Ryan gets .333... + .999... = .121212..., + +

In cutting up a 10x10 = 100 little squares to share with 3 people, Ryan agreed that 1/3 = .33 1/3 = .333... Don asked what 2/3 = ? When Don showed him that 2/3 = 1/3 + 1/3, he said 2/3 = .666..., then 3/3 = .999... just by adding,  and 3/3 = 1 = .999... Then to Don's surprise he wrote 4/3 = .121212... He explained his work: "I added 1/3 + 3/3 =  .333... + .999... = .121212... because 3 + 9 = 12". Don told Ryan he had a good idea, and is correct enough that Don understand what he did, but the answer has to be 1 + something, because 4/3 > 1. So it should be 1. something, and the one's (the carry-overs ), are added to the 2's to get 3's. So we get 4/3 = 1.333... Then he wrote 5/3 = 1.515151... and he fixed this to 1.666... So in the end, it is a case of adding 6 tenths and 9 tenths to get 15 tenths which is 1 and 5 tenths. Then 6 hundredths and 9 hundredths to get 15 hundredths which is 1 tenth and 5 hundredth. and we add the 1 tenth to the 5 tenths to get 6 tenths and so on, to get 1.666.. = 5/3. Or  5/3 = 3/3 + 2/3 = 1 + .666...= 1.666...

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