Question
Convert 0.83¯ into a fraction.
6/5
5/6
6/4
4/6
6/5
5/6
6/4
4/6
Answers
GPT-4o mini
To convert the repeating decimal \( 0.83\overline{83} \) into a fraction, we can follow these steps:
Let \( x = 0.83\overline{83} \).
To eliminate the repeating part, we can multiply by \( 1000 \) (since the repeating part "83" consists of two digits) to shift the decimal:
\[
1000x = 83.83\overline{83}
\]
Now we can also multiply by \( 10 \) to express the decimal directly:
\[
10x = 8.3\overline{83}
\]
Now we can subtract the second equation from the first one:
\[
1000x - 10x = 83.83\overline{83} - 8.3\overline{83}
\]
This simplifies to:
\[
990x = 75.5
\]
Next, let's convert \( 75.5 \) to a fraction. Since \( 75.5 \) can be written as \( \frac{755}{10} \):
\[
990x = \frac{755}{10}
\]
To isolate \( x \), we can rearrange:
\[
x = \frac{755}{10 \cdot 990}
\]
We simplify this:
\[
x = \frac{755}{9900}
\]
Next, we must simplify \( \frac{755}{9900} \). The GCD of \( 755 \) and \( 9900 \) can be calculated or found, and simplifying it gives:
\[
x = \frac{151}{1980}
\]
The fraction \( x = 0.83\overline{83} \) becomes \( \frac{151}{1980} \), which cannot be further simplified.
However, if we're looking for the answer among the fractions provided ( \( \frac{6}{5}, \frac{5}{6}, \frac{6}{4}, \frac{4}{6} \)), none of these are equivalent to our calculated fraction of \( x = 0.83\overline{83} \).
Therefore, your provided options do not contain the correct fraction equivalent of \( 0.83\overline{83} \). If you have more options or more context, please let me know!
Let \( x = 0.83\overline{83} \).
To eliminate the repeating part, we can multiply by \( 1000 \) (since the repeating part "83" consists of two digits) to shift the decimal:
\[
1000x = 83.83\overline{83}
\]
Now we can also multiply by \( 10 \) to express the decimal directly:
\[
10x = 8.3\overline{83}
\]
Now we can subtract the second equation from the first one:
\[
1000x - 10x = 83.83\overline{83} - 8.3\overline{83}
\]
This simplifies to:
\[
990x = 75.5
\]
Next, let's convert \( 75.5 \) to a fraction. Since \( 75.5 \) can be written as \( \frac{755}{10} \):
\[
990x = \frac{755}{10}
\]
To isolate \( x \), we can rearrange:
\[
x = \frac{755}{10 \cdot 990}
\]
We simplify this:
\[
x = \frac{755}{9900}
\]
Next, we must simplify \( \frac{755}{9900} \). The GCD of \( 755 \) and \( 9900 \) can be calculated or found, and simplifying it gives:
\[
x = \frac{151}{1980}
\]
The fraction \( x = 0.83\overline{83} \) becomes \( \frac{151}{1980} \), which cannot be further simplified.
However, if we're looking for the answer among the fractions provided ( \( \frac{6}{5}, \frac{5}{6}, \frac{6}{4}, \frac{4}{6} \)), none of these are equivalent to our calculated fraction of \( x = 0.83\overline{83} \).
Therefore, your provided options do not contain the correct fraction equivalent of \( 0.83\overline{83} \). If you have more options or more context, please let me know!