One bottle is half-full of milk and another bottle with twice the capacity is one-quarter full of milk. If water is added so that both the bottles are full and the contents of both are then poured into a third bottle that is empty and large enough to hold the contents of both, what fraction of the contents in the third bottle is milk?

## Concept & Logic This is a mixture problem involving different base volumes. To find the final fraction of milk, you need to calculate the total volume of milk and divide it by the combined total volume of both bottles. Assume a convenient value for the capacity of the first bottle to make fractions easy to work with. ## Step-by-step Solution * Let the capacity of the first bottle be $V$. * Therefore, the capacity of the second bottle is $2V$. * **Bottle 1 Contents:** * It is half-full of milk, so Milk in Bottle 1 = $\frac{V}{2}$. * It is filled with water, so total volume is $V$. * **Bottle 2 Contents:** * It is one-quarter full of milk, so Milk in Bottle 2 = $\frac{1}{4} \times 2V = \frac{V}{2}$. * It is filled with water, so total volume is $2V$. * **Combining in Bottle 3:** * Total Milk = Milk from Bottle 1 + Milk from Bottle 2 $$ \text{Total Milk} = \frac{V}{2} + \frac{V}{2} = V $$ * Total Volume = Volume of Bottle 1 + Volume of Bottle 2 $$ \text{Total Volume} = V + 2V = 3V $$ * **Final Fraction:** * Fraction of Milk = $\frac{\text{Total Milk}}{\text{Total Volume}}$ $$ \text{Fraction} = \frac{V}{3V} = \frac{1}{3} $$ ## Exam Strategy & Shortcut Use the "Assume a Value" method. Let the capacity of the first bottle be $4$ liters (a multiple of $2$ and $4$). Bottle 1: Capacity = $4$L. Milk = $2$L. Bottle 2: Capacity = $8$L. Milk = $1/4 \times 8 = 2$L. Total Milk = $2 + 2 = 4$L. Total Capacity = $4 + 8 = 12$L. Fraction of milk = $4 / 12 = 1/3$. This completely bypasses algebraic variables. ## Common Pitfall A very common mistake is to average the fractions ($\frac{1}{2}$ and $\frac{1}{4}$) without accounting for the different capacities of the bottles. This leads to an incorrect weighted average. Always ground fractions in their actual base values. ## Final Answer Therefore, the correct answer is **1/3**.