Question

So sánh hai phân số sau :  \(\frac{2009.2009+2008}{2009.2009+2009}\)và \(\frac{2009.2009+2009}{2009.2009+2010}\)

 

Answer 1

\(\frac{2009.2009+2008}{2009.2009+2009}=\frac{2009.2009+2009}{2009.2009+2009}-\frac{1}{2009.2009+2009}=1-\frac{1}{2009.2009+2009}\)

\(\frac{2009.2009+2009}{2009.2009+2010}=\frac{2009.2009+2010}{2009.2009+2010}-\frac{1}{2009.2009+2010}=1-\frac{1}{2009.2009+2010}\)

\(\text{Vì }2009.2009+2009\frac{1}{2009.2009+2010}\)

\(\text{Hay }1-\frac{1}{2009.2009+2009}

Answer 2

\(\frac{2009.2009+2009}{2009.2009+2010}=\frac{2009.2009+2008+1}{2009.2009+2009+1}\)

Đặt 2009.2009+2008 là a; 2009.2009+2009 là b. Ta so sánh \(\frac{a}{b}\)và \(\frac{a+1}{b+1}\)

Qui đồng mẫu số 2 phân số trên

\(\frac{a}{b}=\frac{a\left(b+1\right)}{b\left(b+1\right)}=\frac{a.b+a}{b.\left(b+1\right)}\)

\(\frac{a+1}{b+1}=\frac{\left(a+1\right).b}{b\left(b+1\right)}=\frac{a.b+b}{b\left(b+1\right)}\)

Vì 2008 < 2009

=> 2009.2009+2008 < 2009.2009+2009

=> a < b

=> a.b+a < a.b+b

=> \(\frac{a.b+a}{b.\left(b+1\right)}

Answer 3

\(\frac{2009.2009+2008}{2009.2009+2009}=\frac{2008}{2009}\)

\(1-\frac{2008}{2009}=\frac{1}{2009}\)

\(\frac{2009.2009+2009}{2009.2009+2010}=\frac{2009}{2010}\)

\(1-\frac{2009}{2010}=\frac{1}{2010}\)