The monthly incomes of P and Q are in the ratio 4 : 5 and their monthly expenditures are in the ratio 3 : 4. If each of them saves Rs. 10,000 per month, then their monthly incomes are
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → Rs. 40,000 and Rs. 50,000
Let the incomes be Rs. 4x and Rs. 5x, and the expenditures Rs. 3y and Rs. 4y. Since saving = income − expenditure:
\(4x - 3y = 10000\)
\(5x - 4y = 10000\)
In matrix form AX = B:
\( \displaystyle A = \begin{bmatrix}4 & -3\\ 5 & -4\end{bmatrix},\quad X = \begin{bmatrix}x\\ y\end{bmatrix},\quad B = \begin{bmatrix}10000\\ 10000\end{bmatrix} \)
\(|A| = -16 + 15 = -1\)
Since \(|A| \ne 0\), \(X = A^{-1}B\):
\( \displaystyle A^{-1} = \frac{1}{-1}\begin{bmatrix}-4 & 3\\ -5 & 4\end{bmatrix} = \begin{bmatrix}4 & -3\\ 5 & -4\end{bmatrix} \)
\( \displaystyle X = \begin{bmatrix}40000 - 30000\\ 50000 - 40000\end{bmatrix} = \begin{bmatrix}10000\\ 10000\end{bmatrix} \)
So x = 10000, and the incomes are \(4x\) = Rs. 40,000 and \(5x\) = Rs. 50,000.
Check: the expenditures are Rs. 30,000 and Rs. 40,000, and each saving is Rs. 10,000.
Why the other options are incorrect
Option (1) Rs. 30,000 and Rs. 40,000 — Incorrect. These are the expenditures 3y and 4y, not the incomes.
Option (2) Rs. 20,000 and Rs. 25,000 — Incorrect. The expenditures would then be Rs. 10,000 and Rs. 15,000, which are not in the ratio 3 : 4.
Option (4) Rs. 80,000 and Rs. 1,00,000 — Incorrect. The expenditures would then be Rs. 70,000 and Rs. 90,000, which are not in the ratio 3 : 4.