Question

A train travels at a certain average speed for a distance of 63 km and then travels a distance of 72 km at an average speed of 6 km per hour more than its original speed. If it takes 3 hours to complete the total journey, what is its original average speed ?

Hint- Speed is equal to distance upon time and ${t_1} + {t_2} = 3$ hours where ${t_1}\& {t_2}$ is time taken for complete journey
Let the initial speed be $x$ Km per hour.
As per the question,
Time taken at a speed of $x$ km per hour $\left( {{t_1}} \right){\text{ = }}\dfrac{{63}}{x}$ hr
Time taken at a speed of $x + 6$ km per hour $\left( {{t_2}} \right) = \dfrac{{72}}{{x + 6}}$ hr
Now, according to the question it is given that the total journey takes 3 hr.
${t_1} + {t_2} = 3$
Put the values of ${t_{_1}}$and ${t_2}$
$\dfrac{{63}}{x} + \dfrac{{72}}{{x + 6}} = 3$
Take LCM
$\Rightarrow \dfrac{{63\left( {x + 6} \right) + 72x}}{{x\left( {x + 6} \right)}} = 3$
With the help of cross multiplication, solve the above equation
$\Rightarrow 63x + 378 + 72x = 3x\left( {x + 6} \right) \\ \Rightarrow 378 + 135x = 3{x^2} + 18x \\$
Bring RHS term to LHS, we get
$\Rightarrow 135x + 378 - \left( {3{x^{^2}} + 18x} \right) = 0$
Taking minus sign common, we get
$\Rightarrow 3{x^{^2}} + 18x - 135x - 378 = 0 \\ \Rightarrow 3{x^2} - 117x - 378 = 0 \\$
Taking $3$common, we get
$\Rightarrow 3\left( {{x^2} - 39x - 126} \right) = 0 \\ \Rightarrow {x^2} - 39x - 126 = 0 \\$
Now factorise the above equation
$\Rightarrow {x^2} - 42x + 3x - 126 = 0 \\ \Rightarrow x\left( {x - 42} \right) + 3\left( {x - 42} \right) = 0 \\ \Rightarrow \left( {x - 42} \right)\left( {x + 3} \right) = 0 \\$
Now either $x = 42$ or $x = - 3$ and since speed cannot be negative.
So the speed of Train is $42$Km per hour.
Note - For such a type of question always begin with assuming the initial speed be $x$km per hour. And we know that speed is equal to distance upon time. So Calculate the time taken when speed is $x$ and $x + 6$ km per hour respectively.