2001 SMT/Algebra Problems/Problem 2

Problem

Each valve $A$ , $B$ , and $C$ , when open, releases water into a tank at its own constant rate. With all three valves open, the tank fills in 1 hour, with only valves $A$ and $C$ open it takes 1.5 hours, and with only valves $B$ and $C$ open it takes 2 hours. How many hours will it take to fill the tank with only valves $A$ and $B$ open?

Solution

Let $a$ be the portion of the tank filled by $A$ in one hour, and define $b$ and $c$ similarly. Then we are given $a+b+c=1$ $1.5(a+c)=1$ $2(b+c)=1$ Then, $a+c=\frac{2}{3}$ $b+c=\frac{1}{2}$ Since $2a+2b+2c=2$ , subtracting the above two equations yields $a+b=\frac{5}{6}$ . Thus it would take $\boxed{\frac{6}{5}}$ hours.

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2001 SMT/Algebra Problems/Problem 2

Problem

Solution