How do you convert 29.2987 (987 being repeated) to a fraction?

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How do you convert 29.2987 (987 being repeated) to a fraction?

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Answer 1 · 2016-03-13T18:56:00

$29.2 ¯ ¯¯¯¯ ¯ 987 = \frac{19513}{666}$ $29.2bar(987) = 19513/666$

Explanation:

Multiply by $10 (1000 - 1) = 10000 - 10$ $10(1000-1) = 10000-10$ to get an integer:

$(10000 - 10) \cdot 29.2 ¯ ¯¯¯¯ ¯ 987$ $(10000-10)*29.2bar(987)$

$= 292987 . ¯ ¯¯¯¯ ¯ 987 - 292 . ¯ ¯¯¯¯ ¯ 987$ $=292987.bar(987) - 292.bar(987)$

$= 292695$ $=292695$

The factor $10$ $10$ is to shift the original number so that the repeating portion starts just after the decimal point. The factor $(1000 - 1)$ $(1000-1)$ is to further shift the number $3$ $3$ places and subtract the original to cancel out the pattern that repeats every $3$ $3$ digits.

Then dividing both ends by $(10000 - 10)$ $(10000-10)$ we find:

$29.2bar(987) = 292695/(10000-10) = 292695/9990 = (color(red)(cancel(color(black)(15)))*19513) / (color(red)(cancel(color(black)(15)))*666) = 19513/666$

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How do you convert 29.2987 (987 being repeated) to a fraction?

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