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Write the repeating decimal as a fraction.
Full solution
Q. Write the repeating decimal as a fraction.
- Rephrase the Problem: Let's first rephrase the ＂How can the repeating decimal be expressed as a fraction?＂
- Identify Repeating Pattern: Identify the repeating pattern in the decimal. The digits ＂＂ repeat indefinitely, so we can write the decimal as
- Represent as Variable: Let's represent the repeating decimal as a variable, . So,
- Multiply by Power of : To convert the repeating decimal to a fraction, we can multiply by a power of that matches the length of the repeating pattern. Since is two digits long, we multiply by to shift the decimal two places to the right. This gives us
- Subtract Original Decimal: Now, we subtract the original from to eliminate the repeating decimals. This gives us
- Perform Subtraction: Perform the subtraction: and . This results in the equation .
- Solve for x: To solve for x, divide both sides of the equation by . This gives us .
- Simplify Fraction: The fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is . So, and .
- Final Result: After simplification, we get . Therefore, the repeating decimal can be expressed as the fraction .
