Question

A goods train leaves a station at 6pm, followed by an express train which leaves at 8pm and travels 20 km/hr. faster than the goods train. The express train arrives at a station, 1040 km away, 36 min. before the goods train. Assuming that the speeds of both the trains remain constant between the two stations, calculate the speeds.
A. 80 km/hr and 100 km/hr
B. 20 km/hr and 40 km/hr
C. 60 km/hr and 80 km/hr
D. 120 km/hr 100 km/hr

Answer 1

Hint- To solve this question, we use the formula for speed s = $\dfrac{{\text{d}}}{{\text{t}}}$,which means speed equals distance divided by time. And similarly, to solve for time use the formula for time, t = $\dfrac{{\text{d}}}{{\text{s}}}$ which means time equals distance divided by speed.
Formula use- time = $\dfrac{{{\text{distance}}}}{{{\text{speed}}}}$

Complete step by step solution:
Let the speed of the goods train be x km/hr.
∴ speed of the express will be (x+20) km/hr.
And also
Let the time taken by goods train to travel 1040 km be ${{\text{t}}_{\text{g}}}$
and that taken by express train be ${{\text{t}}_e}$
As per the time of departure,
${{\text{t}}_e}$= ${{\text{t}}_g}$- 2 and D=1040 km
As per the time of arrival,
${{\text{t}}_e}$ = ${{\text{t}}_g}$ - $\dfrac{{36}}{{60}}$
∴ Total time taken by express train
${{\text{t}}_e}$= ${{\text{t}}_g}$- 2 -$\dfrac{{36}}{{60}}$​
 $\dfrac{{1040}}{{x + 20}} = \dfrac{{1040}}{x} - \dfrac{{120 + 36}}{{60}}$
$\dfrac{{1040}}{{x + 20}} - \dfrac{{1040}}{x} = - \dfrac{{13}}{5}$
$\dfrac{{x - \left( {x + 20} \right)}}{{\left( {x + 20} \right)x}} = - \dfrac{{13}}{{5 \times 1040}}$
$\dfrac{{ - 20}}{{\left( {x + 20} \right)x}} = - \dfrac{{13}}{{5 \times 1040}}$
$\dfrac{{20}}{{\left( {x + 20} \right)x}} = \dfrac{{13}}{{5 \times 1040}}$
$\dfrac{{20}}{{\left( {x + 20} \right)x}} = \dfrac{1}{{5 \times 80}}$
$\dfrac{1}{{\left( {x + 20} \right)x}} = \dfrac{1}{{8000}}$
$ \Rightarrow $${x^2} + 20x = 8000$
finally, we get
$ \Rightarrow $${x^2} + 20x - 8000 = 0$
$ \Rightarrow $${x^2} + 100x - 80x - 8000 = 0$
$ \Rightarrow $$\left( {x + 100} \right)\left( {x - 80} \right) = 0$
$\therefore $$x = - 100{\text{ or x = 80}}$
Since speed cannot be negative,
Therefore, we consider only x=80 km/hr
Speed of goods train is 80 km/hr and speed of express is 80+20=100 km/hr.
Thus, option (A) is the correct answer.

Note- Speed and time are inversely proportional (when distance is constant)
                                    ⟹ speed ∝ $\dfrac{{\text{1}}}{{{\text{time}}}}$(when distance is constant)
That means If the ratio of the speeds of A and B is a: b, then, the ratio of the time taken by them to cover the same distance is
$\dfrac{{\text{1}}}{{\text{a}}}{\text{:}}\dfrac{{\text{1}}}{{\text{b}}}$= b: a