Question

Cho a ; b thuộc N* hãy so sánh:
\(\frac{a}{b}\) và \(\frac{a+2015}{b+2015}\)

Answer 1

\(\frac{a}{b}=\frac{a\left(b+2015\right)}{b\left(b+2015\right)}=\frac{ab+2015a}{b\left(b+2015\right)}\)

\(\frac{a+2015}{b+2015}=\frac{b\left(a+2015\right)}{b\left(b+2015\right)}=\frac{ab+2015b}{b\left(b+2015\right)}\)

TH1: a = b

=> ab+2015a = ab+2015b

=> \(\frac{a}{b}=\frac{a+2015}{b+2015}\)

TH2: a > b

=> ab+2015a > ab+2015b

=> \(\frac{a}{b}>\frac{a+2015}{b+2015}\)

TH3: a < b

=> ab+2015a < ab+2015b

=> \(\frac{a}{b}

Answer 2

Ta có:\(\frac{a}{b}=\frac{a.\left(b+2015\right)}{b.\left(b+2015\right)}=\frac{ab+a.2015}{b.\left(b+2015\right)}\)

         \(\frac{a+2015}{b+2015}=\frac{b.\left(a+2015\right)}{b.\left(b+2015\right)}=\frac{ab+b.2015}{b.\left(b+2015\right)}\)

Xét a>b=>a.2015>b.2015

=>\(\frac{ab+a.2015}{b.\left(b+2015\right)}>\frac{ab+b.2015}{b.\left(b+2015\right)}\)

=>\(\frac{a}{b}>\frac{a+2015}{b+2015}\)

Xét a=b=>a.2015=b.2015

=>\(\frac{ab+a.2015}{b.\left(b+2015\right)}=\frac{ab+b.2015}{b.\left(b+2015\right)}\)

=>\(\frac{a}{b}=\frac{a+2015}{b+2015}\)

Xét a<b=>a.2015<b.2015

=>\(\frac{ab+a.2015}{b.\left(b+2015\right)}