Answer
405.3k+ views
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.
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.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)