The ratio of the speed of a boat in still water to its speed upstream is $4 : 3$. The boat covers $560 ext{ km}$ downstream in $7$ hours. What is the time taken by the boat to cover $480 ext{ km}$ upstream?

Question

The ratio of the speed of a boat in still water to its speed upstream is $4 : 3$. The boat covers $560 	ext{ km}$ downstream in $7$ hours. What is the time taken by the boat to cover $480 	ext{ km}$ upstream?

Accepted Answer

**Solution:**

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

2. Given the ratio of the speed of the boat in still water to the speed upstream is:
   $$\frac{x}{x - y} = \frac{4}{3}$$

3. From this, we get:
   $$3x = 4(x - y) \implies 3x = 4x - 4y \implies 4y = x \implies y = \frac{x}{4}$$

4. The downstream speed is:
   $$x + y = x + \frac{x}{4} = \frac{5x}{4}$$

5. The boat covers $560 	ext{ km}$ downstream in $7$ hours, so:
   $$\frac{560}{7} = 80 = \frac{5x}{4}$$

6. Solving for $x$:
   $$80 = \frac{5x}{4} \implies x = \frac{80 	imes 4}{5} = 64 	ext{ km/h}$$

7. Speed of stream $y$:
   $$y = \frac{x}{4} = \frac{64}{4} = 16 	ext{ km/h}$$

8. Speed upstream is:
   $$x - y = 64 - 16 = 48 	ext{ km/h}$$

9. Time taken to cover $480 	ext{ km}$ upstream:
   $$	ext{Time} = \frac{	ext{Distance}}{	ext{Speed}} = \frac{480}{48} = 10 	ext{ hours}$$

**Answer:** $10 	ext{ hours}$

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