The ratio of the upstream speed to the downstream speed of a boat is $3 : 5$. The approximate time taken to cover $60 ext{ km}$ downstream and $60 ext{ km}$ upstream is $16$ hours $10$ minutes. Find the approximate time taken by the boat to cover $37 ext{ km}$ upstream.

Question

The ratio of the upstream speed to the downstream speed of a boat is $3 : 5$. The approximate time taken to cover $60 	ext{ km}$ downstream and $60 	ext{ km}$ upstream is $16$ hours $10$ minutes. Find the approximate time taken by the boat to cover $37 	ext{ km}$ upstream.

Accepted Answer

**Solution:**

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

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

3. Cross-multiplying:
   $$5(x - y) = 3(x + y)$$
   $$5x - 5y = 3x + 3y$$
   $$5x - 3x = 3y + 5y$$
   $$2x = 8y$$
   $$x = 4y$$

4. The total time to cover $60$ km downstream and $60$ km upstream is $16$ hours $10$ minutes, which is approximately $16.17$ hours.

5. Downstream speed = $x + y = 4y + y = 5y$ km/h
   Upstream speed = $x - y = 4y - y = 3y$ km/h

6. Total time:
   $$\frac{60}{5y} + \frac{60}{3y} = 16.17$$

7. Simplify:
   $$\frac{60}{5y} = \frac{12}{y}, \quad \frac{60}{3y} = \frac{20}{y}$$
   So,
   $$\frac{12}{y} + \frac{20}{y} = \frac{32}{y} = 16.17$$

8. Solving for *y*:
   $$y = \frac{32}{16.17} \approx 2$$

9. Therefore, *x* = $4y = 8$ km/h.

10. Time taken to cover $37$ km upstream:
    $$	ext{Time} = \frac{	ext{Distance}}{	ext{Speed}} = \frac{37}{x - y} = \frac{37}{8 - 2} = \frac{37}{6} \approx 6.17 	ext{ hours}$$

**Answer:** Approximately $6$ hours

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