Question

Shyam went from Delhi to Shimla via Chandigarh by car. The distance from Delhi to Chandigarh is $ \dfrac{3}{4} $ times the distance from Chandigarh to Shimla. The average speed from Delhi to Chandigarh was one and a half times that from Chandigarh to Shimla. If the average speed for the entire journey was 49 km/hr what was the average speed from Chandigarh to Shimla? \[\]
A.39.2 km/hr\[\]
B.63 km/hr\[\]
C.42 km/hr \[\]
D.35 km/hr \[\]

Answer 1

Hint: We assume that the distance between from Chandigarh to Shimla as $ {{d}_{2}}=x $ km and the average speed from Chandigarh to Shimla as $ {{v}_{2}}=y $ km/hr. We use the given data to find the distance between Delhi to Chandigarh as $ {{d}_{2}} $ and average speed from Delhi to Chandigarh as $ {{v}_{2}} $ . We use the formula for average of whole journey as $ v=\dfrac{{{d}_{1}}+{{d}_{2}}}{{{t}_{1}}+{{t}_{2}}} $ to given 49 km/hr where $ {{t}_{1}} $ is the time taken from Delhi to Chandigarh and $ {{t}_{2}} $ is the time taken from Chandigarh to Delhi. \[\]

Complete step by step answer:
Let us denote the distance, average speed and time take for the journey between Delhi to Chandigarh as $ {{d}_{1}} $ km, $ {{v}_{1}} $ km/hr, $ {{t}_{1}} $ respectively. Similarly we denote the distance, average speed and time take for the journey between Chandigarh to Shimla as $ {{d}_{2}} $ km, $ {{v}_{2}} $ km/hr, $ {{t}_{2}} $ respectively.
So the total distance travelled during thw whole jorney is $ \left( {{d}_{1}}+{{d}_{2}} \right) $ km. The total time taken for the whole journey is $ \left( {{t}_{1}}+{{t}_{2}} \right) $ hr. So the average speed say $ v $ of the whole journey is
\[v=\dfrac{{{d}_{1}}+{{d}_{2}}}{{{t}_{1}}+{{t}_{2}}}\text{ km/hr}\]
Let us assume $ {{d}_{2}}=x $ km. We are given in the question that the distance from Delhi to Chandigarh is $ \dfrac{3}{4} $ times the distance from Chandigarh to Shimla. So we have;
 $ {{d}_{1}}=\dfrac{3}{4}{{d}_{2}}=\dfrac{3}{4}x\text{ km} $
Let us assume $ {{v}_{2}}=y $ km/hr. We are given in the question that the average speed from Delhi to Chandigarh was one and a half times that from Chandigarh to Shimla. SO we have
\[{{v}_{1}}=\left( 1\dfrac{1}{2} \right){{v}_{2}}=\dfrac{3}{2}{{v}_{2}}=\dfrac{3}{2}y\text{ km/hr}\]
We use the relation for time taken in terms of distance and average speed find the taken as
\[\begin{align}
  & {{t}_{1}}=\dfrac{{{d}_{1}}}{{{v}_{1}}}=\dfrac{\dfrac{3}{4}x}{\dfrac{3}{2}y}=\dfrac{3}{4}x\times \dfrac{2}{3y}=\dfrac{x}{2y}\text{ hr} \\
 & {{\text{t}}_{2}}=\dfrac{{{d}_{2}}}{{{t}_{2}}}=\dfrac{x}{y}\text{ hr} \\
\end{align}\]
We are given that average speed for the whole journey is 49 km/hr. So the average speed for the whole journey is
\[\begin{align}
  & v=\dfrac{{{d}_{1}}+{{d}_{2}}}{{{t}_{1}}+{{t}_{2}}}=49 \\
 & \Rightarrow \dfrac{\dfrac{3}{4}x+x}{\dfrac{x}{2y}+\dfrac{x}{y}}=49 \\
 & \Rightarrow \dfrac{\dfrac{7x}{4}}{\dfrac{3x}{2y}}=49 \\
 & \Rightarrow \dfrac{7y}{6}=49 \\
 & \Rightarrow y=49\times \dfrac{6}{7}=42\text{ km/hr} \\
\end{align}\]
So the average speed from Chandigarh to Shimla was 42 km/hr.\[\]

Note:
We note that h relation between average speed $ v $ , the time taken $ t $ and total distance travelled $ d $ of an object is given by $ v=\dfrac{d}{t} $ . If an object covers equal distance with speed $ {{v}_{1}},{{v}_{2}} $ successively then the average speed is given for the whole distance is given by $ \dfrac{2{{v}_{1}}{{v}_{2}}}{{{v}_{1}}+{{v}_{2}}} $ km/hr. If an object covered a certain distance at a speed of $ {{v}_{1}} $ and it had moved with more $ {{v}_{1}} $ faster, it would have taken $ {{t}_{1}} $ hours less. If it had moved with less speed $ {{v}_{2}} $ slower, it would have taken time $ {{t}_{2}} $ more. Then average speed is given by $ v=\dfrac{{{v}_{1}}{{v}_{2}}\left( {{t}_{1}}+{{t}_{2}} \right)}{{{v}_{1}}{{t}_{2}}-{{v}_{2}}{{t}_{1}}} $ .