Question

The ratio of expenditures of Lakshmi and Meenakshi is $2 : 3$, and the ratio of income of Lakshmi to expenditure of Meenakshi is $6 : 7$. If excess of income over expenditure is saved by Lakshmi and Meenakshi, and the ratio of their savings is $4 : 9$, then the ratio of their incomes is:

• A. \(7 : 8\)
• B. \(3 : 5\)
• C. \(2 : 1\)
• D. \(5 : 6\)

Hint:

When working with income–expenditure–saving problems: \begin{itemize} \item Introduce variables for unknown incomes and expenditures. \item Use the given ratios step by step to express all quantities in terms of a single variable. \item Translate the savings ratio into an equation and solve for the remaining unknowns. \end{itemize}

Answer 1

The Correct Option is B

Step 1: Let the expenditures be \[ E_L = 2x,\qquad E_M = 3x \] for Lakshmi and Meenakshi respectively.
Step 2: Use the ratio of income of Lakshmi to expenditure of Meenakshi: \[ \frac{I_L}{E_M} = \frac{6}{7} \;\Rightarrow\; \frac{I_L}{3x} = \frac{6}{7} \;\Rightarrow\; I_L = \frac{18x}{7}. \]
Step 3: Find savings. Savings = Income $-$ Expenditure. Lakshmi’s savings: \[ S_L = I_L - E_L = \frac{18x}{7} - 2x = \frac{18x - 14x}{7} = \frac{4x}{7}. \] Let $I_M$ be Meenakshi’s income. Then her savings: \[ S_M = I_M - E_M = I_M - 3x. \]
Step 4: Use the savings ratio $S_L : S_M = 4 : 9$: \[ \frac{S_L}{S_M} = \frac{4}{9} \;\Rightarrow\; \frac{\frac{4x}{7}}{I_M - 3x} = \frac{4}{9}. \] Divide both sides by $4$: \[ \frac{\frac{x}{7}}{I_M - 3x} = \frac{1}{9} \;\Rightarrow\; 9\cdot\frac{x}{7} = I_M - 3x \;\Rightarrow\; I_M = \frac{9x}{7} + 3x = \frac{9x + 21x}{7} = \frac{30x}{7}. \]
Step 5: Ratio of incomes: \[ \frac{I_L}{I_M} = \frac{\frac{18x}{7}}{\frac{30x}{7}} = \frac{18}{30} = \frac{3}{5}. \] Therefore, the ratio of Lakshmi’s income to Meenakshi’s income is \(3 : 5\).