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: Define the speeds using the given ratio

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 = $5x$ km/h

Let the speed of the wind = $3x$ km/h

Step 2: Calculate the effective speed flying with and against the wind

When flying with the wind, the wind aids the plane's speed:

km/h

When flying against the wind, the wind opposes the plane's speed:

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

Step 3: Set up the equation for total time to solve for $x$

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

Time taken to cover 240 km flying against the wind:

$t_1=\frac{240}{2x}=\frac{120}{x}$ hours

Time taken to cover 240 km flying with the wind:

$t_2=\frac{240}{8x}=\frac{30}{x}$ hours

The total time for both journeys combined is 5 hours:

$t_1+t_2=5$

$\frac{120}{x}+\frac{30}{x}=5$

$\frac{150}{x}=5$

$x=\frac{150}{5}=30$

Step 4: Find the time taken to travel 640 km with the wind

Now, substitute $x=30$ to find the speed of the plane flying with the wind:

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

Finally, calculate the time required to travel a distance of 640 km with the wind:

$\text{Time}=\frac{640}{240}=\frac{8}{3}=2\frac{2}{3}$ hours

Thus, the plane takes $2\frac{2}{3}$ hours to travel 640 km with the wind.