Question : What is the value of $111\frac{1}{2} + 111\frac{1}{6} + 111\frac{1}{12} + 111\frac{1}{20} + 111\frac{1}{30}?$
Option 1: $111\frac{1}6$
Option 2: $111\frac{5}6$
Option 3: $555\frac{5}6$
Option 4: $555\frac{1}6$
Correct Answer: $555\frac{5}6$
Solution :
$111 \frac{1}{2} + 111 \frac{1}{6} + 111\frac{1}{12}+111 \frac{1}{20} + 111\frac{1}{30}$
= $5\times 111+( \frac{1}{2} + \frac{1}{6} + \frac{1}{12}+\frac{1}{20} + \frac{1}{30})$
= $555+( \frac{1}{1\times 2} + \frac{1}{2\times 3} + \frac{1}{3\times 4}+\frac{1}{4\times 5} + \frac{1}{5\times 6})$
= $555+(\frac{1}{1}-\frac{1}{2} + \frac{1}{2} -\frac{1}{3}+ \frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5} + \frac{1}{5}-\frac{1}{6})$
= $555 +(\frac{1}1-\frac{1}6)$
= $555\frac{5}6$
Hence, the correct answer is $555\frac{5}6$.
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