Question : A boat takes 4 hours to travel from a place X to Y downstream and back from Y to X upstream. If the distance from X to Y is 10.5 km, and the speed of the current is 9 km/hr, then the speed of the boat in still water, in km/hr, is:
Option 1: $10 \frac{1}{2}$
Option 2: $15$
Option 3: $12$
Option 4: $12 \frac{1}{2}$
Correct Answer: $12$
Solution :
Let the speed of the boat be $s$ km/hr.
Speed of current = 9 km/hr
Distance between X and Y = 10.5 km
Upstream speed = $(s-9)$ km/hr
Downstream speed = $(s+9)$ km/hr
According to the question,
$\frac{10.5}{(s-9)} + \frac{10.5}{(s+9)} = 4$
⇒ $\frac{10.5(s+9 + s-9)}{(s^2-9^2)} = 4$
⇒ $\frac{10.5×(2s)}{s^2-81} = 4$
⇒ $10.5×(2s) = 4(s^2-81)$
⇒ $21s = 4s^2-324$
⇒ $4s^2-21s-324=0$
⇒ $4s^2-48s+27s-324=0$
⇒ $4s(s-12)+27(s-12)=0$
⇒ $(4s+27)(s-12)=0$
⇒ $s = 12$ [since speed can't be negative]
Hence, the correct answer is $12$.
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