(0.\overline72) (x = 0.727272\ldots) (100x = 72.727272\ldots) Subtract: (100x - x = 72 \Rightarrow 99x = 72 \Rightarrow x = \frac7299 = \frac811) Case 3: Mixed Repeating Decimal Let (x) be the decimal. Multiply by a power of 10 to move the decimal point to just before the repeating block, and by another power to include the whole repeating part. Subtract.
(0.\overlineabc\ldots = \frac\textrepeating block10^n - 1) where (n) = number of digits in the block. exercicios sobre fração geratriz
The decimal (0.333\ldots) (or (0.\overline3)) is generated by the fraction (\frac13). Therefore, (\frac13) is the geratrix fraction of (0.\overline3). 2. Types of Decimals | Type | Description | Example | |------|-------------|---------| | Terminating decimal | Ends after finite digits | (0.25 = \frac14) | | Pure repeating decimal | All digits after the decimal point repeat | (0.\overline142857 = \frac17) | | Mixed repeating decimal | Some non-repeating digits followed by a repeating block | (0.1\overline6 = \frac16) | exercicios sobre fração geratriz
(0.3\overline18) (x = 0.3181818\ldots) Multiply by 10: (10x = 3.181818\ldots) (now pure repeating: (3.\overline18)) (1000x = 318.181818\ldots) (since (10x \times 100 = 1000x)) Wait — better method: Let (x = 0.3\overline18) Multiply by 10: (10x = 3.\overline18) (pure repeating) Now (10x = 3 + 0.\overline18) (0.\overline18 = \frac1899 = \frac211) So (10x = 3 + \frac211 = \frac33+211 = \frac3511) Thus (x = \frac35110 = \frac722) exercicios sobre fração geratriz