Question

A train leaves New Delhi for Ludhiana, 324 km away, at 9AM. One hour later another train leaves Ludhiana for New Delhi. They meet at noon. If the second train had started at 9AM and first at 10.30AM. They would still have met at noon ? Find the speed of each train.

Answer 1

Hint: Equation in two variable - Equation in the form of $ ax + by = c $
It is a linear equation with two variables x and y.
Solution of linear equations of two variables - Solution of equation are the values of x and y, which can satisfy the given equation.
We can solve two equations by any one of the following methods.
a.Graphical method
b.Elimination method
c.Substitution method
d.Cross-multiplication method
Formula use
Speed, distance and time relation - Distance $ = $ speed $ \times $ time

Complete step-by-step answer:
When the two trains meet sum of their distance covered is 324km.
Let speed of I train $ = $ x
Speed of II train $ = $ y
Total distance $ = $ 324 km
(i) Train I
Time taken by train $ I = Noon - 9AM = 3hours $
So, distance travelled by $ {T_1} = speed \times time $
 $ = 3x $
Train II
Time taken by train $ {T_2} = Noon = 10AM $
Distance travelled by $ {T_2} = 2y $
So, total distance $ = {D_1} + {D_2} $
 $ 324 - 3x + 2y $ …..(1)
(ii) According to question
Time taken by train 1 $ = Noon - 10.30AM $
 $ = 1.5 $ hours
Distance travelled by $ {T_1} = 1.5x $
Time taken by $ {T_2} = Noon - 9AM $
 $ = 3 $ hours
Distance travelled by $ {T_2} = 3y $
Total Distance $ = {D_1} + {D_2} $
 $ 324 = 1.5x + 3y $
Rearranging and solving above equation in simplex form
 $ x + 2y = 216 $ …..(2)
Now we have two equations and two variables. Solving these by elimination method.
3x+2y=324
x+2y=16
subtracting the two equations we get,

Or, $ x = \dfrac{{108}}{2} $
 $ x = 54km/hr $
Substituting this value of x in equation 2
 $ 2y = 216 - x $
 $ y = \dfrac{{216 - 54}}{2} $
 $ y = \dfrac{{162}}{2} = 81 km/hr $
So, speed of first train is 54 km per hour
Speed of first train is 81 km per hour

Note: We can solve these equations by any other method also, substitution method is also an easy method after elimination method.
Let’s have a look from above question value of x from equation 1 is
 $ x = \dfrac{{324 - 2y}}{3} $
And substitute it in equation 2
 $ \dfrac{{324 - 2y}}{3} + 2y = 216 $
Just solve this equation, get the value of y and put it back in the above equation to get the value of x.