AMC 10 · 2023 · #8
Grade 6 arithmeticPick an answer.
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.
Turn the first condition into an equation
The first gives a simple relation.
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.
A mean is a total divided by a count, so one clue becomes one simple equation.
▸ Why?
Multiplying an average back by its count returns the total it came from.
▸ Why?
That total is the individual scores added together, so the unknown one is the leftover.
List the candidates
That leaves only a few candidates.
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 condition
Write the second as a test.
Same definition-of-mean check, now applied to the "three more quizzes" scenario. Grade 6 statistics.
6.SP.B.5Guess And CheckFilter the candidates
Only one passes the test.
Plug-in arithmetic on five small candidates — Grade 3 multi-step calculation, fast in your head.
3.OA.D.8Guess And CheckMatch the choice
The current mean is 7.
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 12 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.