I sent this via PM to Economist, but thought it might be useful to others:
When you have workers working in sequence, you'd want to add the work done by each. It's when you have workers working simultaneously that you'd want to add their rates.
I can illustrate with three different questions, all based on the same initial setup:
Say, worker A can do 1 job in 10 hours, and worker B can do 1 job in 15 hours.
a) If they work together for 30 hours, how many jobs will they complete?
Here, you'd add their rates to determine how many jobs they can finish together. You should find they can complete one job every six hours working together, so in 30 hours they'd finish five jobs.
b) If worker A works alone for 15 hours, and then stops, and then worker B works alone for 15 hours, how many jobs will they complete in the 30 hours?
Here, you don't want to add their rates, because they never work simultaneously. Instead you'd want to add together the work each does during the 15 hours each works - in 15 hours, A will do 1.5 jobs, and in 15 hours, B will do 1 job, so combined they'll complete 2.5 jobs.
c) If worker A works alone for 15 hours, and then B and A work together for 15 hours, how many jobs will they complete in the 30 hours?
This question most resembles the one from the post above. A will complete 1.5 jobs in the first 15 hours. For the remaining 15 hours, A+B work together. We need to work out their combined rate, and we find they will complete 1 job every 6 hours, and will therefore complete 2.5 jobs in 15 hours. In total, over the 30 hours, 4 jobs will therefore be completed.
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There are other approaches to the problems above, of course, but I've confined the discussion to work/rate principles to illustrate the differences between each question setup.
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