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	Sales of Units			Total commission drawn (in ₹)
	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

Given : - 

Record of sales during three months 

Let, rates of commissions on items A,B and C be x, y and z respectively. 

Now, 

We can arrange this model in linear equation system 

Thus, 

We have 

90x + 100y + 20z = 800 

130x + 50y + 40z = 900 

60x + 100y + 30z = 850

Here,

Solving determinant, expanding along 2^nd column 

⇒ D = 50[( – 50)( – 170) – ( – 200)( – 60)] 

⇒ D = 50[8500 – 12000] 

⇒ D = – 175000 

Again, 

Solve D₁ formed by replacing 1^st column by B matrices 

Here,

Solving determinant, expanding along 2^nd column 

⇒ D₁ = 50[( – 1000)( – 500) – ( – 950)( – 60)] 

⇒ D₁ = 50[50000 – 57000] 

⇒ D₁ = – 350000 

Again, 

Solve D₂ formed by replacing 2^nd column by B matrices 

Here,

Solving determinant, expanding along 1^st Row 

⇒ D₂ = 20[17500 – 52500] 

⇒ D₂ = – 700000 

And, 

Solve D₃ formed by replacing 3^rd column by B matrices 

Here,

Solving determinant, expanding along 1st Row 

⇒ D₃ = 50[161500 – 200000] 

⇒ D₃ = – 1925000 

Thus by Cramer’s Rule, we have

z = 11

Thus rates of commission of items A, B and C are 2%, 4% and 11% respectively.