Question

In an envelope there are some $5$ rupee notes and some $10$ rupee notes. Total amount of these notes together is $350$ rupee. Number of $5$ rupee notes is $10$ less than twice the number of $10$ rupee notes. Then find the number of $5$ rupee and $10$ rupee notes.

Answer 1

Hint: First let us assume that the number of $5$ rupee notes are $x$ and the number of $10$ rupee notes be $y$ and then form the equations. The equations formed are linear equations in two variables and we can solve these equations by using elimination methods to get values.

Complete step by step answer:
We have given that, in an envelope there are some $5$ rupee notes and some $10$ rupee notes.
We have to find the number of $5$ rupee and $10$ rupee notes.
So, let us assume that the number of $5$ rupee notes be $x$ and the number of $10$ rupee notes be $y$.
Also we have given that the total amount of these notes together is $350$ rupee.
So the equation becomes $5x+10y=350...........(i)$
Also, we have given that Number of $5$ rupee notes is $10$ less than twice the number of $10$ rupee notes.
So, the equation becomes
$\begin{align}
  & x=2y-10 \\
 & x-2y=-10............(ii) \\
\end{align}$
As we analyze that the equations formed are linear equations. To solve the two linear equations, we use elimination methods. For this first, we multiply the second equation by $5$ to make the coefficients of $x$ equal in both equations.
Now, the equation (ii) becomes-
$5x-10y=-50............(iii)$
Now, when we add equation (i) and equation (iii), we get
$\begin{align}
  & 10x=300 \\
 & x=30 \\
\end{align}$
Now substitute the value of $x$ in equation (ii), we get
$\begin{align}
  & 30-2y=-10 \\
 & -2y=-10-30 \\
 & -2y=-40 \\
 & y=20 \\
\end{align}$
So, the number of $5$ rupee notes are $30$ and the number of $10$ rupee notes are $20$.

Note: Linear equations in two variables can be solved by graphical method also. To solve the equations by graphical method, first we have to plot a line graph by using the coordinates of variables in the equation. The solution will be the coordinates where both the lines intersect. But, here we use elimination methods to solve the equations.