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Write the repeating decimal as a fraction.\newline0.333333330.33333333

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Full solution

Q. Write the repeating decimal as a fraction.\newline0.333333330.33333333
  1. Identify Repeating Pattern: Identify the repeating pattern in the decimal.\newlineThe repeating decimal is 0.33333333…0.33333333\ldots, where the digit 33 repeats indefinitely.
  2. Set Equal to Decimal: Let xx equal the repeating decimal.\newlineSo, we set x=0.33333333…x = 0.33333333\ldots
  3. Multiply by 1010: Multiply xx by 1010 to shift the decimal point to the right, which will help us set up an equation to solve for xx.10x=3.33333333…10x = 3.33333333\ldots
  4. Subtract to Eliminate Decimals: Subtract the original xx from 10x10x to eliminate the repeating decimals.\newline10x−x=3.33333333...−0.33333333...10x - x = 3.33333333... - 0.33333333...\newlineThis gives us 9x=39x = 3.
  5. Divide by 99: Divide both sides of the equation by 99 to solve for xx.x=39x = \frac{3}{9}
  6. Simplify Fraction: Simplify the fraction. 3/93 / 9 can be simplified to 1/31 / 3, since both the numerator and the denominator are divisible by 33.

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