AMC 10 · 2023 · #2
Grade 7 arithmeticPick an answer.
Two combined weights are stated to be equal — this is exactly the trigger for Tool #13 (Convert to Algebra): let p be the pizza weight and turn the balance sentence into an equation. Tool #7 (Identify Subproblems) splits the work cleanly: first reduce the orange-slice quantities to numerical pounds (just arithmetic with 1/4), then solve the resulting linear equation in p alone. This keeps each step a one-thought move and avoids the common error of carrying mixed numbers through algebra.
Write the balance
Write the balance as an equation.
Letting the unknown weight be p and writing each pan's total as an expression turns a word balance into the Grade 6 equation form stuff in p = stuff in p.
6.EE.B.7Convert To AlgebraConvert cups to pounds
Convert the cups into pounds.
Multiplying two fractions a/b · c/d = ac/bd is the Grade 5 fraction-times-fraction skill — no variables involved here, just numbers.
5.NF.B.4Identify SubproblemsGather like terms
Gather the pizza terms and the constants.
Moving like terms to opposite sides of an equation is the standard Grade 7 "solve a linear equation" move.
Moving like terms to opposite sides of the balance keeps it level while gathering the unknown.
▸ Why?
Doing the same thing to both sides keeps the equation true, so the move never disturbs the balance.
▸ Why?
Like terms combine because the shared factor can be lifted out of the sum.
Tidy the coefficient
Tidy the pizza coefficient.
Subtracting fractions with unlike denominators by rewriting with a common denominator is the Grade 5 fraction-add/subtract standard.
5.NF.A.1Convert To AlgebraDivide for the weight
Dividing gives one and four fifths pounds.
Dividing both sides by 5/12 (i.e. multiplying by 12/5) finishes the Grade 6 "px = q" equation in one move.
6.EE.B.7Convert To AlgebraThis AMC 12 problem only needs Grade 7 "build and solve a linear equation" — call the pizza p, write each pan as pounds, and balance.