AMC8 2023

AMC8 2023 · Q10

AMC8 2023 · Q10. It mainly tests Word problems (algebra), Fractions.

Harold made a plum pie to take on a picnic. He was able to eat only $\frac{1}{4}$ of the pie, and he left the rest for his friends. A moose came by and ate $\frac{1}{3}$ of what Harold left behind. After that, a porcupine ate $\frac{1}{3}$ of what the moose left behind. How much of the original pie still remained after the porcupine left?

Harold 做了一个李子派带去野餐。他只吃了派的三分之一，将剩下的留给了朋友。一只驼鹿过来吃了 Harold 留下的三分之一。在此之后，一只豪猪吃了驼鹿留下三分之一的三分之一。豪猪离开后，原派还剩下多少？

(A) \ \frac{1}{12} \ \frac{1}{12}

(B) \ \frac{1}{6} \ \frac{1}{6}

(D) \ \frac{1}{3} \ \frac{1}{3}

(E) \ \frac{5}{12} \ \frac{5}{12}

Answer

Correct choice: (D)

正确答案：（D）

Solution

Note that: - Harold ate $\frac14$ of the pie. After that, $1-\frac14=\frac34$ of the pie was left behind. - The moose ate $\frac13\cdot\frac34 = \frac14$ of the pie. After that, $\frac34 - \frac14 = \frac12$ of the pie was left behind. - The porcupine ate $\frac13\cdot\frac12 = \frac16$ of the pie. After that, $\frac12 - \frac16 = \boxed{\textbf{(D)}\ \frac{1}{3}}$ of the pie was left behind. More simply, we can condense the solution above into the following equation: \[\left(1-\frac14\right)\left(1-\frac13\right)\left(1-\frac13\right) = \frac34\cdot\frac23\cdot\frac23 = \frac13.\]

注意： - Harold 吃了 $\frac14$ 的派。之后，剩下 $1-\frac14=\frac34$ 的派。 - 驼鹿吃了 $\frac13\cdot\frac34 = \frac14$ 的派。之后，剩下 $\frac34 - \frac14 = \frac12$ 的派。 - 豪猪吃了 $\frac13\cdot\frac12 = \frac16$ 的派。之后，剩下 $\frac12 - \frac16 = \boxed{\textbf{(D)}\ \frac{1}{3}}$ 的派。更简单地，我们可以将上面的解浓缩为以下方程： \[\left(1-\frac14\right)\left(1-\frac13\right)\left(1-\frac13\right) = \frac34\cdot\frac23\cdot\frac23 = \frac13.\]

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