Question 282360: This is the problem and they want an algebra solution.
Three tired and hungry men went to sleep with a bag of apples. One man woke up, ate 1/3 of the apples, then went back to sleep. Later a second man woke up and ate 1/3 of the remaining apples, then went back to sleep.Finally,the third man woke and ate 1/3 of the remaining apples. When he was finished there were 8 apples left. How many apples were in the bag originally?
Found 3 solutions by richwmiller, edjones, Theo:
Answer by richwmiller(17219)   (Show Source):
You can put this solution on YOUR website! This problem appears every couple weeks.
x-x/3-1/3(x-x/3)-1/3(x-x/3-1/3(x-x/3))=8
x=27
first guy ate 1/3*27=9 leaving 18
next guy ate 18/3=6 leaving 12
3rd guy at 12/3=4 leaving 8
A simpler way to calculate by working backwards
2/3x=8
x=12
2/3x=12
x=18
2/3x=18
x=27
Answer by edjones(8007)  (Show Source):
You can put this solution on YOUR website! Let x=original amount of apples.
after 1st man eats 2/3 remain
.
after 2nd man eats 2/3 * 1/3 = 2/9 x eaten
x - (1/3)x - (2/9)x
=x-(3/9)x-(2/9)x
=x-(5/9)x= (4/9)x remains
.
after the 3rd man eats 4/9 * 1/3 = (4/27)x eaten
x-(5/9)x-(4/27)x
x-(15/27)x-(4/27)x=8
x-(19/27)x=8
(8/27)x=8
x=27 apples in the bag originally
.
Ed
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! Let x = the number of apples there were originally.
First man eats 1/3 * x, so you are left with 2/3 * x
Second man eats 1/3 * 2/3 * x which equals 2/9 * x.
2/3 * x - 2/9 * x equals 6/9 * x - 2/9 * x equsls 4/9 * x
Third man eats 1/3 * 4/9 * x which equals 4/27 * x.
4/9 * x - 4/27 * x equals 12/9 * x - 4/27 * x equals 8/27 * x
After everybody gets their fill, you are left with 8/27 * x
Since there are 8 apples left, this means that 8/27 * x = 8
Multiply both sides of this equation by 27 to get:
8 * x = 27 * 8
Divide both sides of this equation by 8 to get:
x = 27
The original number of apples is 27.
To prove, substitute 27 for x and go through the original problem again.
There are 27 apples.
First man eats 1/3 * 27 = 9.
27 - 9 = 18
Second man eats 1/3 * 18 = 6.
18 - 6 = 12.
Third man eats 1/3 * 12 = 4.
12 - 4 = 8
There are 8 apples left when the original number is 27.
That's your answer.
I also looked at it again and solved it this way:
x - (1/3)x = (2/3)x = a
a - (1/3)a = (2/3)a = b
b - (1/3)b = (2/3)b = c
I got:
(2/3)x = a
(2/3)a = b
(2/3)b = c
Working back from (2/3)b = c, I then substituted (2/3)a for b and got:
(2/3)*(2/3)*a = c
I then substituted x for a to get:
(2/3)*((2/3)*2/3)*x = c
I simplified this to get:
(8/27)x = c
Since c = 8 which was the number of apples left after everybody had their fill, I got:
(8/27)x = 8
I multiplied both sides of this equation by 27 to get:
8x = 8*27
I divided both sides of this equation by 8 to get:
x = 27.
That's your answer again looking at it a slightly different way.
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