$\frac{pz + qz + p + q}{pt + qt + p + q} = \frac{(pz + qz) + (p + q)}{(pt + qt) + (p + q)} = \frac{z(p + q) + (p + q)}{t(p + q) + (p + q)} = \frac{(p + q)(z + 1)}{(p + q)(t + 1)} = \frac{z + 1}{t + 1}$
при p = 2,5, q = 0,5, z = 25, t = 12:
$\frac{z + 1}{t + 1} = \frac{25 + 1}{12 + 1} = \frac{26}{13} = 2$

$\frac{c - d + c^2 - d^2}{c - d + c^2 - 2cd + d^2} = \frac{(c - d) + (c^2 - d^2)}{(c - d) + (c^2 - 2cd + d^2)} = \frac{(c - d) + (c - d)(c + d)}{(c - d) + (c - d)^2} = \frac{(c - d)(1 + c + d)}{(c - d)(1 + c - d)} = \frac{1 + c + d}{1 + c - d}$
при c = 8, d = −2:
$\frac{1 + 8 + (-2)}{1 + 8 - (-2)} = \frac{9 - 2}{9 + 2} = \frac{7}{11}$

$\frac{m - n + mx - nx}{m - n + my - ny} = \frac{(m - n) + (mx - nx)}{(m - n) + (my - ny)} = \frac{(m - n) + x(m - n)}{(m - n) + y(m - n)} = \frac{(m - n)(1 + x)}{(m - n)(1 + y)} = \frac{1 + x}{1 + y}$
при $x = \frac{1}{2}, y = \frac{1}{3}, m = 1256, n = 4516$:
$\frac{1 + \frac{1}{2}}{1 + \frac{1}{3}} = \frac{1\frac{1}{2}}{1\frac{1}{3}} = \frac{\frac{3}{2}}{\frac{4}{3}} = \frac{3}{2} * \frac{3}{4} = \frac{9}{8} = 1\frac{1}{8}$

$\frac{a + b + a^2 - b^2}{a - b + a^2 - 2ab + b^2} = \frac{(a + b) + (a^2 - b^2)}{(a - b) + (a^2 - 2b + b^2)} = \frac{(a + b) + (a - b)(a + b)}{(a - b) + (a - b)^2} = \frac{(a + b)(1 + a - b)}{(a - b)(1 + a - b)} = \frac{a + b}{a - b}$
при a = 3, b = 5:
$\frac{3 + 5}{3 - 5} = \frac{8}{-2} = -4$