Question

The monthly incomes of A and B are in the ratio 8 : 7 and their expenditures are in the ratio 19 : 16. If each saves Rs 2500 per month, find the monthly income of each

Answer 1

Step 1: Let the incomes and expenditures be:
\[\text{Income of } A = 8x, \quad \text{Income of } B = 7x.\]
\[\text{Expenditure of } A = 19y, \quad \text{Expenditure of } B = 16y.\]
Step 2: Use the savings equation \textit{Savings = Income $-$ Expenditure}:
\[8x - 19y = 2500 \quad \text{(for A)},\]
\[7x - 16y = 2500 \quad \text{(for B)}.\]
Step 3: Solve the equations. From the first equation:
\[8x = 19y + 2500 \implies x = \frac{19y + 2500}{8}.\]
Substitute into the second equation:
\[7 \left(\frac{19y + 2500}{8}\right) - 16y = 2500.\]
Simplify:
\[\frac{133y + 17500}{8} - 16y = 2500.\]
Multiply through by 8:
\[133y + 17500 - 128y = 20000.\]
\[5y = 2500 \implies y = 500.\]
Step 4: Find $x$:
\[x = \frac{19(500) + 2500}{8} = \frac{9500 + 2500}{8} = \frac{12000}{8} = 1500.\]
Step 5: Find incomes:
\[\text{Income of } A = 8x = 8(1500) = 12000.\]
\[\text{Income of } B = 7x = 7(1500) = 10500.\]
Correct Answer: Incomes are 12000 and 10500.