A man undertakes a journey of 240 km. He travels some of the way by train at 48 km/hour and the rest by car in $x$ hours at 72 km/hour. If the total time is 4 hours, calculate what distance he travels by train and by car.

1. Problem Description

A man undertakes a journey of 240 km. He travels some of the way by train at 48 km/hour and the rest by car in $x$ hours at 72 km/hour. If the total time is 4 hours, calculate what distance he travels by train and by car.

2. Solution Steps

Let $t$ be the time spent traveling by train in hours.

The time spent traveling by car is given as $x$ hours. Since the total time is 4 hours, we have $x+t = 4$. Therefore, $t=4-x$.

The distance traveled by train is $d_{train} = 48t = 48(4-x)$ km.

The distance traveled by car is $d_{car} = 72x$ km.

The total distance is 240 km, so $d_{train} + d_{car} = 240$.

Substituting the expressions for $d_{train}$ and $d_{car}$, we have:

$48(4-x) + 72x = 240$

$192 - 48x + 72x = 240$

$24x = 240 - 192$

$24x = 48$

$x = \frac{48}{24}$

$x = 2$

The time spent traveling by car is $x = 2$ hours.

The time spent traveling by train is $t = 4-x = 4-2 = 2$ hours.

The distance traveled by train is $d_{train} = 48t = 48(2) = 96$ km.

The distance traveled by car is $d_{car} = 72x = 72(2) = 144$ km.

3. Final Answer

Distance traveled by train is 96 km.

Distance traveled by car is 144 km.