Three glasses of equal volumes are $\frac{1}{2}$, $\frac{2}{3}$ and $\frac{3}{4}$ full of milk respectively. The remaining portion of all the glasses is filled up with water. The mixture in the three glasses is poured into a container. The ratio of milk and water in the container is

### Concept & Fractional Volumes When equal volumes are filled to different fractional amounts, the final composition ratio can be calculated by summing the fractions of each component across all vessels, assuming each vessel has a total volume of 1 unit. ### Step-by-Step Solution * **Given:** - Three glasses of equal total volume. Let the volume of each glass be $1$ unit. - Glass 1 is $\frac{1}{2}$ full of milk. - Glass 2 is $\frac{2}{3}$ full of milk. - Glass 3 is $\frac{3}{4}$ full of milk. - The rest is filled with water. * **Calculation:** 1. Determine the total amount of milk when poured together: - Total Milk = $\frac{1}{2} + \frac{2}{3} + \frac{3}{4}$ - The LCM of the denominators ($2, 3, 4$) is $12$. - Total Milk = $\frac{6}{12} + \frac{8}{12} + \frac{9}{12} = \frac{23}{12}$ 2. Determine the total amount of water. Since there are $3$ glasses, the total combined volume is $3$ units. - Total Water = Total Volume - Total Milk - Total Water = $3 - \frac{23}{12} = \frac{36}{12} - \frac{23}{12} = \frac{13}{12}$ 3. The ratio of milk to water in the final container is the ratio of their total amounts: - Ratio = $\frac{23}{12} : \frac{13}{12}$ 4. Multiplying both sides by $12$ yields $23 : 13$. ### Exam Strategy & Shortcut Instead of dealing with fractions, assume the volume of each glass is the LCM of $2, 3, \text{ and } 4$, which is $12$ units. Glass 1: Milk = $6$, Water = $6$ Glass 2: Milk = $8$, Water = $4$ Glass 3: Milk = $9$, Water = $3$ Total Milk = $6 + 8 + 9 = 23$. Total Water = $6 + 4 + 3 = 13$. The ratio is directly $23 : 13$. This integer-based approach saves calculation time. ### Common Pitfall A common mistake is forgetting that there are three glasses and mistakenly subtracting the total milk fraction from $1$ instead of $3$ when trying to find the total amount of water. ### Final Answer Therefore, the correct answer is **$23 : 13$**.