Bunuel wrote:
Which of the following is the best approximation for y?
\(\frac{1}{2}- \frac{1}{3} + \frac{1}{6} - \frac{1}{10} + \frac{1}{12} - \frac{1}{14} + \frac{1}{16}\)
A. 0.1
B. 0.31
C. 0.35
D. 0.4
E. 0.6
In general, it pays to know the decimal equivalents for fractions \(\frac{1}{x}\) for \(x = 1, 2,...,10\) because you will see them everywhere. And once you've mastered those, you will have essentially mastered fractions like \(\frac{1}{14}\), because it's just \(\frac{1}{7}*\frac{1}{2}\), in other words, one of the aforementioned fractions divided by two. Btw, 1/7 = 0.1428.
As for this question, it asks for an approximation, and the answer choices are relatively spread out, so I just crunched this the old fashioned way. I hate subtraction, so I summed the positive terms, summed the negative terms, and then found the difference. Though I knew the exact decimal equivalents, to save time I just went out to two decimals, noting that the first sum will slightly understate the actual sum.
\(\frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{16} = 0.5 + 0.16 + 0.08 + 0.06 = 0.8 < actual\)
\(\frac{1}{3} + \frac{1}{10} + \frac{1}{14} = .033 + 0.1 + 0.07 = 0.5\)
\(0.8 - 0.5 = 0.3\), and since this is slightly smaller than the actual sum, the best answer choice is answer B.