Juice
Alice has 100 mL of apple juice, and Bob has 100 mL of orange juice.
Both of them want a mixture of apple and orange juice, so they come up with a plan:
- Alice pours mL of apple juice into Bob's orange juice.
- Bob then pours mL of his mixture into Alice's apple juice.
The goal is for them both to have equal amounts of apple juice and orange juice in their mixture.
What is the value of
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1 solution
Assume that Alice pours some amount like 50 ml of apple juice into Bob's OJ. Then Bob mixes his 150 ml and pours back 16.6667 ml of apple and 33.3333 ml of OJ so he now has 33.3333 ml of apple and the remainder 66.6667 ml is his original OJ.
One can see that the ratio of apple:orange he obtained initially is 1/2: (50 ml apple / 100 ml OJ) and that ratio is unchanged when he pours back the amount X as a mixture. So the initial ratio of apple:orange was 1:2 and that is his final ratio as 33.3333 : 66.6667
So in order to have a 1:1 ratio, Alice must pour ALL her juice into Bob's juice and then Bob mixes and pours 100 ml back.
Alice: starts with 1 0 0 , then has 1 0 0 − X ... Bob starts with 1 0 0 gets 1 0 0 + X with X / ( 1 0 0 + X ) apple and 1 0 0 / ( 1 0 0 + X ) as OJ, with ratio A:OJ = X : 1 0 0 Bob then pours back X, ending with X 2 / ( 1 0 0 + X ) Apple and X ∗ 1 0 0 / ( 1 0 0 + X ) as OJ, with ratio of apple:orange X : 1 0 0
Thus for a 1 : 1 ratio the amount poured as X should be 1 0 0 ml.