Seeta and Geeta have two glasses of equal volumes. Both have some milk in their glasses. Seeta says to Geeta. "Give me half the milk in your glass so that my glass will be full of milk." Geeta says to Seeta, "Instead you give me one-fourth of the milk in your glass so that my glass will be full of milk." Find the ratio of volumes of milk in their glasses. (S.S.C., 2006)

### Concept & Simultaneous Equations Translate conversational conditions into algebraic equations by equating the sum of the volumes exchanged to the total capacity of the glass. ### Step-by-Step Solution * **Define Variables:** Let the total volume of each glass be $V$. Let the initial volume of milk with Seeta be $S$. Let the initial volume of milk with Geeta be $G$. * **Translate Seeta's Statement:** "Give me half the milk in your glass so that my glass will be full..." $S + \frac{G}{2} = V$ --- (Equation 1) * **Translate Geeta's Statement:** "Instead you give me one-fourth of the milk in your glass so that my glass will be full..." $G + \frac{S}{4} = V$ --- (Equation 2) * **Equate and Solve:** Since both equations equal $V$, set them equal to each other: $S + \frac{G}{2} = G + \frac{S}{4}$ Multiply the entire equation by $4$ to remove fractions: $4S + 2G = 4G + S$ Rearrange terms to group $S$ and $G$: $4S - S = 4G - 2G$ $3S = 2G$ * **Find the Ratio:** $\frac{S}{G} = \frac{2}{3}$ ### Exam Strategy & Shortcut Plug the options in as test cases. Let the glass capacity be $V=1$. Test Option (d) 2:3 -> Let $S = 2x, G = 3x$. Seeta takes half of Geeta's: $2x + 1.5x = 3.5x = V$. Geeta takes $1/4$ of Seeta's: $3x + 0.5x = 3.5x = V$. Both yield the same capacity $V = 3.5x$, so the ratio $2:3$ satisfies both conditions perfectly. ### Common Pitfall Misinterpreting "half the milk in your glass" as half the *capacity* of the glass rather than half the *current volume* of milk the other person possesses. ### Final Answer Therefore, the correct answer is **2 : 3**.