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.

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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 \\[-1em]\text{Before } & \dfrac{n}{2} & \dfrac{2n}{3} \\[0.75em] \hline \\[-1em]\text{After } & \dfrac{n}{2} + \dfrac{1}{4} \left( \dfrac{2n}{3} \right) = \dfrac{n}{2} + \dfrac{n}{6} = \dfrac{2n}{3} & \dfrac{2n}{3} - \dfrac{1}{4} \left( \dfrac{2n}{3} \right) = \dfrac{2n}{3} - \dfrac{n}{6} = \dfrac{n}{2}\end{array}

Therefore,

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