- Convert repeating decimals to fractions by setting up linear equations: multiply the decimal by powers of \(10\) to align the repeating segments and subtract to eliminate the infinite tail.
- Simplify each resulting fraction by dividing the numerator and denominator by their greatest common divisor.
- Results:
A\(2.\overline{6} = \boxed{\frac{8}{3}}\)
B\(0.\overline{14} = \boxed{\frac{14}{99}}\)
C\(0.7\overline{16} = \boxed{\frac{709}{990}}\)
D\(1.32\overline{12} = \boxed{\frac{218}{165}}\)
E\(-0.532\overline{13} = \boxed{-\frac{52681}{99000}}\)
Explanation
- Conversion Method for Repeating Decimals
To convert a repeating decimal into a fraction, we identify the non-repeating part and the repeating part. For a decimal \(d\), we shift the decimal point to isolate the repeating portion and use subtraction to eliminate the infinite repetition. For example, if \(d = 0.\overline{r}\) where \(r\) has \(n\) digits, then \(10^n d - d = r\), giving \(d = \frac{r}{10^n - 1}\). For mixed decimals with a non-repeating part, we shift the decimal point so that the repeating part begins immediately after it, allowing for the same subtraction process. All resulting fractions are simplified by dividing both the numerator and the denominator by their greatest common divisor.
- Step-by-Step Conversion and Simplification
A\(2.\overline{6}\): Let \(d = 2.666...\). Multiplying by \(10\) gives \(10d = 26.666...\). Subtracting \(d\) from \(10d\) yields \(9d = 24\), so \(d = \frac{24}{9}\). Dividing both terms by \(3\), we get \(d = \frac{8}{3}\).
B\(0.\overline{14}\): Let \(d = 0.141414...\). Multiplying by \(100\) gives \(100d = 14.1414...\). Subtracting \(d\) from \(100d\) yields \(99d = 14\), so \(d = \frac{14}{99}\).
C\(0.7\overline{16}\): Let \(d = 0.71616...\). Multiplying by \(10\) gives \(10d = 7.1616...\) and multiplying by \(1000\) gives \(1000d = 716.1616...\). Subtracting the two expressions gives \(990d = 709\), so \(d = \frac{709}{990}\).
D\(1.32\overline{12}\): Let \(d = 1.321212...\). Multiplying by \(100\) gives \(100d = 132.1212...\) and multiplying by \(10000\) gives \(10000d = 13212.1212...\). Subtracting gives \(9900d = 13080\), so \(d = \frac{13080}{9900} = \frac{1308}{990} = \frac{218}{165}\).
E\(-0.532\overline{13}\): Let \(d = -0.5321313...\). Multiplying by \(1000\) gives \(1000d = -532.1313...\) and multiplying by \(100000\) gives \(100000d = -53213.1313...\). Subtracting gives \(99000d = -52681\), so \(d = -\frac{52681}{99000}\).
- Final Answer Summary
The simplified fractional forms are:
A\(\frac{8}{3}\)
B\(\frac{14}{99}\)
C\(\frac{709}{990}\)
D\(\frac{218}{165}\)
E\(-\frac{52681}{99000}\)
These represent the simplest fractional forms for each given repeating decimal.
Examples
Repeating decimals frequently appear in financial modeling and scientific measurements where precision is key. For example, if a company reports a consistent growth rate that repeats as \(0.0\overline{3}\) annually, representing this as the fraction \(\frac{1}{30}\) allows for much easier and more precise calculations when predicting future revenue growth, as it avoids rounding errors inherent in using decimal approximations. Converting repeating decimals into fractions is a standard technique to maintain exact values throughout algebraic and financial analysis.