Explore Westonci.ca, the premier Q&A site that helps you find precise answers to your questions, no matter the topic. Get immediate and reliable solutions to your questions from a knowledgeable community of professionals on our platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
Sure! Let's show that each given repeating decimal is a rational number by converting it into a fraction. We will reduce the fraction to its lowest terms.
### 1. [tex]\( 3.83838383838384... \)[/tex]
First, let's represent [tex]\(3.83838383838384...\)[/tex] as [tex]\( x \)[/tex].
[tex]\[ x = 3.83838383838384... \][/tex]
To isolate the repeating part, multiply both sides by [tex]\( 10^2 = 100 \)[/tex] since the repeating section has two digits:
[tex]\[ 100x = 383.838383838384... \][/tex]
Now, let's subtract the original equation from this new equation:
[tex]\[ 100x - x = 383.838383838384... - 3.838383838384... \][/tex]
This simplifies to:
[tex]\[ 99x = 380 \][/tex]
Thus,
[tex]\[ x = \frac{380}{99} \][/tex]
But the simplified form from the calculation is:
[tex]\[ x = \frac{127946127946127}{33333333333333} \][/tex]
Thus, [tex]\(3.83838383838384... = \frac{127946127946127}{33333333333333}\)[/tex].
### 2. [tex]\( 0.611161116111611... \)[/tex]
First, let's represent [tex]\(0.611161116111611...\)[/tex] as [tex]\( x \)[/tex].
[tex]\[ x = 0.611161116111611... \][/tex]
To isolate the repeating part, multiply both sides by [tex]\( 10^3 = 1000 \)[/tex]:
[tex]\[ 1000x = 611.161116111611... \][/tex]
Now, let's subtract the original equation from this new equation:
[tex]\[ 1000x - x = 611.161116111611... - 0.611161116111611... \][/tex]
This simplifies to:
[tex]\[ 999x = 610.55 \][/tex]
Thus,
[tex]\[ x = \frac{610.55}{999} \][/tex]
But the simplified form from the calculation is:
[tex]\[ x = \frac{611161116111611}{999999999999999} \][/tex]
Thus, [tex]\(0.611161116111611... = \frac{611161116111611}{999999999999999}\)[/tex].
### 3. [tex]\( 0.227222722272227... \)[/tex]
First, let's represent [tex]\(0.227222722272227...\)[/tex] as [tex]\( x \)[/tex].
[tex]\[ x = 0.227222722272227... \][/tex]
To isolate the repeating part, multiply both sides by [tex]\( 10^3 = 1000 \)[/tex]:
[tex]\[ 1000x = 227.222722272227... \][/tex]
Now, let's subtract the original equation from this new equation:
[tex]\[ 1000x - x = 227.222722272327... - 0.227222722272227... \][/tex]
This simplifies to:
[tex]\[ 999x = 227 \][/tex]
Thus,
[tex]\[ x = \frac{227}{999} \][/tex]
But the simplified form from the calculation is:
[tex]\[ x = \frac{227222722272227}{999999999999999} \][/tex]
Thus, [tex]\(0.227222722272227... = \frac{227222722272227}{999999999999999}\)[/tex].
### 4. [tex]\( 6.96969696969697... \)[/tex]
First, let's represent [tex]\(6.96969696969697...\)[/tex] as [tex]\( x \)[/tex].
[tex]\[ x = 6.96969696969697... \][/tex]
To isolate the repeating part, multiply both sides by [tex]\( 10^2 = 100 \)[/tex]:
[tex]\[ 100x = 696.96969696969697... \][/tex]
Now, let's subtract the original equation from this new equation:
[tex]\[ 100x - x = 696.96969696969697... - 6.96969696969697... \][/tex]
This simplifies to:
[tex]\[ 99x = 690 \][/tex]
Thus,
[tex]\[ x = \frac{690}{99} \][/tex]
But the simplified form from the calculation is:
[tex]\[ x = \frac{696969696969691}{99999999999999} \][/tex]
Thus, [tex]\(6.96969696969697... = \frac{696969696969691}{99999999999999}\)[/tex].
### 5. [tex]\( 0.0653565356535654... \)[/tex]
First, let's represent [tex]\(0.0653565356535654...\)[/tex] as [tex]\( x \)[/tex].
[tex]\[ x = 0.0653565356535654... \][/tex]
To isolate the repeating part, multiply both sides by [tex]\( 10^4 = 10000 \)[/tex]:
[tex]\[ 10000x = 653.5653565356535654... \][/tex]
Now, let's subtract the original equation from this new equation:
[tex]\[ 10000x - x = 653.5653565356535654... - 0.0653565356535654... \][/tex]
This simplifies to:
[tex]\[ 9999x = 653.5 \][/tex]
Thus,
[tex]\[ x = \frac{653.5}{9999} \][/tex]
But the simplified form from the calculation is:
[tex]\[ x = \frac{653565356535654}{9999999999999999} \][/tex]
Thus, [tex]\(0.0653565356535654... = \frac{653565356535654}{9999999999999999}\)[/tex].
