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A man travels 600 km partly by train and partly by car. If he covers 400 km by train and the rest by car, it takes him 6 hours and 30 minutes. But, if he 200 km travels by train and the rest by car, he takes half an hour longer. Find the speed of the train and that of the car.

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Last updated date: 20th Jun 2024
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Answer
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Hint: In this question remember to use the given information and remember to use the formula of distance which is given as; $ D = speed \times time $ , using this information will help you to approach the solution.

Complete step-by-step answer:
Case 1:
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Case 2:
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According to the question, we are given that when man covers 400 km by train and rest by car of total 600 km, he takes 6 hours 30 minutes
Let $ {V_t} $ velocity of train and $ {V_C} $ time taken by car and \[{T_t}\] time taken by train and \[{T_c}\] time taken by car
We know that by given information it is given that \[{T_t} + {T_c} = 6.5hrs\]
Since we know that formula of distance i.e. $ D = speed \times time $ or $ time = \dfrac{D}{{speed}} $ here D is the distance covered by the object or body in time “T” with speed “S”
For case 1: Time taken= $ \dfrac{{400}}{{{V_t}}} + \dfrac{{200}}{{{V_c}}} = 6.5hrs $ (equation 1)
For case 2: Time taken $ \dfrac{{200}}{{{V_t}}} + \dfrac{{400}}{{{V_c}}} = 7hrs $ (equation 2)
Multiplying equation (1) by $ {V_t}{V_C} $ on both sides, we get
 $ {\text{400}}{V_C} + 200{V_t} = 6.5{V_t}{V_C} $ (equation 3)
Now, multiplying equation (2) by $ {\text{2}}{V_t}{V_C} $ on both sides, we get
 $ {\text{400}}{V_C} + 800{V_t} = 14{V_t}{V_C} $ (equation 4)
Subtracting equation (3) from (4), we get
 $ {\text{400}}{V_C} - 4{\text{0}}{V_C} + 800{V_t} - 200{V_t} = 14{V_t}{V_C} - 6.5{V_t}{V_C} $
 $ \Rightarrow $ $ 600{V_t} = 7.5{V_t}{V_C} $
 $ \Rightarrow $ $ 600 = 7.5{V_C} $
 $ \Rightarrow $ \[{V_C} = \dfrac{{600}}{{7.5}}\]
 $ \Rightarrow $ $ {V_C} = 80km/hr $
Now substituting the value of $ {V_C} $ in the equation 4 we get
 $ {\text{400}}\left( {80} \right) + 800{V_t} = \left( {14} \right){V_t}\left( {80} \right) $
 $ {\text{32000}} + 800{V_t} = 1120{V_t} $
 $ {\text{32000}} = 1120{V_t} - 800{V_t} $
 $ {\text{32000}} = 320{V_t} $
 $ {V_t} = \dfrac{{32000}}{{320}} $
 $ {V_t} = 100km/hr $
Therefore, the speed of the train and car is 110 km/ hr and 80 km/ hr
So, this is the required solution.

Note: In the above solution we got the time taken by the man travelling 600 km when 400 km travelled by train and 200 by car whereas in the second case man travelled 200km by train and rest by car so to form both of the equation we used the relation between the distance, speed and time and then we can simply the equations and compare both the equation.