AMC 8 · 2016 · #11
Grade 6 algebranumber-theoryPick an answer.
AMC 8 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The condition is a statement about digits, so Tool #5 (Use Variables) lets us name the tens digit t and the units digit u and write N = 10t + u in place-value form. Adding N to its reversal 10u + t collapses neatly to 11(t+u) = 132, which simplifies to t + u = 12. Once the condition reduces to a single equation in two digits, Tool #13 (Work Systematically) takes over: list every (t,u) with t + u = 12, 1 ≤ t ≤ 9, 0 ≤ u ≤ 9, and count what survives.
Let t be the tens digit and u the units digit, so N = 10t + u and its reversal is 10u + t.
Place value (Grade 4) says the tens digit contributes ten times its face value, and the ones digit contributes its face value.
4.NBT.A.1Look For A PatternAdding N and its reversal and setting the sum to 132 gives (10t + u) + (10u + t) = 132.
Writing a word condition as an algebraic equation is Grade 6 expressions-and-equations work.
6.EE.A.2Look For A PatternCombine like terms to 11(t + u) = 132, then divide by 11 to get t + u = 12.
Grouping 11t + 11u as 11(t+u) is the distributive property — a Grade 6 "equivalent expressions" move.
6.EE.A.3Look For A PatternWith t from 1 to 9 and u = 12 - t, keep only the pairs where u is also a valid digit (0 to 9).
Substituting each candidate t into t + u = 12 and checking the digit range is the Grade 6 idea of "which values make the equation true."
6.EE.B.5Convert To AlgebraEach valid pair gives one number N = 10t + u, so counting them gives 7 numbers — answer (B).
Counting the items in a generated list is the Grade 4 "analyze a pattern" finishing step.
4.OA.C.5Convert To AlgebraOnce you write a two-digit number as 10t + u, the whole problem turns into a Grade 6 equation t + u = 12 — then you just list the digit pairs that work!