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]
