AMC 10 · 2023 · #4

Easy mode Grade 5

Problem

Picture Jackson painting a long, thin stripe on a piece of paper.

The stripe is the width of his paintbrush: $6.5$ millimeters wide. He has just enough paint to make the stripe $25$ meters long.

The stripe forms a long rectangle. How many square centimeters of paper does it cover?

$\textbf{(A) } 162,500 \qquad\textbf{(B) } 162.5 \qquad\textbf{(C) }1,625 \qquad\textbf{(D) }1,625,000 \qquad\textbf{(E) } 16,250$

Pick an answer.

(A)

162,500

(B)

162.5

(C)

1,625

(D)

1,625,000

(E)

16,250

View mode:

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?

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

#8 Analyze the Units 5.MD.A.1 Step 1

Convert the width from millimeters to centimeters.
Since $10$ mm $= 1$ cm, divide by $10$.

$$6.5 \text{ mm} = \dfrac{6.5}{10} \text{ cm} = 0.65 \text{ cm}$$

💡 Going from a smaller unit (mm) to a larger one (cm), the number gets smaller — Grade 5 unit conversion.

#8 Analyze the Units 5.MD.A.1 Step 2

Convert the length from meters to centimeters.
Since $1$ m $= 100$ cm, multiply by $100$.

$$25 \text{ m} = 25 \times 100 \text{ cm} = 2500 \text{ cm}$$

💡 Going from a larger unit (m) to a smaller one (cm), the number gets bigger.

#1 Draw a Diagram 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.

$$\text{Area} = 2500 \times 0.65 \text{ cm}^2$$

💡 Length times width for a rectangle — Grade 4 area formula.

#8 Analyze the Units 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 $25 \times 65 = 25 \times (60 + 5) = 1500 + 125 = 1625$.

$$2500 \times 0.65 = 25 \times 65 = 1625 \;\Rightarrow\; \textbf{(C)}$$

💡 Splitting $0.65$ into $\frac{65}{100}$ turns the decimal multiplication into clean whole-number arithmetic — Grade 5.

[1] #8 5.MD.A.1 Convert the width from millimeters to centimeters. Since $10$ mm $= 1$ cm, divid

[2] #8 5.MD.A.1 Convert the length from meters to centimeters. Since $1$ m $= 100$ cm, multiply

[3] #1 4.MD.A.3 Sketch the stripe as a long rectangle with width $0.65$ cm and length $2500$ cm.

[4] #8 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.3 Apply 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.1 Convert 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.7 Add, 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$^2$.