Two runners cover the same distance at the rate of 10 km/h and 15 km/h respectively. Find the distance travelled when one takes 16 mintes longer than the other ?

Answer

To solve the problem, we need to find the distance traveled by the two runners given their speeds and the time difference. Let’s break it down step by step.

### Step 1: Define the Variables
Let the distance covered by both runners be \( D \) kilometers.

### Step 2: Calculate the Time Taken by Each Runner
- The time taken by the first runner (speed = 10 km/h) is given by:
\[ \text{Time}_1 = \frac{D}{10} \text{ hours} \]

- The time taken by the second runner (speed = 15 km/h) is given by:
\[ \text{Time}_2 = \frac{D}{15} \text{ hours} \]

### Step 3: Set Up the Equation for Time Difference
According to the problem, one runner takes 16 minutes longer than the other. We need to convert 16 minutes into hours:
\[ 16 \text{ minutes} = \frac{16}{60} \text{ hours} = \frac{4}{15} \text{ hours} \]

Now, we can set up the equation based on the time difference:
\[ \text{Time}_1 = \text{Time}_2 + \frac{4}{15} \]

Substituting the expressions for time:
\[ \frac{D}{10} = \frac{D}{15} + \frac{4}{15} \]

### Step 4: Solve the Equation
To eliminate the fractions, we can multiply the entire equation by 30 (the least common multiple of 10 and 15):
\[ 30 \cdot \frac{D}{10} = 30 \cdot \left(\frac{D}{15} + \frac{4}{15}\right) \]
This simplifies to:
\[ 3D = 2D + 8 \]

Now, subtract \( 2D \) from both sides:
\[ 3D - 2D = 8 \]
\[ D = 8 \]

### Step 5: Conclusion
The distance travelled by both runners is \( D = 8 \) kilometers.

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