Question
To convert 0.82¯¯¯¯¯
to a rational number, you would set x
equal to 0.82¯¯¯¯¯
and then multiply both sides of the equation by what number?
to a rational number, you would set x
equal to 0.82¯¯¯¯¯
and then multiply both sides of the equation by what number?
Answers
Answer
wrong
Answered by
GPT-4o mini
To convert the repeating decimal \(0.82\overline{2}\) (which means \(0.822222...\)) to a rational number, you can follow these steps:
1. Let \( x = 0.82\overline{2} \).
2. Notice that the part of the decimal that repeats is a single digit (the '2'). To eliminate the repeating part, you multiply both sides of the equation by \(10\):
\[
10x = 8.22222...
\]
3. Next, since the repeating part is one digit long, you will multiply again by \(10\) to align the decimals:
\[
100x = 82.22222...
\]
4. Now, you will set up an equation by subtracting:
\[
100x - 10x = 82.22222... - 8.22222...
\]
which simplifies to:
\[
90x = 74
\]
5. Finally, divide both sides by \(90\):
\[
x = \frac{74}{90}
\]
Thus, the repeating decimal \(0.82\overline{2}\) can be expressed as the rational number \(\frac{74}{90}\), which can also be simplified further if needed.
In summary, you should multiply both sides of the equation by \(10\) to begin the conversion process. For rounding to another step to align the repetition, \(10\) appears twice, but the crucial first step is multiplying by \(10\).
1. Let \( x = 0.82\overline{2} \).
2. Notice that the part of the decimal that repeats is a single digit (the '2'). To eliminate the repeating part, you multiply both sides of the equation by \(10\):
\[
10x = 8.22222...
\]
3. Next, since the repeating part is one digit long, you will multiply again by \(10\) to align the decimals:
\[
100x = 82.22222...
\]
4. Now, you will set up an equation by subtracting:
\[
100x - 10x = 82.22222... - 8.22222...
\]
which simplifies to:
\[
90x = 74
\]
5. Finally, divide both sides by \(90\):
\[
x = \frac{74}{90}
\]
Thus, the repeating decimal \(0.82\overline{2}\) can be expressed as the rational number \(\frac{74}{90}\), which can also be simplified further if needed.
In summary, you should multiply both sides of the equation by \(10\) to begin the conversion process. For rounding to another step to align the repetition, \(10\) appears twice, but the crucial first step is multiplying by \(10\).