A boat travels $22.4 ext{ km}$ 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 $22.4 	ext{ km}$ 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 downstream time into hours:
   - $48$ minutes = $\frac{48}{60} = 0.8$ hours

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

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

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

5. Speed of stream = $0.4 	imes 20 = 8 	ext{ km/h}$

6. Calculate upstream speed:
   - Upstream speed = speed of boat - speed of stream = $20 - 8 = 12 	ext{ km/h}$

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

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

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

**Answer:** 3 : 5

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