Question

A salesman has the following record of sales during three months for three items A, B and C which have different rates of commission

Month	Sale of units		Total commission drawn (in Rs)
	A	B	C
Jan	90	100	20	800
Feb	130	50	40	900
March	60	100	30	850

Find out the rates of commission on items A, B and C by using determinant method.

Answer 1

Let x, y and z be the rates of commission on items A, B and C respectively. Based on the given data, we get

$$\begin{gathered} 90x+100y+20z=800 \\ 130x+50y+40z=900 \\ 60x+100y+30z=850 \end{gathered}$$

Dividing all the equations by 10 on both sides, we get

$$\begin{gathered} 9x+10y+2z=80 \\ 13x+5y+4z=90 \\ 6x+10y+3z=85 \end{gathered}$$


$$\begin{gathered} D=\left|\begin{array}{ccc}9& 10& 2\\ 13& 5& 4\\ 6& 10& 3\end{array}\right|\left[\mathrm{Expressing}\mathrm{the}\mathrm{equation}\mathrm{as}\mathrm{a}\mathrm{determinant}\right] \\ =9(15-40)-10(39-24)+2(130-30) \\ =9(-25)-10\left(15\right)+2\left(100\right) \\ =-175 \\ {D}_1=\left|\begin{array}{ccc}80& 10& 2\\ 90& 5& 4\\ 85& 10& 3\end{array}\right| \\ =80(15-40)-10(270-340)+2(900-425) \\ =80(-25)-10(-70)+2\left(475\right) \\ =-350 \\ {D}_2=\left|\begin{array}{ccc}9& 80& 2\\ 13& 90& 4\\ 6& 85& 3\end{array}\right| \\ =9(270-340)-80(39-24)+2(1105-540) \\ =9(-70)-80\left(15\right)+2\left(565\right) \\ =-700 \\ {D}_3=\left|\begin{array}{ccc}9& 10& 80\\ 13& 5& 90\\ 6& 10& 85\end{array}\right| \\ =9(425-900)-10(1105-540)+80(130-30) \\ =9(-475)-10\left(565\right)+80\left(100\right) \\ =-1925 \\ \mathrm{Thus}, \\ x=\frac{D_1}{D}=\frac{-350}{-175}=2 \\ y=\frac{D_2}{D}=\frac{-700}{-175}=4 \\ z=\frac{D_3}{D}=\frac{-1925}{-175}=11 \end{gathered}$$

Therefore, the rates of commission on items A, B and C are 2, 4 and 11, respectively.