The downstream speed of boat A is $21 ext{ km/hr}$. The speed of boat B in still water is $12 ext{ km/hr}$, and the ratio of the speed of boat A to boat B in still water is $5:4$, respectively. Given that the speed of the stream is the same for both boats, find the time taken by boat B to cover $81 ext{ km}$ upstream.

Question

The downstream speed of boat A is $21 	ext{ km/hr}$. The speed of boat B in still water is $12 	ext{ km/hr}$, and the ratio of the speed of boat A to boat B in still water is $5:4$, respectively. Given that the speed of the stream is the same for both boats, find the time taken by boat B to cover $81 	ext{ km}$ upstream.

Accepted Answer

**Solution:**

1. Let the speed of boat A in still water be $5x$ and the speed of boat B in still water be $4x$ as per the given ratio $5:4$.

2. Given speed of boat B in still water is $12 	ext{ km/hr}$, so:
   $$4x = 12 \implies x = 3$$

3. Therefore, speed of boat A in still water:
   $$5x = 5 	imes 3 = 15 	ext{ km/hr}$$

4. The downstream speed of boat A is given as $21 	ext{ km/hr}$. Downstream speed is the sum of boat speed in still water and stream speed:
   $$	ext{Downstream speed of A} = 15 + s = 21$$
   $$\implies s = 21 - 15 = 6 	ext{ km/hr}$$

5. Speed of stream $s = 6 	ext{ km/hr}$ for both boats.

6. Speed of boat B upstream is:
   $$	ext{Upstream speed of B} = 12 - 6 = 6 	ext{ km/hr}$$

7. Time taken by boat B to cover $81 	ext{ km}$ upstream:
   $$	ext{Time} = \frac{	ext{Distance}}{	ext{Speed}} = \frac{81}{6} = 13.5 	ext{ hours}$$

**Answer:** $13.5 	ext{ hours}$

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