Try Converting Into Fractions First

Let us convert these decimals into fractions instead, which will make the calculation much easier.

$0.\overline{3}$ represents the repeating decimal number, $0.33333\ldots = 0.3 + 0.03 + 0.003 + \cdots$ , which represents a geometric progression sum , with first term of $a = 0.3$ and common ratio of $r = 0.1$ , thus the sum of this geometric progression is $\dfrac a{1-r} = \dfrac{0.3}{1-0.1} = \dfrac{0.3}{0.9} = \dfrac{3}{9} = \dfrac13.$

This means that $0.\overline{3} = 0.33333\ldots = 0.3 + 0.03 + 0.003 + \cdots = \dfrac13$ .

Similarly, we can also show that $0.\overline{6} = 0.66666\ldots = 0.6 + 0.06 + 0.006 + \cdots = \dfrac{0.6}{1 -0.1} = \dfrac{0.6}{0.9} = \dfrac69 = \dfrac23 .$

In short, what we have obtained is that $0.\overline{3}=\dfrac 13$ and $0.\overline{6} =\dfrac 23$ .

And we just need to add these two fractions up:

$0.\overline{3}+0.\overline{6}=\dfrac 13 +\dfrac 23 =\dfrac{1+2}3 = \dfrac 33 =1$

And so, our answer is $\boxed1$ .

Footnote :

We can write $0.\overline{3}$ as $0.333 \ldots$ and $0.\overline{6}$ as $0.666 \ldots$ . Therefore adding these two values we get $0.333 \ldots + 0. 666 \ldots = 0.999 \ldots$ . But the provided answer is $1$ . To check $0.999 \ldots =1$ , you might want to visit is 0.999... = 1? .

To know how to convert repeating decimals into fractions, you may visit the wiki page: Converting repeating decimals into fractions .