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.

Months	Sales of units			Total Commission drawn (in ₹)
	A	B	C
January	90	100	20	800
February	130	50	40	900
March	60	100	30	850

Find out the rate of commission on the items A, B and C by using Cramer’s rule

Answer 1

Let x, y, and z be the commission on items A, B, and C respectively.

90x + 100y + 20z = 800

⇒ 9x + 10y + 2z = 80  ........(1)

130x + 50y + 40z = 900

⇒ 13x + 5y + 4z = 90  .......(2)

60x + 100y + 30z = 850

⇒ 6x + 10y + 3z = 85  ........(3)

Here `Delta = |(9, 10, 2),(13, 5, 4),(6, 10, 3)|`

= 9(15 – 40) – 10(39 – 24) + 2(130 – 30)

= 9(– 25) – 10(15) + 2(100)

= – 225 – 150 + 200

= – 175 ≠ 0

∴ We can apply Cramer’s Rule and the system is consistent and it has unique solution.

`Delta_x = |(80, 10, 2),(90, 5, 4),(85, 10, 3)|`

= 80(15 – 40) – 10(270 – 340) + 2(900 – 425)

= 80(– 25) – 10(– 70) + 2(475)

= – 2000 + 700 + 950

= – 350

`Delta_y = |(9, 80, 2),(13, 90, 4),(6, 85, 3)|`

= 9(270 – 340) – 80(39 – 24) + 2(1105 – 540)

= 9(– 70) – 80(15) + 2(565)

= – 630 – 1200 + 1130

= – 700

`Delta_x = |(9, 10, 80),(13, 5, 90),(6, 10, 85)|`

= 9(425 – 900) – 10(1105 – 540) + 80(130 – 30)

= 9(– 475) – 10(565) + 80(100)

= – 4275 – 5650 + 8000

= – 1925

∴ By cramer’s rule

x = `Delta_x/Delta = (-350)/(- 175)` = 2

y = `Delta_y/Delta = (-700)/(-175)` = 4

z = `Delta_z/Delta = (- 1925)/(-175)` = 11

The solution is (x, y, z) = (2, 4, 11)

∴ The rates of commission for A, B and C are ₹ 2, ₹ 4 and ₹ 11 respectively.