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AMC 8 · 2023 · #10

Grade 5 arithmetic
Archetype Arithmetic Expression Decomposition
fraction-arithmeticcomplementary-counting complementary-countingidentify-subproblems ↑ Prerequisites: fraction-arithmeticfraction-multiplication
📏 Short solution 💡 2 insights
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Problem

Harold made a plum pie to take on a picnic. He was able to eat only 14\frac{1}{4} of the pie, and he left the rest for his friends. A moose came by and ate 13\frac{1}{3} of what Harold left behind. After that, a porcupine ate 13\frac{1}{3} of what the moose left behind. How much of the original pie still remained after the porcupine left?

Pick an answer.

(A)
$\frac{1}{12}$
(B)
$\frac{1}{6}$
(C)
$\frac{1}{4}$
(D)
$\frac{1}{3}$
(E)
$\frac{5}{12}$
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Toolkit + CCSS Solution

Understand

Restated: Harold baked one whole plum pie and ate $\tfrac{1}{4}$ of it. A moose then ate $\tfrac{1}{3}$ of what Harold left. A porcupine then ate $\tfrac{1}{3}$ of what the moose left. What fraction of the ORIGINAL pie is still there after the porcupine leaves?

Givens: The pie starts whole, i.e., $1$; Harold eats $\tfrac{1}{4}$ of the original pie; The moose eats $\tfrac{1}{3}$ of what Harold leaves; The porcupine eats $\tfrac{1}{3}$ of what the moose leaves; Answer choices: (A) $\tfrac{1}{12}$, (B) $\tfrac{1}{6}$, (C) $\tfrac{1}{4}$, (D) $\tfrac{1}{3}$, (E) $\tfrac{5}{12}$

Unknowns: The fraction of the ORIGINAL pie that remains after the porcupine

Understand

Restated: Harold baked one whole plum pie and ate $\tfrac{1}{4}$ of it. A moose then ate $\tfrac{1}{3}$ of what Harold left. A porcupine then ate $\tfrac{1}{3}$ of what the moose left. What fraction of the ORIGINAL pie is still there after the porcupine leaves?

Givens: The pie starts whole, i.e., $1$; Harold eats $\tfrac{1}{4}$ of the original pie; The moose eats $\tfrac{1}{3}$ of what Harold leaves; The porcupine eats $\tfrac{1}{3}$ of what the moose leaves; Answer choices: (A) $\tfrac{1}{12}$, (B) $\tfrac{1}{6}$, (C) $\tfrac{1}{4}$, (D) $\tfrac{1}{3}$, (E) $\tfrac{5}{12}$

Plan

Primary tool: #16 Change Focus / Count the Complement

Secondary: #7 Identify Subproblems

The problem keeps asking "how much was eaten?" but what we actually want is "how much is LEFT." Tool #16 says: flip the question — if an eater takes $\tfrac{1}{3}$, they LEAVE $1-\tfrac{1}{3}=\tfrac{2}{3}$, so we can just multiply leftover fractions instead of tracking eaten amounts and subtracting. Tool #7 then splits the chain of events into three small subproblems (Harold's leftover, moose's leftover, porcupine's leftover), and we multiply the three "leftover" fractions to get the final answer.

Execute — Answer: D

#16 Change Focus / Count the Complement 5.NF.A.1 Step 1
  • Flip Harold's bite into a leftover.
  • Harold eats $\tfrac{1}{4}$ of the original pie, so he LEAVES $1-\tfrac{1}{4}=\tfrac{3}{4}$ of the pie.
  • This is the "complement" move from Tool #16: track what stays instead of what goes.
$$1-\dfrac{1}{4}=\dfrac{3}{4}$$

💡 Subtracting fractions to find what's left of a whole is Grade 5 fraction subtraction with unlike (here, like) denominators.

#16 Change Focus / Count the Complement 5.NF.B.4 Step 2
  • Do the same trick for the moose.
  • The moose eats $\tfrac{1}{3}$ of the pie in front of him, so he LEAVES $\tfrac{2}{3}$ of that pie.
  • To find what fraction of the ORIGINAL pie remains after the moose, multiply $\tfrac{2}{3}$ by the $\tfrac{3}{4}$ Harold left behind.
  • This is the second subproblem (Tool #7).
$$\dfrac{2}{3}\times\dfrac{3}{4}=\dfrac{6}{12}=\dfrac{1}{2}$$

💡 "$\tfrac{2}{3}$ of $\tfrac{3}{4}$" is a Grade 5 fraction-times-fraction calculation.

