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

- $39.2$ kmph
- $63$ kmph
- $42$ kmph
- None of the above

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Let’s draw the diagram for a better understanding. Let the distance from Chandigarh to Shimla be $x$ km. Then the distance from Delhi to Chandigarh $= \frac{3}{4}x$ km.

DIAGRAM

We know that, $\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total TIme}}$

(OR)

$\text{Speed} = \frac{\text{Distance}}{\text{TIme}}$

Let the average speed from chandigarh to shimla be $y \; \frac{km}{h}.$ Then the average speed from Delhi to Chandigarh $= \frac{3}{2} \; y \; \frac{km}{hr}.$

Now, $\frac{x + \frac{3x}{4}}{\frac{x}{y} + \frac{\frac{3x}{4}}{\frac{3y}{2}} } = 49 $

$ \Rightarrow \frac{\frac{4x+3x}{4}}{\frac{x}{y} + \frac{3x}{4} \times \frac{2}{3y}} = 49$

$ \Rightarrow \frac{\frac{7x}{4}}{\frac{x}{y} + \frac{x}{2y}} = 49$

$ \Rightarrow \frac{\frac{7x}{4}}{\frac{2x+x}{2y}} = 49$

$\Rightarrow \frac{7x}{4} \times \frac{2y}{3x} =49$

$\Rightarrow \frac{7y}{6} =49$

$\Rightarrow \boxed{ y =42 \; \frac{km}{hr}}$

$\therefore$ The average speed from Chandigarh to Shimla $y= 42 \; \frac{km}{hr}.$

Correct Answer $: \text{C} $

DIAGRAM

We know that, $\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total TIme}}$

(OR)

$\text{Speed} = \frac{\text{Distance}}{\text{TIme}}$

Let the average speed from chandigarh to shimla be $y \; \frac{km}{h}.$ Then the average speed from Delhi to Chandigarh $= \frac{3}{2} \; y \; \frac{km}{hr}.$

Now, $\frac{x + \frac{3x}{4}}{\frac{x}{y} + \frac{\frac{3x}{4}}{\frac{3y}{2}} } = 49 $

$ \Rightarrow \frac{\frac{4x+3x}{4}}{\frac{x}{y} + \frac{3x}{4} \times \frac{2}{3y}} = 49$

$ \Rightarrow \frac{\frac{7x}{4}}{\frac{x}{y} + \frac{x}{2y}} = 49$

$ \Rightarrow \frac{\frac{7x}{4}}{\frac{2x+x}{2y}} = 49$

$\Rightarrow \frac{7x}{4} \times \frac{2y}{3x} =49$

$\Rightarrow \frac{7y}{6} =49$

$\Rightarrow \boxed{ y =42 \; \frac{km}{hr}}$

$\therefore$ The average speed from Chandigarh to Shimla $y= 42 \; \frac{km}{hr}.$

Correct Answer $: \text{C} $