A boat covers a distance of $240 ext{ km}$ upstream and the same distance downstream in a total of $5$ hours. The ratio of the speed of the boat in still water to the speed of the stream is $5:3$. Calculate the time taken by the boat to travel $640 ext{ km}$ downstream.

Question

A boat covers a distance of $240 	ext{ km}$ upstream and the same distance downstream in a total of $5$ hours. The ratio of the speed of the boat in still water to the speed of the stream is $5:3$. Calculate the time taken by the boat to travel $640 	ext{ km}$ downstream.

Accepted Answer

**Solution:**

1. Let the speed of the boat in still water be $5x$ and the speed of the stream be $3x$.

2. Then, the upstream speed = speed of boat - speed of stream = $5x - 3x = 2x$.

3. The downstream speed = speed of boat + speed of stream = $5x + 3x = 8x$.

4. Given that the boat travels $240 	ext{ km}$ upstream and $240 	ext{ km}$ downstream in $5$ hours, we can write the equation:
   $$\frac{240}{2x} + \frac{240}{8x} = 5$$

5. Simplify the equation:
   $$\frac{240}{2x} = \frac{120}{x}, \quad \frac{240}{8x} = \frac{30}{x}$$

So,
   $$\frac{120}{x} + \frac{30}{x} = 5 \implies \frac{150}{x} = 5$$

6. Solve for $x$:
   $$x = \frac{150}{5} = 30$$

7. The downstream speed is $8x = 8 	imes 30 = 240 	ext{ km/h}$.

8. Time taken to travel $640 	ext{ km}$ downstream:
   $$	ext{Time} = \frac{	ext{Distance}}{	ext{Speed}} = \frac{640}{240} = \frac{8}{3} = 2\frac{2}{3} 	ext{ hours}$$

**Answer:** $2\frac{2}{3}$ hours

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