A motorcyclist covers a distance from point A to point B at an average speed of 32 kmph and return journey from point B to point A at an average speed of 60 kmph. If he takes 11.5 hours for the whole journey, then find the distance between point A and point B.

Options

A. A)480 km
B. B)360 km
C. C)240 km
D. D)180 km

Answer

To solve the problem step by step, we will denote the distance between point A and point B as \( D \) kilometers.

### Step 1: Define the variables
Let:
- \( D \) = distance from A to B (in km)
- \( t \) = time taken to travel from A to B (in hours)
- The time taken to travel from B to A will then be \( 11.5 - t \) hours.

### Step 2: Write the equations for time
Using the formula for time, which is \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \), we can express the time taken for each leg of the journey:
- From A to B:
\[ t = \frac{D}{32} \]
- From B to A:
\[ 11.5 - t = \frac{D}{60} \]

### Step 3: Substitute \( t \) in the second equation
Substituting the expression for \( t \) from the first equation into the second equation:
\[ 11.5 - \frac{D}{32} = \frac{D}{60} \]

### Step 4: Clear the fractions
To eliminate the fractions, we can multiply the entire equation by the least common multiple of 32 and 60, which is 960:
\[ 960 \left( 11.5 - \frac{D}{32} \right) = 960 \left( \frac{D}{60} \right) \]
This simplifies to:
\[ 11040 - 30D = 16D \]

### Step 5: Combine like terms
Now, we can combine the terms involving \( D \):
\[ 11040 = 30D + 16D \]
\[ 11040 = 46D \]

### Step 6: Solve for \( D \)
Now, divide both sides by 46 to find \( D \):
\[ D = \frac{11040}{46} = 240 \text{ km} \]

### Conclusion
The distance between point A and point B is \( 240 \) km.

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