The speed of the stream is $16\frac{2}{3}\%$ of the downstream speed of a boat. The boat covers $120 \text{ km}$ upstream in $5$ hours. If the boat covers $D - 3$ km downstream and $D + 30$ km at the speed of the boat in still water in $5.5$ hours, find the value of $D$.

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Accepted Answer

**Solution:**

1. Let the downstream speed of the boat be $6x$ km/hr.

2. Given, the speed of the stream is $16\frac{2}{3}\% = \frac{1}{6}$ of the downstream speed, so:
   
   $$
   \text{Speed of stream} = \frac{1}{6} \times 6x = x \text{ km/hr}
   $$

3. The speed of the boat in still water is the downstream speed minus the speed of the stream:

$$
   6x - x = 5x \text{ km/hr}
   $$

4. The boat covers $120 \text{ km}$ upstream in $5$ hours, so upstream speed is:

$$
   \frac{120}{5} = 24 \text{ km/hr}
   $$

5. Upstream speed is the speed of the boat in still water minus the speed of the stream:

$$
   5x - x = 4x = 24 \implies x = 6
   $$

6. Therefore:
   - Speed of stream = $6 \text{ km/hr}$
   - Speed of boat in still water = $5x = 30 \text{ km/hr}$

7. According to the question, the boat covers $D - 3$ km downstream and $D + 30$ km at the speed of the boat in still water in $5.5$ hours. So, the total time is:

$$
   \frac{D - 3}{\text{downstream speed}} + \frac{D + 30}{\text{speed in still water}} = 5.5
   $$

Downstream speed = speed of boat in still water + speed of stream = $30 + 6 = 36 \text{ km/hr}$

Substitute values:

$$
   \frac{D - 3}{36} + \frac{D + 30}{30} = 5.5
   $$

8. Multiply both sides by $180$ (LCM of 36 and 30) to clear denominators:

$$
   5(D - 3) + 6(D + 30) = 5.5 \times 180
   $$

$$
   5D - 15 + 6D + 180 = 990
   $$

$$
   11D + 165 = 990
   $$

$$
   11D = 825
   $$

$$
   D = 75
   $$

**Answer:** $75$

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