AMC 10 · 2002 · #7
Grade 8 arithmeticPick an answer.
Naming the middle integer n (Tool #4) is the smart move because the three consecutive numbers become n-1, n, n+1 — a symmetric setup that makes both the product and the sum collapse into clean expressions. Tool #13 (Convert to Algebra) then turns the sentence 'the product is 8 times the sum' into one equation in n. That equation shrinks to n²=25, and Tool #3 (Eliminate Possibilities) uses the word 'positive' to keep n=5 and discard n=-5.
Name the middle number
Naming the middle number n makes the trio n-1, n, n+1, symmetric about n.
For evenly spaced numbers, naming the middle one lets the outer two balance around it.
6.EE.B.6Introduce A VariableWrite the product and the sum
The sum collapses to 3n and the product to n³-n.
The symmetric n-1 and n+1 make the sum's ends cancel and the product fold into a difference of squares.
Naming the middle number makes the outer two mirror images, so the product folds into a difference of squares.
▸ Why?
A sum times a difference of the same two things cancels the middle terms and leaves two squares.
▸ Why?
Evenly spaced numbers lean equally far above and below the middle, so their outer terms pair off around it.
Solve for the middle number
Setting product equal to 24n gives n² = 25, so n = 5 since the numbers are positive.
Dividing out the shared factor n collapses a cubic into n²=25, and 'positive' picks the single valid root.
8.EE.A.2Eliminate PossibilitiesAdd up the squares
The trio is 4,5,6 and their squares add to 77, choice (B).
Once the numbers are pinned down, the answer is just three squares added together.
6.EE.A.1Convert To AlgebraFor three consecutive numbers, name the middle one n: the sum becomes 3n and the product becomes n³-n, and the equation shrinks to n²=25.
- Name the middle number
- Write the product and the sum
- Solve for the middle number
- Add up the squares