1973 AMC 12 Problem 14

Question

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. The number of hours required with only valves

A

and

B

open is

1.1

1.15

1.2

1.25

1.75

Let

a,

b,

c

be the fractions of the tank filled per hour by the three valves

Combine

a+b+c=1,

a+c=\frac{2}{3},

and

b+c=\frac{1}{2}

to find

a+b

Let

a,

b,

c

be the hourly rates in tankfuls. Then

\begin{aligned} a+b+c&=1,\\ a+c&=\frac23,\\ b+c&=\frac12. \end{aligned}

Twice the first equation minus the other two gives

a+b=2-\frac23-\frac12=\frac56.

Thus valves

A

and

B

fill the tank in

\frac1{a+b}=\frac65=1.2

hours.

Therefore, the correct answer is C.

Problem 14 in other years