Shyam went from Delhi to Simla via Chandigarh by car. The distance from Delhi to Chandigarh is 3/4 times the distance from Chandigarh to Simla. The average speed from Delhi to Chandigarh was half as much again as that from Chandigarh to Simla. If the average speed for the entire journey was 49 kmph. What was the average speed from Chandigarh to Simla?

Options

A. A) 39.2 kmph
B. B) 63 kmph
C. C) 42 kmph
D. D) 21 kmph

Answer

To find the average speed from Chandigarh to Simla, we can break down the problem step by step.

### Step 1: Define the Distances
Let the distance from Chandigarh to Simla be \( x \) km.
According to the problem, the distance from Delhi to Chandigarh is \( \frac{3}{4} \) times the distance from Chandigarh to Simla. Therefore, the distance from Delhi to Chandigarh is:
\[ \text{Distance from Delhi to Chandigarh} = \frac{3}{4}x \text{ km} \]

### Step 2: Define the Speeds
Let the average speed from Chandigarh to Simla be \( v \) km/h.
The average speed from Delhi to Chandigarh is stated to be "half as much again" as that from Chandigarh to Simla. This means:
\[ \text{Speed from Delhi to Chandigarh} = \frac{3}{2}v \text{ km/h} \]

### Step 3: Calculate Total Distance
The total distance for the journey from Delhi to Simla via Chandigarh is:
\[ \text{Total Distance} = \text{Distance from Delhi to Chandigarh} + \text{Distance from Chandigarh to Simla} = \frac{3}{4}x + x = \frac{3}{4}x + \frac{4}{4}x = \frac{7}{4}x \text{ km} \]

### Step 4: Calculate Time for Each Segment
The time taken to travel from Delhi to Chandigarh is given by:
\[ \text{Time from Delhi to Chandigarh} = \frac{\text{Distance}}{\text{Speed}} = \frac{\frac{3}{4}x}{\frac{3}{2}v} = \frac{3}{4}x \cdot \frac{2}{3v} = \frac{x}{2v} \text{ hours} \]

The time taken to travel from Chandigarh to Simla is:
\[ \text{Time from Chandigarh to Simla} = \frac{x}{v} \text{ hours} \]

### Step 5: Calculate Total Time
The total time for the journey is:
\[ \text{Total Time} = \text{Time from Delhi to Chandigarh} + \text{Time from Chandigarh to Simla} = \frac{x}{2v} + \frac{x}{v} = \frac{x}{2v} + \frac{2x}{2v} = \frac{3x}{2v} \text{ hours} \]

### Step 6: Use Average Speed to Relate Total Distance and Total Time
The average speed for the entire journey is given as 49 km/h. We can use the formula:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]
Substituting the values we have:
\[ 49 = \frac{\frac{7}{4}x}{\frac{3x}{2v}} \]

### Step 7: Simplify the Equation
Cross-multiplying gives:
\[ 49 \cdot \frac{3x}{2v} = \frac{7}{4}x \]
Cancelling \( x \) from both sides (assuming \( x \neq 0 \)):
\[ 49 \cdot \frac{3}{2v} = \frac{7}{4} \]

### Step 8: Solve for \( v \)
Rearranging gives:
\[ \frac{3 \cdot 49}{2v} = \frac{7}{4} \]
Cross-multiplying:
\[ 3 \cdot 49 \cdot 4 = 7 \cdot 2v \]
\[ 588 = 14v \]
\[ v = \frac{588}{14} = 42 \text{ km/h} \]

### Conclusion
The average speed from Chandigarh to Simla is \( \boxed{42} \) km/h.