Alice, Bob and Charlie Math Problem

I ran across this on YT the other day.

It takes Alice and Bob 2 hours to do a task; Alice and Charlie 3 hours; and Bob and Charlie 4 hours. How long will it take for all three of them to do it?

It happens to tie in with something else I'm working on, so...

Almost everyone who didn't behave like a good pupil gave Alice and Bob the job. That way you pay four hours labour, but with three people, one of whom is all but useless by inspection, you're going to pay more than five and less than six hours. Or, Charlie would just get in the way and slow Alice and Bob down, which is pretty standard experience for anyone who has worked in teams.

To give the good pupil answer, you have to make a number of assumptions: a) you are not concerned with cost; b) three people can work together without getting in each other's way; c) the work rate of each person is not affected by a third person on the team. Since we are given no costs, we have to assume a). b) and c) have to be stated in the question, but are not.

I've always felt these questions are either a) posed by people who would not pass Logic 101, and/or b) are designed to see if you can make the "right" assumptions, in order to c) exclude awkward logical / lawyer-brained people who won't work well the Normies.

If you are going to use the good pupil assumptions, the correct reasoning starts by noticing that if you're going to "solve" for A, B, C, then those have to be the workrate (per hour) in order for the arithmetical operations to be meaningful.

Thus we get 1. a+b = ½ job / hr 

2. a+c = ⅓ job /hr 

3. b+c = ¼ job /hr

From which we get

4. b-c = 1/6 (eq 1- eq2) 

5. b = 5/24 (eq 3+eq 4, divided by 2) 

6. c = 1/24 (substitute 5 in eq 2) 

7. a = 7/24 (substitute 6 in eq 1)

so a+b+c = 13/24 (workrate / hr) time = 24/13 = 1.85 hrs.

Adding Charlie saves 9 mins, and costs a whole extra man-hour.

The math is fine, given the assumptions. it's the assumptions that are wonky.

If you have experience of the world.

If you're a high-school math problem-setter, you don't. So you make wonky assumptions.