$\frac{pz+qz+p+q}{pt+qt+p+q}=\frac{(pz+qz)+(p+q)}{(pt+qt)+(p+q)}=\frac{z(p+q)+(p+q)}{t(p+q)+(p+q)}=\frac{(p+q)(z+1)}{(p+q)(t+1)}=\frac{z+1}{t+1}$
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 $x=\frac{1}{2},y=\frac{1}{3},m=1256,n=4516$:
$\frac{1+\frac{1}{2}}{1+\frac{1}{3}}=\frac{1\frac{1}{2}}{1\frac{1}{3}}=\frac{\frac{3}{2}}{\frac{4}{3}}=\frac{3}{2}\ast \frac{3}{4}=\frac{9}{8}=1\frac{1}{8}$

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