1962 AMC 12 Problem 24

Attempt Problem 24 of the 1962 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1962 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

24.

Three machines

P,

Q,

and

R,

working together, can do a job in

x

hours. When working alone,

P

needs an additional

6

hours to do the job;

Q,

one additional hour; and

R,

x

additional hours. The value of

x

is:

\dfrac23

\dfrac{11}{12}

\dfrac32

2

3

The individual completion times are

x+6,

x+1,

and

2x

Set the sum of the individual hourly rates equal to

\frac{1}{x}

The rates satisfy

\frac1{x+6}+\frac1{x+1}+\frac1{2x}=\frac1x.

Thus

\frac1{x+6}+\frac1{x+1}=\frac1{2x},

which simplifies to

3x^2+7x-6=0,

(3x-2)(x+3)=0.

A time must be positive, so

x=\frac{2}{3}.

Therefore, the correct answer is A.

Problem 24 in other years