AMC 10 · 2016 · #2
Grade 6 arithmeticPick an answer.
AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The heart symbol is a brand-new operation, so the whole problem is really about reading its rule n♥ m = n³ m² correctly and substituting the right number into the right slot. Tool #4 (Introduce a Variable) is exactly that: treat n and m as labeled slots, plug in the given numbers, and respect that the first slot is cubed while the second is squared. Tool #7 (Identify Subproblems) splits the work into two clean calculations — the top 2♥ 4 and the bottom 4♥ 2 — so they never get tangled. Tool #3 (Eliminate Possibilities) then matches the reduced fraction to the one correct choice.
Read the rule's two slots
The rule n♥ m = n³ m² cubes the first slot, squares the second, so order matters and 2♥ 4 ≠ 4♥ 2.
A new operation is just a recipe with labeled blanks — fill the right number into the right blank before doing anything else.
6.EE.A.2Use Matrix LogicCompute the numerator
Top 2♥ 4: cube the first, square the second — 2³ = 8 and 4² = 16, so the numerator is 128.
Evaluating one input combination is a small self-contained job — finish it completely before touching the other.
6.EE.A.1Identify SubproblemsCompute the denominator
Bottom 4♥ 2 swaps the slots: 4³ = 64 and 2² = 4, so the denominator is 256 — larger than the top.
Swapping the inputs sends the bigger number to the cube, which is why the bottom comes out larger than the top.
4.NBT.B.5Identify SubproblemsDivide and simplify
The fraction is 128/256, and since 256 = 2 × 128 it reduces to 1/2 — matching choice (B).
When the bottom is exactly twice the top, the fraction is one-half — no long division needed.
4.NF.A.1Eliminate PossibilitiesWhen a problem invents a new symbol, treat it as a recipe with labeled blanks — fill in the right number, and remember the order can change the answer.
- Read the rule's two slots
- Compute the numerator
- Compute the denominator
- Divide and simplify