how to write a repeating decimal as a fraction

1 answer

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).

  1. Set up the equation: Let \( x = 0.666...\).

  2. Multiply by a power of 10: Since the repeating part is one digit long, multiply both sides of the equation by 10: \[ 10x = 6.666... \]

  3. Set up a second equation: Now you have two equations: \[ x = 0.666... \] \[ 10x = 6.666... \]

  4. Subtract the first equation from the second: \[ 10x - x = 6.666... - 0.666... \] \[ 9x = 6 \]

  5. Solve for \( x \): \[ x = \frac{6}{9} \]

  6. 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 \( 0.23333...\)).

  1. Set up the equation: Let \( x = 0.23333...\).

  2. Identify the non-repeating and repeating parts:

    • The non-repeating part is 0.2.
    • The repeating part is 0.03333...
  3. Multiply for the non-repeating part: Since the non-repeating part is one digit long, multiply by 10: \[ 10x = 2.3333... \]

  4. Multiply for the repeating part: Multiply by 100 because the repeating part (3) has one digit: \[ 100x = 23.3333... \]

  5. Subtract the two equations: \[ 100x - 10x = 23.3333... - 2.3333... \] \[ 90x = 21 \]

  6. Solve for \( x \): \[ x = \frac{21}{90} \]

  7. 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!