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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 problem into a single ＂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
- Denote Decimal as : Let's denote the repeating decimal as : To convert this into a fraction, we will multiply by a power of that moves the repeating digits to the left of the decimal point. Since there are three repeating digits, we multiply by (which is ):
- Multiply by Power of : Now we have two equations:) ) Subtract the first equation from the second to eliminate the repeating decimals:
- Subtract Equations: Perform the subtraction:
- Solve for x: Now, solve for x by dividing both sides of the equation by :
- Check Fraction: Check the fraction to ensure it is in simplest form. Since and have no common factors other than , the fraction is already in its simplest form.
