To convert a repeating decimal into a fraction, follow these steps. Let's use the example of the repeating decimal \( 0.666...\) (which repeats the digit 6).
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Set up the equation: Let \( x = 0.666...\).
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Multiply by a power of 10: Since the repeating part is one digit long, multiply both sides of the equation by 10: \[ 10x = 6.666... \]
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Set up a second equation: Now you have two equations: \[ x = 0.666... \] \[ 10x = 6.666... \]
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Subtract the first equation from the second: \[ 10x - x = 6.666... - 0.666... \] \[ 9x = 6 \]
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Solve for \( x \): \[ x = \frac{6}{9} \]
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Simplify the fraction: Divide both the numerator and the denominator by their greatest common divisor (3): \[ x = \frac{2}{3} \]
So, \( 0.666... \) as a fraction is \( \frac{2}{3} \).
Another Example
Let's consider another repeating decimal, \( 0.2\overline{3} \) (which is \( 0.23333...\)).
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Set up the equation: Let \( x = 0.23333...\).
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Identify the non-repeating and repeating parts:
- The non-repeating part is 0.2.
- The repeating part is 0.03333...
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Multiply for the non-repeating part: Since the non-repeating part is one digit long, multiply by 10: \[ 10x = 2.3333... \]
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Multiply for the repeating part: Multiply by 100 because the repeating part (3) has one digit: \[ 100x = 23.3333... \]
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Subtract the two equations: \[ 100x - 10x = 23.3333... - 2.3333... \] \[ 90x = 21 \]
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Solve for \( x \): \[ x = \frac{21}{90} \]
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Simplify the fraction: Divide both the numerator and the denominator by 3: \[ x = \frac{7}{30} \]
So, \( 0.2\overline{3} \) as a fraction is \( \frac{7}{30} \).
This method can be applied to any repeating decimal!