#16 Change Focus / Count the Complement 5.NF.B.6 Step 3
  • And again for the porcupine.
  • The porcupine eats $\tfrac{1}{3}$ of the pie he finds, so he LEAVES $\tfrac{2}{3}$ of it.
  • Multiply by the $\tfrac{1}{2}$ the moose left behind to get the final fraction of the ORIGINAL pie still on the plate.
$$\dfrac{2}{3}\times\dfrac{1}{2}=\dfrac{2}{6}=\dfrac{1}{3}$$

💡 Solving a real-world word problem by chaining fraction multiplications is a Grade 5 application standard.

#7 Identify Subproblems 5.NF.B.4 Step 4
  • Bundle the three subproblems into one product.
  • We can also write the whole chain at once: the final leftover is the product of each eater's "leftover fraction." This is Tool #7's combine step.
$$\left(1-\dfrac{1}{4}\right)\left(1-\dfrac{1}{3}\right)\left(1-\dfrac{1}{3}\right)=\dfrac{3}{4}\cdot\dfrac{2}{3}\cdot\dfrac{2}{3}=\dfrac{12}{36}=\dfrac{1}{3}\;\Rightarrow\;\textbf{(D)}$$

💡 Multiplying three fractions in a row is still just the Grade 5 fraction-times-fraction skill, repeated.

[1] #16 5.NF.A.1 Flip Harold's bite into a leftover. Harold eats $\tfrac{1}{4}$ of the original p
[2] #16 5.NF.B.4 Do the same trick for the moose. The moose eats $\tfrac{1}{3}$ of the pie in fro
[3] #16 5.NF.B.6 And again for the porcupine. The porcupine eats $\tfrac{1}{3}$ of the pie he fin
[4] #7 5.NF.B.4 Bundle the three subproblems into one product. We can also write the whole chain

Review

Reasonableness: Does $\tfrac{1}{3}$ make sense? Harold alone leaves $\tfrac{3}{4}$. Each animal then keeps $\tfrac{2}{3}$ of what arrives, so the pie shrinks by a factor of $\tfrac{2}{3}$ twice: $\tfrac{3}{4}\to\tfrac{1}{2}\to\tfrac{1}{3}$. The values are decreasing as expected and end above $\tfrac{1}{4}$ (since Harold left $\tfrac{3}{4}$ and the animals together couldn't eat all of that). Answer (D) $\tfrac{1}{3}$ is consistent.

Alternative: Use Tool #9 (Solve an Easier Related Problem) by picking a friendly total — imagine the pie was cut into $12$ equal slices. Harold eats $3$, leaving $9$. The moose eats $\tfrac{1}{3}$ of $9 = 3$, leaving $6$. The porcupine eats $\tfrac{1}{3}$ of $6 = 2$, leaving $4$. So $\tfrac{4}{12}=\tfrac{1}{3}$ of the pie remains — same answer (D).

CCSS standards used (min grade 5)

  • 5.NF.A.1 Add and subtract fractions with unlike denominators (Computing each "leftover" fraction by subtracting the eaten part from $1$ (e.g., $1-\tfrac{1}{4}=\tfrac{3}{4}$, $1-\tfrac{1}{3}=\tfrac{2}{3}$).)
  • 5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction (Multiplying the leftover fractions together: $\tfrac{2}{3}\times\tfrac{3}{4}=\tfrac{1}{2}$ and $\tfrac{3}{4}\cdot\tfrac{2}{3}\cdot\tfrac{2}{3}=\tfrac{1}{3}$.)
  • 5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers (Modeling the picnic word problem as a chain of fraction-of-a-fraction multiplications to find the remaining pie.)

⭐ This AMC 8 problem only needs Grade 5 fraction-times-fraction multiplication you already know!

⭐ This AMC 8 problem only needs Grade 5 fraction-times-fraction multiplication you already know!

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