AMC 10 · 2002 · #2
Grade 6 arithmeticPick an answer.
Both of the first two terms carry the same factor (3x-2), so instead of multiplying everything out, Tool #15 (Organize Information in More Ways) rewrites the expression to pull that shared factor to the front: (3x-2)[(4x+1)-4x]+1. The bracket collapses to a single number, which turns a messy problem into a one-line one. Tool #7 (Identify Subproblems) then handles the leftover in tidy steps — simplify the bracket, add the stray +1, substitute x=4. Tool #3 (Eliminate Possibilities) guards the built-in traps: dropping the outside +1 gives 10 (kills (C)), and mishandling the leftover constants -2+1 as 0 gives 12 (kills (E)).
Pull out the shared factor
The first two terms share (3x-2), so factor it out and hold the lone +1 aside.
When two terms share a factor, you can lift it out front and subtract what is left inside.
The two terms share a factor, so it lifts out front and leaves a short bracket behind.
▸ Why?
So many of this less so many of that is that many of the difference, so a common factor comes out of a subtraction too.
▸ Why?
The expression is its terms combined, so simplifying the bracket simplifies the whole thing.
Simplify the bracket
Inside the bracket the 4x terms cancel, leaving 1, so the products reduce to 3x-2.
Adding then subtracting the same 4x leaves nothing behind, so the bracket is just 1.
6.EE.A.3Identify SubproblemsAdd the leftover +1
Bringing back the waiting term gives 3x-1, not 3x-2.
The stray +1 nudges the constant from -2 up to -1; it must not be forgotten.
6.EE.A.3Identify SubproblemsSubstitute x=4
Substituting x=4 gives 12-1 = 11, choice (D).
Once the expression is boiled down to 3x-1, one substitution finishes it.
6.EE.A.2Eliminate PossibilitiesWhen two terms share the same factor, pull it out front first — the mess often cancels and leaves a tiny expression to plug into.
- Pull out the shared factor
- Simplify the bracket
- Add the leftover +1
- Substitute x=4