Equal amounts of water were poured into two empty jars of different capacities, which made one jar $\frac{1}{4}$ full and the other jar $\frac{1}{3}$ full. If the water in the jar with lesser capacity is, then poured into the jar with the greater capacity, what fraction of the larger jar will be filled with water?

### Concept & Logic Equal quantities of volume fill different ratios of two distinct capacities. The jar that becomes $\frac{1}{3}$ full with the same water must have a smaller capacity than the jar that becomes $\frac{1}{4}$ full. Thus, the jar that is $\frac{1}{4}$ full is the larger jar. ### Step-by-Step Solution * **Given:** * Volume of water added to both jars is identical ($W$). * Jar 1 (Larger Capacity $C_1$): $\frac{1}{4}$ full $\implies W = \frac{1}{4}C_1 \implies C_1 = 4W$ * Jar 2 (Smaller Capacity $C_2$): $\frac{1}{3}$ full $\implies W = \frac{1}{3}C_2 \implies C_2 = 3W$ * **Calculation:** * The question states that the water from the smaller capacity jar ($C_2$) is poured into the larger capacity jar ($C_1$). * Note that the smaller jar already contained exactly $W$ amount of water. * The larger jar also already contained exactly $W$ amount of water. * When combined into the larger jar, the total volume of water becomes $W + W = 2W$. * The fraction of the larger jar filled is: $$ \text{Fraction} = \frac{\text{Total Volume of Water}}{\text{Capacity of Larger Jar}} = \frac{2W}{C_1} = \frac{2W}{4W} = \frac{1}{2} $$ ### Exam Strategy & Shortcut **Direct Value Trick:** Assume the equal water amount poured into both jars is 1 unit. * Larger jar capacity = 4 units (since 1 unit fills $\frac{1}{4}$ of it). * Smaller jar capacity = 3 units (since 1 unit fills $\frac{1}{3}$ of it). * Combining both water contents means we have $1 + 1 = 2$ units of water inside the larger jar. * Fraction of larger jar filled = $\frac{2}{4} = \frac{1}{2}$. ### Common Pitfall A common point of confusion is parsing "poured into the jar with the greater capacity". Students sometimes assume the smaller jar's total volume capacity ($3W$) is being poured over, or they forget that the larger jar already contains its initial portion of water. ### Final Answer **Therefore, the correct answer is 1/2.**