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

The correct option is 3:5.

Let's break down the given information step-by-step to find the ratio of the incomes of Lakshmi and Meenakshi.

Step 1: Define variables for expenditures
We are given that the ratio of expenditures of Lakshmi and Meenakshi is 2 : 3.
Let the expenditure of Lakshmi be $E_L$ and the expenditure of Meenakshi be $E_M$.
So, we can write:
$E_L=2x$
$E_M=3x$

Step 2: Express Lakshmi's income
We are given that the ratio of Lakshmi's income ($I_L$) to Meenakshi's expenditure ($E_M$) is 6 : 7.
$\frac{I_L}{E_M}=\frac{6}{7}$

Substitute $E_M=3x$ into this equation:
$I_L=\frac{6}{7}\times 3x=\frac{18x}{7}$

Step 3: Calculate Lakshmi's savings
Savings are defined as income minus expenditure ($S=I-E$).
Lakshmi's savings ($S_L$):
$S_L=I_L-E_L=\frac{18x}{7}-2x=\frac{4x}{7}$

Step 4: Find Meenakshi's savings and income
The ratio of the savings of Lakshmi to Meenakshi is given as 4 : 9.
$\frac{S_L}{S_M}=\frac{4}{9}$

Substitute $S_L=\frac{4x}{7}$:
$\frac{\frac{4x}{7}}{S_M}=\frac{4}{9}$
$S_M=\frac{9x}{7}$

Now, calculate Meenakshi's income ($I_M$):
$I_M=E_M+S_M=3x+\frac{9x}{7}=\frac{21x+9x}{7}=\frac{30x}{7}$

Step 5: Determine the ratio of their incomes
The ratio of their incomes ($I_L:I_M$) is:
$\frac{I_L}{I_M}=\frac{\frac{18x}{7}}{\frac{30x}{7}}=\frac{18}{30}=\frac{3}{5}$

Thus, the ratio of the incomes of Lakshmi and Meenakshi is 3:5.