Question

A train travelled a certain distance at a uniform speed. Had the speed been 6 km per hour more, it would have needed 4 hours less. Had the speed been 6 km per hour less,it would have needed 6 hours more. The distance, in km, travelled by the train is

• A. 800
• B. 640
• C. 720
• D. 780

Answer 1

The Correct Option is C

The problem involves finding the distance traveled by a train given varying speeds and time conditions. Let's denote the original speed of the train as \(v\) km/h and the distance it traveled as \(d\) km. 

According to the problem, the time taken to travel the distance at the original speed is \(\frac{d}{v}\) hours.

Condition 1: If the speed is increased by 6 km/h, the journey takes 4 hours less.

\(\frac{d}{v+6} = \frac{d}{v} - 4\)

Condition 2: If the speed is decreased by 6 km/h, the journey takes 6 hours more.

\(\frac{d}{v-6} = \frac{d}{v} + 6\)

We have a system of equations:

(1) \(\frac{d}{v+6} = \frac{d}{v} - 4\)

(2) \(\frac{d}{v-6} = \frac{d}{v} + 6\)

Let's solve equation (1):

\(\Rightarrow d\left(\frac{1}{v+6} - \frac{1}{v}\right) = -4\)

 

\(\Rightarrow d\frac{v - (v+6)}{v(v+6)} = -4\)

\(\Rightarrow d\left(\frac{-6}{v(v+6)}\right) = -4\)

\(\Rightarrow \frac{d}{v(v+6)} = \frac{2}{3}\)

Let's solve equation (2):

\(\Rightarrow d\left(\frac{1}{v-6} - \frac{1}{v}\right) = 6\)

 

\(\Rightarrow d\frac{v - (v-6)}{v(v-6)} = 6\)

\(\Rightarrow d\left(\frac{6}{v(v-6)}\right) = 6\)

\(\Rightarrow \frac{d}{v(v-6)} = 1\)

We have the following:

\[\frac{d}{v(v+6)} = \frac{2}{3}\]

\[\frac{d}{v(v-6)} = 1\]

From these,:

\[\frac{1}{v(v+6)} = \frac{2}{3d}\]

\[\frac{1}{v(v-6)} = \frac{1}{d}\]

Divide these equations:

\[\frac{2}{3d} \div \frac{1}{d} = \frac{1 \cdot d}{v(v+6) \cdot 3d} \cdot \frac{v(v-6)}{d} = \frac{2}{3}\]

\[\Rightarrow \frac{v(v-6)}{v(v+6)} = \frac{2}{3}\]

Simplify:

\[\left(\frac{v-6}{v+6}\right) = \frac{2}{3}\]

\[3(v-6) = 2(v+6)\]

\[3v - 18 = 2v + 12\]

\[v = 30\]

Plug \(v\) into \(\frac{d}{v(v-6)} = 1\):

\[\frac{d}{30(24)} = 1\]

\[d = 720\]

Thus, the distance traveled by the train is 720 km.