Question

The cost of 8kg apple and 3kg is Rs 70. The cost of 10kg apple and 6kg orange is 90. Find the cost of each item if x is the cost of apples per kg and y is the cost of oranges per kg.

(a) x=2, y=3

(b) x=3, y=2

(c) x=2, y=2

(d) x=3, y=3

This question was addressed to me in homework.

Question is taken from Applications of Determinants and Matrices topic in section Determinants of Mathematics – Class 12

Answer 1

Correct choice is (b) x=3, y=2

To elaborate: The given situation can be represented by a system of equations as:

8x+3y=70

10x+5y=90

Consider, A=\(\begin{bmatrix}8&3\\10&5\end{bmatrix}\), X=\(\begin{bmatrix}x\\y\end{bmatrix}\), B=\(\begin{bmatrix}70\\90\end{bmatrix}\)

It can be expressed in the form of AX=B

⇒X=A^-1 B

We know that, A^-1=\(\frac{1}{|A|}\) adj A=\(\frac{1}{10} \begin{bmatrix}5&-3\\-10&8\end{bmatrix}\)

∴X=\(\frac{1}{10} \begin{bmatrix}5&-3\\-10&8\end{bmatrix}\begin{bmatrix}70\\90\end{bmatrix}\)=\(\frac{1}{10} \begin{bmatrix}5×70-3×90\\-10×70+8×90\end{bmatrix}\)=\(\frac{1}{10} \begin{bmatrix}300-270\\-700+720\end{bmatrix}\)=\(\frac{1}{10} \begin{bmatrix}30\\20\end{bmatrix}\)

∴x=3, y=2.

i.e. The cost of apples is Rs 3 per kg and the cost of oranges is Rs 2 per kg.