AMC 10 · 2003 · #3
Grade 6 geometry-3dPick an answer.
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.
Volume of the whole box
The box's volume is 15 × 10 × 8 = 1200 cubic centimetres.
The amount of space in a box is just its three side lengths multiplied together.
5.MD.C.5Identify SubproblemsCount the corners and one cube
A box has eight corners, and each cut cube holds 27 cubic centimetres.
A box has one corner for each of the four bottom edges and four top edges — eight in all.
A box has exactly eight corners, so eight identical cubes are removed.
▸ Why?
Each corner is where three faces meet, and a box has one such meeting per corner and no more.
▸ Why?
The box's own volume is its flat base repeated all the way up, so both volumes are plain products of side lengths.
Total volume removed
Eight of them remove 8 × 27 = 216 in total.
Eight identical pieces removed means eight times the size of one piece.
5.NBT.B.5Identify SubproblemsTurn the ratio into a percent
Dividing gives 216 over 1200 = 18%, choice (D).
"What percent" means the part divided by the whole, then read as a number out of 100.
6.RP.A.3Analyze The Units"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
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