paradox: two machines making chairs

Here's an interesting mathematical paradox.

In a factory, there are two independent machines. The first machine produces one chair in 50 days. The second (much faster) machine produces one chair in 2 days. I'm wondering: how many days will it take the factory to produce 20 chairs?

It's simple: the first machine has a productivity of 0.02 chairs per day. The second machine has a productivity of 0.5 chairs per day. So, in total, the factory has a productivity of 0.52 chairs per day. I calculate: 20 chairs / 0.52 chairs per day = 38.46154 days. I round it up, which gives 39 days. I check: a factory with a productivity of 0.52 chairs per day will produce 0.52 * 39 = 20.28 chairs in 39 days. Good, everything matches up.

But now let's check it another way: the first machine (the one making a chair in 50 days) will produce nothing in 39 days. The second machine (the one making a chair in two days) will produce 19 chairs in 39 days. Conclusion: in 39 days, the factory will only produce 19 chairs. So, contrary to the previous calculations, 39 days will not be enough to produce 20 chairs!

comments:

2020.09.19 09:58 P.

Można też to zobaczyć na prostszych danych: mam maszynę (tylko jedną), która w 100 dni produkuje 100 elementów - tak, że przez pierwszych 99 dni nie ma wyprodukowanego żadnego elementu, a setnego dnia pojawia się sto gotowych elementów. Ile dnia zajmie wyprodukowanie 50 elementów?

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