How do you write $0.\overline{8}$ as a fraction?
$\frac{8}{9}$
$\frac{3}{5}$
$\frac{5}{8}$
$\frac{2}{3}$
Answer (1)
Let x = 0. 8 .
Multiply by 10: 10 x = 8. 8 .
Subtract the equations: 9 x = 8 .
Solve for x : 9 8 .
Explanation
Understanding the Problem We are given the repeating decimal 0. 8 and asked to express it as a fraction. The overline indicates that the digit 8 repeats infinitely: 0. 8 = 0.8888 …
Setting up the Equations Let x = 0. 8 = 0.8888 … Then, we multiply both sides of the equation by 10: 10 x = 8.8888 …
Subtracting the Equations Now, we subtract the first equation from the second equation: 10 x − x = 8.8888 ⋯ − 0.8888 … This simplifies to 9 x = 8 .
Solving for x Finally, we solve for x : x = 9 8 .
Final Answer Therefore, the repeating decimal 0. 8 can be written as the fraction 9 8 .
Examples
Repeating decimals often appear when dealing with measurements or ratios that don't divide evenly. For example, if you divide a pizza into 9 slices and take 8 of those slices, each slice represents approximately 0. 1 of the pizza, and the 8 slices you took represent 0. 8 of the pizza. Converting this repeating decimal to a fraction, 9 8 , allows for more precise calculations and understanding of the proportion.
