AMC 10 · 2023 · #4
Grade 5 geometry-2dProblem
Jackson's paintbrush makes a narrow strip with a width of 6.5 millimeters. Jackson has enough paint to make a strip 25 meters long. How many square centimeters of paper could Jackson cover with paint?
Pick an answer.
AMC 10 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: A paintbrush stripe is $6.5$ mm wide. Jackson has enough paint for a stripe $25$ m long. How many square centimeters of paper can be covered?
Givens: Width of stripe: $6.5$ mm; Length of stripe: $25$ m; Stripe shape is a rectangle (uniform width); Answer choices: (A) $162{,}500$, (B) $162.5$, (C) $1{,}625$, (D) $1{,}625{,}000$, (E) $16{,}250$
Unknowns: Area of paper that can be covered, in $\text{cm}^2$
Understand
Restated: A paintbrush stripe is $6.5$ mm wide. Jackson has enough paint for a stripe $25$ m long. How many square centimeters of paper can be covered?
Givens: Width of stripe: $6.5$ mm; Length of stripe: $25$ m; Stripe shape is a rectangle (uniform width); Answer choices: (A) $162{,}500$, (B) $162.5$, (C) $1{,}625$, (D) $1{,}625{,}000$, (E) $16{,}250$
Plan
Primary tool: #8 Analyze the Units
Secondary: #1 Draw a Diagram
The whole problem is a units trap. Tool #8 (Analyze Units) forces both length and width into centimeters before multiplying — the only safe path to $\text{cm}^2$. The choice list is spread across factors of $10$ on purpose ($162.5$, $1{,}625$, $16{,}250$, $162{,}500$, $1{,}625{,}000$), so an off-by-a-factor-of-$10$ conversion mistake lands on a wrong choice. A quick diagram (Tool #1) of a very long thin rectangle reminds the student which dimension is which and that the formula is just length $\times$ width.
Execute — Answer: C
5.MD.A.1 Step 1 - Convert the width from millimeters to centimeters.
- Since $10$ mm $= 1$ cm, divide by $10$.
💡 Going from a smaller unit (mm) to a larger one (cm), the number gets smaller — Grade 5 unit conversion.
5.MD.A.1 Step 2 - Convert the length from meters to centimeters.
- Since $1$ m $= 100$ cm, multiply by $100$.
💡 Going from a larger unit (m) to a smaller one (cm), the number gets bigger.
4.MD.A.3 Step 3 - Sketch the stripe as a long rectangle with width $0.65$ cm and length $2500$ cm.
- The area is length $\times$ width.
💡 Length times width for a rectangle — Grade 4 area formula.
5.NBT.B.7 Step 4 - Multiply.
- The cleanest path is to clear the decimal first: $0.65 = \frac{65}{100}$, so $2500 \times 0.65 = 2500 \times \frac{65}{100} = 25 \times 65$.
- Then $25 \times 65 = 25 \times (60 + 5) = 1500 + 125 = 1625$.
💡 Splitting $0.65$ into $\frac{65}{100}$ turns the decimal multiplication into clean whole-number arithmetic — Grade 5.
5.MD.A.1 Convert the width from millimeters to centimeters. Since $10$ mm $= 1$ cm, divid 5.MD.A.1 Convert the length from meters to centimeters. Since $1$ m $= 100$ cm, multiply 4.MD.A.3 Sketch the stripe as a long rectangle with width $0.65$ cm and length $2500$ cm. 5.NBT.B.7 Multiply. The cleanest path is to clear the decimal first: $0.65 = \frac{65}{100 Review
Reasonableness: Cross-check the order of magnitude. A $25$ m $\times$ $6.5$ mm stripe is roughly $25 \text{ m} \times \tfrac{1}{150}\text{ m} \approx 0.16$ $\text{m}^2$. Since $1\text{ m}^2 = 10{,}000$ $\text{cm}^2$, that is about $1{,}600$ $\text{cm}^2$ — right in the neighborhood of $1{,}625$. So (C) is the right answer; the other choices are off by powers of ten.
Alternative: Tool #15 (Reorganize): convert everything to millimeters instead. Width $= 6.5$ mm, length $= 25{,}000$ mm, area $= 6.5 \times 25{,}000 = 162{,}500$ $\text{mm}^2$. Then convert: $1$ $\text{cm}^2 = 100$ $\text{mm}^2$, so divide by $100$ to get $1{,}625$ $\text{cm}^2$. Same answer; this path also surfaces (A) $162{,}500$ as the classic distractor.
CCSS standards used (min grade 5)
4.MD.A.3Apply area and perimeter formulas for rectangles in real-world problems (Using $\text{area} = \text{length} \times \text{width}$ for the rectangular stripe.)5.MD.A.1Convert among different-sized standard measurement units within a given system (Converting $6.5$ mm to $0.65$ cm and $25$ m to $2500$ cm before multiplying.)5.NBT.B.7Add, subtract, multiply, and divide decimals to hundredths (Computing $2500 \times 0.65 = 1625$ via $25 \times 65$.)
⭐ This AMC 10 problem only needs Grade 5 unit conversion — get both sides into centimeters first, then multiply $2500 \times 0.65$ to land on $1{,}625$ cm$^2$.
⭐ This AMC 10 problem only needs Grade 5 unit conversion — get both sides into centimeters first, then multiply $2500 \times 0.65$ to land on $1{,}625$ cm$^2$.