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 answer is 3:5.

Let's define variables to represent the incomes and expenditures of Lakshmi and Meenakshi. Let the expenditure of Lakshmi be denoted as $E_L$ and the expenditure of Meenakshi be $E_M$.

The problem states that the ratio of their expenditures is 2:3. We can represent these expenditures using a common variable, $x$:

$E_L=2x$

$E_M=3x$

We are also given that the ratio of the income of Lakshmi ($I_L$) to the expenditure of Meenakshi ($E_M$) is 6:7. This can be expressed as a fraction:

$\frac{I_L}{E_M}=\frac{6}{7}$

Now, substitute the value we defined for $E_M$ ($3x$) into this equation:

$\frac{I_L}{3x}=\frac{6}{7}$

By solving for $I_L$, we find Lakshmi's income in terms of $x$:

$I_L=3x\times \frac{6}{7}=\frac{18x}{7}$

Savings are equal to income minus expenditure. Let's find the savings for Lakshmi ($S_L$):

$S_L=I_L-E_L$

$S_L=\frac{18x}{7}-2x$

$S_L=\frac{18x-14x}{7}=\frac{4x}{7}$

Let Meenakshi's income be $I_M$. Her savings ($S_M$) will be her income minus her expenditure:

$S_M=I_M-3x$

We are given that the ratio of Lakshmi's savings to Meenakshi's savings is 4:9. We can set up the following equation:

$\frac{S_L}{S_M}=\frac{4}{9}$

Substitute the expressions we found for their savings:

$\frac{\left(\frac{4x}{7}\right)}{I_M-3x}=\frac{4}{9}$

We can solve for $I_M$ by cross-multiplying:

$9\times \left(\frac{4x}{7}\right)=4\times \left(I_M-3x\right)$

Divide both sides of the equation by 4 to simplify:

$\frac{9x}{7}=I_M-3x$

Add $3x$ to both sides to isolate $I_M$:

$I_M=\frac{9x}{7}+3x$

$I_M=\frac{9x+21x}{7}=\frac{30x}{7}$

Finally, we need to determine the ratio of their incomes, $I_L:I_M$:

$I_L:I_M=\frac{18x}{7}:\frac{30x}{7}$

By multiplying both sides of the ratio by $\frac{7}{x}$, we remove the denominators and the variable $x$:

$18:30$

We can simplify this ratio by dividing both numbers by their greatest common divisor, which is 6:

$3:5$

Therefore, the ratio of their incomes is 3:5.