Question

The distance between school and metro station is $300m$. Kartikay starts running from school towards the metro station, while Ashu starts running from the metro station to school. They meet after 4 minutes. Had Kartikay doubled his speed and Ashu reduced his speed to a third of the original they would have met one minute earlier. Find their speeds.

Answer 1

Hint: For solving this question we will assume that the speed of Kartikay is $x$ and Ashu as $y$. They both are traveling in opposite directions so they will have a negative direction of speed with respect to the other person. The distance traveled by the person=$time\times speed$.

Complete step-by-step solution
Let the speed of Kartikay be $x$ and the speed of Ashu be $y$.
They are traveling in opposite directions.
The distance traveled by Kartikay in 4 minutes = $4x$
The distance traveled Ashu in 4 minutes = $4y$
Since the distance between the school and metro station is $300m$.
The total distance traveled by both of them will be $300m$.
$\begin{align}
  & 4x+4y=300 \\
 & x+y=\dfrac{300}{4} \\
\end{align}$
$x+y=75$……(A)
It is given in the question that if Kartikay had doubled his speed that is 2x and Ashu reduced his speed to a third of the original that is $\dfrac{y}{3}$, then they would have met one minute earlier.
This implies that $2x\left( 3 \right)+\dfrac{y}{3}\left( 3 \right)=300$
 Since the distance between the school and metro station is $300m$.
The total distance traveled by both of them will be $300m$.
 Since distance covered is equal to $\text{time}\times \text{speed}$.
$\begin{align}
  & 2x\left( 3 \right)+\dfrac{y}{3}\left( 3 \right)=300 \\
 & \Rightarrow 6x+y=300 \\
\end{align}$………(B)
Now from (A) and (B)
$\begin{align}
  & 6x+y=300 \\
 & \Rightarrow 5x+x+y=300 \\
 & \Rightarrow 5x+75=300 \;\;\; [\because x+y =75 \text{from eq. A}]\\
 & \Rightarrow 5x=300-75 \\
 & \Rightarrow 5x=225 \\
 & \Rightarrow x=45 \\
\end{align}$
Since $x+y=75$, y will be 30 in the opposite direction.

Note: Here we should take care that while two bodies are traveling in opposite directions one body will have a negative direction of speed with respect to the other body. If traveling in the same direction they will both be positive with respect to the other. Here we are considering speed so there is no negative sign for direction as it has only magnitude.