what is the value of 0.\\overline{4}+0.\\overline{2}? write your answer as a fraction in lowest terms.

Answer:

$\frac{2}{3}$

Explanation:

Step1: Convert repeating decimals to fractions

Let (x = 0.\overline{4}), then (10x=4.\overline{4}). Subtract (x) from (10x): (10x - x=4.\overline{4}-0.\overline{4}), so (9x = 4), and (x=\frac{4}{9}). Let (y = 0.\overline{2}), then (10y = 2.\overline{2}). Subtract (y) from (10y): (10y - y=2.\overline{2}-0.\overline{2}), so (9y=2), and (y = \frac{2}{9}).

Step2: Add the fractions

(0.\overline{4}+0.\overline{2}=\frac{4}{9}+\frac{2}{9}). Since the denominators are the same, we add the numerators: (\frac{4 + 2}{9}=\frac{6}{9}).

Step3: Simplify the fraction

Divide both the numerator and denominator by their greatest - common divisor. The GCD of 6 and 9 is 3. So (\frac{6\div3}{9\div3}=\frac{2}{3}).