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: Understanding the problem:
We are given the following ratios:
- The monthly incomes of A and B are in the ratio $8 : 7$.
- The monthly expenditures of A and B are in the ratio $19 : 16$.
- Both A and B save Rs 2500 per month.
We need to find the monthly income of each person.

Step 2: Represent the incomes and expenditures of A and B:
Let the monthly incomes of A and B be $8x$ and $7x$ respectively.
Let the monthly expenditures of A and B be $19y$ and $16y$ respectively.

Step 3: Set up the equations for savings:
Savings = Income - Expenditure.
Since both A and B save Rs 2500 per month, we can write the equations for their savings as follows:
- For A: $8x - 19y = 2500$
- For B: $7x - 16y = 2500$

Step 4: Solve the system of equations:
We have the system of equations: \[ 8x - 19y = 2500 \tag{1} \] \[ 7x - 16y = 2500 \tag{2} \] We will use the method of elimination to solve this system.

Multiply equation (1) by 7 and equation (2) by 8 to make the coefficients of $x$ the same: \[ 7(8x - 19y) = 7 \times 2500 \quad \Rightarrow \quad 56x - 133y = 17500 \tag{3} \] \[ 8(7x - 16y) = 8 \times 2500 \quad \Rightarrow \quad 56x - 128y = 20000 \tag{4} \] Now, subtract equation (4) from equation (3): \[ (56x - 133y) - (56x - 128y) = 17500 - 20000 \] \[ -5y = -2500 \] \[ y = 500 \]

Step 5: Find the value of $x$:
Substitute $y = 500$ into equation (1): \[ 8x - 19(500) = 2500 \] \[ 8x - 9500 = 2500 \] \[ 8x = 2500 + 9500 = 12000 \] \[ x = \frac{12000}{8} = 1500 \]

Step 6: Calculate the incomes of A and B:
- The monthly income of A is $8x = 8 \times 1500 = 12000$. - The monthly income of B is $7x = 7 \times 1500 = 10500$.

Conclusion:
The monthly income of A is Rs 12000, and the monthly income of B is Rs 10500.