Competition · AMC preparation · step 4 of 4

AMC 10 · 2023B · #4

Grade 5 geometry-2d
area-rectanglesunit-conversiondecimal-arithmetic dimensional-analysisidentify-subproblems ↑ Prerequisites: area-rectanglesunit-conversion
📏 Short solution 💡 2 insights
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Problem
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?

Pick an answer.

(A)
162,500
(B)
162.5
(C)
1,625
(D)
1,625,000
(E)
16,250

AMC 10 2023 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Analyze the Units

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 cm². 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 × width.

1STEP 1

Change millimeters to centimeters

Width: divide by 10 to go from mm to cm, giving 0.65 cm.

6.5 mm = 6.5/10 cm = 0.65 cm
2STEP 2

Change meters to centimeters

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

25 m = 25 × 100 cm = 2500 cm
3STEP 3

See the stripe as a rectangle

Sketch the stripe as a long rectangle with width 0.65 cm and length 2500 cm. The area is length × width.

Area = 2500 × 0.65 cm²
4STEP 4

Multiply the two sides

Clear the decimal — 2500 × 0.65 = 25 × 65 = 1,625 cm².

2500 × 0.65 = 25 × 65 = 1625 → (C)
Answer
1,625
Cross-check the order of magnitude. A 25 m × 6.5 mm stripe is roughly 25 m × 1150\frac{1}{150} m ≈ 0.16 m². Since 1 m² = 10,000 cm², that is about 1,600 cm² — right in the neighborhood of 1,625. So (C) is the right answer; the other choices are off by powers of ten.
💡Key takeaway

This AMC 10 problem only needs Grade 5 unit conversion — get both sides into centimeters first, then multiply 2500 × 0.65 to land on 1,625 cm².

  • Change millimeters to centimeters
  • Change meters to centimeters
  • See the stripe as a rectangle
  • Multiply the two sides

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