1962 AHSME Problems/Problem 24
Problem
Three machines working together, can do a job in
hours. When working alone,
needs an additional
hours to do the job;
, one additional hour; and
,
additional hours. The value of
is:
Solution
Machine P takes hours, machine Q takes
hours, and machine R takes
hours.
We also know that all three working together take
hours.
Now the time it takes for all three machines to complete the job is the harmonic mean of the times the three machines take individually; that is,
Obviously, the number of hours is positive, so the answer is
.