Zeallous, I don't think I understand the problem you're having. When you write (k+m+p)/24, what do k, m and p represent?
Two concepts are needed to solve this algebraically
- 1) Rate is always the inverse of the time it takes to do 1 job. For instance if I take 5 days to do a job, then my rate is 1/5 of the job per day.
- 2) When machines work together simultaneously, their combined rate is the sum of their individual rates.
The machines are indeed working simultaneously. In the official explanation, k, m and p represent the number of minutes each machine takes. Because of concept #1 above, if machine K takes k minutes to do the job, its rate must be 1/k (jobs/minute). The same logic is used to drive that the rates of the other two machines are 1/m and 1/p jobs per minute.
when the machines work simultaneously, according to concept #2, their combined rate will be the sum of their individual rate. Thus their combined rate is 1/k + 1/m + 1/p. We know that their combined time is 24 minutes, so according to concept #1, their combined rate can also be expressed as the inverse, or 1/24 jobs per minute. This is how the book derives the equation
Since we are interested in k, Statement 1 is sufficient because it allows us to replace 1/m + 1/p with 1/36, leaving us with a single-variable equation that could be solved for k:
Was that helpful?
-Patrick
