Question

Two schools $P$ and $Q$ want to award their selected students on the values of Discipline, Politeness, and Punctually. The school $P$ wants to award $Rs.x$ each, $y$ each and $Rs.z$ each for the three respective values to its $3,2$ and $1$ students with a total award money of $Rs.1,000$. School Q wants to spend $Rs.1,500$ to award its $4,1$ and $3$ students on the respective values (by giving the same award money for the three values as before). If the total amount of awards for one prize on each value is $Rs.600,$ using matrices, find the sum of the award money for each value.

• A. 900
• B. 700
• C. 800
• D. 600

Answer 1

The correct option is D 600

The equations obtained from given data are $3x+2y+z=1000$ , $4x+y+3z=15000$ and $x+y+z=600$

By using matrix method , we get $AX=B$

Where $A=\left[\begin{array}{ccc}3& 2& 1\\ 4& 1& 3\\ 1& 1& 1\end{array}\right]$ , $X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$ and $B=\left[\begin{array}{c}1000\\ 1500\\ 600\end{array}\right]$

$\Rightarrow X=A^{-1}B=-\frac{1}{5}\left[\begin{array}{ccc}-2& -1& 5\\ -1& 2& -5\\ 3& -1& -5\end{array}\right]\left[\begin{array}{c}1000\\ 1500\\ 600\end{array}\right]=\left[\begin{array}{c}100\\ 200\\ 300\end{array}\right]$

Therefore $x=100$ , $y=200$ and $z=300$