$(\frac{a + x}{a} - \frac{x - y}{x}) * \frac{a^2}{x^2 + ay} = \frac{x(a + x) - a(x - y)}{ax} * \frac{a^2}{x^2 + ay} = \frac{ax + x^2 - ax + ay}{x} * \frac{a}{x^2 + ay} = \frac{x^2 + ay}{x} * \frac{a}{x^2 + ay} = \frac{a}{x}$

$(\frac{a}{a - 1} + 1) : (1 - \frac{a}{a - 1}) = \frac{a + a - 1}{a - 1} : \frac{a - 1 - a}{a - 1} = \frac{2a - 1}{a - 1} : \frac{-1}{a - 1} = \frac{2a - 1}{a - 1} * \frac{a - 1}{-1} = -(2a - 1) = 1 - 2a$

$(m - \frac{1}{1 + m}) * \frac{m + 1}{1 - m - m^2} = \frac{m(1 + m) - 1}{1 + m} * \frac{m + 1}{1 - m - m^2} = \frac{m + m^2 - 1}{1 + m} * \frac{m + 1}{1 - m - m^2} = \frac{-(1 - m - m^2)}{1 - m - m^2} = -1$

$(a + \frac{a^2}{c}) : (b + \frac{bc}{a}) = \frac{ac + a^2}{c} : \frac{ab + bc}{a} = \frac{a(c + a)}{c} * \frac{a}{b(a + c)} = \frac{a^2}{bc}$

$(\frac{a + x}{x} - \frac{2x}{x - a}) : \frac{a^2 + x^2}{x - a} = \frac{(x - a)(x + a) - 2x^2}{x(x - a)} * \frac{x - a}{a^2 + x^2} = \frac{x^2 - a^2 - 2x^2}{x(a + x^2)} = \frac{-a^2 - x^2}{x(a^2 + x^2)} = \frac{-(a^2 + x^2)}{x(a^2 + x^2)} = -\frac{1}{x}$

$(\frac{x^2 + 1}{2x - 1} - \frac{x}{2}) * \frac{1 - 2x}{x + 2} = \frac{2(x^2 + 1) - x(2x - 1)}{2(2x - 1)} * \frac{1 - 2x}{x + 2} = \frac{2x^2 + 2 - 2x^2 + x}{2(2x - 1)} * \frac{-1(2x - 1)}{x + 2} = \frac{(2 + x) * (-1)}{2(x + 2)} = -\frac{1}{2}$

$(\frac{n}{n + x} - \frac{n}{n - x}) : (\frac{n}{n - x} + \frac{n}{n + x}) = \frac{n(n - x) - n(n + x)}{(n + x)(n - x)} : \frac{n(n + x) + n(n - x)}{(n - x)(n + x)} = \frac{n^2 - nx - n^2 - nx}{(n + x)(n - x)} * \frac{(n - x)(n + x)}{n^2 + nx + n^2 - nx} = \frac{-2nx}{2n^2} = -\frac{x}{n}$

$\frac{3}{5x} - \frac{3}{x + y} * (\frac{x + y}{5x} - x - y) = \frac{3}{5x} - \frac{3}{x + y} * \frac{x + y - 5x^2 - 5xy}{5x} = \frac{3}{5x} - \frac{3(x + y - 5x^2 - 5xy)}{5x(x + y)} = \frac{3(x + y) - 3(x + y - 5x^2 - 5xy)}{5x(x + y)} = \frac{3x + 3y - 3x - 3y + 15x^2 + 15xy}{5x(x + y)} = \frac{15x^2 + 15xy}{5x(x + y)} = \frac{15(x + y)}{5x(x + y)} = 3$