Rakesh runs `1(1)/(3)` times as fast as Mukesh. In a race , If Rakesh gives a head start of 60 m to Mukesh, then after running for how many metres does Rakesh meet Mukesh ?

Answer

To solve the problem, we need to determine how far Rakesh runs before he meets Mukesh, given that Rakesh runs \( \frac{4}{3} \) times as fast as Mukesh and gives Mukesh a head start of 60 meters.

### Step-by-Step Solution:

1. **Define Speeds**:
Let the speed of Mukesh be \( v \) meters per second. Then, the speed of Rakesh will be \( \frac{4}{3}v \) meters per second.

2. **Head Start**:
Mukesh starts 60 meters ahead of Rakesh. This means when Rakesh starts running, Mukesh is already at the 60-meter mark.

3. **Relative Speed**:
The relative speed at which Rakesh is catching up to Mukesh is the difference in their speeds:
\[ \text{Relative Speed} = \text{Speed of Rakesh} - \text{Speed of Mukesh} = \frac{4}{3}v - v = \frac{4}{3}v - \frac{3}{3}v = \frac{1}{3}v \]

4. **Time to Meet**:
To find out how long it takes for Rakesh to catch up to Mukesh, we can use the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
The distance Rakesh needs to cover to catch up is the 60 meters head start. Thus:
\[ \text{Time} = \frac{60 \text{ meters}}{\frac{1}{3}v} = 60 \times \frac{3}{v} = \frac{180}{v} \text{ seconds} \]

5. **Distance Rakesh Runs**:
Now, we can find out how far Rakesh runs in that time:
\[ \text{Distance Rakesh runs} = \text{Speed of Rakesh} \times \text{Time} = \frac{4}{3}v \times \frac{180}{v} \]
Simplifying this:
\[ \text{Distance Rakesh runs} = \frac{4}{3} \times 180 = 240 \text{ meters} \]

### Conclusion:
Rakesh meets Mukesh after running **240 meters**.