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 Statement:
Given the following information:
- A and B's monthly incomes are in the ratio $8 : 7$.
- A and B's monthly expenditures are in the ratio $19 : 16$.
- Both A and B have a monthly saving of Rs 2500.
Determine the monthly income for both A and B.

Variable Representation:
Let A's monthly income be $8x$ and B's monthly income be $7x$.
Let A's monthly expenditure be $19y$ and B's monthly expenditure be $16y$.

Savings Equations:
Savings is defined as Income - Expenditure.
Based on the given savings, the equations are:
- For A: $8x - 19y = 2500$
- For B: $7x - 16y = 2500$

System of Equations:
The system to solve is:
\[8x - 19y = 2500 \tag{1}\]\[7x - 16y = 2500 \tag{2}\]Using the elimination method.

To equalize the coefficients of $x$, multiply equation (1) by 7 and equation (2) by 8:
\[56x - 133y = 17500 \tag{3}\]\[56x - 128y = 20000 \tag{4}\]Subtract equation (4) from equation (3):\[(56x - 133y) - (56x - 128y) = 17500 - 20000\]\[-5y = -2500\]\[y = 500\]

Solving for x:
Substitute $y = 500$ into equation (1):\[8x - 19(500) = 2500\]\[8x - 9500 = 2500\]\[8x = 12000\]\[x = 1500\]

Monthly Incomes:
- A's monthly income: $8x = 8 \times 1500 = 12000$.- B's monthly income: $7x = 7 \times 1500 = 10500$.

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