Income & Expenditure (Matrix Method)
Question:
The monthly incomes of Aryan and Babban are in the ratio 3:4 and their expenditures are in the ratio 5:7. If each saves ₹15000 per month, find their monthly income using matrix method. Also state the value reflected.
The monthly incomes of Aryan and Babban are in the ratio 3:4 and their expenditures are in the ratio 5:7. If each saves ₹15000 per month, find their monthly income using matrix method. Also state the value reflected.
Solution:
Step 1: Assume incomes
\[ \text{Aryan’s income} = 3x,\quad \text{Babban’s income} = 4x \]Step 2: Assume expenditures
\[ \text{Aryan’s expenditure} = 5y,\quad \text{Babban’s expenditure} = 7y \]Step 3: Use savings formula
\[ \text{Income} – \text{Expenditure} = \text{Saving} \] For Aryan: \[ 3x – 5y = 15000 \quad …(1) \] For Babban: \[ 4x – 7y = 15000 \quad …(2) \]Step 4: Matrix form
\[ \begin{bmatrix} 3 & -5 \\ 4 & -7 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 15000 \\ 15000 \end{bmatrix} \]Step 5: Solve equations
Multiply (1) by 4 and (2) by 3: \[ 12x – 20y = 60000 \] \[ 12x – 21y = 45000 \] Subtract: \[ y = 15000 \] Substitute into (1): \[ 3x – 5(15000) = 15000 \] \[ 3x = 90000 \Rightarrow x = 30000 \]Step 6: Find incomes
\[ \text{Aryan’s income} = 3x = 90000 \] \[ \text{Babban’s income} = 4x = 120000 \]Final Answer:
Aryan’s income = ₹90,000
Babban’s income = ₹1,20,000
Value Reflected:
Savings habit, financial discipline, and responsible money management.