Two runner start runing together for a certain distance one at 5 km/h and another at 3 km/h the former arrives one and half an hour before the latter.the distance( km)

Options

A. 12
B. 20
C. 25
D. 36

Answer

To solve the problem step by step, we can follow these calculations:

### Step 1: Define Variables
Let the speeds of the two runners be:
- Speed of Runner A (faster runner) = 5 km/h
- Speed of Runner B (slower runner) = 3 km/h

Let the distance they run be \( D \) km.

### Step 2: Calculate Time Taken by Each Runner
The time taken by each runner to cover the distance \( D \) can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]

For Runner A:
\[ \text{Time taken by A} = \frac{D}{5} \text{ hours} \]

For Runner B:
\[ \text{Time taken by B} = \frac{D}{3} \text{ hours} \]

### Step 3: Set Up the Equation Based on the Given Information
According to the problem, Runner A arrives 1.5 hours (or \( \frac{3}{2} \) hours) before Runner B. Therefore, we can set up the equation:
\[ \frac{D}{3} - \frac{D}{5} = \frac{3}{2} \]

### Step 4: Find a Common Denominator and Simplify
To solve the equation, we need a common denominator for the fractions on the left side. The least common multiple of 3 and 5 is 15.

Rewriting the equation:
\[ \frac{5D}{15} - \frac{3D}{15} = \frac{3}{2} \]
This simplifies to:
\[ \frac{2D}{15} = \frac{3}{2} \]

### Step 5: Cross-Multiply to Solve for D
Cross-multiplying gives:
\[ 2D \cdot 2 = 3 \cdot 15 \]
\[ 4D = 45 \]

### Step 6: Solve for D
Now, divide both sides by 4:
\[ D = \frac{45}{4} \text{ km} \]

### Final Answer
The distance \( D \) is \( \frac{45}{4} \) km or 11.25 km.

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