A cargo ferry travels at a speed of 60 km/h in calm water, and the river current flows at a speed equal to one-third of the ferry's speed in calm water. If the difference between the time taken by the ferry to travel a distance of D km upstream and the time taken to travel the same distance downstream is 5 hours, find the value of D.

The correct answer is 400.

Given Data:

Speed of the cargo ferry in calm water ($v_b$) = 60 km/h

Speed of the river current ($v_c$) is equal to one-third of the ferry's speed in calm water:

$v_c=\frac{1}{3}\times 60=20$ km/h

Step 1: Calculate Downstream and Upstream Speeds

When travelling downstream, the river current assists the ferry. Therefore, downstream speed ($v_d$) is the sum of the ferry's speed in calm water and the river current's speed:

$v_d=v_b+v_c=60+20=80$ km/h

When travelling upstream, the river current opposes the ferry. Therefore, upstream speed ($v_u$) is the difference between the ferry's speed in calm water and the river current's speed:

$v_u=v_b-v_c=60-20=40$ km/h

Step 2: Express the Time Taken for Each Journey

Let $D$ be the total distance in kilometers.

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

Time taken to travel downstream ($t_d$):

$t_d=\frac{D}{80}$ hours

Time taken to travel upstream ($t_u$):

$t_u=\frac{D}{40}$ hours

Step 3: Set up the Difference Equation and Solve for D

The difference between the time taken to travel upstream and downstream is 5 hours:

$t_u-t_d=5$

Substitute the expressions for $t_u$ and $t_d$ into the equation:

$\frac{D}{40}-\frac{D}{80}=5$

Take a common denominator of 80 to combine the fractions:

$\frac{2D-D}{80}=5$

Simplify the numerator:

$\frac{D}{80}=5$

Multiply both sides by 80:

$D=5\times 80=400$

Thus, the value of $D$ is 400 km.