AMC 8 · 2023 · #10
Easy mode Grade 5Problem
Imagine one whole plum pie sitting on a picnic blanket.
First, Harold eats 41 of the pie. He leaves the rest behind.
Then a moose wanders over and eats 31 of what Harold left behind.
After the moose leaves, a porcupine comes by and eats 31 of what the moose left behind.
How much of the original whole pie is still there?
Pick an answer.
AMC 8 2023 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
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
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.
💡 Subtracting fractions to find what's left of a whole is Grade 5 fraction subtraction with unlike (here, like) denominators.
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).
💡 "$\tfrac{2}{3}$ of $\tfrac{3}{4}$" is a Grade 5 fraction-times-fraction calculation.
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.
💡 Solving a real-world word problem by chaining fraction multiplications is a Grade 5 application standard.
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.
💡 Multiplying three fractions in a row is still just the Grade 5 fraction-times-fraction skill, repeated.
5.NF.A.1 Flip Harold's bite into a leftover. Harold eats $\tfrac{1}{4}$ of the original p 5.NF.B.4 Do the same trick for the moose. The moose eats $\tfrac{1}{3}$ of the pie in fro 5.NF.B.6 And again for the porcupine. The porcupine eats $\tfrac{1}{3}$ of the pie he fin 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.1Add 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.4Apply 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.6Solve 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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