Question
To convert the repeating decimal 0.82 to a rational number, you would set x equal to the repeating decimal of 0.82 and then multiply both sides of the equation by what number?
Answers
Answer
wrong, try again, it isnt 82/100
Answer
82/99 is still wrong
Answered by
GPT-4o mini
To convert the repeating decimal \( 0.82\overline{82} \) (where "82" is the repeating part) to a rational number, you can set:
\[
x = 0.828282\ldots
\]
Next, since the repeating block has two digits ("82"), you multiply both sides of the equation by \( 100 \) (which moves the decimal point two places to the right):
\[
100x = 82.828282\ldots
\]
Now, you can set up an equation to eliminate the repeating decimal:
\[
100x = 82.828282\ldots
\]
\[
x = 0.828282\ldots
\]
By subtracting the second equation from the first, you have:
\[
100x - x = 82.828282\ldots - 0.828282\ldots
\]
This simplifies to:
\[
99x = 82
\]
Now, solve for \( x \):
\[
x = \frac{82}{99}
\]
So, to summarize, you set \( x = 0.82\overline{82} \) and then multiplied both sides of the equation by \( 100 \).
\[
x = 0.828282\ldots
\]
Next, since the repeating block has two digits ("82"), you multiply both sides of the equation by \( 100 \) (which moves the decimal point two places to the right):
\[
100x = 82.828282\ldots
\]
Now, you can set up an equation to eliminate the repeating decimal:
\[
100x = 82.828282\ldots
\]
\[
x = 0.828282\ldots
\]
By subtracting the second equation from the first, you have:
\[
100x - x = 82.828282\ldots - 0.828282\ldots
\]
This simplifies to:
\[
99x = 82
\]
Now, solve for \( x \):
\[
x = \frac{82}{99}
\]
So, to summarize, you set \( x = 0.82\overline{82} \) and then multiplied both sides of the equation by \( 100 \).
Answered by
GPT-4o mini
I apologize for the misunderstanding in your original statement. Let's handle the decimal \(0.82\overline{82}\) correctly this time.
To convert the repeating decimal \(0.828282...\) to a rational number, we can set:
\[
x = 0.828282\ldots
\]
Since the repeating part "82" has two digits, we multiply both sides by \(100\) to shift the decimal point two places to the right:
\[
100x = 82.828282\ldots
\]
Now, we have:
\[
100x = 82 + x
\]
Next, we subtract \(x\) from both sides:
\[
100x - x = 82
\]
This simplifies to:
\[
99x = 82
\]
Now, we solve for \(x\):
\[
x = \frac{82}{99}
\]
Thus, the correct rational representation of the repeating decimal \(0.828282...\) is indeed \(\frac{82}{99}\). Thank you for your patience, and I hope this clarifies the conversion!
To convert the repeating decimal \(0.828282...\) to a rational number, we can set:
\[
x = 0.828282\ldots
\]
Since the repeating part "82" has two digits, we multiply both sides by \(100\) to shift the decimal point two places to the right:
\[
100x = 82.828282\ldots
\]
Now, we have:
\[
100x = 82 + x
\]
Next, we subtract \(x\) from both sides:
\[
100x - x = 82
\]
This simplifies to:
\[
99x = 82
\]
Now, we solve for \(x\):
\[
x = \frac{82}{99}
\]
Thus, the correct rational representation of the repeating decimal \(0.828282...\) is indeed \(\frac{82}{99}\). Thank you for your patience, and I hope this clarifies the conversion!
Answered by
GPT-4o mini
I apologize for the confusion in the previous messages. Let’s clarify and correct this step by step.
To convert the repeating decimal \(0.82\overline{82}\) to a rational number:
1. **Set up the equation**:
\[
x = 0.828282\ldots
\]
2. **Identify the repeating part**: The repeating part "82" has two digits, so multiply both sides by \(100\):
\[
100x = 82.828282\ldots
\]
3. **Now, we subtract the first equation from the second equation**:
\[
100x - x = 82.828282\ldots - 0.828282\ldots
\]
This results in:
\[
99x = 82
\]
4. **Now solve for \(x\)**:
\[
x = \frac{82}{99}
\]
However, if you're referring to \(0.82\) (which is actually \(0.82\) with no repeating part and can be directly converted to a fraction), that would be:
1. Set \(x = 0.82\).
2. Multiply both sides by \(100\):
\[
100x = 82
\]
3. Solve for \(x\):
\[
x = \frac{82}{100} = \frac{41}{50}
\]
If it's indeed \(0.82\overline{82}\) that you're asking about, I've correctly stated that it is \(\frac{82}{99}\). If you meant just \(0.82\), the answer is \(\frac{41}{50}\). Please specify if you have a different repeating pattern or clarification in mind, and I'll be glad to assist further!
To convert the repeating decimal \(0.82\overline{82}\) to a rational number:
1. **Set up the equation**:
\[
x = 0.828282\ldots
\]
2. **Identify the repeating part**: The repeating part "82" has two digits, so multiply both sides by \(100\):
\[
100x = 82.828282\ldots
\]
3. **Now, we subtract the first equation from the second equation**:
\[
100x - x = 82.828282\ldots - 0.828282\ldots
\]
This results in:
\[
99x = 82
\]
4. **Now solve for \(x\)**:
\[
x = \frac{82}{99}
\]
However, if you're referring to \(0.82\) (which is actually \(0.82\) with no repeating part and can be directly converted to a fraction), that would be:
1. Set \(x = 0.82\).
2. Multiply both sides by \(100\):
\[
100x = 82
\]
3. Solve for \(x\):
\[
x = \frac{82}{100} = \frac{41}{50}
\]
If it's indeed \(0.82\overline{82}\) that you're asking about, I've correctly stated that it is \(\frac{82}{99}\). If you meant just \(0.82\), the answer is \(\frac{41}{50}\). Please specify if you have a different repeating pattern or clarification in mind, and I'll be glad to assist further!
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.