How do you write \( 896.0 \overline{5} \) as a fraction?

To convert the repeating decimal \( 896.0 \overline{5} \) into a fraction, we can follow these steps: 1. **Define the repeating decimal**: Let \( x = 896.0 \overline{5} \). 2. **Eliminate the repeating part**: Since the repeating part is one digit long (the digit 5), we can multiply \( x \) by 10 to shift the decimal point one place to the right: \[ 10x = 8960.5 \overline{5} \] 3. **Set up the equation**: Now we have two equations: \[ x = 896.0 \overline{5} \] \[ 10x = 8960.5 \overline{5} \] 4. **Subtract the first equation from the second**: \[ 10x - x = 8960.5 \overline{5} - 896.0 \overline{5} \] This simplifies to: \[ 9x = 8960.5 - 896.0 \] 5. **Calculate the right side**: \[ 8960.5 - 896.0 = 8064.5 \] 6. **Now we have**: \[ 9x = 8064.5 \] 7. **Solve for \( x \)**: \[ x = \frac{8064.5}{9} \] 8. **Convert \( 8064.5 \) to a fraction**: \[ 8064.5 = \frac{80645}{10} \] Therefore, \[ x = \frac{80645}{10 \cdot 9} = \frac{80645}{90} \] 9. **Simplify the fraction**: We can simplify \( \frac{80645}{90} \) by finding the greatest common divisor (GCD) of 80645 and 90. Let's calculate the GCD and simplify the fraction. First, we can check if 80645 is divisible by 5 (since 90 is divisible by 5): - The last digit of 80645 is 5, so it is divisible by 5. Now, let's divide both the numerator and the denominator by 5: \[ \frac{80645 \div 5}{90 \div 5} = \frac{16129}{18} \] Thus, the fraction representation of \( 896.0 \overline{5} \) is: \[ \frac{16129}{18} \] This is the final answer.