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

Problem Analysis:
Given:
- Ratio of monthly incomes (A:B) = 8:7.
- Ratio of monthly expenditures (A:B) = 19:16.
- Monthly savings for A = Rs 2500.
- Monthly savings for B = Rs 2500.
Objective: Determine the monthly incomes of A and B.

Variable Assignment:
Let A's monthly income = \( 8x \), and B's monthly income = \( 7x \).
Let A's monthly expenditure = \( 19y \), and B's monthly expenditure = \( 16y \).
Savings equation for A: \( 8x - 19y = 2500 \)
Savings equation for B: \( 7x - 16y = 2500 \)

System of Equations:
1. \( 8x - 19y = 2500 \)
2. \( 7x - 16y = 2500 \)

Equation Solution:
To eliminate \( x \), multiply equation (1) by 7 and equation (2) by 8:
- \( 7 \times (8x - 19y = 2500) \Rightarrow 56x - 133y = 17500 \)
- \( 8 \times (7x - 16y = 2500) \Rightarrow 56x - 128y = 20000 \)
Subtract the second modified equation from the first:
\[(56x - 133y) - (56x - 128y) = 17500 - 20000\]\[-5y = -2500\]Solving for \( y \):
\[y = \frac{-2500}{-5} = 500\]

Determination of \( x \):
Substitute \( y = 500 \) into equation (1):
\[8x - 19(500) = 2500\]\[8x - 9500 = 2500\]\[8x = 2500 + 9500\]\[8x = 12000\]\[x = \frac{12000}{8} = 1500\]

Monthly Income Calculation:
- A's monthly income = \( 8x = 8 \times 1500 = 12000 \) Rs.
- B's monthly income = \( 7x = 7 \times 1500 = 10500 \) Rs.

Final Result:
A's monthly income is Rs 12000 and B's monthly income is Rs 10500.