AMC12 2002 A

AMC12 2002 A · Q10

AMC12 2002 A · Q10. It mainly tests Word problems (algebra), Fractions.

Sarah places four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then pours half the coffee from the first cup to the second and, after stirring thoroughly, pours half the liquid in the second cup back to the first. What fraction of the liquid in the first cup is now cream?

Sarah 将四盎司咖啡倒入一个八盎司的杯子中，并将四盎司奶油倒入另一个同样大小的杯子中。然后她把第一个杯子中一半的咖啡倒入第二个杯子，充分搅拌后，再把第二个杯子中一半的液体倒回第一个杯子。现在第一个杯子中的液体有多少分数是奶油？

(A) $\frac{1}{4}$ $\frac{1}{4}$

(B) $\frac{1}{3}$ $\frac{1}{3}$

(D) $\frac{2}{5}$ $\frac{2}{5}$

(E) $\frac{1}{2}$ $\frac{1}{2}$

Answer

Correct choice: (D)

正确答案：（D）

Solution

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 $4/2=2$ 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 $2/2=1$ ounce of coffee and $4/2=2$ 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 $\boxed{\text{(D) } \frac 25}$. Let's consider this in steps. We have 4 ounces of coffee in the first cup. We hace 4 ounces of cream in the second cup. We take half of the coffee in the first cup(2 ounces), and add it to the second cup, yielding 6 ounces in total in the second cup(in a 1:2 ratio between coffee and cream, respecitvely). We then take half of the second cup and pour it into the first cup. $6/2=3$, so there is now 5 ounces in the first cup, 2 coffee and 3 the mixture. Remember that the mixture is in a 1:2 ratio between coffee and cream. So, coffee has one ounce and cream has 2 ounces. In total, there is 2 ounces in the 5 ounce first cup. Putting 2 over 5, we get the answer. Therefore, the answer is $\boxed{\text{(D) } \frac 25}$.

我们分步模拟该过程。开始时： - 杯 1 中有 $4$ 盎司咖啡 - 杯 2 中有 $4$ 盎司奶油第一步，从杯 1 倒出 $4/2=2$ 盎司咖啡到杯 2，得到： - 杯 1 中有 $2$ 盎司咖啡 - 杯 2 中有 $2$ 盎司咖啡和 $4$ 盎司奶油第二步，从杯 2 倒回杯 1：咖啡倒出 $2/2=1$ 盎司，奶油倒出 $4/2=2$ 盎司，得到： - 杯 1 中有 $2+1=3$ 盎司咖啡和 $0+2=2$ 盎司奶油 - 其余留在杯 2 中因此最后杯 1 中共有 $3+2=5$ 盎司液体，其中 $2$ 盎司是奶油。故答案为 $\boxed{\text{(D) } \frac 25}$。

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