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

AMC 8 2023 Problem 10

AMC8 2023 年第 10 题

表达式与运算★★☆☆☆

Harold made a plum pie to take on a picnic. He was able to eat only

\displaystyle \frac{1}{4}

of the pie, and he left the rest for his friends. A moose came by and ate

\displaystyle \frac{1}{3}

of what Harold left behind. After that, a porcupine ate

\displaystyle \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。他只吃了这个派的

\displaystyle \frac{1}{4}

，剩下的留给他的朋友们。一只驼鹿经过，吃掉了 Harold 剩下的

\displaystyle \frac{1}{3}

。在那之后，一只豪猪吃掉了驼鹿剩下的派的

\displaystyle \frac{1}{3}

。问豪猪走后，原来的派还剩下多少？

正确答案 · Correct Answer

Solution 1

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,

\displaystyle \frac12 - \frac16 = \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.

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