A boat travels from point $X$ to point $Y$ downstream and returns from $Y$ to $X$ upstream in a total time of $5$ hours. The speed of the current is $4 ext{ km/hr}$. What is the speed of the boat in still water (in km/hr)?

Question

A boat travels from point $X$ to point $Y$ downstream and returns from $Y$ to $X$ upstream in a total time of $5$ hours. The speed of the current is $4 	ext{ km/hr}$. What is the speed of the boat in still water (in km/hr)?

Accepted Answer

**Solution:**

1. Let the speed of the boat in still water be $b$ km/hr.

2. Given the speed of the current is $4$ km/hr.

3. The downstream speed is $(b + 4)$ km/hr and the upstream speed is $(b - 4)$ km/hr.

4. The total time for the round trip is $5$ hours, but the distance between $X$ and $Y$ is not given.

5. The time taken downstream plus the time taken upstream is:
   $$\frac{d}{b+4} + \frac{d}{b-4} = 5$$
   where $d$ is the distance between $X$ and $Y$.

6. Simplifying:
   $$d \left( \frac{1}{b+4} + \frac{1}{b-4} ight) = 5$$
   $$d 	imes \frac{(b-4) + (b+4)}{(b+4)(b-4)} = 5$$
   $$d 	imes \frac{2b}{b^2 - 16} = 5$$

7. Rearranged:
   $$d = \frac{5(b^2 - 16)}{2b}$$

8. Since $d$ is unknown and there is only one equation with two variables ($b$ and $d$), it is impossible to find a unique value for $b$.

**Answer:** Can’t be determined

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