Work Together

A hose fills a pool in 3 hours, a second hose in 6. How long do they take together? Think in rates. In one hour the first hose fills $\frac{1}{3}$ of the pool and the second $\frac{1}{6}$. If together they take t hours, they fill $\frac{1}{t}$ of it each hour, so $\frac{1}{3}+\frac{1}{6} = \frac{1}{t}$.

Add the left side: $\frac{1}{3}+\frac{1}{6}$ is $\frac{1}{2}$, so $\frac{1}{2} = \frac{1}{t}$. Half the pool in one hour means the whole pool in 2 hours: t is the reciprocal of $\frac{1}{2}$, so $t = 2$.

Rates add; times do not. 3 hours plus 6 hours is 9, slower than either hose alone. Working together has to beat the faster hose, so the answer is always below the smaller time.