A ferry can travel a distance of 270 km downstream and 100 km upstream in 5 hours. Find the time taken by the ferry to cover a distance of 315 km in still water if the ratio of the speed of the ferry in still water to the speed of the river current is 7:2 respectively.

The correct answer is 4.5 hours.

Step 1: Define the ratio of speeds.
Let the speed of the ferry in still water be represented by $u$ and the speed of the river current be represented by $v$.
The ratio of the speed of the ferry in still water to the speed of the river current is 7:2.
We can express these speeds in terms of a common variable $x$:

$u=7x$

$v=2x$

Step 2: Determine downstream and upstream speeds.
Downstream speed occurs when traveling in the direction of the river current, so the speeds add up:

$s_d=u+v=7x+2x=9x\text{km/h}$

Upstream speed occurs when traveling against the river current, so the current's speed is subtracted:

$s_u=u-v=7x-2x=5x\text{km/h}$

Step 3: Set up the total time equation.
The relationship between time, distance, and speed is given by:

$\text{Time}=\frac{\text{Distance}}{\text{Speed}}$

The total time taken to travel 270 km downstream and 100 km upstream is 5 hours. Therefore:

$\frac{270}{s_d}+\frac{100}{s_u}=5$

Substitute $s_d=9x$ and $s_u=5x$ into the equation:

$\frac{270}{9x}+\frac{100}{5x}=5$

Simplify each fraction:

$\frac{30}{x}+\frac{20}{x}=5$

Combine the fractions:

$\frac{50}{x}=5$

Solve for $x$:

$x=\frac{50}{5}=10$

Step 4: Find the speed of the ferry in still water.
Substitute $x=10$ back into the equation for $u$:

$u=7\times 10=70\text{km/h}$

Step 5: Calculate the time taken to cover 315 km in still water.
Now, calculate the time required to travel 315 km at a speed of 70 km/h in still water:

$\text{Time}=\frac{315}{70}=4.5\text{hours}$

Hence, the ferry takes 4.5 hours to cover a distance of 315 km in still water.