7th grade pre algebra word problem

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rangerstud

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This is the problem -

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. One of the men awoke very hungry, so he ate 1/3 of the pineapples - more than his proper share. He went back to sleep. The second man awoke and 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?

My work -

I worked on it awhile, and I think the answer is 216 pineapples. I came to this conclusion by thinking that if there were 6 pineapples left, and the fourth man had his equal 1/4 of what was remaining. Then that means his equal share is 2 pineapples. Then I multiplied 8 by 3 to get 24 pineapples which would have been the amount of pineapples that were there when the third man woke up. Then I multiplied 24 times 3 to get 72 pineapples which would have been the amount of pineapples that were there when the second man woke up. Then I multiplied 72 by 3 and got 216 pineapples which would have been the amount of pineapples that the men collected.

My question -

First, Im not sure my answer is correct, so I wanted to know what you thought. Second, I have no actual math problem to back up my work, it is mostly me thinking it through. I only know how to answer it by using the paragraph above, and not with numbers to back it up.


Thank you very much for your help!
Rangerstud
 
216 is wrong
man 1 took 72 left 144
man2 took 48 left 96
man3 took 32 left 64
man4 took 16 left 48 should be 6 left

27 is the answer

let 1st man take n1, second man n2 third man n3 fourth man n4 and 6 left over
n1+n2+n3+n4+6=total
n1= total/3

n2 +n3 +n4 +6= 2total/3
n2 =[2total/3]/3

n3+n4+6 =2total/3 - 2total/9
n3+n4+6=4total/9

n3 =[4total/9]/3
n4+6= 12 total/27-4total/27
n4+6= 8total/27

n4=[8total/27]4
n4=2total/27
6= 8 total/27-2total/27
6[27]=6total
total=27 answer

Arthur
6=32 total
 
disregard the 6=32 don't know where that came from 27 is the answer check it
27
1st man took 9 left 18
2nd 6 12
3rd 4 8
4th 2 6
Arthur
 
The 4th guy's "rightful share" is 1/4 of what was left; so:
total: 27
1:-9 : 18
2:-6 : 12
3:-4 : 8
4:-2 : 6

It cannot be 1/4 of the original number, because less than 1/4
are left after 3 consecutive takes of 1/3 of what's left:
let t = total pineapples
t
2/3 t (left after 1st guy)
2/3(2/3 t) = 4/9 t (after 2nd guy)
2/3(4/9 t) = 8/27 t (after 3rd guy)

1/4 t > 8/27 t : so not enough left

That's quite a 7th grade problem 8-)
 
I don't understand the comment.
if we start with 27 and each of three men take 1/3 of what they find and the fourth man takes 1/4 of what he finds we have"
start 27 pineapples
man 1 takes 9 leaving 18
man 2 takes 6 leaving 12
man 3 takes 4 leaving 8
mn 4 takes 2 leaving 6

do you mean 4th man takes 1/4 of the original total?
let T be total
let n1,n2,n3,n4 be the amount each man takes
n1+n2+n3+n4+6=T
n1=T/3
n2+n3+n4+6=2T/3
n2=2T/9
n3+n4+6=4T/9
n3=4T/27
n4+6=8T/27
n4=T/4
6= 8T/27 -T/4
6[27][4]=32T-27T
648=5T
T=129.6 no integer solutions

I stand by my answer of 27 the 4th man took 1/4 of the amount present when he awoke
Arthur
 
I had not seen your post when I posted mine, Arthur;
but anyway, I'm the one that now does not understand your latest one:
I have the exact solution you have : 27 :shock:

The next part is where I prove it CANNOT be 1/4 of the original number.
 
I agree, it cannot be 1/4 of the total, but 1/4 of the remainder he see's.There are no integer solutions if the 4th man takes 1/4 of the total.
Arthur
 
arthur ohlsten said:
I agree, it cannot be 1/4 of the total, but 1/4 of the remainder he see's.There are no integer solutions if the 4th man takes 1/4 of the total.Arthur
Click to expand...

There is NO solution whatsoever (integer or not) because it is IMPOSSIBLE for 4th man
to take 1/4 of total: there is always LESS than 1/4 of total left.
That's what I'm proving with this:

It cannot be 1/4 of the total, because less than 1/4
are left after 3 consecutive deductions of 1/3:
let t = total pineapples
t
2/3 t (left after 1st guy)
2/3(2/3 t) = 4/9 t (after 2nd guy)
2/3(4/9 t) = 8/27 t (after 3rd guy)

1/4 t > 8/27 t : so not enough left
 
for integer solutions 27
for non integer solutions 129.6

let T=129.6 and then the last man can take 1/4 of this

man 1 takes 43.2 leaves 86.4
man 2 takes 28.8 leaves 57.6
man 3 takes 19.2 leaves 38.4
man4 takes 32.4 leaves 6

derivation
n1+n2+n3+n4+6=T
n1=T/3 substitute and subtract

n2+n3+n4+6=2T/3
n2=2T/9

n3+n4+6=4T/9
n3=4T/27

n4+6=8T/27
n4=T/4

6=8T/27 -T/4
6[27][4]=32T-27T648=5T
T=129.6
Arthur
 
Correct, Arthur.
This was kinda stupid of me: 1/4 t > 8/27 t :cry:
 
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