⚡ AZUCATION SHORTCUT:
Expenditure ratio $2:3$, but we need $E_M$ to be divisible by 7. Let $E_L = 14x$, $E_M = 21x$.
Lakshmi's Income ($I_L$) $= \frac{6}{7} \times 21x = 18x$.
Lakshmi's Savings ($S_L$) $= 18x - 14x = 4x$.
Savings ratio $4:9 \implies S_M = 9x$.
Meenakshi's Income ($I_M$) $= E_M + S_M = 21x + 9x = 30x$.
Income Ratio $= 18x : 30x = \mathbf{3 : 5}$.

1. Define Variables:
Let $E_L = 2k$ and $E_M = 3k$.
From $I_L : E_M = 6 : 7 \implies I_L = \frac{6}{7} \times 3k = \frac{18k}{7}$.

2. Find Savings:
$S_L = I_L - E_L = \frac{18k}{7} - 2k = \frac{4k}{7}$.
Since $S_L : S_M = 4 : 9$, we have $\frac{4k/7}{S_M} = \frac{4}{9} \implies S_M = \frac{9k}{7}$.

3. Calculate $I_M$ and Ratio:
$I_M = E_M + S_M = 3k + \frac{9k}{7} = \frac{21k + 9k}{7} = \frac{30k}{7}$.
$I_L : I_M = \frac{18k}{7} : \frac{30k}{7} = 18 : 30 = \mathbf{3 : 5}$.

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