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.

Options

A. a)`2:4`
B. b)`8:6`
C. c)`4:3`
D. d)`2:3`

Explanation

To find the ratio of the volumes of milk in Seeta and Geeta's glasses, we can set up equations based on the information given in the problem.

### Step-by-Step Solution:

1. **Define Variables**:
Let the amount of milk in Seeta's glass be \( S \) and the amount of milk in Geeta's glass be \( G \).

2. **Set Up the Equations**:
- According to Seeta, if Geeta gives her half of the milk in her glass, her glass will be full. This can be expressed as:
\[ S + \frac{G}{2} = V \]
where \( V \) is the volume of the glass (which is equal for both).

- According to Geeta, if Seeta gives her one-fourth of the milk in her glass, her glass will be full. This can be expressed as:
\[ G + \frac{S}{4} = V \]

3. **Rearranging the Equations**:
From the first equation:
\[ \frac{G}{2} = V - S \implies G = 2(V - S) \implies G = 2V - 2S \]

From the second equation:
\[ \frac{S}{4} = V - G \implies S = 4(V - G) \implies S = 4V - 4G \]

4. **Substituting**:
Now, we can substitute \( G \) from the first equation into the second equation:
\[ S = 4V - 4(2V - 2S) \]
Simplifying this gives:
\[ S = 4V - 8V + 8S \implies S - 8S = -4V \implies -7S = -4V \implies S = \frac{4V}{7} \]

5. **Finding G**:
Substitute \( S \) back into the equation for \( G \):
\[ G = 2V - 2S = 2V - 2\left(\frac{4V}{7}\right) = 2V - \frac{8V}{7} = \frac{14V}{7} - \frac{8V}{7} = \frac{6V}{7} \]

6. **Finding the Ratio**:
Now we have:
\[ S = \frac{4V}{7} \quad \text{and} \quad G = \frac{6V}{7} \]
The ratio of the volumes of milk in Seeta's and Geeta's glasses is:
\[ \frac{S}{G} = \frac{\frac{4V}{7}}{\frac{6V}{7}} = \frac{4}{6} = \frac{2}{3} \]

### Final Answer:
The ratio of the volumes of milk in Seeta's and Geeta's glasses is \( 2:3 \).