狼狗那么长的舌头,还整天在外面# pets - 心有所宠
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You have two jugs, the first one has 10L water and the second one has 10L
alcohol. Now you pour some liquid from the first one to the second one, mix
well and pour some liquid from the second one to the first one. You do this
many times, in the end both jugs contains 10L liquid. Now the question is:
Let r1 be alcohol/water in the first one and r2 be water/alcohol in the
second one, is r1 > r2 or r1 = r2 or r1 < r2 or it depends on how you do the
above pouring operation?
The answer is r1=r2, but why?
Let's try to prove it. Let
w1: the amount of water in the first jug
a1: the amount of alcohol in the first jug
w2: the amount of water in the second jug
a2: the amount of alcohol in the second jug
Now our goal is to show a1/w1 = w2/a2. Since a1+w1 = a2+w2, the goal is
equivalent to a1 = w2.
Proof by contradiction. Suppose a1 > w2. Then from a1+w1 = a2+w2, we have w1
< a2. Therefore we have a1+a2 > w1+w2, but this contradicts with the fact
that a1+a2 = w1+w2. The case a1 < w2 can be proved similarly. Therefore we
have a1 = w2 and hence r1 = r2.
alcohol. Now you pour some liquid from the first one to the second one, mix
well and pour some liquid from the second one to the first one. You do this
many times, in the end both jugs contains 10L liquid. Now the question is:
Let r1 be alcohol/water in the first one and r2 be water/alcohol in the
second one, is r1 > r2 or r1 = r2 or r1 < r2 or it depends on how you do the
above pouring operation?
The answer is r1=r2, but why?
Let's try to prove it. Let
w1: the amount of water in the first jug
a1: the amount of alcohol in the first jug
w2: the amount of water in the second jug
a2: the amount of alcohol in the second jug
Now our goal is to show a1/w1 = w2/a2. Since a1+w1 = a2+w2, the goal is
equivalent to a1 = w2.
Proof by contradiction. Suppose a1 > w2. Then from a1+w1 = a2+w2, we have w1
< a2. Therefore we have a1+a2 > w1+w2, but this contradicts with the fact
that a1+a2 = w1+w2. The case a1 < w2 can be proved similarly. Therefore we
have a1 = w2 and hence r1 = r2.