Question

Rahul takes $6$ hours more than Pathak to cover a distance of 540km. If instead, Rahul doubles his speed, he would reach the destination one and a half hours before Pathak. Find Pathak’s speed.
(A) $36\;{\rm{kmph}}$
(B) $60\;{\rm{kmph}}$
(C) $45\;{\rm{kmph}}$
(D) $40\;{\rm{kmph}}$

Answer 1

Hint:
When an object travels some distance(d) in any direction and if it takes some time(t) then speed(S) of that object will be the division of distance $d$ covered by the object and time $t$ taken by the object to complete this journey. The mathematical expression of the speed $S$ is given as follows,
$S = \dfrac{d}{t}$
Complete step-by-step solution:
Let the time taken by the Pathak to cover the distance of $d = 540\;{\rm{km}}$ is :$y$
Rahul takes $6\;{\rm{h}}$ more than the time taken by the Pathak to cover the distance $d$, so the time taken by the Rahul to cover the distance is : $y + 6$
The speed of the Rahul during the journey is
$\begin{array}{l}
S = \dfrac{d}{t}\\
S = \dfrac{{540}}{{y + 6}}............{\rm{(1)}}
\end{array}$
It is given that if Rahul doubles its speed, Rahul completes the journey $1\dfrac{1}{2}\;{\rm{h}}$ before the Pathak.
So the new time taken by the Rahul to complete the journey is:$\left( {y - 1\dfrac{1}{2}} \right) = \left( {y - \dfrac{3}{2}} \right)$
Now the relation between speed and time is
$2S = \dfrac{{540}}{{y - \dfrac{3}{2}}}................{\rm{(2)}}$
Divide the equation (1) from the equation (2),
$\begin{array}{l}
\dfrac{S}{{2S}} = \dfrac{{\dfrac{{540}}{{y + 6}}}}{{\dfrac{{540}}{{y - \dfrac{3}{2}}}}}\\
\dfrac{1}{2} = \dfrac{{y - \dfrac{3}{2}}}{{y + 6}}
\end{array}$
Evaluate further
$\begin{array}{l}
y + 6 = 2\left( {y - \dfrac{3}{2}} \right)\\
\Rightarrow y + 6 = 2y - 3\\
\Rightarrow y = {\rm{9}}\;{\rm{h}}
\end{array}$
So the time taken by the Pathak to complete the journey is $9\;{\rm{h}}$ .
Now we will calculate the speed of the Pathak during the journey.
The expression for the speed of the Pathak is given as follows,
${S_p} = \dfrac{d}{y}$
Here, ${S_p}$ is the speed of the Pathak during the journey and $y$ is the time taken by the Pathak to cover the distance.
Substitute all the values in the above expression,
$\begin{array}{l}
{S_p} = \dfrac{{540\;{\rm{km}}}}{{9\;{\rm{h}}}}\\
\Rightarrow {S_p} = 60\;{\rm{kmph}}
\end{array}$
Therefore, the speed of the Pathak is $60\;{\rm{kmph}}$ and the correct answer is ${\rm{option}}\;{\rm{(B)}}{\rm{.}}$

Note:
In such types of questions use the relations between the speed, time and the distance with the correct units. Speed and velocity are two different things because speed does not depend on the direction but velocity changes with the change in the direction of the motion.