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# Ram travels from P to Q at 10km/hr and returns at 15km/hr. Shyam travels from P to Q and returns at 12.5km/hr if he takes 12 minutes less than Ram then what is the distance between P and Q?(A) 60 km(B) 45 km(C) 36 km(D) 30 km

Last updated date: 13th Jul 2024
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Hint: Assume that the distance from P to Q be $x$ km. Use the formula, Time = $\dfrac{\text{Distance}}{\text{Speed}}$ and calculate the time taken by Ram to travel from P to Q and then, from Q to P. Similarly, calculate the time taken by Shyam to travel from P to Q and then, from Q to P. Use the information that Shyam takes 12 minutes less than Ram and form an equation. Now, solve it further and get the value of x.

Complete step-by-step solution
According to the question, we are given that Ram travels from P to Q at 10km/hr and returns at 15km/hr. Shyam travels from P to Q and returns at 12.5km/hr and he takes 12 minutes less than Ram.
First of all, let us assume that the distance from P to Q is $x$ km.
The speed of Ram while traveling from P to Q = 10 km/hr …………………………………..(1)
Distance covered by Ram while traveling from P to Q = $x$ km ………………………………………….(2)
We know the formula that, Time = $\dfrac{\text{Distance}}{\text{Speed}}$ …………………………………………..(3)
Now, using equation (1), equation (2), and equation (3), we get
Time is taken by Ram to travel from P to Q = $\dfrac{x}{10}$ hr …………………………………………(4)
The speed of Ram while returning from Q to P = 15 km/hr …………………………………..(5)
Distance covered by Ram while traveling from P to Q = $x$ km ………………………………………….(6)
Now, from equation (3), equation (5), and equation (6), we get
Time is taken by Ram to return from Q to P = $\dfrac{x}{15}$ hr ……………………………………………(7)
Total time taken by Ram to travel from P to Q and then return from Q to P = $\dfrac{x}{10}+\dfrac{x}{15}=\dfrac{5x}{30}=\dfrac{x}{6}$ hr …………………………………………………(8)
The speed of Shyam while traveling from P to Q = 12.5 km/hr …………………………………..(9)
Distance covered by Shyam while traveling from P to Q = $x$ km ………………………………………….(10)
Now, using equation (3), equation (9), and equation (10), we get
Time is taken by Shyam to travel from P to Q = $\dfrac{x}{12.5}$ hr …………………………………………(11)
The speed of Shyam while returning from Q to P = 12.5 km/hr …………………………………..(12)
Distance covered by Shyam while traveling from P to Q = $x$ km ………………………………………….(13)
Now, from equation (3), equation (11), and equation (12), we get
Time is taken by Shyam to return from Q to P = $\dfrac{x}{12.5}$ hr ……………………………………………(14)
Total time taken by Ram to travel from P to Q and then return from Q to P = $\dfrac{x}{12.5}+\dfrac{x}{12.5}=\dfrac{2x}{12.5}$ hr …………………………………………………(15)
We know the relation between hour and minutes, $60\min =1hr\Rightarrow 1\min =\dfrac{1}{60}hr$ ……………………………………………..(16)
Now, using equation (16), let us convert 12 minutes into an hour.
$\Rightarrow 12\min =\dfrac{1}{60}\times 12hr$
$\Rightarrow 12\min =\dfrac{1}{5}hr$ ……………………………………………(17)
We are also given that Shyam takes 12 minutes less than Ram …………………………………………(18)
Now, from equation (8), equation (15), equation (17), and equation (18), we get
\begin{align} & \Rightarrow \dfrac{x}{6}=\dfrac{2x}{12.5}+\dfrac{1}{5} \\ & \Rightarrow \dfrac{x}{6}-\dfrac{4x}{25}=\dfrac{1}{5} \\ & \Rightarrow \dfrac{25x-24x}{6\times 25}=\dfrac{1}{5} \\ & \Rightarrow \dfrac{x}{6\times 25}=\dfrac{1}{5} \\ \end{align}
$\Rightarrow x=30$
Therefore, the distance between P to Q is 30km. Hence, the correct option is (D).

Note: In this question, one might make a silly mistake while forming the equation using the information that Shyam takes 12 minutes less than Ram. Here, one might write the equation as $\dfrac{x}{6}+\dfrac{1}{5}=\dfrac{2x}{12.5}$ . This is wrong. Since the time taken by Shyam is 12 minutes less than the time taken by Ram so, we have to add $\dfrac{1}{5}$ hr to the time taken by Shyam to make it equal to the time taken by Ram.