AMC 10 · 2003 · #3

Grade 6 geometry-3d

Archetype Solid Volume by Decomposition └ Sub-archetype Fraction or Difference of a Bounding Solid

volume-rectangular-prismpercentagespatial-visualization identify-subproblems ↑ Prerequisites: volume-rectangular-prism

📏 Medium solution 💡 2 insights

Problem

A solid box measures 15 cm by 10 cm by 8 cm. A 3 cm cube is cut away from each corner of the box. Find what percent of the box's original volume is cut away.

Pick an answer.

(A)

$4.5\%$

(B)

$9\%$

(C)

$12\%$

(D)

$18\%$

(E)

$24\%$

AMC 10 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The word "percent" hides three smaller jobs, so Tool #7 (Identify Subproblems) breaks it into: find the whole box's volume, find the volume taken away, then turn the ratio into a percent. The one place a solver can slip is counting the corners, so Tool #17 (Visualize Spatial Relationships) pins down that a box has exactly 8 corners — not 4 or 6. Tool #8 (Analyze the Units) keeps every volume in cm³ so the final ratio is a clean number-over-number percent.

1STEP 1

Volume of the whole box

The box is a rectangular solid, so multiply 15 × 10 × 8 = 1200 cubic centimeters.

V_box = 15 × 10 × 8 = 1200 cm³

2STEP 2

Count the corners and one cube

Picture the box as a room: it has 8 corners, and one 3 cm cube fills 3 × 3 × 3 = 27 cubic centimeters.

8 corners, V_cube = 3³ = 27 cm³

3STEP 3

Total volume removed

All 8 cut-out cubes are the same size, so the total removed is 8 × 27 = 216 cubic centimeters.

V_removed = 8 × 27 = 216 cm³

4STEP 4

Turn the ratio into a percent

Divide removed by original: 216 ÷ 1200 = 0.18, so 18% of the box is gone — choice (D).

V_removed/V_box = 216/1200 = 0.18 = 18% → (D)

Answer

18%

The removed volume 216 cm³ is a small slice of the 1200 cm³ box, so the answer should be well under a quarter — and 18% is. A quick sanity check: 216/1200 is a bit under 240/1200 = 1/5 = 20%, so 18% sits right where it should. If a solver had wrongly used 6 corners they would get 162/1200 = 13.5% (no such choice), and using 4 corners gives 9% (choice B) — both are traps for miscounting corners, which confirms getting the count right is the whole game.

💡Key takeaway

"What percent is removed" is just the volume taken away divided by the whole volume — and remember a box has eight corners, not four.

Volume of the whole box
Count the corners and one cube
Total volume removed
Turn the ratio into a percent

Volume of the whole box

Count the corners and one cube

Total volume removed

Turn the ratio into a percent

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