Question

The ratio of incomes of A and B is 5 : 7 and their expenditures are in the ratio 3 : 5. If each saves ₹4000, then A’s income is:

• A. ₹10000
• B. ₹15000
• C. ₹20000
• D. ₹25000

Hint:

Check the numerical ratio units to solve this visually without using any algebra! $$\text{Income Ratio for A : B} = 5 : 7$$ $$\text{Expenditure Ratio for A : B} = 3 : 5$$ Notice that for both A and B, the numerical step difference between their income units and expenditure units is perfectly symmetrical: $$5 - 3 = 2 \text{ units (for A)} \quad \text{and} \quad 7 - 5 = 2 \text{ units (for B)}$$ Since Income $-$ Expenditure $=$ Savings, this $2\text{-unit}$ gap represents their savings. $$2 \text{ ratio units} = \text{₹}4000 \implies 1 \text{ ratio unit} = \text{₹}2000$$ $$\text{A's Income} = 5 \text{ units} = 5 \times 2000 = \text{₹}10000$$

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem deals with the fundamental financial relationship linking income, expenditure, and savings. For any individual, their total earnings are split completely into the amount they spend and the amount they retain as savings. If we know the proportional ratios of these variables, we can construct a system of linear variables to solve for the absolute values.

Step 2: Key Formula or Approach:
$$\text{Income} - \text{Savings} = \text{Expenditure}$$ Let the income values of A and B be represented as $5x$ and $7x$ respectively.

Step 3: Detailed Explanation:
Let's translate the ratio values into an algebraic equation setup: Income of A = $5x$ Income of B = $7x$ Both individuals save an identical amount of money, which is ₹$4000$. Subtracting this uniform saving from their respective incomes yields their expenditure values: $$\text{Expenditure of A} = 5x - 4000$$ $$\text{Expenditure of B} = 7x - 4000$$ The problem states that the ratio of their expenditures is exactly $3 : 5$. Let's set up the proportional fraction: $$\frac{5x - 4000}{7x - 4000} = \frac{3}{5}$$ Cross-multiply to solve for the common variable $x$: $$5(5x - 4000) = 3(7x - 4000)$$ $$25x - 20000 = 21x - 12000$$ Isolate the variable $x$ terms on the left side: $$25x - 21x = 20000 - 12000$$ $$4x = 8000$$ $$x = \frac{8000}{4} = 2000$$ Now that we have the value of our ratio constant $x = 2000$, we can calculate A's absolute total income: $$\text{A's Income} = 5x = 5 \times 2000 = \text{₹}10000$$

Step 4: Final Answer:
A's income is ₹10000.