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

2003 AMC 12B Problem 1

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Which of the following is the same as

\frac{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 common constant out of the numerator and the denominator

The two parenthesized sums

1 - 2 + 3 - \cdots + 7

are identical

Factor

2

from the numerator and

3

from the denominator:

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

The equal parenthesized sums cancel, leaving

\dfrac{2}{3}.

Thus, the correct answer is C.