To solve the problem of finding the sum [tex]\(a + b\)[/tex] where [tex]\(a = 0.\overline{3}\)[/tex] and [tex]\(b = 0.\overline{5}\)[/tex], we need to follow these detailed steps:
1. Convert Recurring Decimals to Fractions:
- First, convert [tex]\(a = 0.\overline{3}\)[/tex] to a fraction.
[tex]\[
0.\overline{3} = \frac{1}{3}
\][/tex]
- Similarly, convert [tex]\(b = 0.\overline{5}\)[/tex] to a fraction.
[tex]\[
0.\overline{5} = \frac{5}{9}
\][/tex]
2. Sum the Fractions:
- To add the fractions [tex]\(\frac{1}{3}\)[/tex] and [tex]\(\frac{5}{9}\)[/tex], we need a common denominator. The least common denominator of 3 and 9 is 9.
- Convert [tex]\(\frac{1}{3}\)[/tex] to a fraction with a denominator of 9:
[tex]\[
\frac{1}{3} = \frac{3}{9}
\][/tex]
- Now add the fractions:
[tex]\[
\frac{3}{9} + \frac{5}{9} = \frac{3 + 5}{9} = \frac{8}{9}
\][/tex]
3. Choose the Correct Option:
- The fraction representing [tex]\(a + b\)[/tex] is [tex]\(\frac{8}{9}\)[/tex].
Therefore, the correct answer is:
B. [tex]\(\frac{8}{9}\)[/tex]
