What is the easiest method for adding and subtracting rational numbers

1. Make sure the denominators (bottom numbers) of the two fractions you're adding/subtracting are the same

2. Add/Subtract the numerators (the top numbers) and put the answer over the common denominator you've found

And if needed,

3. Simplify the fraction by dividing both top and bottom by the same amount, to keep the value of the fraction the same

 

So, the standard way of adding / subtracting numbers is to first identify a ``least common denominator". So, say I had:
##\frac{1}{2} + \frac{3}{7}##
Well, what is the smallest number that is a multiple of both 2 and 7? It's 14. So now, how do we write the original fractions with denominators of 14? Well, we multiply 2 by 7 to get 14, so we do the same to the numberator, 1. Likewise, we multiply 7 by 2 to get 14, so we multiply 3 by 2 to get 6. So we may write
##\frac{1}{2} + \frac{3}{7} = \frac{7}{14} + \frac{6}{14} = \frac{13}{14}##
Which doesn't reduce. So, very roughly speaking,
##\frac{a}{bc} + \frac{d}{ef} = \frac{(aef)+(bcd)}{bcef}##