2023 AMC 8

Complete problem set with solutions and individual problem pages

Problem 10 Easy

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?

  • A.

    \frac{1}{12}

  • B.

    \frac{1}{6}

  • C.

    \frac{1}{4}

  • D.

    \frac{1}{3}

  • E.

    \frac{5}{12}

Answer:D

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.

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