Question

Which of the following statement(s) is/are correct?

I: A boat can travel 156 km in downstream in 26 hours and it takes same time to cover 52 km in upstream. Speed of the stream is 2 km/h.

II: The speed of train M is 96 km/h, and it is 380 meters long, then the time taken by train M to cross a train N which is 580 meters long and running in the opposite direction at 64 km/h is 21.6 seconds?

• A. Only I
• B. Both I and II
• C. Only II
• D. Neither I nor II

Answer 1

Answer: (B)

Both I and II

Answer 2

Statement I Analysis:

Let the speed of the boat in still water be $v_b$ km/h and the speed of the stream be $v_s$ km/h.

Downstream speed = $v_b + v_s$ Upstream speed = $v_b - v_s$

Given, the boat travels 156 km downstream in 26 hours. Downstream speed $=$ $\frac{\text{Distance}}{\text{Time}} = \frac{156 \text{ km}}{26 \text{ hours}} = 6 \text{ km/h}$. So, $v_b + v_s = 6$ (Equation 1)

Given, it takes the same time (26 hours) to cover 52 km in upstream. Upstream speed $=$ $\frac{\text{Distance}}{\text{Time}} = \frac{52 \text{ km}}{26 \text{ hours}} = 2 \text{ km/h}$. So, $v_b - v_s = 2$ (Equation 2)

Add Equation 1 and Equation 2: $(v_b + v_s) + (v_b - v_s) = 6 + 2$ $2v_b = 8$ $v_b = 4 \text{ km/h}$

Substitute $v_b = 4$ into Equation 1: $4 + v_s = 6$ $v_s = 6 - 4$ $v_s = 2 \text{ km/h}$

The speed of the stream is 2 km/h, which matches the statement. Therefore, Statement I is correct.

Statement II Analysis:

Speed of train M ($S_M$) = 96 km/h Length of train M ($L_M$) = 380 meters Speed of train N ($S_N$) = 64 km/h Length of train N ($L_N$) = 580 meters

Since the trains are running in opposite directions, their relative speed is the sum of their individual speeds. Relative speed ($S_{rel}$) = $S_M + S_N = 96 \text{ km/h} + 64 \text{ km/h} = 160 \text{ km/h}$.

To calculate time in seconds, convert the relative speed from km/h to m/s. $1 \text{ km/h} = \frac{5}{18} \text{ m/s}$ $S_{rel} = 160 \times \frac{5}{18} \text{ m/s} = \frac{800}{18} \text{ m/s} = \frac{400}{9} \text{ m/s}$.

When one train crosses another, the total distance to be covered is the sum of their lengths. Total distance ($D$) = $L_M + L_N = 380 \text{ m} + 580 \text{ m} = 960 \text{ m}$.

Time taken ($T$) = $\frac{\text{Total Distance}}{\text{Relative Speed}}$ $T = \frac{960 \text{ m}}{\frac{400}{9} \text{ m/s}} = \frac{960 \times 9}{400} \text{ s}$ $T = \frac{96 \times 9}{40} \text{ s} = \frac{12 \times 9}{5} \text{ s} = \frac{108}{5} \text{ s}$ $T = 21.6 \text{ seconds}$.

The time taken by train M to cross train N is 21.6 seconds, which matches the statement. Therefore, Statement II is correct.

Since both Statement I and Statement II are correct, the correct option is B.