(A) 0
(B) \(\frac{2}{15}\)
(C) \(\frac{2}{5}\)
(D) \(\frac{9}{20}\)
(E) \(\frac{5}{6}\)
This is a direct calculation. The key in these kinds of problems is to take 10 – 20 second pause and find the most efficient way to complete the calculation. In general, the most straightforward approach is NOT optimal. Here the obvious approach is to find the least common multiple of 2, 3, 4, 5, 6, and 20, and use that as a common denominator. Of course this would work and it’s not that difficult, but a close look reveals that we can group the fractions in such a way that we already have obvious common denominators:
\(\frac{1}{3} + \frac{1}{2} – \frac{5}{6} + \frac{1}{5} + \frac{1}{4} – \frac{9}{20} = \)
\((\frac{1}{3} + \frac{1}{2} – \frac{5}{6}) + (\frac{1}{5} + \frac{1}{4} – \frac{9}{20}) = \)
We’ll use a common denominator of 6 for the first group, and 20 for the second group:
\((\frac{2}{6} + \frac{3}{6} – \frac{5}{6}) + (\frac{4}{20} + \frac{5}{20} – \frac{9}{20}) = \)
\(0 + 0 = 0\)
The correct answer is A.