3650.21 – Two Barrels

Of two identical barrels, one is half full and one is two-thirds full. One quarter of the liquid in the second barrel (the barrel that's two-thirds full) is poured into the first. The first barrel now contains 25 more gallons than the second. Find the capacity in gallons of the barrels.

Solution

Let the barrels be barrel A and barrel B, each with a capacity of nn gallons. Here is what happens.

 A  B Before n22n3After n2+14(2n3)=n2+n6=2n32n3−14(2n3)=2n3−n6=n2 \begin{array}{l|c|c} & \text{ A } & \text{ B } \\ \hline \text{Before } & \frac{n}{2} & \frac{2n}{3} \\ \hline \text{After } & \frac{n}{2} + \frac{1}{4} \left( \frac{2n}{3} \right) = \frac{n}{2} + \frac{n}{6} = \frac{2n}{3} & \frac{2n}{3} - \frac{1}{4} \left( \frac{2n}{3} \right) = \frac{2n}{3} - \frac{n}{6} = \frac{n}{2} \end{array}

Therefore,

2n3−n2=25n6=25n=150 gal. \begin{aligned} \frac{2n}{3} - \frac{n}{2} &=& 25 \\ \frac{n}{6} &=& 25 \\ n &=& 150 \text{ gal.} \end{aligned}