Question

A man travels 370 km partly by train and partly by car. If he covers 250 km by train and the rest by car it takes him 4 hours. but , if he travels 130 km by train and the rest by car, he takes 18 minutes. find the speed of the train and that of the car.

Answer 1

let the speed of the train be X km/h
let speed of the car be y km/h
time=distance/speed
in situation 1:time spent by train=250/X
and by car=120/y
total time taken=250/X+120/y
but time of journey in 4hr,
250/X+120/y=4
cancel by 2 we get.125/X+60/y=2........eq1
hope this helps you.....

Answer 2

$\fcolorbox{green}{Pink}{ \huge{solution}}$

Let the speed of the train be x km/hr and that of the car by y km/hr.

$\bold{case \: 1}. \bold{ \underline{when \: he \: travels \: 250km \: by \: train \: and \: rest }} \\ \bold{ \underline {by \: car.}}$

Time taken by the man to travel 250 km by train = hours

Time taken by the man to travel (370 - 250)
= 120 km by car = 120/y hours.

$total \: time \: taken \: by \: the \: man \: to \: cover \: 370 \: \\ km \: = \frac{250}{x} + \frac{120}{y} hours$

Given, Total time = 4 hrs.

$\frac{250}{x} + \frac{120}{y} = 4 \\ \implies \: \frac{125}{x} + \frac{60}{y} = 2$

$\bold{case \: 2.} \bold{ \underline{when \: he \: travels \: 130km \: by \: train \: and \: the \: rest \: }} \\ \bold{ \underline{by \: car.}}$

Time taken by the man to travel 130 km by train = 130/x hrs

Time taken by the man to travel (370 - 130) = 240 km by train = 240/y hours

∴ Total time of the journey is 4 hours 18 minutes.

$\implies \: \frac{130}{x} + \frac{240}{y} = 4 \: hours \: 18 \: min or \\ \frac{130}{x} + \frac{240}{y} = 4 \frac{18}{60} = \frac{43}{10}$

Thus, we obtain the following system of equations:

$\frac{125}{x} + \frac{60}{y} = 2 \\ \frac{130}{x} + \frac{240}{y} = \frac{43}{10}$

Let 1/x = a and 1/y = b, then the above reduces to

125 a + 60 b = 2

130 a + 240 b = 43/10

multiplying equation (3) by 4, the above equation become

500 a + 240 b = 8

130 a + 240 b = 43/10

subtracting equation (6) from equation (5), we have

$370 \: a \: = 8 - \frac{43}{10} = \frac{80 - 43}{10} = \frac{37}{10} \\ \implies \: a \: = \frac{1}{100}$

Putting the value of a in (5) we have

$500 \times \frac{1}{100} + 240b = 8$

$\implies \: 5 + 240b = 8 \\ \implies240b = 8 - 5 = 3 \\ b \: = \frac{3}{240} = \frac{1}{80}$

Now
$a = \frac{1}{x} = \frac{1}{100} \:and \: b \: = \frac{1}{y} = \frac{1}{80} \\ \implies \: x = 100 \implies \: y = 80.$

$\implies \bold{ \mathfrak{ \underline{speed \: of \: train \: = 100km/hr \: speed \: of \: car 80 km/h}}}$

$\mathfrak{ \red{glad \: help \: you}}$
$\mathfrak{ \pink{it \: helps \: \: you}}$
$\mathfrak{thanks}$

$\fcolorbox{red}{aqua}{@vaibhavhoax}$