Question

On her birthday Seema decide to donate some money to children of an orphanage home. if there were 8 children less. everyone would have got Rs. 10 more.however if there were 16 children more everyone would have got Rs. 10 less. using matrix multiplication find the number of children and amount distributed by Seema.

Answer 1

Let x be no. of children, y is the amount per child.
1) (x - 8) children will (y + 10) each
$\Rightarrow$ Total (x - 8) (y + 10) = xy
$\text{/}xy-8y+10x-80=\text{/}xy$
$ar10x-8y=80\dots \dots \left(1\right)$

2) (x + 16) children will get (y - 10) each
(x + 16) (y - 10) = xy
xy + 16y - 10x - 160 = xy
- 10x + 16y = 160 ...... (2)
This linear pair of equations can be solved using matrices
Let $A=\left[\begin{array}{cc}10& -8\\ -10& 16\end{array}\right]$ (coefficients of x and y) $\left[x\right].\left[\begin{array}{c}x\\ y\end{array}\right]$ and $B=\left[\begin{array}{c}80\\ 160\end{array}\right]$
$A^{-1}=\frac{1}{160-80}\left[\begin{array}{cc}16& 8\\ 10& 10\end{array}\right]=\frac{1}{80}\left[\begin{array}{cc}16& 8\\ 10& 10\end{array}\right]$
$\left[x\right]=A^{-1}B=\frac{1}{80}\left[\begin{array}{cc}16& 8\\ 10& 10\end{array}\right]\left[\begin{array}{c}80\\ 160\end{array}\right]=\left[\begin{array}{c}\frac{16}{\text{/}80}\times \text{/}80+\frac{8}{\text{/}80\times \text{/}{160}^2}\\ \frac{10}{\text{/}80}\times \text{/}80+\frac{10}{\text{/}80\times \text{/}{160}^2}\end{array}\right]=\left[\begin{array}{c}16+16\\ 10+20\end{array}\right]=\left[\begin{array}{c}32\\ 30\end{array}\right]$
$\therefore$ x = No. of children = 32 and y = money in each child = Rs. 30