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Distance between the two cities is \[750km\]. Ram and Gopal started their journey by scooter. Speed of Ram’s scooter is more than Gopal’s scooter by $10km$ per hour. Though Gopal started $2\dfrac{1}{2}$ hours earlier than Ram, both of them reached at the same time. Find the speed of Ram’s Scooter.

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Last updated date: 13th Jun 2024
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Answer
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Hint: For this problem we have to find the speed of the Ram’s scooter. The given problem is that there is a two people going to the city on their scooter. There is a difference between their speeds of the scooter. To solve this question we have to find out the time taken by Gopal and Ram respectively. Let the speed of Ram be $v$, therefore the speed of Gopal would be $v - 10$.

Complete step-by-step answer:
It is given in the question that the distance between two cities is $750$$km$.
We have consider Speed of Ram be $v$$km/hr$ and Speed of Gopal be $v - 10$$km/hr$
We know that $speed = \dfrac{{Dis\tan ce}}{{Time}}$ or $Time = \dfrac{{Dis\tan ce}}{{Speed}}$
It is mentioned in the question that time taken by Ram $ = $ time taken by Gopal $ - \dfrac{5}{2}$ since Gopal has started his journey $2\dfrac{1}{2}$ hours or $\dfrac{5}{2}$ hours.
Time taken by Ram $ = \dfrac{{750}}{v}$
Time taken by Gopal $ = \dfrac{{750}}{{v - 10}}$
Therefore we can write that-
$\Rightarrow$$\dfrac{{750}}{v}$$ = \dfrac{{750}}{{v - 10}} - \dfrac{5}{2}$
By doing L.C.M we get-
$\Rightarrow$$\dfrac{{750}}{v}$$ = \dfrac{{1500 - 5(v - 10)}}{{2(v - 10)}}$
By cancelling $v$ from both the sides of the denominator we get-
$\Rightarrow$$\dfrac{{750}}{v}$$ = \dfrac{{1500 - 5v + 50}}{{2v - 20}}$
By doing cross-multiplication we get-
$\Rightarrow$$1500v - 15000 = 1550v - 5{v^2}$
Simplifying the terms,
$\Rightarrow$$5{v^2} - 50v - 15000 = 0$
Divided by $5$ we get,
$\Rightarrow$${v^2} - 10v - 3000 = 0$
By doing factorisation we get-
$\Rightarrow$${v^2} - 60v + 50v - 3000 = 0$
Taking out the common terms,
$\Rightarrow$$v(v - 60) + 50(v - 60) = 0$
Hence,
$\Rightarrow$$(v - 60)(v + 50) = 0$
Therefore either $v - 60 = 0$ or, $v = 60$ or $v + 50 = 0$ so, $v = - 50$
Since the value of the speed cannot be negative therefore the value of $v = 60$

Hence the speed of Ram’s scooter is $60km/hr$

Note: Speed is measured by dividing distance with respect to time. In order to solve the questions related to speed time distance there is only one formula which you need to keep in mind always for solving the questions.
If you have to find out the distance then you need to multiply speed and time.