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Write the repeating decimal as a fraction.
Full solution
Q. Write the repeating decimal as a fraction.
- Rephrase Problem: Let's first rephrase the ＂Convert the repeating decimal to a fraction.＂
- Identify Repeating Pattern: Identify the repeating pattern in the decimal. The digits repeat indefinitely, so we can express the decimal as
- Define Variable: Let be the value of the repeating decimal:
- Shift Decimal Point: Multiply by to shift the decimal point three places to the right, since the repeating block is three digits long:
- Subtract Decimals: Subtract the original from the to eliminate the repeating decimals:
- Solve for x: Perform the subtraction:
- Simplify Fraction: Divide both sides of the equation by to solve for :
- Find GCD: Simplify the fraction by finding the greatest common divisor (GCD) of and . The GCD of and is .
- Divide by GCD: Divide both the numerator and the denominator by the GCD to simplify the fraction:
