A boat can row to a place 56 km away and come back in 22 hours. The time to row 21 km with the stream is same as the time to row 12 km against the stream. Find the speed of boat in still water?

The correct answer is 5.5 kmph.

Let us solve the problem step-by-step.

Let the speed of the boat in still water be $u$ km/h and the speed of the stream be $v$ km/h.

Then, the speed downstream (with the stream) is given by:

$s_{\text{down}}=u+v$

And the speed upstream (against the stream) is given by:

$s_{\text{up}}=u-v$

According to the question, the time taken to row 21 km downstream is equal to the time taken to row 12 km upstream.

Using the formula $\text{Time}=\frac{\text{Distance}}{\text{Speed}}$:

$\frac{21}{u+v}=\frac{12}{u-v}$

Simplifying the fraction by dividing both numerators by 3:

$\frac{7}{u+v}=\frac{4}{u-v}$

Cross-multiplying gives:

$7(u-v)=4(u+v)$

$7u-7v=4u+4v$

$3u=11v$

$v=\frac{3u}{11}$

Now, let us express the downstream and upstream speeds in terms of $u$ only:

$s_{\text{down}}=u+\frac{3u}{11}=\frac{14u}{11}$

$s_{\text{up}}=u-\frac{3u}{11}=\frac{8u}{11}$

We are given that the total time to row 56 km away and come back (total distance 56 km each way) is 22 hours.

${\text{Time}}_{\text{downstream}}+{\text{Time}}_{\text{upstream}}=22$

$\frac{56}{s_{\text{down}}}+\frac{56}{s_{\text{up}}}=22$

Substituting the expressions for $s_{\text{down}}$ and $s_{\text{up}}$:

$\frac{56}{\left(\frac{14u}{11}\right)}+\frac{56}{\left(\frac{8u}{11}\right)}=22$

$\frac{56\times 11}{14u}+\frac{56\times 11}{8u}=22$

Simplifying each fraction:

$\frac{4\times 11}{u}+\frac{7\times 11}{u}=22$

$\frac{44}{u}+\frac{77}{u}=22$

$\frac{121}{u}=22$

Solving for $u$:

$u=\frac{121}{22}=5.5$

Thus, the speed of the boat in still water is 5.5 kmph.