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 greater capacity, then the part of the larger jar filled with water is

Amounts of water in two jars are equal: the jar with the greater capacity is $\frac{1}{4}$ full, and the Jar with lesser capacity is $\frac{1}{3}$ full. 

$\therefore$ When the water in smaller jar is poured into the larger Jar, the addition of an equal amount of water will double the amount in the larger jar, which will then be $2\times \frac{1}{4}=\frac{1}{2}$ full.