The monthly incomes of Aryan and Babban are in the ratio 3 : 4 and their monthly expenditures are in the ratio 5 : 7. If each saves Rs 15,000 per month, find their monthly incomes using matrix method. This problem reflects which value?

#### Solution

Let the monthly incomes of Aryan and Babban be 3*x* and 4*x*, respectively.

Suppose their monthly expenditures are 5*y* and 7y, respectively.

Since each saves Rs 15,000 per month

Monthly saving of Aryan: 3x−5y=15,000

Monthly saving of Babban: 4x−7y=15,000

The above system of equations can be written in the matrix form as follows:

`[(3,5),(4,-7)][(x),(y)]=[(15000),(15000)]`

or

AX = B, where

`A=[(3,-5),(4,-7)],X=[(x),(y)]`

Now,

`|A|=|(3,-5),(4,-7)|=-21-(-20)=-1`

Adj `A=[(-7,-4),(5,3)]^T=[(-7,5),(-4,3)]`

So,

`A^(-1)=1/|A|adjA=-1[(-7,5),(-4,3)]=[(7,-5),(4,-3)]`

∴ X = A^{-1}B

`=>[(x),(y)]=[(7,-5),(4,-3)][(15000),(15000)]`

`=>[(x),(y)]=[(105000,-75000),(60000,-45000)]`

`=>[(x),(y)]=[(30000),(15000)]`

⇒ x=30,000 and y=15,000

Therefore,

Monthly income of Aryan = 3×Rs 30,000=Rs 90,000

Monthly income of Babban = 4×Rs 30,000= Rs 1,20,000

From this problem, we are encouraged to understand the power of savings. We should save certain part of our monthly income for the future