Question

Five years ago, Shikha was three times as old as Rani. 10 years later Shikha will be twice as old as Rani. How old are they now?​
A) Shikha =20 and Rani=50
B) Shikha =50 and Rani=10
C) Shikha =10 and Rani=50
D) Shikha =50 and Rani=20

Answer 1

Hint: For solving the above question, we would be requiring the knowledge of solving the system of linear equations in two variables. In this question we would be using the elimination method.
In elimination method, we first try to make the coefficient of any one variable of the two as equal and then subtract or add the new equations accordingly.
Then, we will get the equation which will be having only one variable.
Then we can solve the equation to get the value of that variable which is left and after getting the value of any one variable, we can plug in that value in any of the equations and then get the value of the other variable as well.

Complete step-by-step answer:
As mentioned in the question,
Let the age of Shikha be x and the age of Rani be y.
Now 5 years ago, according to the question, we get
\[\begin{align}
  & \left( x-5 \right)=3\left( y-5 \right) \\
 & x-3y=-10\ \ \ \ \ ...(a) \\
\end{align}\]
Now, after 10 years, according to the question, we get
\[\begin{align}
  & \left( x+10 \right)=2\left( y+10 \right) \\
 & x-2y=10\ \ \ \ \ ...(b) \\
\end{align}\]
Now, on subtracting equations (a) and (b), we get
\[\begin{align}
  & \ \ \ \ x-3y=-10 \\
 & \dfrac{-(x-2y=10)}{-y=-20} \\
 & y=20 \\
\end{align}\]
Now, putting this value in (b), we get
\[\begin{align}
  & x-40=10 \\
 & x=50 \\
\end{align}\]
Hence, the present age of Shikha is 50 years and Rani’s age is 20 years.
     \[\]
So option (D) is correct.

Note: For questions in which there are more than 2 variables, in order to know whether the equations are solvable or whether we will be able to get the values of the variables by just counting the number of variables and number of the equations. If the number of equations and the number of variables involved in the question is equal then we can surely say that every variable will be having a unique value. If these numbers are not equal, then we do not comment on that.
There are two other methods of solving a 2 variable system of equations:-
1) substitution method
2) cross multiplication method