Question

A boat covers certain distance between two spots in a river taking $t_1$ hrs going downstream and $t_2$ hrs going upstream. What time will be taken by boat to cover same distance in still water?

• A. $\frac{t_1+t_2}{2}$
• B. $2\left(t_2-t_1\right)$
• C. $\frac{2t_1{t}_2}{t_1+t_2}$
• D. $\sqrt{t_1{t}_2}$

Answer 1

Answer: (C)

For upstream, Speed $\Rightarrow v-u$
For downstream, Speed $\Rightarrow v+$
$t_{up}=\frac{d}{v-u}$
$t_2=\frac{d}{v-u}$
$\Rightarrow d=(v-u)t_2$ ...(i)
(where $v\to$ man and $u\to$ water)
$t_{\text{down }}=\frac{d}{v+u}$
$t_1=\frac{d}{v+u}$
$\Rightarrow d=(v+u)t_1$ ...(ii)
$t_{\text{still }}=\frac{d}{v}$
$-t_{\text{still}}=\frac{2t_1{t}_2}{t_1+t_2}$
On equating (i) and (ii)
$(v-u)t_2=(v+u)t_1$
$\Rightarrow v{t}_2-u{t}_2=v{t}_1+u{t}_1$
$\Rightarrow v\left(t_2-t_1\right)=u\left(t_1+t_2\right)$
$\Rightarrow u=\frac{v\left(t_2-t_1\right)}{t_2+t_1}$
So, $d=\left(v-\frac{v\left(t_2-t_1\right)}{t_1+t_2}\right)t_2$
$=v{t}_2\left(\frac{t_1+t_2-t_2+t_1}{t_1+t_2}\right)$
$\frac{d}{v}=\frac{2t_1{t}_2}{t_1+t_2}\mid \to$ Remember as shortcut