The speed of a stream is 20\% of the downstream speed of a boat. The boat covers $180 ext{ km}$ upstream in $6$ hours. If the boat covers $(D - 5) ext{ km}$ downstream and $(D + 60) ext{ km}$ upstream in $5.9$ hours, find the value of $D$.

Question

The speed of a stream is 20\% of the downstream speed of a boat. The boat covers $180 	ext{ km}$ upstream in $6$ hours. If the boat covers $(D - 5) 	ext{ km}$ downstream and $(D + 60) 	ext{ km}$ upstream in $5.9$ hours, find the value of $D$.

Accepted Answer

**Solution:**

1. Let the downstream speed of the boat be $5a$ km/h.

2. Given, the speed of the stream is 20\% of the downstream speed, so stream speed = $\frac{20}{100} 	imes 5a = a$ km/h.

3. The speed of the boat in still water is downstream speed minus stream speed:
   $$	ext{Boat speed in still water} = 5a - a = 4a 	ext{ km/h}$$

4. The boat covers $180$ km upstream in $6$ hours. Upstream speed is boat speed in still water minus stream speed:
   $$\frac{180}{6} = 4a - a = 3a$$
   $$30 = 3a \implies a = 10$$

5. Now, downstream speed = $5a = 5 	imes 10 = 50$ km/h, and upstream speed = $3a = 3 	imes 10 = 30$ km/h.

6. According to the problem, the boat covers $(D - 5)$ km downstream and $(D + 60)$ km upstream in $5.9$ hours:
   $$\frac{D - 5}{50} + \frac{D + 60}{30} = 5.9$$

7. Multiply both sides by the least common multiple of denominators (150):
   $$3(D - 5) + 5(D + 60) = 5.9 	imes 150$$
   $$3D - 15 + 5D + 300 = 885$$

8. Simplify:
   $$8D + 285 = 885$$
   $$8D = 885 - 285 = 600$$
   $$D = \frac{600}{8} = 75$$

**Answer:** $75$

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