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 information:
- 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 each per month.
We need to find the monthly incomes of A and B.

Step 2: Let the incomes and expenditures be:
Let the monthly incomes of A and B be \( 8x \) and \( 7x \) respectively (since their incomes are in the ratio 8:7).
Let the monthly expenditures of A and B be \( 19y \) and \( 16y \) respectively (since their expenditures are in the ratio 19:16).
We are also told that each saves Rs 2500 per month. Therefore:
- A's savings = Income of A - Expenditure of A = \( 8x - 19y = 2500 \)
- B's savings = Income of B - Expenditure of B = \( 7x - 16y = 2500 \)

Step 3: Set up the system of equations:
From the savings equations for A and B, we have the system of equations:
1. \( 8x - 19y = 2500 \)
2. \( 7x - 16y = 2500 \)

Step 4: Solve the system of equations:
We will solve these two equations by elimination or substitution. First, let's multiply equation (1) by 7 and equation (2) by 8 to align the coefficients of \( x \):
- Equation (1) multiplied by 7:
\[ 56x - 133y = 17500 \] - Equation (2) multiplied by 8:
\[ 56x - 128y = 20000 \]
Now subtract the second equation from the first equation to eliminate \( x \):
\[ (56x - 133y) - (56x - 128y) = 17500 - 20000 \] Simplifying this gives:
\[ -5y = -2500 \] Solving for \( y \):
\[ y = \frac{2500}{5} = 500 \]

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

Step 6: Find the monthly incomes of A and B:
- A's income = \( 8x = 8 \times 1500 = 12000 \) Rs.
- B's income = \( 7x = 7 \times 1500 = 10500 \) Rs.

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