Samwong wrote:

What is the 101st digit after the decimal point in the decimal representation of \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{37} ?

A) 0

B) 1

C) 5

D) 7

E) 8

Official Explanation:

\frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{37}=\frac{333}{999} + \frac{111}{999} + \frac{37}{999} + \frac{27}{999}=\frac{508}{999}=0.508508... .

102nd digit will be 8, thus 101st digit will be 0.

The correct answer is A.

It took me a while to see that 1/27 is equal to 37/999 and that 1/37 is equal to 27/999. What clue in the question indicates that you need to make that conversion? What is the shortcut for that conversion?

\frac{1}{3} =\frac{1*333}{3*333}=\frac{27}{999}.

\frac{1}{9} =\frac{1*111}{9*111}=\frac{27}{999}.

\frac{1}{27} =\frac{1*37}{27*37}=\frac{27}{999}.

\frac{1}{37} =\frac{1*27}{37*27}=\frac{27}{999}.

Following link might help for this problem:

math-number-theory-88376.html (check Converting Fractions chapter).

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