Question

A train travels from A to B. while travelling from A to B its average speed was 60 km/h and while returning from B to A its average speed was se 90 km/h. Find the average speed of the train during the whole journey.

Answer 1

Let the Distance between A and B be : D

Given : Train travels from A to B with a speed 60 kmph

$\bigstar\;\;\textsf{We know that : \boxed{\mathsf{Speed = \dfrac{Distance\;traveled}{Time\;taken}}}}$

$\mathsf{\implies \dfrac{D}{Time\;taken} = 60}$

$\implies \mathsf{Time\;taken\;to\;travel\;from\;A\;to\;B\;with\;speed\;60\;kmph = \dfrac{D}{60}}$

Given : Train travels from B to A with a speed 90 kmph

$\implies \mathsf{Time\;taken\;to\;travel\;from\;B\;to\;A\;with\;speed\;90\;kmph = \dfrac{D}{90}}$

We know that :

$\bigstar\;\;\boxed{\mathsf{Average\;Speed = \dfrac{Total\;Distance\;traveled}{Total\;Time\;taken}}}$

★  Total Distance traveled by the Train = (D + D) = 2D

★  $\mathsf{Total\;Time\;taken\;by\;the\;Train = \bigg(\dfrac{D}{60} + \dfrac{D}{90}\bigg)}$

$\mathsf{\implies \dfrac{D}{10}\bigg(\dfrac{1}{6} + \dfrac{1}{9}\bigg)}$

$\mathsf{\implies \dfrac{D}{10}\bigg(\dfrac{3 + 2}{18}\bigg)}$

$\implies \mathsf{Total\;Time\;taken\;by\;the\;Train = \dfrac{D}{10}\bigg(\dfrac{5}{18}\bigg)}}$

$\mathsf{\implies Average\;Speed = \dfrac{2D}{\dfrac{D}{10}\bigg(\dfrac{5}{18}\bigg)}}$

$\mathsf{\implies Average\;Speed = \bigg(\dfrac{2 \times 180}{5}\bigg)}$

$\mathsf{\implies Average\;Speed = (2 \times 36)}$

$\mathsf{\implies Average\;Speed = 72}$

Answer : Average Speed of the Train = 72 kmph

Answer 2

$\mathfrak{Answer:}$

= 72 km h⁻¹.

$\mathfrak{Step-by-Step\;Explanation:}$

$\underline{\underline{\bold{Given\;in\;the\;Question:}}}$

• While traveling A to B its average speed = 60 km h⁻¹.
• And while travelling B to A its average speed = 90 km h⁻¹.

Let the distance from A to B be x km.

In First case ,

• Distance = x km.
• Speed = 60 km h⁻¹.

$\boxed{\bold{Time=\dfrac{Distance}{Speed}}}\\\\\\\tt{=\dfrac{x\;km}{60\;km\;h^{-1}}}\\\\\\\tt{=\dfrac{x}{60}\;hrs.}$

In second case,

• Distance = x km.
• Speed = 90 km h⁻¹.

$\bold{Time=\dfrac{x\;km}{90\;km\;h^{-1}}}\\\\\\\tt{=\dfrac{x}{90}\;hrs.}\\\\\\\boxed{\bold{Average\;Speed=\dfrac{Total\;distance}{Total\;time}}}\\\\\\\tt{=\dfrac{x+x}{\frac{x}{60}+\frac{x}{90}}}\\\\\\\tt{=\dfrac{2x}{\frac{3x+2x}{180}}}\\\\\\\tt{=\dfrac{2x}{5x}\times 180}\\\\\\\tt{=72\;km\;h^{-1}}\\\\\\\\\boxed{\boxed{\bold{Average\;Speed=72\;km\;h^{-1}.}}}$