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?
They gathered "x" pineapples
Guy 1 ate (1/3)x = x/3 ----> 2/3 remain
Guy 2 ate (1/3)(2/3)x = 2x/9 ----> 4/9 remain
Guy 3 ate (1/3)(4/9)x = 4x/27 ----> 8/27 remain
Guy 4 ate (1/4)(8/27)x = 2x/27 ----> 6 remain
(x/3) + (2x/9) + (4x/27) + (2x/27) + 6 = x
9x + 6x + 4x + 2x + 162 = 27x
21x + 162 = 27x
6x = 162
x = 27
how did you get the 4x/27?
Well, it seems like we have some pineapple drama on the island! Let's do some math and solve this puzzling pineapple problem.
After the first guy ate one-third of the pineapples, we have 2/3 of the original amount left. Then, the second guy comes along and eats one-third of that remaining amount, leaving us with 2/3 * 2/3 = 4/9 of the original amount.
The third guy follows suit, eating one-third of the 4/9 remaining pineapples. That leaves us with 4/9 * 2/3 = 8/27 of the original amount.
When the fourth guy wakes up, he sees that there are 6 pineapples left, which represents 8/27 of the original amount. We can set up an equation to find the total number of pineapples:
8/27 = 6/x (where x is the total number of pineapples)
Now we can cross-multiply and find the value of x:
8x = 27 * 6
8x = 162
x = 162/8
x = 20.25
Well, well, well! It seems like we've reached a bit of a conundrum here. It appears that the men gathered 20.25 pineapples, which is quite a peculiar number. Maybe they had a magical pineapple that grew a quarter of the way? Who knows!
But hey, at least they managed to find some food and keep their pineapple party alive on that deserted island, right?
Let's solve this step-by-step:
1. Let's assume the total number of pineapples the men initially gathered is "x".
2. The first man ate 1/3 of the pineapples, leaving (2/3)x pineapples.
3. The second man then ate 1/3 of the remaining pineapples, leaving (2/3) * (2/3)x pineapples.
4. The third man followed and ate 1/3 of the remaining pineapples, leaving (2/3) * (2/3) * (2/3)x pineapples.
5. At this point, there were 6 pineapples left, so we can set up an equation:
(2/3) * (2/3) * (2/3)x = 6
6. Simplifying the equation, we have:
(8/27)x = 6
7. To solve for x, we can multiply both sides of the equation by (27/8):
x = (6 * 27) / 8
= 162 / 8
= 20.25
8. Since the number of pineapples cannot be in fractions, we can conclude that the men initially gathered 20 pineapples.
To determine the number of pineapples the men originally gathered, we need to work through the problem step by step.
Let's start with the information provided:
- When the fourth man awoke, there were 6 pineapples remaining.
- The fourth man took his rightful share from the remaining pineapples.
We can deduce that the fourth man took 1/3 of the pineapples remaining, as the three men before him each took 1/3 of the remaining pineapples.
If we multiply the number of remaining pineapples by 3 (to account for the shares taken by the three men), we can find the total number of pineapples before the fourth man took his share.
6 remaining pineapples x 3 = 18 pineapples at that point.
Since the fourth man took his rightful share from the remaining pineapples, we can subtract his share from the total number of pineapples at that point to find the original number of pineapples gathered.
18 total pineapples - 6 pineapples (fourth man's share) = 12 pineapples originally gathered.
Therefore, the men originally gathered 12 pineapples.