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?

Answer 1

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 \( 100 \).

Answer 2

wrong, try again, it isnt 82/100

Answer 3

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!

Answer 4

82/99 is still wrong

Answer 5

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!