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2003 AMC 12B Problem 1

Problem 1 of 25EasierAlgebra

Which of the following is the same as 2−4+6−8+10−12+143−6+9−12+15−18+21?\frac{2 - 4 + 6 - 8 + 10 - 12 + 14}{3 - 6 + 9 - 12 + 15 - 18 + 21}?

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Solution

Factor 22 from the numerator and 33 from the denominator: 2(1−2+3−4+5−6+7)3(1−2+3−4+5−6+7).\frac{2(1 - 2 + 3 - 4 + 5 - 6 + 7)}{3(1 - 2 + 3 - 4 + 5 - 6 + 7)}. The equal parenthesized sums cancel, leaving 23.\dfrac{2}{3}. Thus, the correct answer is C.

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Concepts: factoring · fraction

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.