Often people who don't know how to add fractions do so as if the rule is
a
b
 + c
d
 = a+c
b+d
where a and b are coprime and c and d are coprime. Does this ever give the right answer?

As Andrei Lazanu from Tudor Vianu National College, Bucharest, Romania says "The only solution for the problem is a = -c and b = d so that the sum
a
b
 + c
d
is zero and
a+c
b+d
 =0,
so the relation is true".

Why is this the only possibility? The relation


a
b
 + c
d
 = a+c
b+d
holds if and only if
ad+bc
bd
 = a+c
b+d
that is
(ad+bc)(b+d) = (a+c)bd
which holds if and only if
ad2 + b2c = 0,
or equivalently
ad2=-b2c.
We know that any whole number can be written as the unique product of prime factors. As a and b are coprime and c and d are coprime ad2=-b2c is true only if a divides c and c divides a and they are of opposite sign, that is a = -c. Thus the original formula holds if and only if d = ±b and b+d ¹ 0 that is b=d so the formula holds if and only if
a/b = -c
d

and their sum is zero.