AMC 10 · 2003 · #1
Grade 3 countingPick an answer.
Adding two lists of 2003 numbers and subtracting is huge and error-prone. Tool #5 (Look for a Pattern) spots that the two lists run in lockstep: 2 sits right after 1, 4 right after 3, and so on. So instead of two giant sums, pair the lists position by position and look at one pair at a time. Tool #7 (Identify Subproblems) turns the whole thing into 2003 tiny subtractions, each of which is the same. Tool #3 (Eliminate Possibilities) then checks the size of the answer against the choices.
Pair the two lists position by position
Line the lists up and subtract in matching positions rather than adding each in full.
Every even number sits exactly one step past the odd number in the same position, so compare them one pair at a time.
3.OA.D.9Look For A PatternEvery pair differs by exactly 1
Each even number is exactly one more than its partner, so every pair gives 1.
Consecutive whole numbers are always 1 apart, so each even-minus-odd pair is just 1.
Each even number sits exactly one step past the odd number in the same position, so every pair differs by 1.
▸ Why?
Consecutive whole numbers march up by a fixed step of one, so neighbouring terms are always a single unit apart.
▸ Why?
Each term of one list has exactly one partner in the other, so the two lists can be compared pair by pair with none left over.
Add up the 2003 ones
Adding 2003 copies of one gives 2003, choice (D).
Adding one 2003 times is the same as multiplying, and 2003 × 1 is just 2003.
3.OA.A.1Identify SubproblemsInstead of adding two long lists and subtracting, pair them up: each even number is just 1 past its odd partner, so 2003 pairs give a difference of 2003.
- Pair the two lists position by position
- Every pair differs by exactly 1
- Add up the 2003 ones