#### Question

A train travels at a certain average speed for a distance of 54 km and then travels a distance of 63 km at an average speed of 6 km/h more than the first speed. If it takes 3 hours to complete the total journey, what is its first speed?

#### Solution

Let x be the first speed of the train.

We know that `t=3 hours` Thus,we have,

`54/x+63/(x+6)=3 hours`

`54(x+6)+63x/(x(x+6))=3`

`54(x+6)+63x=3(x(x+6))`

`54x+324+63x=3x^2+18x`

`117x+324=3x^2+18x`

`3x^2-117x-324+18x=0`

`3x^2-99x-324=0`

`x^3-33x-108=0`

`x(x-36)+3(x-36)=0`

`(x+3)(x-36)=0`

`(x+3)=0 or (x-36)=0`

`x=-3 " or " x=36`

Speed cannot be negative and hence intial speed of the train is 36 km/hour.

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Solution A train travels at a certain average speed for a distance of 54 km and then travels a distance of 63 km at an average speed of 6 km/h more than the first speed. If it takes 3 hours to complete the total journey, what is its first speed? Concept: Algebraic Methods of Solving a Pair of Linear Equations - Cross - Multiplication Method.