Question

Se considera fractiile echivalente a'b si c\d. Aratati ca a)a+b\b=c+d\d b)a\a+b=c\c+d c)a-b\b=c-d\d, a>b si c>d d)a\b-a=c\d-c, pentru a

Answer 1

Stiim ca $\frac{a}{b} = \frac{c}{d}$ => a*d = b*c ( * inseamna ori )
a) $\frac{a+b}{b} = \frac{c+d}{d}$ | *b*d (inmultim totul cu b*d)
=? ad + bd = bc + bd => ad-bc+bd-bd=0 si cum ad=bc => ad-bc=0 si bd=bd => bd-bd = 0 => 0+0 = 0 => $\frac{a}{b} = \frac{c}{d}$ Adevarat

b)$\frac{a}{a+b} = \frac{c}{c+d}$ (facem produs de mezi si extremi si egalam)
=> a*(c+d)=c*(a+b) => ac+ad=ac+bc => ac-ac+ad-bc=0 cum ac=ac => ac-ac=0 si stiim ca ad=bc => ad-bc=0 => 0+0=0 => $\frac{a}{a+b} = \frac{c}{c+d}$ Adevarat

c) $\frac{a-b}{b} = \frac{c-d}{d}$ (facem produs de mezi si extremi si egalam)
=> (a-b)*d=(c-d)*b => ad-bd=bc-bd => ad-bc+bd-bd=0 cum bd=bd => bd-bd=0 si stiim ca ad=bc => ad-bc=0 => 0+0=0 => $\frac{a-b}{b} = \frac{c-d}{d}$ Adevarat pt a>b si c>d

d) $\frac{a}{b-a} = \frac{c}{d-c}$ (facem produs de mezi si extremi si egalam)
=>a*(d-c)=c*(b-a) => ad-ac=bc-ac => ad-bc+ac-ac=0 cum ac=ac => ac-ac=0 si stiim ca ad=bc => ad-bc=0 => $\frac{a}{b-a} = \frac{c}{d-c}$ Adevarat