The ratio of the time taken by a boat to cover a certain distance downstream to the time taken to cover another distance upstream is $3 : 5$. If the boat covers $90 ext{ km}$ downstream in $t$ hours and $72 ext{ km}$ upstream in $(t+1)$ hours, find the downstream speed of the boat in km/hr.

Question

The ratio of the time taken by a boat to cover a certain distance downstream to the time taken to cover another distance upstream is $3 : 5$. If the boat covers $90 	ext{ km}$ downstream in $t$ hours and $72 	ext{ km}$ upstream in $(t+1)$ hours, find the downstream speed of the boat in km/hr.

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 time downstream to upstream is $3 : 5$. Since time = distance/speed, the ratio of speeds upstream to downstream will be the inverse, i.e.,
   $$\frac{x - y}{x + y} = \frac{3}{5}$$

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

4. The downstream speed is $x + y = 4y + y = 5y$.

5. The boat covers $90$ km downstream in $t$ hours, so:
   $$t = \frac{90}{x + y} = \frac{90}{5y} = \frac{18}{y} \quad 	ext{(i)}$$

6. The boat covers $72$ km upstream in $(t + 1)$ hours, so:
   $$t + 1 = \frac{72}{x - y} = \frac{72}{4y - y} = \frac{72}{3y} = \frac{24}{y} \quad 	ext{(ii)}$$

7. Substitute $t$ from (i) into (ii):
   $$\frac{18}{y} + 1 = \frac{24}{y}$$

8. Multiply both sides by $y$:
   $$18 + y = 24$$

9. Solve for $y$:
   $$y = 6$$

10. Calculate $x$:
    $$x = 4y = 4 	imes 6 = 24$$

11. Finally, downstream speed:
    $$x + y = 24 + 6 = 30 	ext{ km/hr}$$

**Answer:** $30$ km/hr

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