AMC 10 · 2023 · #10
Grade 6 arithmeticPick an answer.
AMC 10 2023 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Five small whole-number choices (4, 5, 6, 7, 8) and a clean integer relationship make Tool #6 (Guess and Check) the fastest path: for each candidate M, find the n forced by the first condition, then test whether the second condition holds. Tool #3 (Eliminate Possibilities) catches us when a candidate forces a non-positive-integer n. Tool #13 (Convert to Algebra) becomes the verification path — set up both conditions as equations and solve the 2 × 2 linear system to confirm.
Translate the first clue with the mean definition; expanding and canceling nM leaves the clean relation n + M = 10.
The definition of mean — total ÷ count — turns the first clue into one simple sentence: n + M = 10. Grade 6 "summarize a data set with mean" thinking.
6.SP.B.5Convert To AlgebraSweep the five choices: each candidate M reads off n = 10 - M, and every one gives a positive integer n.
Each guess picks a candidate M and reads n off the first clue — Grade 6 "use a variable to record an unknown" worked one number at a time.
6.EE.B.6Guess And CheckSet up the second clue: three more 11's give total nM + 33 on n + 3 quizzes, which must equal (n+3)(M+2).
Same definition-of-mean check, now applied to the "three more quizzes" scenario. Grade 6 statistics.
6.SP.B.5Guess And CheckRun the check on all five: only M = 7 (with n = 3) gives nM + 33 = 54 = (n+3)(M+2); every other candidate misses.
Plug-in arithmetic on five small candidates — Grade 3 multi-step calculation, fast in your head.
3.OA.D.8Guess And CheckOnly M = 7 survives both clues, with n = 3 quizzes taken — matching choice (D).
One candidate survives the second clue; eliminate the rest — Grade 6 "solve for the value that satisfies all conditions".
6.EE.B.6Eliminate PossibilitiesThis AMC 10 problem only needs Grade 6 understanding of mean you already know — "total = number of quizzes × mean" turns each clue into one equation, and trying each answer choice locates M = 7 in under a minute.