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in Linear Equations by (56.3k points)

Ramesh travels 760 km to his home partly by train and partly by car. He takes 8 hours if he travels 160 km by train and the rest by car. He takes 12 minutes more if he travels 240 km by train and the rest by car. Find the speed of the train and car respectively.

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Best answer

Let’s assume, 

The speed of the train be x km/hr 

The speed of the car = y km/hr 

From the question, it’s understood that there are two parts 

# Part 1: When Ramesh travels 160 Km by train and the rest by car. 

# Part 2: When Ramesh travels 240 Km by train and the rest by car. 

Part 1, 

Time taken by Ramesh to travel 160 km by train = 160/x hrs [∵ time = distance/ speed] 

Time taken by Ramesh to travel the remaining (760 – 160) km i.e., 600 km by car =600/y hrs 

So, the total time taken by Ramesh to cover 760Km = 160/x hrs + 600/y hrs 

It’s given that, 

Total time taken for this journey = 8 hours 

So, by equations its 

160/x + 600/y = 8 

20/x + 75/y = 1 [on dividing previous equation by 8] …………………… (i) 

Part 2, 

Time taken by Ramesh to travel 240 km by train = 240/x hrs 

Time taken by Ramesh to travel (760 – 240) = 520 km by car = 520/y hrs 

For this journey, it’s given that Ramesh will take a total of is 8 hours and 12 minutes to finish. 

240/x + 520/y = 8 hrs 12 mins = 8 + (12/60) = 41/5 hr 

240/x + 520/y = 41/5 

6/x + 13/y = 41/200 ………. (ii) 

Solving (i) and (ii), we get the required solution 

Let’s take 1/x = u and 1/y = v, 

So, (i) and (ii) becomes, 

20u + 75v = 1 ……….. (iii) 

6u + 13v = 41/200 ……. (iv) 

On multiplying (iii) by 3 and (iv) by 10, 

60u + 225v = 3 

60u + 130v = 41/20 

Subtracting the above two equations, we get 

(225 – 130)v = 3 – 41/20 

95v = 19/ 20 

⇒ v = 19/ (20 x 95) = 1/100 

⇒ y = 1/v = 100 

Using v = 1/100 in (iii) to find v, 

20u + 75(1/100) = 1 

20u = 1 – 75/100 

⇒ 20u = 25/100 = 1/4 

⇒ u = 1/80 

⇒ x = 1/u = 80 

So, the speed of the train is 80 km/hr and the speed of car is 100 km/hr.

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