How do you write $0.6 \overline{1}$ as a fraction?
A. $\frac{7}{18}$
B. $\frac{11}{18}$
C. $\frac{6}{11}$
D. $\frac{13}{16}$
Answer (1)
Let x = 0.6 1 .
Multiply by 10 and 100 to get 10 x = 6.1111... and 100 x = 61.1111... .
Subtract the equations: 100 x − 10 x = 61.1111... − 6.1111... , which simplifies to 90 x = 55 .
Solve for x and reduce the fraction: x = 90 55 = 18 11 .
Explanation
Understanding the Problem We are asked to convert the repeating decimal 0.6\[\]overline{1} into a fraction. This means the decimal is 0.611111... where the digit 1 repeats indefinitely.
Setting up Equations Let x = 0.61111... . To eliminate the repeating part, we can multiply by powers of 10. First, multiply by 10: 10 x = 6.1111... Next, multiply by 100: 100 x = 61.1111...
Eliminating the Repeating Decimal Now, subtract the first equation from the second equation to eliminate the repeating decimal part: 100 x − 10 x = 61.1111... − 6.1111... This simplifies to: 90 x = 55
Solving for x and Reducing the Fraction Solve for x by dividing both sides by 90: x = 90 55 Now, reduce the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 5: x = 90 ÷ 5 55 ÷ 5 = 18 11
Final Answer Therefore, the repeating decimal 0.6 1 can be written as the fraction 18 11 .
Examples
Repeating decimals often appear when dealing with measurements or ratios that aren't perfectly divisible. For example, if you divide a task into 18 equal parts and complete 11 of those parts, you've completed approximately 0.6111... of the task. Converting this repeating decimal to a fraction, 18 11 , allows for more precise calculations and better understanding of proportions in various real-world scenarios, such as resource allocation or project management.
