To solve the problem of dividing [tex]\( 3. \overline{21} \)[/tex] by [tex]\( 0. \overline{5} \)[/tex], we need to first represent these repeating decimals as fractions.
1. Step 1: Convert the repeating decimals to fractions
- For [tex]\( 3. \overline{21} \)[/tex]:
[tex]\[
x = 3.21212121\ldots
\][/tex]
Let’s multiply both sides by 100 (because there are two repeating digits):
[tex]\[
100x = 321.21212121\ldots
\][/tex]
Now, subtract the original [tex]\( x \)[/tex]:
[tex]\[
100x - x = 321.21212121\ldots - 3.21212121\ldots
\][/tex]
This simplifies to:
[tex]\[
99x = 318
\][/tex]
So,
[tex]\[
x = \frac{318}{99} = \frac{106}{33} \quad (\text{after simplification})
\][/tex]
Therefore,
[tex]\[
3. \overline{21} = 3 + \frac{21}{99} = 3 + \frac{7}{33} = \frac{106}{33}
\][/tex]
- For [tex]\( 0. \overline{5} \)[/tex]:
[tex]\[
y = 0.555555\ldots
\][/tex]
Let’s multiply both sides by 10 (because there's one repeating digit):
[tex]\[
10y = 5.555555\ldots
\][/tex]
Now, subtract the original [tex]\( y \)[/tex]:
[tex]\[
10y - y = 5.555555\ldots - 0.555555\ldots
\][/tex]
This simplifies to:
[tex]\[
9y = 5
\][/tex]
So,
[tex]\[
y = \frac{5}{9}
\][/tex]
Therefore,
[tex]\[
0. \overline{5} = \frac{5}{9}
\][/tex]
2. Step 2: Perform the division
- We need to divide [tex]\( \frac{106}{33} \)[/tex] by [tex]\( \frac{5}{9} \)[/tex]
[tex]\[
\frac{106}{33} \div \frac{5}{9} = \frac{106}{33} \times \frac{9}{5} = \frac{106 \times 9}{33 \times 5} = \frac{954}{165}
\][/tex]
Simplify [tex]\( \frac{954}{165} \)[/tex]:
[tex]\[
\frac{954 \div 3}{165 \div 3} = \frac{318}{55}
\][/tex]
Finally, dividing [tex]\( 318 \)[/tex] by [tex]\( 55 \)[/tex]:
[tex]\[
318 \div 55 \approx 5.781818181818181
\][/tex]
Therefore, the value of [tex]\( 3. \overline{21} \div 0. \overline{5} \)[/tex] is approximately [tex]\( 5.781818181818181 \)[/tex].
