What are the necessary conditions on integers $a, b$ for the sum of the fractions below to be reduced? 
*

*$a/3+b/6$  

*$a/20+b/12$  

*$a/10+b/12$
For 1., $2a+b$ would have to be a multiple of 2 or 3. $2a$ must be even, and for $2a+b$ to be divisible 3, then $b$ would have to be odd. I can show other obvious relations between $b$ and $a$, but I have trouble proving the properties of $b$ and $a$ specifically, as well as generalizing it to different cases as in the ones in 2 and 3.  


Notate the first fraction's denominator as $c$ and the second fraction's denominator as $d$. Then set $g = \gcd(c, d)$. If $g > 1$ and, then if $\gcd(a + b, g) > 1$, you'll be able to reduce the sum of the two fractions.
To work through your second example: $g = \gcd(20, 12) = 4$. Then suppose $a = 9$ and $b = 7$. Then $$\frac{9}{20} + \frac{7}{12} = \frac{27}{60} + \frac{35}{60} = \frac{62}{60} = \frac{31}{30}.$$
