Tanks and taps

Statement

A tap $A$ fills a tank in $4$ h, a tap $B$ in $6$ h. Opened together, in how long do they fill it?

Solution

Unknown. $t=$ time in hours with both taps open.

Idea. We reason in terms of flow rates (fractions of a tank per hour). $A$ has a flow rate of $1/4$ tanks/h, $B$ has $1/6$. Together the flow rate is $1/4+1/6$. The equation requires that in time $t$ exactly one tank is filled: $\Bigl(\frac{1}{4}+\frac{1}{6}\Bigr)t = 1 \implies \frac{5}{12}\,t=1 \implies t=\boxed{\dfrac{12}{5}}=2{,}4\text{ h}$ That is, $2$ hours and $24$ minutes. Note that $t<\min(4,6)$: two taps open together are always faster than the faster of the two (the same structure as the law for resistors in parallel).

Connections

Topics: First-degree equations
Concepts: Equation · Flow rate
Methods: Tank problem
Skills: Modelling · Solving equations
Exercise type: Word problem