Four men were shipwrecked on an island. Having no food, they went to work gathering pineapples. After gathering pineapples, they were tired and all fell asleep. After another while, one of the men awoke and was very hungry so he ate 1/3 of the pineapples - more than his proper share. He then went back to sleep. The second man awoke and being hungry, ate 1/3 of the remaining pineapples and went back to sleep. The third man did the same. When the fourth man awoke, he took only his rightful share of the remaining pineapples. Then there were 6 pineapples left. How many pineapples did the men gather?
Let x be the total number of pineapples gathered.
After the first man eats 1/3 of the pineapples, there are 2/3 left: (2/3)x.
After the second man eats 1/3 of the remaining pineapples, there are 2/3 of 2/3 left: (2/3)^2x.
After the third man eats 1/3 of the remaining pineapples, there are 2/3 of 2/3 of 2/3 left: (2/3)^3x.
The fourth man took his rightful share, which is 1/3 of what was left: (1/3)(2/3)^3x.
We know that there were 6 pineapples left after the fourth man took his share:
(1/3)(2/3)^3x = 6
Simplifying:
(8/27)x = 6
x = (27/8) * 6
x = 20.25
So the men gathered 20.25 pineapples. However, since they can't gather a fractional pineapple, we can assume that they gathered 20 pineapples and the fourth man took 2 pineapples (his rightful share).