AMC 8 · 2000 · #22

Grade 7 geometry-3d

Archetype Compound Area by Decomposition / Difference └ Sub-archetype Rectilinear Frame / Box Border

surface-areaspatial-visualizationpercentagearea-rectangles area-differenceidentify-subproblems ↑ Prerequisites: surface-areapercentage

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

A cube with edge length 2 has a smaller cube with edge length 1 glued on its top face so that one full face of the small cube rests on the big cube's top. Count the new total surface area (top, bottom, all sides) and compare it to the original cube's surface area. Which choice is closest to the percent increase?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2000 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Change Focus / Count the Complement

Recomputing the surface area of the bumpy new solid face by face is slow and error-prone. Tool #16 (Change Focus) flips the question: don't recount everything, just count the net change at the seam. Gluing the small cube on top removes the 1 × 1 contact patch from the outside (two faces actually — the small cube's bottom and the matching patch on the big cube's top), and adds the small cube's other five 1 × 1 faces. Tool #1 (Draw a Diagram) makes the contact patch and the five newly exposed faces easy to see, so the net change is a simple count: +5 minus -1 on the top face = +4 square units. The percent step is then one division.

1STEP 1

The original big cube has 6 congruent faces of area 2² = 4, so its surface area is 6 × 4 = 24.

Original = 6 × 2² = 24 square units

2STEP 2

At the seam a 1 × 1 patch on the big top is hidden while the small cube reveals its top and four sides — five new 1 × 1 faces.

Removed from outside = 1 (the 1 × 1 patch on the big cube's top) Added to outside = 5 × 1 = 5 (small cube's top + 4 sides)

3STEP 3

Combine gain and loss: +5 - 1 = +4 square units net.

Δ Surface area = +5 - 1 = +4 square units

4STEP 4

Percent change is the increase over the original: $\frac{4}{24}$ = $\frac{1}{6}$ ≈ 16.67%.

Δ/Original × 100% =

\frac{4}{24}

× 100% =

\frac{1}{6}

× 100% ≈ 16.67%

5STEP 5

16.67% sits between 15% and 17%, but it is far closer to 17% — choice (C).

|16.67 - 17| = 0.33 ＜ |16.67 - 15| = 1.67 → 17% → (C)

Answer

Cross-check with the long way. New total surface area = 24 + 4 = 28, so the increase is 28 - 24 = 4, and 4 ÷ 24 = 0.16 = 16.6%. That matches the complement count, and rounding to the listed options pins (C) 17. A sanity check against the choices: 10% would mean an increase of 2.4 (impossible — the change at the seam is a whole number of unit squares), and 25% would mean an increase of 6 (more than the entire small cube can offer to the outside, since the small cube only exposes 5 new faces). So (C) is the only choice that even fits the geometry.

💡Key takeaway

Don't recount the whole new solid. Gluing a 1 × 1 × 1 cube on top hides one 1 × 1 patch and exposes five 1 × 1 faces, so the surface area grows by 4. Then $\frac{4}{24}$ ≈ 16.7%, which rounds to 17% — answer (C).

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