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? |
Answer:
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.
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