Sure, let's find the value of the division [tex]\( 3 . \overline{21} \div 0 . \overline{5} \)[/tex] step-by-step.
1. Convert the repeating decimals to fractions:
- [tex]\( 3 . \overline{21} \)[/tex]:
To convert [tex]\( 3 . \overline{21} \)[/tex] to a fraction, recognize that the repeating part "21" can be represented as a fraction:
[tex]\[
3 . \overline{21} = 3 + 0 . \overline{21}
\][/tex]
To convert [tex]\( 0 . \overline{21} \)[/tex] to a fraction, note that:
[tex]\[
0 . \overline{21} = \frac{21}{99} = \frac{7}{33}
\][/tex]
Thus:
[tex]\[
3 . \overline{21} = 3 + \frac{7}{33}
\][/tex]
- [tex]\( 0 . \overline{5} \)[/tex]:
To convert [tex]\( 0 . \overline{5} \)[/tex] to a fraction, recognize that:
[tex]\[
0 . \overline{5} = \frac{5}{9}
\][/tex]
2. Compute the value of the fractions:
- Calculate [tex]\( 3 + \frac{7}{33} \)[/tex]:
[tex]\[
3 + \frac{7}{33} \approx 3 . 212121212121\ldots \approx 3 . 2121
\][/tex]
- Recognize [tex]\( \frac{5}{9} \)[/tex]:
[tex]\[
\frac{5}{9} \approx 0 . 555555555555 \ldots \approx 0 . 5556
\][/tex]
3. Perform the division of the two obtained values:
- Calculate [tex]\( \frac{3.212121212121212}{0.5555555555555556} \)[/tex]:
[tex]\[
3.212121212121212 \div 0.5555555555555556 \approx 5 . 781818181818\ldots \approx 5 . 7818
\][/tex]
Therefore, the value of [tex]\( 3 . \overline{21} \div 0 . \overline{5} \)[/tex] is approximately [tex]\( 5.7818 \)[/tex].
