How do you write $0.\overline{63}$ as a fraction?
A. $\frac{4}{11}$
B. $\frac{7}{11}$
C. $\frac{7}{9}$
D. $\frac{7}{15}$
Answer (2)
Let x = 0. 63 .
Multiply by 100: 100 x = 63. 63 .
Subtract the equations: 99 x = 63 .
Solve and simplify: x = 99 63 = 11 7 .
The fraction is 11 7 .
Explanation
Understanding the Problem We are given the repeating decimal 0. 63 and we want to express it as a fraction. The repeating block is '63'. Let's call the decimal x , so x = 0. 63 = 0.636363...
Multiplying by 100 To convert the repeating decimal to a fraction, we multiply x by 100, since the repeating block has two digits. This gives us 100 x = 63.636363...
Subtracting the Equations Now, we subtract the original equation x = 0.636363... from the new equation 100 x = 63.636363... . This eliminates the repeating part of the decimal: 100 x − x = 63.636363... − 0.636363... , which simplifies to 99 x = 63 .
Solving for x Next, we solve for x by dividing both sides of the equation by 99: x = 99 63 .
Simplifying the Fraction Finally, we simplify the fraction by finding the greatest common divisor (GCD) of 63 and 99. The GCD of 63 and 99 is 9. We divide both the numerator and the denominator by 9: 99 ÷ 9 63 ÷ 9 = 11 7 . Therefore, 0. 63 = 11 7 .
Final Answer The repeating decimal 0. 63 can be written as the fraction 11 7 .
Examples
Repeating decimals often appear when dealing with measurements or ratios that aren't perfectly divisible. For example, when splitting a task that takes 2 hours and 21 minutes equally between 11 people, you need to divide 21 minutes by 11, which results in a repeating decimal. Converting this repeating decimal to a fraction helps in accurately representing and understanding the exact division of time or resources. This skill is useful in everyday situations like cooking, budgeting, and project management, where precise calculations are essential.
The repeating decimal 0. 63 can be expressed as the fraction 11 7 . This is done by defining the decimal, multiplying to eliminate the repeating part, and simplifying the resulting fraction. The correct choice is B. 11 7 .
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