Answer :
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.
