Repeat#3

The general approach is as follows: Let the given decimal number $x$ consist of

• whole part $w$ ;

• non-repeating part of $d$ digits, with integer value $n$ ;

• repeating part of $e$ digits, with integer value $r$ .

Then the fraction is equal to $x = w + \frac{(10^e - 1)n + r}{10^d(10^e - 1)}.$ The fraction must be further be reduced by dividing out common factors (typically powers of 2 and 5, or 3, 11, 13, 37.)

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Application

In this problem, $w = 42$ , $d = 2$ , $n = 34$ , $e = 2$ , $r = 78$ , so that $x = 42 + \frac{99\cdot 34 + 78}{100\cdot 99} = 42\frac{3444}{9900}.$ After dividing out the common factor $2^2\cdot 3 = 12$ , we find $x = 42\frac{287}{825},$ making the final answer $42 + 287 + 825 = \boxed{1154}$ .