Question

[Trần Thu Phương]

Tìm a,b thuộc N sao cho :

\(\frac{11}{7}>\frac{a}{b}>\frac{23}{29}\) và \(8a-9b=31\)

Answer 1

[Trần Thùy Dương]

P/s : 8b-9a=31

Vì \(\frac{11}{7}>\frac{a}{b}>\frac{23}{29}\)

\(8b-9a=31\)(1)

\(\Rightarrow9a=8b-31\)

\(a=\frac{8b-31}{9}\)vì \(a\in N\)

\(8b-31\ge9\)

\(\Leftrightarrow8b\ge40\Leftrightarrow b\ge5\)

\(\Rightarrow\frac{11}{7}>\frac{8b-31}{9b}>\frac{23}{29}\)

\(\Leftrightarrow\frac{11}{7}>\frac{8}{9}>\frac{23}{29}\)

Mà  \(7>\frac{8}{9}-\frac{31}{9b}< \frac{11}{7}\)

     \(\frac{8}{9}-\frac{11}{7}< \frac{31}{9b}\)

      ...... \(\frac{-43}{63}< \frac{31}{9b}\)

\(\frac{-43}{7}< \frac{31}{b}\)

\(\Leftrightarrow-43b< 31.7\)

\(b>\frac{31.7}{-43}=\frac{-217}{43}\)

\(\Rightarrow b\in N\Leftrightarrow b>0\)

Mà \(\frac{8}{9}-\frac{31}{9b}>\frac{23}{29}\Leftrightarrow\frac{8}{9}-\frac{23}{29}>\frac{31}{9b}\)

\(\Leftrightarrow\frac{25}{261}>\frac{31}{9b}\Rightarrow25.9b>31.261\)

\(\Leftrightarrow b>\frac{31.261}{25.9}=\frac{899}{25}=35,9\)

Vậy \(5< b< \frac{899}{25}\)

\(\Rightarrow5< b< 35\)

Đến đây bạn lập bảng .