Which of the following is the same as (2-4+6-8+10-12+14)/(3-6+9-12+15-18+21)?

2003 AMC 10B Problem 1

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Which of the following is the same as

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

-1

-\dfrac{2}{3}

\dfrac{2}{3}

1

\dfrac{14}{3}

Factor a

2

out of the numerator and a

3

out of the denominator

Both share the factor

1-2+3-4+5-6+7

Factoring gives

\dfrac{2(1-2+3-4+5-6+7)}{3(1-2+3-4+5-6+7)}.

The identical alternating sums cancel, leaving

\dfrac{2}{3}.

Thus, the correct answer is C.