Question

In covering a distance of 30km, Amit takes 2hrs more than Suresh. If Amit doubles his speed, he would take one hour less than Suresh. Amit's speed is
(A). 5 km/hr
(B). 7.5 km/hr
(C). 6 km/hr
(D). 6.25 km/hr

Answer 1

Hint: In this question, we have to use out the formula for the time taken to cover a given distance with a fixed speed. Now, if we assume the speeds of Amit and Suresh to be two variables x and y respectively, we can use that formula to obtain two equations which can be solved for x to give the required answer to this question.

Complete step-by-step solution -
We have been given that the distance covered by Amit and Suresh is 30km.
Now, the time taken to cover a distance d at a speed v is given by the following formula:
$\text{Time Taken=}\dfrac{\text{Distance Travelled }}{\text{Speed}}...........(1.1)$
Let the original speed of Amit and Suresh be x km/hr and y km/hr respectively. It is given that in this case time taken by Amit will be two hours more than Suresh. Thus, by using equation (1.1), we get
$\begin{align}
  & \text{Time taken by Amit- Time taken by Suresh=2hours} \\
 & \Rightarrow \dfrac{30km}{x\text{ km/hr}}-\dfrac{30km}{y\text{ km/hr}}=2\text{hr = }\dfrac{y-x}{xy}\text{=}\dfrac{1}{15}\Rightarrow 15(y-x)=xy.................\text{(1}\text{.2) } \\
\end{align}$

Now, if Amit doubles his speed the speed of Amit and Suresh will be 2x km/hr and y km/hr respectively. It is given that in this case, Amit will take one hour less than Suresh. Therefore, using equation (1.1)
$\begin{align}
  & \text{Time taken by Amit- Time taken by Suresh=}-1\text{hours} \\
 & \Rightarrow \dfrac{30km}{2x\text{ km/hr}}-\dfrac{30km}{y\text{ km/hr}}=-1\text{hr = }\dfrac{y-2x}{2xy}\text{=}\dfrac{-1}{30}\Rightarrow 30(y-2x)=-2xy.................\text{(1}\text{.3) } \\
\end{align}$
Now, we can use xy from (1.2) in (1.3) to obtain
$\begin{align}
  & 30(y-2x)=-2xy=-2\times 15(y-x) \\
 & \Rightarrow y-2x=-(y-x)\Rightarrow 2y=3x\Rightarrow y=\dfrac{3}{2}x............(1.4) \\
\end{align}$
Using this value in (1.2), we get
\[\begin{align}
  & 15\left( \dfrac{3}{2}x-x \right)=x\times \dfrac{3}{2}x\Rightarrow 15x=3{{x}^{2}}\Rightarrow x(15-3x)=0 \\
 & \Rightarrow x=15\text{ or }x=\dfrac{15}{3}=5 \\
\end{align}\]

Thus we obtain the speed of Amit to be 5km/hr which matches option (a). Therefore, option(a) is the correct answer to this question.

Note: We should note that we could also have substituted x in terms of y in equation (1.4). However, then we would have to again use the relation to obtain x as in the question, we are asked to find Amit's age which is equal to x.