Step 1: Understanding the Concept:
This problem deals with the fundamental financial relationship between earnings, spending, and savings. For any individual:
\[ \text{Income} - \text{Expenditure} = \text{Savings} \]
When we are given ratios instead of absolute values, we use a common ratio variable (typically \(x\)) to represent the absolute numbers. If we know that the savings are equal for both individuals, we can construct a system of linear equations to solve for the unknown variable and then find the specific income values.
Step 2: Key Formula or Approach:
1. Let the income of A be \(5x\) and the income of B be \(7x\).
2. Since both save 4000, their expenditures can be expressed as \((5x - 4000)\) and \((7x - 4000)\).
3. Set the ratio of these expressions equal to the given expenditure ratio (\(3/5\)).
Step 3: Detailed Explanation:
Let's translate the ratios into an algebraic setup:
Income of A = \(5x\); Income of B = \(7x\).
Subtracting the uniform saving (4000) from their respective incomes gives their expenditures:
Expenditure of A = \(5x - 4000\)
Expenditure of B = \(7x - 4000\)
The problem states that the ratio of their expenditures is \(3 : 5\). We set up the proportional fraction:
\[ \frac{5x - 4000}{7x - 4000} = \frac{3}{5} \]
Now, we cross-multiply to solve for \(x\):
\[ 5(5x - 4000) = 3(7x - 4000) \]
\[ 25x - 20000 = 21x - 12000 \]
Isolate the \(x\) terms on one side:
\[ 25x - 21x = 20000 - 12000 \]
\[ 4x = 8000 \]
\[ x = \frac{8000}{4} = 2000 \]
Now that we have the value of the ratio constant \(x = 2000\), we calculate A’s absolute income:
\[ \text{A’s Income} = 5x = 5 \times 2000 = 10000 \]
Step 4: Final Answer:
A’s income is 10000, which is option (a).