Решения:

1. $$rac{a+b}{5}-rac{a}{5} = rac{a+b-a}{5} = rac{b}{5}$$
2. $$rac{m-n}{a}+rac{m+n}{a} = rac{m-n+m+n}{a} = rac{2m}{a}$$
3. $$rac{a^2}{a+b}-rac{b^2}{a+b} = rac{a^2-b^2}{a+b} = rac{(a-b)(a+b)}{a+b} = a-b$$
4. $$rac{c^2+d^2}{c-d}-rac{2cd}{c-d} = rac{c^2+d^2-2cd}{c-d} = rac{(c-d)^2}{c-d} = c-d$$
5. $$rac{x}{x^2-y^2}-rac{y}{x^2-y^2} = rac{x-y}{x^2-y^2} = rac{x-y}{(x-y)(x+y)} = rac{1}{x+y}$$
6. $$rac{a+b}{4}-rac{b}{4} = rac{a+b-b}{4} = rac{a}{4}$$
7. $$rac{x-y}{3c}+rac{x+y}{3c} = rac{x-y+x+y}{3c} = rac{2x}{3c}$$
8. $$rac{x^2}{x-y}-rac{y^2}{x-y} = rac{x^2-y^2}{x-y} = rac{(x-y)(x+y)}{x-y} = x+y$$
9. $$rac{m^2+n^2}{m+n}+rac{2mn}{m+n} = rac{m^2+n^2+2mn}{m+n} = rac{(m+n)^2}{m+n} = m+n$$
10. $$rac{a}{a^2-b^2}-rac{b}{a^2-b^2} = rac{a-b}{a^2-b^2} = rac{a-b}{(a-b)(a+b)} = rac{1}{a+b}$$
11. $$rac{x-y}{2}+rac{y}{2} = rac{x-y+y}{2} = rac{x}{2}$$
12. $$rac{c+d}{2a}-rac{c-d}{2a} = rac{c+d-(c-d)}{2a} = rac{c+d-c+d}{2a} = rac{2d}{2a} = rac{d}{a}$$
13. $$rac{a^2}{a-b}-rac{b^2}{a-b} = rac{a^2-b^2}{a-b} = rac{(a-b)(a+b)}{a-b} = a+b$$
14. $$rac{a^2+c^2}{a-c}-rac{2ac}{a-c} = rac{a^2+c^2-2ac}{a-c} = rac{(a-c)^2}{a-c} = a-c$$
15. $$rac{p}{p^2-q^2}+rac{q}{p^2-q^2} = rac{p+q}{p^2-q^2} = rac{p+q}{(p-q)(p+q)} = rac{1}{p-q}$$