Question

Se considera fractiile echivalente a supra b si c supra d. Aratati ca:
A) a+b supra b= c+d supra d.
B) a supra a+b=c supra c+d
C) a-b supra b=c-d supra d,pentru a mai mare sau egal cu b si c mai mare sau egal d
D)a supra b-a=c supra d-c,pentru a mai mic decat b si c mai mic decat d
E)a la puterea n supra b la puterea n,pentru orice n apartine N
F) daca a+d=b+c,atunci a+n supra b+n = c+n supra d+n,pentru orice n apartine N de

Answer 1

$\frac{a}{b} = \frac{c}{d}$

A) $\frac{a+b}{b} = \frac{a}{b} + \frac{b}{b} =\frac{a}{b} + 1=\frac{c}{d} + 1 =\frac{c}{d} + \frac{d}{d} =\frac{c+d}{d}$

B) $\frac{a}{a+b} = \frac{b*a}{b(a+b)}= \frac{ \frac{a}{b}}{ \frac{a+b}{b}}=\frac{ \frac{a}{b}}{ \frac{a}{b}+1}=\frac{ \frac{c}{d}}{ \frac{c}{d}+1}= \frac{ \frac{c}{d}}{ \frac{c+d}{d}}= \frac{c*d}{d(c+d)}= \frac{c}{c+d}$

C) $\frac{a-b}{b} = \frac{a}{b} - \frac{b}{b} =\frac{a}{b} - 1=\frac{c}{d} - 1 =\frac{c}{d} - \frac{d}{d} =\frac{c-d}{d}$
 pentru a ≥ b si c≥ d


D) $\frac{a}{b-a} = \frac{b*a}{b(b-a)}= \frac{ \frac{a}{b}}{ \frac{b-a}{b}}=\frac{ \frac{a}{b}}{1- \frac{a}{b}}=\frac{ \frac{c}{d}}{1- \frac{c}{d}}= \frac{ \frac{c}{d}}{ \frac{d-c}{d}}= \frac{c*d}{d(d-c)}= \frac{c}{d-c}$
pentru a < b si c < d


E)$\frac{a^n}{b^n} = (\frac{a}{b})^n= (\frac{c}{d})^n= \frac{c^n}{d^n}$
pentru orice n ∈ N

F) a+d=b+c
$\frac{a}{b} = \frac{c}{d}$=>ad=bc

$\frac{a+n}{b+n} = \frac{a+n}{b+n}* \frac{d+n}{d+n}=\frac{ad+an+dn+n^2}{(b+n)(d+n)}=\frac{ad+(a+d)n+n^2}{(b+n)(d+n)}=$

$=\frac{bc+(b+c)n+n^2}{(b+n)(d+n)}=\frac{bc+bn+cn+n^2}{(b+n)(d+n)}=\frac{(b+n)(c+n)}{(b+n)(d+n)}=\frac{c+n}{d+n}$

pentru orice n ∈ N de