An express train, Train A, having a length of M meters passes a signal post in 15 seconds. Another train, Train B, of length N meters passes the same signal post in 10 seconds. If Train A completely passes Train B while Train B is stationary at a depot in 25 seconds, determine the ratio of the length of Train A to Train B.

The correct option is 3:2.

Step-by-step Explanation:

Step 1: Express the speed of Train A using the signal post crossing time.

When a train passes a signal post, the distance covered is equal to the length of the train itself.

Length of Train A =

$M$

Time taken to pass the signal post = 15 seconds

Speed is calculated as distance divided by time. Therefore, the speed of Train A,

$v_A$

is given by the expression:

$v_A=\frac{M}{15}$

Step 2: Express the speed of Train A when passing stationary Train B.

When Train A completely passes Train B while Train B is stationary at a depot, the total distance covered is the sum of the lengths of both trains:

$M+N$

Time taken to pass Train B = 25 seconds

Using the speed formula, the speed of Train A can also be written as:

$v_A=\frac{M+N}{25}$

Step 3: Equate the two speed expressions to calculate the ratio M:N.

Since the speed of Train A remains constant, we set the two expressions equal to each other:

$\frac{M}{15}=\frac{M+N}{25}$

Cross-multiply to eliminate the denominators:

$25\times M=15\times (M+N)$

Divide both sides of the equation by 5 to simplify:

$5M=3(M+N)$

Expand the right side:

$5M=3M+3N$

Subtract 3M from both sides:

$2M=3N$

Rearrange the equation to find the ratio of M to N:

$\frac{M}{N}=\frac{3}{2}$

Thus, the ratio of the length of Train A to Train B is 3:2.