A boat travels a distance of $22.4 ext{ km}$ downstream in $48$ minutes. The speed of the stream is $40\%$ of the speed of the boat in still water. What is the ratio of the time taken by the boat to cover $54 ext{ km}$ upstream to the time taken to cover $210 ext{ km}$ downstream?

Question

A boat travels a distance of $22.4 	ext{ km}$ downstream in $48$ minutes. The speed of the stream is $40\%$ of the speed of the boat in still water. What is the ratio of the time taken by the boat to cover $54 	ext{ km}$ upstream to the time taken to cover $210 	ext{ km}$ downstream?

Accepted Answer

**Solution:**

1. Convert the given time to hours:
   - $48$ minutes = $\frac{48}{60} = 0.8$ hours

2. Calculate the downstream speed:
   - Distance downstream = $22.4 	ext{ km}$
   - Time downstream = $0.8$ hours
   - Speed downstream = $\frac{22.4}{0.8} = 28 	ext{ km/h}$

3. Let the speed of the boat in still water be $b$ km/h.
   - Speed of stream = $40\%$ of $b = 0.4b$

4. Downstream speed = speed of boat + speed of stream:
   $$b + 0.4b = 1.4b$$
   Given downstream speed = $28$ km/h, so:
   $$1.4b = 28 \implies b = \frac{28}{1.4} = 20 	ext{ km/h}$$

5. Upstream speed = speed of boat - speed of stream:
   $$20 - 0.4 	imes 20 = 20 - 8 = 12 	ext{ km/h}$$

6. Calculate time taken to cover $54$ km upstream:
   $$t_1 = \frac{54}{12} = 4.5 	ext{ hours}$$

7. Calculate time taken to cover $210$ km downstream:
   $$t_2 = \frac{210}{28} = 7.5 	ext{ hours}$$

8. Find the ratio of time upstream to time downstream:
   $$\frac{t_1}{t_2} = \frac{4.5}{7.5} = \frac{3}{5}$$

**Answer:** $3 : 5$

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