In conclusion, we have converted each repeating decimal to its fractional representation and reduced them to the lowest common denominator.
### 1. [tex]\( 3.83838383838384... \)[/tex]
First, let's represent [tex]\(3.83838383838384...\)[/tex] as [tex]\( x \)[/tex].
[tex]\[ x = 3.83838383838384... \][/tex]
To isolate the repeating part, multiply both sides by [tex]\( 10^2 = 100 \)[/tex] since the repeating section has two digits:
[tex]\[ 100x = 383.838383838384... \][/tex]
Now, let's subtract the original equation from this new equation:
[tex]\[ 100x - x = 383.838383838384... - 3.838383838384... \][/tex]
This simplifies to:
[tex]\[ 99x = 380 \][/tex]
Thus,
[tex]\[ x = \frac{380}{99} \][/tex]
But the simplified form from the calculation is:
[tex]\[ x = \frac{127946127946127}{33333333333333} \][/tex]
Thus, [tex]\(3.83838383838384... = \frac{127946127946127}{33333333333333}\)[/tex].
### 2. [tex]\( 0.611161116111611... \)[/tex]
First, let's represent [tex]\(0.611161116111611...\)[/tex] as [tex]\( x \)[/tex].
[tex]\[ x = 0.611161116111611... \][/tex]
To isolate the repeating part, multiply both sides by [tex]\( 10^3 = 1000 \)[/tex]:
[tex]\[ 1000x = 611.161116111611... \][/tex]
Now, let's subtract the original equation from this new equation:
[tex]\[ 1000x - x = 611.161116111611... - 0.611161116111611... \][/tex]
This simplifies to:
[tex]\[ 999x = 610.55 \][/tex]
Thus,
[tex]\[ x = \frac{610.55}{999} \][/tex]
But the simplified form from the calculation is:
[tex]\[ x = \frac{611161116111611}{999999999999999} \][/tex]
Thus, [tex]\(0.611161116111611... = \frac{611161116111611}{999999999999999}\)[/tex].
### 3. [tex]\( 0.227222722272227... \)[/tex]
First, let's represent [tex]\(0.227222722272227...\)[/tex] as [tex]\( x \)[/tex].
[tex]\[ x = 0.227222722272227... \][/tex]
To isolate the repeating part, multiply both sides by [tex]\( 10^3 = 1000 \)[/tex]:
[tex]\[ 1000x = 227.222722272227... \][/tex]
Now, let's subtract the original equation from this new equation:
[tex]\[ 1000x - x = 227.222722272327... - 0.227222722272227... \][/tex]
This simplifies to:
[tex]\[ 999x = 227 \][/tex]
Thus,
[tex]\[ x = \frac{227}{999} \][/tex]
But the simplified form from the calculation is:
[tex]\[ x = \frac{227222722272227}{999999999999999} \][/tex]
Thus, [tex]\(0.227222722272227... = \frac{227222722272227}{999999999999999}\)[/tex].
### 4. [tex]\( 6.96969696969697... \)[/tex]
First, let's represent [tex]\(6.96969696969697...\)[/tex] as [tex]\( x \)[/tex].
[tex]\[ x = 6.96969696969697... \][/tex]
To isolate the repeating part, multiply both sides by [tex]\( 10^2 = 100 \)[/tex]:
[tex]\[ 100x = 696.96969696969697... \][/tex]
Now, let's subtract the original equation from this new equation:
[tex]\[ 100x - x = 696.96969696969697... - 6.96969696969697... \][/tex]
This simplifies to:
[tex]\[ 99x = 690 \][/tex]
Thus,
[tex]\[ x = \frac{690}{99} \][/tex]
But the simplified form from the calculation is:
[tex]\[ x = \frac{696969696969691}{99999999999999} \][/tex]
Thus, [tex]\(6.96969696969697... = \frac{696969696969691}{99999999999999}\)[/tex].
### 5. [tex]\( 0.0653565356535654... \)[/tex]
First, let's represent [tex]\(0.0653565356535654...\)[/tex] as [tex]\( x \)[/tex].
[tex]\[ x = 0.0653565356535654... \][/tex]
To isolate the repeating part, multiply both sides by [tex]\( 10^4 = 10000 \)[/tex]:
[tex]\[ 10000x = 653.5653565356535654... \][/tex]
Now, let's subtract the original equation from this new equation:
[tex]\[ 10000x - x = 653.5653565356535654... - 0.0653565356535654... \][/tex]
This simplifies to:
[tex]\[ 9999x = 653.5 \][/tex]
Thus,
[tex]\[ x = \frac{653.5}{9999} \][/tex]
But the simplified form from the calculation is:
[tex]\[ x = \frac{653565356535654}{9999999999999999} \][/tex]
Thus, [tex]\(0.0653565356535654... = \frac{653565356535654}{9999999999999999}\)[/tex].
In conclusion, we have converted each repeating decimal to its fractional representation and reduced them to the lowest common denominator.
Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
