Competition · AMC preparation · step 4 of 4

AMC 10 · 2003B · #1

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

fraction-arithmeticratio-proportionpattern-recognition identify-subproblems ↑ Prerequisites: fraction-arithmetic

📏 Medium solution 💡 1 insight

Problem

The fraction

\frac{2-4+6-8+10-12+14}{3-6+9-12+15-18+21}

lines up seven terms on top with seven terms on the bottom, and both rows carry the same alternating signs +,-,+,-,+,-,+. Which single value is this fraction equal to?

Pick an answer.

(A)

-1

(B)

$-\frac{2}{3}$

(C)

$\frac{2}{3}$

(D)

(E)

$\frac{14}{3}$

AMC 10 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Look for a Pattern

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).

1STEP 1

Line up top and bottom

Read the rows position by position: the top runs 2, 4, 6, …, 14, the bottom 3, 6, 9, …, 21, and the signs match.

top: 2,4,6,8,10,12,14 bottom: 3,6,9,12,15,18,21

2STEP 2

Each top term is 2/3 of its bottom term

Every matched pair reduces the same way: $\frac{4}{6}$ , $\frac{6}{9}$ , …, $\frac{14}{21}$ all equal $\frac{2}{3}$ , negative terms included.

2/3=4/6=6/9=8/12=10/15=12/18=14/21

3STEP 3

Pull the 2/3 out of the whole top

Pull $\frac{2}{3}$ out of the top and the leftover sum is exactly the bottom, so it cancels: the fraction collapses to $\frac{2}{3}$ , choice (C).

(2/3(3-6+9-12+15-18+21))/(3-6+9-12+15-18+21)=2/3 → (C)

Answer

2/3

The top and bottom have identical sign patterns, so the fraction must come out positive — that alone rules out (A) -1 and (B) -2/3. Because the top numbers (2s) are smaller than the matching bottom numbers (3s), the fraction is less than 1, ruling out (D) 1 and (E) 14/3. The only positive value below 1 left is (C) 2/3, matching the term-by-term ratio.

💡Key takeaway

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

A parent dashboard for the family lives at sensimlab.com.

Line up top and bottom

Each top term is 2/3 of its bottom term

Pull the 2/3 out of the whole top

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