speed of A = 10 m/min
- Speed of B = 15 m/min
- Speed of C = 35 m/min

### Step 3: Calculate the relative speeds

Since A and B are running in the opposite direction to C, we need to calculate the relative speeds for A and C, and B and C.

- **Relative speed of A and C**:
\[ \text{Relative Speed (A and C)} = \text{Speed of A} + \text{Speed of C} = 10 + 35 = 45 \text{ m/min} \]

- **Relative speed of B and C**:
\[ \text{Relative Speed (B and C)} = \text{Speed of B} + \text{Speed of C} = 15 + 35 = 50 \text{ m/min} \]

### Step 4: Calculate the time taken for A and C to meet

Using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
The time \( T_1 \) for A and C to meet is:
\[ T_1 = \frac{132 \text{ m}}{45 \text{ m/min}} = \frac{132}{45} = \frac{44}{15} \text{ min} \]

### Step 5: Calculate the time taken for B and C to meet

Similarly, the time \( T_2 \) for B and C to meet is:
\[ T_2 = \frac{132 \text{ m}}{50 \text{ m/min}} = \frac{132}{50} = \frac{66}{25} \text{ min} \]

### Step 6: Find the least common multiple (LCM) of \( T_1 \) and \( T_2 \)

To find when they will collectively meet for the first time, we need to find the LCM of \( T_1 \) and \( T_2 \).

1. Convert \( T_1 \) and \( T_2 \) to a common denominator:
- \( T_1 = \frac{44}{15} \)
- \( T_2 = \frac{66}{25} \)

2. The LCM of the numerators (44 and 66) is 132.
3. The GCD of the denominators (15 and 25) is 5.

Thus, the LCM of \( T_1 \) and \( T_2 \) is:
\[ \text{LCM} = \frac{132}{5} \text{ min} \]

### Step 7: Calculate the time taken for A, B, and C to reach the starting point together

To find the time taken for all three to reach the starting point together, we calculate the time taken by each to complete one lap:

- Time taken by A:
\[ T_A = \frac{132 \text{ m}}{10 \text{ m/min}} = 13.2 \text{ min} \]

- Time taken by B:
\[ T_B = \frac{132 \text{ m}}{15 \text{ m/min}} = 8.8 \text{ min} \]

- Time taken by C:
\[ T_C = \frac{132 \text{ m}}{35 \text{ m/min}} = 3.77 \text{ min} \]

Now, we find the LCM of \( T_A, T_B, \) and \( T_C \):
1. The LCM of the numerators (132, 88, and 132) is 132.
2. The GCD of the denominators (1, 1, and 35) is 1.

Thus, the LCM is:
\[ \text{LCM} = \frac{132}{1} = 132 \text{ min} \]

### Final Answers
1. They will collectively meet for the first time after \( \frac{132}{5} \) minutes.
2. They will reach the starting point together after 132 minutes.