A boat covers $44.8 ext{ km}$ downstream in $1$ hour $36$ 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 $108 ext{ km}$ upstream to the time taken to cover $420 ext{ km}$ downstream?

Question

A boat covers $44.8 	ext{ km}$ downstream in $1$ hour $36$ 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 $108 	ext{ km}$ upstream to the time taken to cover $420 	ext{ km}$ downstream?

Accepted Answer

**Solution:**

1. Convert the time taken downstream to hours:
   - $1$ hour $36$ minutes = $1 + \frac{36}{60} = 1.6$ hours

2. Let the speed of the boat in still water be $b$ km/h.
   The speed of the stream is $40\%$ of $b$, so stream speed = $0.4b$ km/h.

3. Downstream speed = speed of boat + speed of stream = $b + 0.4b = 1.4b$ km/h.

4. Using distance = speed $	imes$ time for downstream:
   $$44.8 = 1.4b 	imes 1.6$$
   $$44.8 = 2.24b$$
   $$b = \frac{44.8}{2.24} = 20 	ext{ km/h}$$

5. Upstream speed = speed of boat - speed of stream = $b - 0.4b = 0.6b = 0.6 	imes 20 = 12 	ext{ km/h}$

6. Calculate time taken upstream to cover $108$ km:
   $$t_{up} = \frac{108}{12} = 9 	ext{ hours}$$

7. Calculate time taken downstream to cover $420$ km:
   $$t_{down} = \frac{420}{1.4b} = \frac{420}{1.4 	imes 20} = \frac{420}{28} = 15 	ext{ hours}$$

8. Find the ratio of time taken upstream to downstream:
   $$\frac{t_{up}}{t_{down}} = \frac{9}{15} = \frac{3}{5}$$

**Answer:** 3 : 5

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