AMC 10 · 2012 · #3

Grade 5 geometry-3drate-ratio
Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Direct Ratio Scale-Up
volume-rectangular-prismratio-proportiondimensional-analysis identify-subproblemsspatial-visualizationdimensional-analysis ↑ Prerequisites: volume-rectangular-prism
📏 Short solution 💡 2 insights
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Problem
A second box is built with two of its dimensions enlarged by whole-number factors. Find how much it holds.

Pick an answer.

(A)
120
(B)
160
(C)
200
(D)
240
(E)
280
How to solve
Strategy Identify Subproblems

The second box is not just "bigger" — its dimensions are whole-number multiples of the first box's, and that is worth much more than a size comparison. Tool #17 makes the payoff visible: because the height doubles and the width triples, the second box can be sliced cleanly into blocks that are exact copies of the first box. Tool #7 then turns the big box into a problem already solved, since each copy's capacity is the 40 grams handed to us. Tool #1 keeps the two boxes and their labelled edges straight while slicing, and Tool #8 does the first job of all: notice that the problem states grams but describes centimetres, and pin down what lets one be read off the other.

1STEP 1

Grams are decided by space

What it holds follows the space inside.

grams held = (grams in one cm³) × (space inside, in cm³)
2STEP 2

Size up the second box

Two dimensions grow, the third stays the same.

2 × 3 × 5 cm ⟶ (2 · 2) × (3 · 3) × (1 · 5) = 4 × 9 × 5 cm
3STEP 3

Slice it into copies

The big box slices into 6 copies of the small one.

4 × 9 × 5 = (2 · 2)×(3 · 3)×(1 · 5) → 2 × 3 × 1 = 6 blocks, each 2 × 3 × 5
4STEP 4

Six copies, six times the clay

Six copies hold 240, choice (D).

n = 40 + 40 + 40 + 40 + 40 + 40₆ blocks = 6 × 40 = 240 → (D)
Answer
240
The answer is only as good as the slicing picture, so check that picture against raw volumes. The first box encloses 2 × 3 × 5 = 30 cubic centimetres and the second encloses 4 × 9 × 5 = 180, and 180 = 6 × 30 exactly — six copies, no remainder, just as the cuts claimed. The weights agree too: 240/180 = 40/30 = 4/3 grams per cubic centimetre in both boxes, which is the one promise the problem made about the clay. Two of the wrong choices are readable: 120 = 3 × 40 is what counting only the tripled width gives, and 280 = 7 × 40 is what "six times bigger" gives if it is read as six copies added on top of the original instead of six copies in total. One coincidence here is worth distrusting: the scale factors 2, 3, 1 multiply to 6 but they also add to 6, so a student who adds them lands on 240 and never finds out the rule is wrong. With twice the height and four times the width, adding would give 7 where the truth is 8. The factors multiply, because each edge carries its own factor into the product of the three edges.
💡Key takeaway

Doubling one edge and tripling another turns the box into six copies of the original, so it holds six times as much clay: 6 × 40 = 240 grams.

  • Grams are decided by space
  • Size up the second box
  • Slice it into copies
  • Six copies, six times the clay

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