Train ‘A’ takes 2 hours more than train ‘B’ to cover certain distance ‘X’ while train ‘A’ can cover (X + 160) is 8 hours. If speed of train ‘B’ is 1.5 times of speed of train ‘A’, then find the speed of train ‘B’.

Correct Answer: 120 km/hr

Let's break down the problem step-by-step to find the speed of train B.

Step 1: Define variables for speeds and distance.
Let the speed of train A be $S_A$ and the speed of train B be $S_B$.
Let the distance be $X$ km.

Step 2: Use the relationship between the speeds of train A and train B.
It is given that the speed of train B is 1.5 times the speed of train A:

$S_B=1.5\times S_A$

$\frac{S_B}{S_A}=\frac{3}{2}$

Since time taken is inversely proportional to speed for a fixed distance $X$, the ratio of time taken by train A ($T_A$) to time taken by train B ($T_B$) to cover distance $X$ is:

$\frac{T_A}{T_B}=\frac{S_B}{S_A}=\frac{3}{2}$

Step 3: Calculate the time taken by train A and train B to cover distance X.
We are given that train A takes 2 hours more than train B to cover distance $X$:

$T_A-T_B=2$

Let $T_{}=3t$ and $T_{}=2t$ hours.

$3t-2t=2$

$t=2$

Therefore, time taken by train A to cover distance $X$ is:

$T_A=3\times 2=6\text{hours}$

Step 4: Express distance X in terms of speed of train A.
Since distance = speed × time:

$X=S_A\times T_A=6S_A$

Step 5: Find the speed of train A using the second condition.
We are given that train A can cover a distance of $(X+160)$ km in 8 hours:

$\frac{X+160}{S_A}=8$

Substitute $X=6S_A$ into the equation:

$6S_A+160=8S_A$

$2S_A=160$

$S_A=80\text{km/hr}$

Step 6: Calculate the speed of train B.
Now, find $S_B$ using $S_B=1.5\times S_A$:

$S_B=1.5\times 80=120\text{km/hr}$

Thus, the speed of train B is 120 km/hr.