Question

The ratio of income of two persons is 9 : 7 and the ratio of their expenditure is 4 : 3. If each of them saves ₹ 2,000 per month, then find their monthly income.

Hint:

When solving problems involving ratios of income and expenditure, use the savings equation to form a system of linear equations, then solve for the unknowns.

Answer 1

Let the monthly incomes of the two persons be \( 9x \) and \( 7x \), where \( x \) is a constant. Let their monthly expenditures be \( 4y \) and \( 3y \), where \( y \) is a constant. Since the savings for each person is ₹ 2,000 per month, we can write the following equations for their savings: For the first person: \[ \text{Income} - \text{Expenditure} = 2000 \quad \Rightarrow \quad 9x - 4y = 2000 \quad \text{(1)}. \] For the second person: \[ \text{Income} - \text{Expenditure} = 2000 \quad \Rightarrow \quad 7x - 3y = 2000 \quad \text{(2)}. \] Now, we have the system of equations: \[ 9x - 4y = 2000 \quad \text{(1)} \] \[ 7x - 3y = 2000 \quad \text{(2)}. \] Multiply equation (1) by 3 and equation (2) by 4: \[ 27x - 12y = 6000 \quad \text{(3)} \] \[ 28x - 12y = 8000 \quad \text{(4)}. \] Now subtract equation (3) from equation (4): \[ (28x - 12y) - (27x - 12y) = 8000 - 6000, \] \[ x = 2000. \] Substitute \( x = 2000 \) into equation (1): \[ 9(2000) - 4y = 2000, \] \[ 18000 - 4y = 2000, \] \[ 4y = 16000, \] \[ y = 4000. \] Thus, the monthly incomes of the two persons are: \[ 9x = 9(2000) = 18000 \quad \text{and} \quad 7x = 7(2000) = 14000. \]
Conclusion:
The monthly incomes of the two persons are ₹ 18,000 and ₹ 14,000.