AMC 10 · 2011 · #7
Grade 6 number-theorylogicPick an answer.
The total cost is (number of buyers) × (pencils each) × (price per pencil), so Tool #4 (Introduce a Variable) is the natural start: name the three unknowns and write one product equation, s · n · c = 1771. That turns the word problem into a number-factoring task. Tool #8 (Analyze the Units) first pins everything to one unit — cents — so the total is a whole-number product. Tool #7 (Identify Subproblems) handles the key sub-step — breaking 1771 into its prime factors, which reveals exactly which whole numbers can appear. Tool #3 (Eliminate Possibilities) then uses the three inequalities (majority, pencils ＞ 1, price ＞ pencils) to lock each factor into its correct role and rule out the trap choices.
Put everything in cents
Everything goes into one unit first.
Working in a single unit means every quantity is a whole number of cents, so the total is just a product of whole numbers.
4.MD.A.2Analyze The UnitsName the three unknowns
Three unknowns multiply to that total.
One equation ties the three unknowns together; the inequalities are the extra clues that will separate them.
6.EE.B.6Introduce A VariableFactor the total
The total is a product of three primes.
Prime factorization lists the only bricks available, so the three answers must be assembled from exactly 7, 11, and 23.
The prime factorization lists the only bricks available, so the three answers are assembled from exactly those.
▸ Why?
Every number has exactly one prime recipe, so no other factors can appear from anywhere.
▸ Why?
Divisors come in pairs that multiply back to the number, so each brick names what it must be paired with.
Assign each factor with the clues
The inequalities assign them, giving 11, choice (B).
Each inequality forces exactly one factor into its slot, so the roles are pinned down with no guessing left.
6.EE.B.5Eliminate PossibilitiesWhen whole numbers multiply to a fixed total, factor the total into primes first — then the extra clues just tell you which factor plays which role.
- Put everything in cents
- Name the three unknowns
- Factor the total
- Assign each factor with the clues