AMC 10 · 2003 · #1
Grade 6 arithmeticPick an answer.
Adding up seven signed numbers on top and seven on the bottom is slow and easy to slip on. Tool #5 (Look for a Pattern) spots that the two lists are the same sequence scaled differently — top is 2,4,6,… and bottom is 3,6,9,… — so the top is a fixed fraction of the bottom, term for term. That fixed fraction is the whole answer, and no negative-number addition is needed. Tool #3 (Eliminate Possibilities) confirms the sign: top and bottom carry identical sign patterns, so the quotient is positive, killing the negative choices (A) and (B).
Line up top and bottom
Read the lists side by side: the top is multiples of 2, the bottom multiples of 3, with the same signs.
Before adding anything, notice the top and bottom are the same pattern built from 2s and 3s.
4.OA.B.4Look For A PatternEach top term is 2/3 of its bottom term
Every matched pair gives the same fraction 2/3, negatives included.
The same ratio 2:3 shows up in every matched pair of terms.
4.NF.A.1Look For A PatternPull the 2/3 out of the whole top
Factoring it out leaves the denominator itself, which cancels, giving 2/3, choice (C).
Multiplying every term of a sum by 2/3 multiplies the whole sum by 2/3, so the ratio is 2/3.
Multiplying every term of the bottom sum by two thirds multiplies the whole sum by two thirds.
▸ Why?
A factor shared by every term can be lifted out of the whole sum at once.
▸ Why?
Scaling every part by the same factor scales the total by that factor, so the ratio survives the addition.
When every top term is the same fraction of the bottom term beneath it, that fraction is the answer for the whole thing — no need to add anything up.
- Line up top and bottom
- Each top term is 2/3 of its bottom term
- Pull the 2/3 out of the whole top