2002年AMC10竞赛A第17题
$$2002-AMC10A-17$$  Sarah pours $$4$$ ounces of coffee into a cup that can hold $$8$$ ounces. Then she pours $$4$$ ounces of cream into a second cup that can also hold $$8$$ ounces. She then pours half of the contents of the first cup into the second cup, completely mixes the contents of the second cup, then pours half of the contents of the second cup back into the first cup. What fraction of the contents in the first cup is cream?  莎拉将 4盎司的咖啡倒入一个可容纳 8盎司的杯子中。 然后她将 4盎司的奶油倒入第二个杯子中，该杯子也可以装 8 盎司。 然后她将第一个杯子的一半内容倒入第二个杯子，将第二个杯子的内容完全混合，然后将第二个杯子的一半内容倒回第一个杯子。 第一杯中的混合物有多少是奶油？
美国AMC10/12->Knowledge Point->Algebra->Application->Equation Word Problems 课内体系->知识点->集合->容斥原理
We will simulate the process in steps.  In the beginning, we have:  ▪ $$4$$ ounces of coffee in cup $$1$$  ▪ $$4$$ ounces of cream in cup $$2$$  In the first step we pour $$\frac 42=2$$ ounces of coffee from cup $$1$$ to cup $$2$$; getting:  ▪ $$2$$ ounces of coffee in cup $$1$$  ▪ $$2$$ ounces of coffee and $$4$$ ounces of cream in cup $$2$$;  In the second step we pour $$\frac 22=1$$ ounce of coffee and $$\frac 42=2$$ ounces of cream from cup $$2$$ to cup $$1$$, getting:  ▪ $$2+1=3$$ ounces of coffee and $$0+2=2$$ ounces of cream in cup $$1$$  ▪ the rest in cup $$2$$  Hence at the end we have $$3+2=5$$ ounces of liquid in cup $$1$$, and out of these $$2$$ ounces is cream. Thus the answer is $$\frac 25$$. 