A plane covers 240 km flying against the wind and 240 km flying with the wind in a total of 5 hours. The ratio of the speed of the plane in still air to the speed of the wind is 5:3. Find the time taken by the plane to travel 640 km with the wind. (In hours)

The correct answer is $2\frac{2}{3}$ hours.

Step 1: Understand the given ratio and define the speeds.
The ratio of the speed of the plane in still air to the speed of the wind is given as 5:3.
Let the speed of the plane in still air be $5x$ km/h.
Let the speed of the wind be $3x$ km/h.

Step 2: Determine effective speeds flying with and against the wind.
When flying with the wind (downwind), the wind aids the plane, so speeds add up:

$\text{Speed with wind}=5x+3x=8x\text{km/h}$
When flying against the wind (upwind), the wind opposes the plane, so speed decreases:

$\text{Speed against wind}=5x-3x=2x\text{km/h}$

Step 3: Set up the equation using total time.
The formula relating time, distance, and speed is:

$\text{Time}=\frac{\text{Distance}}{\text{Speed}}$
Time taken to fly 240 km against the wind:

${\text{Time}}_{\text{against}}=\frac{240}{2x}$
Time taken to fly 240 km with the wind:

${\text{Time}}_{\text{with}}=\frac{240}{8x}$
The total time for both journeys is 5 hours:

$\frac{240}{2x}+\frac{240}{8x}=5$

Step 4: Solve the equation for x.
Simplify each fraction:

$\frac{120}{x}+\frac{30}{x}=5$
Combine the terms on the left side:

$\frac{150}{x}=5$
Multiply both sides by $x$:

$5x=150$
$x=30$

Step 5: Calculate the speed of the plane flying with the wind.
Substitute $x=30$ into the expression for speed with the wind:

$\text{Speed with wind}=8x=8\times 30=240\text{km/h}$

Step 6: Find the time taken to travel 640 km with the wind.
Using the time formula for 640 km at a speed of 240 km/h:

$\text{Time}=\frac{640}{240}=\frac{8}{3}=2\frac{2}{3}\text{hours}$
Therefore, the time taken by the plane to travel 640 km with the wind is $2\frac{2}{3}$ hours.