AMC 8 · 2001 · #1

Grade 4 arithmeticrate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

rateunit-conversionmulti-digit-arithmetic identify-subproblemsdimensional-analysis ↑ Prerequisites: multi-digit-arithmetic

📏 Short solution 💡 2 insights

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Problem

Casey's shop class is making a golf trophy. He has to paint 300 dimples on a golf ball. If it takes him 2 seconds to paint one dimple, how many minutes will it take for him to finish

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2001 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.

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Toolkit + CCSS Solution

Understand

Restated: Casey paints $300$ dimples on a golf ball. Each dimple takes $2$ seconds to paint. How many minutes does the whole job take?

Givens: There are $300$ dimples to paint; Each dimple takes $2$ seconds; Answer choices: (A) $4$, (B) $6$, (C) $8$, (D) $10$, (E) $12$

Unknowns: The total painting time, expressed in minutes

Understand

Restated: Casey paints $300$ dimples on a golf ball. Each dimple takes $2$ seconds to paint. How many minutes does the whole job take?

Givens: There are $300$ dimples to paint; Each dimple takes $2$ seconds; Answer choices: (A) $4$, (B) $6$, (C) $8$, (D) $10$, (E) $12$

Plan

Primary tool: #7 Break into Subproblems

Secondary: #16 Change Representation

The question mixes two ideas: how long the painting takes, and what unit the answer should be in. Tool #7 (Break into Subproblems) splits the work into two clean steps — first find the total time in seconds, then deal with the unit. Tool #16 (Change Representation) handles the second step: rewriting a seconds total as minutes by dividing by $60$. Keeping the two steps separate prevents unit-mixing mistakes.

Execute — Answer: D

#7 Break into Subproblems 4.OA.A.2 Step 1

Find the total painting time in seconds.
Each of the $300$ dimples takes $2$ seconds, so the total is $300$ groups of $2$.

$$300 \times 2 = 600 \text{ seconds}$$

💡 Grade 4 multiplicative comparison: "$300$ dimples, $2$ seconds each" means total $=$ (count) $\times$ (time per item).

#16 Change Representation 4.MD.A.1 Step 2

Change the unit from seconds to minutes.
Since $60$ seconds make $1$ minute, divide the seconds total by $60$.

$$600 \div 60 = 10 \text{ minutes}$$

💡 Grade 4 measurement conversion: dividing by $60$ regroups seconds into whole minutes.

#7 Break into Subproblems 4.OA.A.3 Step 3

Match the result to the answer choices.
$10$ minutes is choice (D).

$$10 \text{ minutes} \;\Rightarrow\; \textbf{(D)}$$

💡 Reading off the choice that matches the computed value is the last step of a multistep word problem.

[1] #7 4.OA.A.2 Find the total painting time in seconds. Each of the $300$ dimples takes $2$ sec

[2] #16 4.MD.A.1 Change the unit from seconds to minutes. Since $60$ seconds make $1$ minute, div

[3] #7 4.OA.A.3 Match the result to the answer choices. $10$ minutes is choice (D).

Review

Reasonableness: Quick sanity check by working backward: $10$ minutes is $10 \times 60 = 600$ seconds, and $600 \div 2 = 300$ dimples. That matches the problem exactly. Also, the size feels right: $2$ seconds per dimple is fast, so painting hundreds of them should still finish in only a few minutes, not hours.

Alternative: Tool #16 (Change Representation) used earlier: first convert the per-dimple time to minutes — $2$ seconds $= \tfrac{2}{60} = \tfrac{1}{30}$ minute per dimple. Then $300 \times \tfrac{1}{30} = 10$ minutes. Same answer (D), reached by switching units before multiplying instead of after.

CCSS standards used (min grade 4)

4.OA.A.2 Multiply or divide to solve word problems involving multiplicative comparison (Computing $300 \times 2 = 600$ seconds as "$300$ dimples, each taking $2$ seconds.")
4.MD.A.1 Know relative sizes of measurement units within one system; convert larger units to smaller units (Using $1 \text{ minute} = 60 \text{ seconds}$ to rewrite $600$ seconds as $10$ minutes.)
4.OA.A.3 Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations (Combining the multiplication step and the unit-conversion step into one final answer.)

⭐ When a problem mixes amounts and units, do them in order: first compute the total in the unit you were given, then convert to the unit the question asks for. Here that means seconds first ($300 \times 2 = 600$), minutes second ($600 \div 60 = 10$).

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Review

CCSS standards used (min grade 4)

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