Question

A train travels at a certain average speed for a distance of 120 km and then travels a distance of 130 km at an average speed of 5 km/hr more than its original speed. If it takes 4 hours to complete the total journey, then the original speed of the train is​

Answer 1

we know that total time taken to complete the journey is 4 hrs ..(1)

let the the original speed be x

case 1

d is 120 km

s = d/t t= 120/ s

case2

d is 130km

s is 5 + x

so t = 130 / 5+x

equation 1 implies

120/s + 130/ 5+x = 4

120/5+x + 130/ 5+x = 4

x = 57.5 which is the original speed

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Answer 2

Solution:

Step 1: Let the original average speed of the train be x km/hr.

As we know,

Speed = Distance / Time taken

Time taken = Distance / Speed            ------------------   (i)

For distance 120 km, using equation (i), we get

Step 2: Time taken at a speed of x km/hr,  t₁ = 120/x hr   [∵ At x km/hr speed, it covered a distance of 120 km]

Step 3: Now average speed is increased by 5 km/hr

∴ Time taken at a speed of (x+5) km/hr, t₂ = 130/(x+5) hr     [∵ At (x+5) km/hr speed, it covered a distance of 130 km]

Step 4: Total time taken to complete the journey, t = 4 hours

t₁ + t₂ = 4

120/x + 130/(x+5) = 4

$\frac{120(x+5) + 130x}{x(x+5)} = 4\\\\\frac{120x + 600 + 130x}{x^2 + 5x} = 4\\\\250x + 600 = 4(x^2 + 5x)\\\\250x + 600 = 4x^2 + 20x\\\\4x^2 + 20x - 250x - 600 = 0\\\\4x^2 - 230x - 600 = 0\\\\4x^2 +10x-240x-600 = 0\\\\2x(2x + 5) - 120x (2x+5) = 0\\\\(2x - 120x)(2x + 5) = 0$

Solution 1:

$2x - 120 = 0\\\\2x = 120\\\\x = \frac{120}{2} \\\\x = 60$

Solution 2:

$2x + 5 = 0\\\\2x = -5\\\\x = \frac{-5}{2}$

We get,

x = 60, (-5/2)

x is the original average speed that can't be negative. So, we neglect (-5/2).

Hence, the original average speed of the train = x = 60 km/hr

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