When a job has a fixed setup time, subtract that time before applying inverse proportion to the work itself. Doubling the number of equally fast machines halves the running time for a fixed job. The setup still takes the same number of minutes.
This SEC G3 Mathematics practice applies inverse proportion to one part of a total. It also shows how to work backwards from a deadline when the number of machines must be a whole number.
- Separate setup from running time.
- Find the constant from the running time.
- Use the constant to calculate the new running time.
- Add setup back before checking the deadline.
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Decide which time changes
Suppose a print job is split evenly among identical machines. They all start together, and the paper and power supplies are sufficient for them to run at the same time. For the same quantity of printing, the number of machines multiplied by running time stays constant.
The shared setup happens once before any printing begins. If setup takes 12 minutes, the total time T is 12 + k/n, where n is the number of machines and k represents the size of the printing job in machine-minutes.
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Worked example | Four machines become eight
Four machines finish in 48 minutes including setup. Their running time is 48 − 12 = 36 minutes, so the constant is 4 × 36 = 144 machine-minutes. With eight machines the running time is 144 ÷ 8 = 18 minutes. The new total is 12 + 18 = 30 minutes.
A direct halving of 48 gives 24 minutes. That calculation halves the setup as well. Check by subtracting the actual setup from 24. The resulting 12 minutes of running time would allow eight machines to complete only 96 machine-minutes of work.
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Work backwards from a deadline
To finish within 35 minutes, there are 35 − 12 = 23 minutes of running time available. The number of machines must be at least 144 ÷ 23, which lies between six and seven. Six machines take 24 minutes of running time and finish in 36 minutes. Seven finish in about 32.6 minutes overall, so seven is the minimum.
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CHAPTER 4 OF 4 · Back to contents
Frequently asked questions
What is the quickest way to recognise the fixed part?
Look for a time that applies once, regardless of how many machines run. In this example the same 12 minutes are added to every running time.
How can a parent check the answer?
Ask which time is halved when the number doubles. Then have the student show how the setup reappears in the final calculation.