If a over b equals c over d then is it possible that a plus b equals c plus d and why?

Answer 1

Yes.

To show the conditions on a, b, c and d given that if a/b = c/d then a+b = c+d. Suppose b != d (and that both b and d are non-zero) then:

d = kb for some number k (!= 0), so

c/d = c/kb = (c/k)/b

so

a/b = (c/k)/b

=> a = c/k

=> c = ka

Thus:

c + d = ka + kb

= k(a + b)

Which means that c + d = a + b only if k = 1.

Thus if a/b = c/d then a + b = c + d only if a = c and b = d.

The condition on b and d both being non-zero prevents the possibility of division by zero. If either is zero, a division by zero will occur and at least one of the fractions is infinite.