If $a=0 . \overline{3}$ and $b=0 . \overline{5}$, what is the value of $a+b$ ?
The problem asks for the sum of two repeating decimals, \(a = 0.\overline{3}\) and \(b = 0.\overline{5}\). A repeating decimal of the form \(0.\overline{d}\) can be expressed as the fraction \(\frac{d}{9}\). Applying this rule, we can convert both \(a\) and \(b\) into their fractional forms to perform the addition.
First, we express \(a = 0.\overline{3}\) as a fraction. According to the conversion rule, \(a = \frac{3}{9}\). Next, we express \(b = 0.\overline{5}\) as a fraction. Similarly, \(b = \frac{5}{9}\). Now, we add the two fractions together:
\[a + b = \frac{3}{9} + \frac{5}{9}\]
Since the denominators are the same, we simply add the numerators:
\[a + b = \frac{3 + 5}{9} = \frac{8}{9}\]
The resulting value is \(\frac{8}{9}\). Comparing this result to the given options, we find it matches the second option.
The conversion of repeating decimals into fractions is a fundamental skill in finance and scientific modeling, where precision is crucial. For example, if a financial account yields an interest rate that is a repeating decimal, like \(0.333...\%\) per month, converting it to the fraction \(\frac{1}{3}\%\) makes calculating annual returns or compound interest much more accurate and easier to handle in complex formulas. By representing these values as fractions, we avoid rounding errors that could accumulate over time.