The ratio of the speed of a boat downstream to upstream is $5 : 3$. If the boat takes $5$ hours to cover $30 { km}$ upstream, what is the time taken by the boat to cover $40 { km}$ downstream?

Question

The ratio of the speed of a boat downstream to upstream is $5 : 3$. If the boat takes $5$ hours to cover $30 	{ km}$ upstream, what is the time taken by the boat to cover $40 { km}$ downstream?

Accepted Answer

**Solution:**

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

2. Given the ratio of downstream speed to upstream speed is $\frac{x+y}{x-y} = \frac{5}{3}$.

3. Cross-multiplying, we get:
   $$3(x + y) = 5(x - y)$$
   $$3x + 3y = 5x - 5y$$

4. Rearranging terms:
   $$3x + 3y - 5x + 5y = 0$$
   $$-2x + 8y = 0$$
   $$2x = 8y$$
   $$x : y = 4 : 1$$

5. Let $x = 4s$ and $y = s$.

6. Upstream speed = $x - y = 4s - s = 3s$.

7. Time taken upstream to cover $30$ km is $5$ hours, so:
   $$\frac{30}{3s} = 5 \implies 3s = \frac{30}{5} = 6 \implies s = 2$$

8. Therefore, downstream speed = $x + y = 4s + s = 5s = 5 	imes 2 = 10$ km/hr.

9. Time taken downstream to cover $40$ km:
   $$\frac{40}{10} = 4 	ext{ hours}$$

**Answer:** $4 	ext{ hours}$

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