AMC 10 · 2016 · #8
Grade 7 geometry-2dPick an answer.
Because the wood, thickness, and density are identical, the weight of each piece rides entirely on its flat area — tool #8 (Analyze the Units) tells us "more area means proportionally more ounces." Both pieces are equilateral triangles, so they are similar, and a flat shape scaled in length by a factor k has its area scaled by k². That turns the whole problem into a single proportion: name the unknown weight w with tool #4 (Introduce a Variable) and set w/12 equal to the area ratio. Tool #3 (Eliminate Possibilities) then matches the computed weight to the closest choice.
Weight tracks area
Same thickness means weight tracks area.
With identical thickness, a flat shape is just a stack of equal-weight layers, so ounces scale exactly with the surface it covers.
7.RP.A.2Analyze The UnitsArea scales by the square of the side
Area scales by the square of the side ratio.
Area is length times length, so scaling each length by 5/3 scales area by 5/3 twice over.
Area is length times length, so scaling every length by a factor scales the area by that factor twice over.
▸ Why?
Area lives in two directions at once, so it picks up the scale factor once for each of them.
▸ Why?
With the same thickness everywhere, the weight rises in step with the area alone.
Set up the proportion
That is a factor of 25/9.
Naming the unknown weight w lets the proportional relationship become one clean equation to solve.
7.RP.A.3Introduce A VariableSolve and pick the closest choice
Solving gives about 33.3, choice (D).
Scaling 12 ounces up by 25/9 — a bit under triple — lands near 33, ruling out every other choice.
7.NS.A.3Eliminate PossibilitiesWhen a flat shape keeps the same thickness, its weight follows its area — and scaling every length by k multiplies the area, and the weight, by k².
- Weight tracks area
- Area scales by the square of the side
- Set up the proportion
- Solve and pick the closest choice