A tram overtakes 2 persons X and Y walking at an average speed of 3 km/hr and 4 km/hr in the same direction and completely passes them in 8 seconds and 9 seconds respectively. What is the length of the tram?

The correct option is 20 m.

Step-by-Step Explanation:

1. Define the Variables:

Let the speed of the tram be $v$ km/hr, and let the length of the tram be $L$ meters.

2. Convert Speeds to meters per second (m/s):

To convert speed from km/hr to m/s, we multiply by $\frac{5}{18}$.

Speed of person X, $v_X=3\times \frac{5}{18}=\frac{5}{6}$ m/s

Speed of person Y, $v_Y=4\times \frac{5}{18}=\frac{10}{9}$ m/s

Let the speed of the tram in m/s be $S=v\times \frac{5}{18}$ m/s.

3. Calculate Relative Speeds:

Since the tram and the persons are moving in the same direction, the relative speed is obtained by subtracting the speed of the person from the speed of the tram.

Relative speed of tram with respect to X $=S-\frac{5}{6}$

Relative speed of tram with respect to Y $=S-\frac{10}{9}$

4. Set up Distance Equations:

Distance covered to completely pass a person is equal to the length of the tram ($L$).

Using the formula $\text{Distance}=\text{Relative Speed}\times \text{Time}$:

For person X (time = 8 seconds):

$L=\left(S-\frac{5}{6}\right)\times 8$

For person Y (time = 9 seconds):

$L=\left(S-\frac{10}{9}\right)\times 9$

5. Solve for Tram's Speed ($S$):

Equating the two expressions for $L$:

$8S-8\times \frac{5}{6}=9S-9\times \frac{10}{9}$

$8S-\frac{20}{3}=9S-10$

$9S-8S=10-\frac{20}{3}$

$S=\frac{30-20}{3}=\frac{10}{3}$ m/s

6. Calculate the Length of the Tram ($L$):

Substitute $S=\frac{10}{3}$ into the equation for person Y:

$L=\left(\frac{10}{3}-\frac{10}{9}\right)\times 9$

$L=\left(\frac{30}{9}-\frac{10}{9}\right)\times 9$

$L=\frac{20}{9}\times 9=20$ m

Thus, the length of the tram is 20 m.