What happens when you add $x$ to $\frac{1}{3}x$? I am dealing with an equation that requires me to add $x$ to $\frac{1}{3}x$:
$x + \frac{1}{3}x$ = ??
I know this might be simple to any of you on this site, because you are all asking questions with symbols I have never seen, but this is confusing to me.
I guess one way of thinking about this is - You are adding $x$ to $yx$, right?
Or just adding another $\frac{1}{3}$? 
The complete equation that I am working on is [- don't laugh at its simplicity ;)]:
$\frac{2}{3}b + 5 = 20 - b$
So, when worked out... I got:
$\frac{2}{3}b + b = 15$
And this is where I get stuck. 

$$
x + \frac{1}{3} x = 1 \cdot x + \frac{1}{3} x = \left( 1 + \frac{1}{3} \right) x
= \frac{4}{3} x
$$
Your full equation is slighty different, but the same rules (distributivity) apply:
$$
15 = \frac{2}{3} b + b = \left( \frac{2}{3} + 1 \right) b
= \frac{5}{3} b \iff \\
b = \frac{3}{5} \cdot 15 = 9
$$
