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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? - Mathematics

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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?

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Solution

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

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

Since each saves Rs 15,000 per month

Monthly saving of Aryan: 3x5y=15,000

Monthly saving of Babban: 4x7y=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-1B

`=>[(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

Concept: Inverse of Matrix - Inverse of a Square Matrix by the Adjoint Method
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