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

Approach: Pin a clean money value on Meenakshi's expenditure to clear all the sevenths at once. Since $I_L:E_M = 6:7$, choosing $E_M = 7$ makes $I_L=6$ a whole number, and every other quantity follows in integers.

Step 1: Let $E_M = 7$. Then from $I_L:E_M = 6:7$ we get $I_L = 6$.

Step 2: Expenditures are in ratio $2:3$, i.e. $E_L:E_M = 2:3$. With $E_M = 7$, \[ E_L = \frac{2}{3}\times 7 = \frac{14}{3}. \]

Step 3: Lakshmi's saving: \[ S_L = I_L - E_L = 6 - \frac{14}{3} = \frac{18-14}{3} = \frac{4}{3}. \]

Step 4: Savings are in ratio $4:9$, so \[ S_M = \frac{9}{4}\,S_L = \frac{9}{4}\times\frac{4}{3} = 3. \] Meenakshi's income is then \[ I_M = E_M + S_M = 7 + 3 = 10. \]

Step 5: Income ratio: \[ I_L : I_M = 6 : 10 = 3 : 5. \] Final answer: $3 : 5$.