SOLUTION: Bottle A contains some water. Bottle B contains an equal amount of pure alcohol. First 1/3 of the water in A is poured into B. Stir Well. Then 1/3 of the mixture in B is poured int

Algebra ->  Percentage-and-ratio-word-problems -> SOLUTION: Bottle A contains some water. Bottle B contains an equal amount of pure alcohol. First 1/3 of the water in A is poured into B. Stir Well. Then 1/3 of the mixture in B is poured int      Log On


   

Question 589674: Bottle A contains some water. Bottle B contains an equal amount of pure alcohol. First 1/3 of the water in A is poured into B. Stir Well. Then 1/3 of the mixture in B is poured into A. Stir Well. Again 1/3 of the mixture in A is poured into B. Stir Well. Finally the correct amount of mixture is poured from B into A so the both bottles contains the same amount of mixture. In what ratio is the pure alcohol divided between A and B now?
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Call the amount of water in A at the start +1+ unit
Call the amount of alcohol in B at the start +1+ unit
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After the 1st step,
A has +2%2F3+ units of water in it
B has +1%2F3+ units of water plus +1+ unit of alcohol
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After the 2nd step,
+%281%2F3%29+%2A+%281%2F3%29+=+1%2F9+ units of water from B is added to A, and
+%281%2F3%29%2A1+=+1%2F3+ units of alcohol from B is added to A
Now A contains +2%2F3+%2B+1%2F9+=+7%2F9+ units of water
and +1%2F3+ units of alcohol
Now B contains +1%2F3+-+%281%2F3%29%2A%281%2F3%29+=+2%2F9+ units of water
and +1+-+1%2F3+=+2%2F3+ units of alcohol
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Note that after each step the total units of both water and alcohol must = 1
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After the next step:
+%281%2F3%29%2A%287%2F9%29+=+7%2F27+ units of water from A is added to B
+%281%2F3%29%2A%281%2F3%29+=+1%2F9+ units of alcohol from A is added to B
Now A contains +7%2F9+-+7%2F27+=+14%2F27+ units of water
and +1%2F3+-+1%2F9+=+2%2F9+ units of alcohol
Now B contains +2%2F9+%2B+7%2F27+=+13%2F27+ units of water
and +2%2F3+%2B+1%2F9+=+7%2F9+ units of alcohol
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check: water = +14%2F27+%2B+13%2F27+=+1+ and
alcohol = +2%2F9+%2B+7%2F9+=+1+
OK
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After the next step:
Some fraction of the mixture in B is poured into A
Let that fraction = +1%2Fx+
Now B contains +13%2F27+-+13%2F%2827x%29+=+%2813%2F27%29%2A%28+1+-+1%2Fx+%29+ units of water
and +7%2F9+-+7%2F%289x%29+=+%287%2F9%29%2A%28+1++-+1%2Fx%29+ units of alcohol
Now A contains +14%2F27+%2B+%281%2Fx%29%2A%2813%2F27%29++ units of water
and +2%2F9+%2B+%281%2Fx%29%2A%287%2F9%29+ units of alcohol
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The mixture in A now = the mixture in B, so



+-42%2F%2827x%29+=+-14%2F27+
+21%2F%2827x%29+=+7%2F27+
+21%2Fx+=+7+
+x+=+3+
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The units of alcohol in A is now
+2%2F9+%2B+%281%2F3%29%2A%287%2F9%29+=+6%2F27+%2B+7%2F27+
+6+%2F27+%2B+7%2F27+=+13%2F27+
The units of alcohol in B is now
+%287%2F9%29%2A%28+1+-+1%2F3+%29+=+%287%2F9%29%2A%282%2F3%29+
+%287%2F9%29%2A%282%2F3%29+=+14%2F27+
The ratio of alcohol is
A/B = 13/14
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Unless I goofed somewhere