Forgot my basic math... So a younger (college student) asked me for my help to solve a basic math question, and to my surprise, I've forgot some basic math rules, rendering me unable to answer the problem.
According to online math-generators the answer should'nt be what I'm getting.. And I am unable to follow the auto-generated steps... I hope this forum can be of help, as I actually really do like maths, (but obviously have been neglecting it)
The problem is a reduction problem, and is as follows:
$\left(\frac{1}{2}\right) \times\left(\frac{2}{2a}\right)+\frac{\left(\frac{2a}{6}\right)}{\left(\frac{6}{5}\right)}$
What I do is this:

$$\left(\frac{1}{2}\right) \times\left(\frac{2}{2a}\right)+\frac{\left(\frac{2a}{6}\right)}{\left(\frac{6}{5}\right)} $$

$$=\left(\frac{2}{6a}\right)+\frac{\left(\frac{2a}{6}\right)}{\left(\frac{6}{5}\right)} $$

$$=\left(\frac{1}{3}\right)+\frac{\left(\frac{1a}{3}\right)}{\left(\frac{6}{5}\right)} $$ (I fear this may be wrong?)

$$=\frac{\left(\frac{4}{4a}\right)}{\left(\frac{6}{5}\right)} $$

$$=\left(\frac{4\times5}{4\times6a}\right) $$

$$=\frac{20}{24a} $$

$$=\frac{5}{6a}$$
 A: $$(\frac{1}{2}) \times(\frac{2}{2a})+\frac{(\frac{2a}{6})}{(\frac{6}{5})}$$
$$=(\frac{1}{2a})+(\frac{2a}{6}) \times (\frac{5}{6})$$
$$=(\frac{1}{2a})+(\frac{5a}{18})$$
$$=\left(\frac{9+5a^2}{18a}\right)$$
A: You did two careless mistakes and one thing I simply don't understand.
1) 2nd line:  $\frac 1 2 \times \frac 2 {2a} = \frac 2 {4a}$ not $\frac 2 {6a}$
2) 3rd line:  "$\frac 2 {6a} = \frac 1 3$.  The $a$ has disappeared into the ether.
so you should have $(\frac{1}{2a})+\frac{(\frac{1a}{3})}{(\frac{6}{5})}$ which is reducible.
What I don't understand is how you jumped from $ (\frac{1}{3})+\frac{(\frac{1a}{3})}{(\frac{6}{5})} $(sic) to $ \frac{(\frac{4}{4a})}{(\frac{6}{5})} $.
A: Small mistake from step 1 to step 2. Notice that: 
$$ \left(\frac{1}{2}\right) \times \left(\frac{2}{2a}\right)  = \left( \frac{1\times 2}{2\times 2a} \right) = \left(\frac{2}{4a} \right) = \left( \frac{1}{2a} \right).$$
You will then end up with 
$$ \left( \frac{1}{2a} \right) + \frac{\left(\frac{2a}{6}\right)}{\left(\frac{6}{5}\right)}.$$
You can simplify $\frac{2a}{6}$ to $\frac{a}{3}$, leaving you with
$$ \left( \frac{1}{2a} \right) + \frac{\left(\frac{a}{3}\right)}{\left(\frac{6}{5}\right)}.$$
From here, simplify $ \frac{\left(\frac{a}{3}\right)}{\left(\frac{6}{5}\right)}$ using the fact that 
$$ \frac{\left(\frac{A}{B}\right)}{\left(\frac{C}{D}\right)} = \left(\frac{A}{B}\right) \times \left(\frac{D}{C}\right).$$
Once you do that, find a common denominator to add the fractions.  
