Welcome to Westonci.ca, where you can find answers to all your questions from a community of experienced professionals. Join our platform to connect with experts ready to provide precise answers to your questions in different areas. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.

What is the equivalent fraction of the non-terminating and repeating decimal [tex]\(0.\overline{72}\)[/tex]?

A. [tex]\(\frac{11}{8}\)[/tex]

B. [tex]\(\frac{11}{9}\)[/tex]

C. [tex]\(\frac{9}{11}\)[/tex]

D. [tex]\(\frac{8}{11}\)[/tex]


Sagot :

To determine the fraction equivalent to the repeating decimal [tex]\(0.72727272\ldots\)[/tex], let's go through the steps to convert it to a fraction.

1. Let [tex]\( x \)[/tex] be the repeating decimal. So, [tex]\( x = 0.72727272\ldots \)[/tex].

2. To eliminate the repeating part, multiply [tex]\( x \)[/tex] by 100 (since the repeating block is two digits long):
[tex]\[ 100x = 72.72727272\ldots \][/tex]

3. Now, we have two equations:
[tex]\[ x = 0.72727272\ldots \quad \text{(Equation 1)} \][/tex]
[tex]\[ 100x = 72.72727272\ldots \quad \text{(Equation 2)} \][/tex]

4. Subtract Equation 1 from Equation 2 to eliminate the repeating decimals:
[tex]\[ 100x - x = 72.72727272\ldots - 0.72727272\ldots \][/tex]
[tex]\[ 99x = 72 \][/tex]

5. Solve for [tex]\( x \)[/tex] by dividing both sides by 99:
[tex]\[ x = \frac{72}{99} \][/tex]

6. Now, we simplify the fraction [tex]\(\frac{72}{99}\)[/tex]:
- Find the greatest common divisor (GCD) of 72 and 99. The GCD is 9.
- Divide the numerator and the denominator by their GCD:
[tex]\[ \frac{72 \div 9}{99 \div 9} = \frac{8}{11} \][/tex]

Therefore, the equivalent fraction of the non-terminating and repeating decimal [tex]\(0.72727272\ldots\)[/tex] is [tex]\(\frac{8}{11}\)[/tex].

So, the correct answer is [tex]\(\frac{8}{11}\)[/tex].
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. We hope this was helpful. Please come back whenever you need more information or answers to your queries. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.