Question

The ratio of monthly incomes of A and B is 7 : 6 and the ratio of their monthly expenditures is 5 : 4. If each of them saves Rupees 600 per month, find the sum of their monthly incomes?

• A. 3600
• B. 2100
• C. 3900
• D. None of these

Hint:

When ratios of income and expenditure are given along with equal savings, always assume variables for incomes and expenditures, form two equations, and solve simultaneously.

Answer 1

The Correct Option is C

Step 1: Assume the incomes. Let the monthly incomes of A and B be \(7x\) and \(6x\) respectively. 

Step 2: Assume the expenditures. Let the monthly expenditures of A and B be \(5y\) and \(4y\) respectively. 

Step 3: Use the saving condition. Savings = Income – Expenditure. 

- For A: \(7x - 5y = 600\) 

- For B: \(6x - 4y = 600\) 

Step 4: Form simultaneous equations. \[ 7x - 5y = 600 \quad ...(1) \] \[ 6x - 4y = 600 \quad ...(2) \] 

Step 5: Simplify equations. From (2): \(6x - 4y = 600 \Rightarrow 3x - 2y = 300 \quad ...(3)\) Multiply (3) by 2: \[ 6x - 4y = 600 \] (Already the same form). Now solve (1): \[ 7x - 5y = 600 \] Multiply (3) by 5: \[ 15x - 10y = 1500 \] Multiply (1) by 2: \[ 14x - 10y = 1200 \] Subtract: \[ (15x - 10y) - (14x - 10y) = 1500 - 1200 \] \[ x = 300 \] 

Step 6: Find incomes. 

- Income of A = \(7x = 7 \times 300 = 2100\) 

- Income of B = \(6x = 6 \times 300 = 1800\) 

Step 7: Find sum. \[ \text{Total income} = 2100 + 1800 = 3900 \] 

Step 8: Final Answer. \[ \text{Sum of incomes} = 2100 + 1800 = \boxed{3900} \]

Answer 2

Instead of solving two simultaneous equations from scratch, this method checks each option directly against the ratio conditions to see which sum of incomes is actually consistent with the given savings.

1. 3600: If the combined income were \(3600\), then A's income would be \(\frac{7}{13}\times 3600 \approx 1938.46\) and B's income would be \(\frac{6}{13}\times 3600 \approx 1661.54\). Their expenditures (after saving Rupees 600 each) would be \(1338.46\) and \(1061.54\), giving a ratio of about \(1.26 : 1\), not the required \(5:4 = 1.25:1\). This option fails the check.
2. 2100: Here A's income would be \(\frac{7}{13}\times 2100 \approx 1130.77\) and B's income \(\approx 969.23\). Subtracting the fixed saving of 600 from each gives expenditures in the ratio \(1.4375:1\), which is far from \(5:4\). This is actually close to A's individual income in the correct case, not the combined total, so it is a distractor value rather than the answer.
3. 3900: Writing the sum as \(S\), A's income is \(\frac{7S}{13}\) and B's is \(\frac{6S}{13}\); their expenditures are \(\frac{7S}{13}-600\) and \(\frac{6S}{13}-600\). Requiring this pair to be in ratio \(5:4\) gives \(4\left(\frac{7S}{13}-600\right)=5\left(\frac{6S}{13}-600\right)\), i.e. \(4(7S-7800)=5(6S-7800)\). For \(S=3900\): left side \(=4(27300-7800)=4\times19500=78000\); right side \(=5(23400-7800)=5\times15600=78000\). Both sides match exactly, so this value satisfies every condition in the question.
4. None of these: Since \(3900\) already checks out perfectly, there is no need to fall back on this option.

Only \(3900\) keeps the income ratio \(7:6\), the expenditure ratio \(5:4\), and the equal saving of Rupees 600 simultaneously true.

So the correct answer is 3900.