AMC 10 · 2003 · #1
Grade 6 arithmeticWhich of the following is the same as
3−6+9−12+15−18+212−4+6−8+10−12+14?
Pick an answer.
AMC 10 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Simplify the single fraction $\dfrac{2-4+6-8+10-12+14}{3-6+9-12+15-18+21}$ and match it to one of the given values.
Givens: Numerator: $2-4+6-8+10-12+14$; Denominator: $3-6+9-12+15-18+21$; Answer choices: (A) $-1$, (B) $-\dfrac{2}{3}$, (C) $\dfrac{2}{3}$, (D) $1$, (E) $\dfrac{14}{3}$
Unknowns: The single value the whole fraction is equal to
Understand
Restated: Simplify the single fraction $\dfrac{2-4+6-8+10-12+14}{3-6+9-12+15-18+21}$ and match it to one of the given values.
Givens: Numerator: $2-4+6-8+10-12+14$; Denominator: $3-6+9-12+15-18+21$; Answer choices: (A) $-1$, (B) $-\dfrac{2}{3}$, (C) $\dfrac{2}{3}$, (D) $1$, (E) $\dfrac{14}{3}$
Plan
Primary tool: #5 Look for a Pattern
Secondary: #7 Identify Subproblems, #3 Eliminate Possibilities
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 $2,4,6,\dots$ and bottom is $3,6,9,\dots$ — 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).
Execute — Answer: C
4.OA.B.4 Step 1 Line up top and bottom
- Read the two lists side by side, position by position.
- The top is $2, 4, 6, 8, 10, 12, 14$ with signs $+,-,+,-,+,-,+$; the bottom is $3, 6, 9, 12, 15, 18, 21$ with the exact same signs.
- The top numbers are the multiples of $2$; the bottom numbers are the multiples of $3$.
- Same seven positions, same sign each time.
💡 Before adding anything, notice the top and bottom are the same pattern built from $2$s and $3$s.
4.NF.A.1 Step 2 Each top term is 2/3 of its bottom term
- Compare each matched pair as a fraction.
- $\dfrac{2}{3}$, then $\dfrac{4}{6}=\dfrac{2}{3}$, then $\dfrac{6}{9}=\dfrac{2}{3}$, and so on all the way to $\dfrac{14}{21}=\dfrac{2}{3}$.
- Every single top number is exactly $\dfrac{2}{3}$ of the bottom number sitting beneath it — the negative terms too, since $-4=\dfrac{2}{3}\cdot(-6)$.
💡 The same ratio $2:3$ shows up in every matched pair of terms.
6.EE.A.3 Step 3 Pull the 2/3 out of the whole top
- Since every term on top is $\dfrac{2}{3}$ of the matching term on the bottom, factor that $\dfrac{2}{3}$ out of the entire numerator.
- The leftover sum in the parentheses is exactly the denominator, so it cancels — you never have to add the seven signed numbers at all.
- The whole fraction collapses to $\dfrac{2}{3}$, which is choice (C).
💡 Multiplying every term of a sum by $\tfrac{2}{3}$ multiplies the whole sum by $\tfrac{2}{3}$, so the ratio is $\tfrac{2}{3}$.
4.OA.B.4 Read the two lists side by side, position by position. The top is $2, 4, 6, 8, 1 4.NF.A.1 Compare each matched pair as a fraction. $\dfrac{2}{3}$, then $\dfrac{4}{6}=\dfr 6.EE.A.3 Since every term on top is $\dfrac{2}{3}$ of the matching term on the bottom, fa Review
Reasonableness: The top and bottom have identical sign patterns, so the fraction must come out positive — that alone rules out (A) $-1$ and (B) $-\tfrac{2}{3}$. Because the top numbers ($2$s) are smaller than the matching bottom numbers ($3$s), the fraction is less than $1$, ruling out (D) $1$ and (E) $\tfrac{14}{3}$. The only positive value below $1$ left is (C) $\tfrac{2}{3}$, matching the term-by-term ratio.
Alternative: If you would rather just add, work out each sum directly: the top is $2-4+6-8+10-12+14=8$ and the bottom is $3-6+9-12+15-18+21=12$. Then $\dfrac{8}{12}=\dfrac{2}{3}$, landing on (C) again.
CCSS standards used (min grade 6)
4.OA.B.4Gain familiarity with factors and multiples (Recognizing the top numbers as multiples of $2$ and the bottom numbers as multiples of $3$, lined up position for position.)4.NF.A.1Explain fraction equivalence a/b = (n×a)/(n×b) (Seeing that each matched pair reduces to the same value, $\tfrac{4}{6}=\tfrac{6}{9}=\dots=\tfrac{2}{3}$.)6.EE.A.3Apply the properties of operations to generate equivalent expressions (Factoring $\tfrac{2}{3}$ out of the whole numerator so the denominator cancels and the fraction equals $\tfrac{2}{3}$.)
⭐ 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.
⭐ 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.
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