• # question_answer 23) A motor boat whose speed is 18 km/hr. in still water takes 1 hr. more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream. OR A train travels at a certain average speed for a distance of 63 km and then travels at a distance of 72 km at an average speed of 6 km/hr. more than its original speed. If it takes 3 hours to complete total journey, what is the original average speed?

 Given, speed of motor boat in still water = 18 km/hr. Let speed of stream $=x\,km/hr$. $\therefore$ Speed of boat downstream $=(18+x)\text{ }km/hr$ And speed of boat upstream $=(18-x)\text{ }km/hr$. Time of the upstream journey =$\frac{24}{(18-x)}$ Time of the downstream journey $\frac{24}{(18+x)}$ According to the question, $\frac{24}{(18-x)}-\frac{24}{(18+x)}=1$ $\frac{24(18+x)-24(18-x)}{(18-x)(18+x)}=1$ $\frac{24\times 18+24x-24\times 18+24x}{324-{{x}^{2}}}=1$ $\frac{48x}{324-{{x}^{2}}}=1$ $48x=324-{{x}^{2}}$ $\Rightarrow$           ${{x}^{2}}+48x-324=0$ $\Rightarrow$   ${{x}^{2}}+54x-6x-324=0$ $\Rightarrow$   $x(x+54)-6(x+54)=0$ $\Rightarrow$             $(x+54)(x-6)=0$ Either                         $x+54=0$ $x=-54$ Rejected, as speed cannot be negative Or                     $x-6=0$ $x=6$ Thus, the speed of the stream is 6 km/hr. OR Let original average speed of train be x km/hr. $\therefore$ Increased speed of train $=(x+6)\text{ }km/hr$. Time taken to cover 63 km with average speed $=\frac{63}{x}hr.$ Time taken to cover 72 km with increased speed $=\frac{72}{(x+6)}hr.$ According to the question, $\frac{63}{x}+\frac{72}{x+6}=3$ $\Rightarrow$       $\frac{63(x+6)+72(x)}{(x)(x+6)}=3$ $\Rightarrow$      $\frac{63x+378+72x}{{{x}^{2}}+6x}=3$ $\Rightarrow$   $135x+378=3({{x}^{2}}+6x)$ $\Rightarrow$   $135x+378=3{{x}^{2}}+18x$ $\Rightarrow$   $3{{x}^{2}}+18x-135x-378=0$ $\Rightarrow$   $3{{x}^{2}}-117x-378=0$ $\Rightarrow$   $3({{x}^{2}}-39x-126)=0$ $\Rightarrow$       ${{x}^{2}}-39x-126=0$ $\Rightarrow$   ${{x}^{2}}-42x+3x-126=0$ $\Rightarrow$   $x(x-42)+3(x-42)=0$ $\Rightarrow$   $(x-42)(x+3)=0$ Either                $x-42=0$ $x=42$ Or                     $x+3=0$ $x=-3$ Rejected (as speed cannot be negative) Thus, average speed of train is 42 km/hr.