Question

A bag contains $3$ types of coins namely Re. $1$, Rs. $2$ and Rs. $5$. There are $30$ coins amounting to Rs. $100$ in total. Find the number of coins in each category.

Answer 1

Let $x,y$ and $z$ be the number of coins respectively in each category Re.$1$, Rs.$2$ and Rs.$5$. From the given information
$x+y+z=30$ ......(1)
$x+2y+5z=100$ ......(2)
Here we have $3$ unknowns but $2$ equations. We assign arbitrary value $k$ to $z$ and solve for $x$ and $y$
(1) and (2) become
$x+y=30-k;x+2y=100-5k$ $k\in R$
$\Delta =\left|\begin{array}{cc}1& 1\\ 1& 2\end{array}\right|=1$
${\Delta }_x=\left|\begin{array}{cc}30-k& 1\\ 100-5k& 2\end{array}\right|=3k-40$
${\Delta }_y=\left|\begin{array}{cc}1& 30-k\\ 1& 100-5k\end{array}\right|=70-4k$
By Cramer's Rule, we have
$x=\frac{{\Delta }_x}{\Delta }=3k-40$
$y=\frac{{\Delta }_y}{\Delta }=70-4k$
The solution is $(x,y,z)=(3k-40,70-4k,k)$ $k\in R$
Since the number of coins is a non-negative integer $k=0,1,2,....$
Moreover $3k-40\ge 0$ and
$70-4k\ge 0$ $\Rightarrow$ $14\le k\le 17$
$\therefore$ The possible solutions are
$(2,14,14),(5,10,15),(8,6,16)$ and $(11,2,17)$