Resposta :
a) [tex]1,\overline{7}=1+\dfrac{7}{9}=\dfrac{16}{9}[/tex] e [tex]0,\overline{1}=\dfrac{1}{9}[/tex]
Desta maneira, [tex]\dfrac{1,\overline{7}}{0,\overline{1}}=\dfrac{\frac{16}{9}}{\frac{1}{9}}=\dfrac{16}{9}\cdot\dfrac{9}{1}=16[/tex]
b) Observe que:
[tex]\text{A}=\dfrac{2}{3}-1,\overline{4}[/tex]
[tex]\text{A}=\dfrac{2}{3}-1+\dfrac{4}{9}=\dfrac{2}{3}-\dfrac{13}{9}=\dfrac{6-13}{9}=\dfrac{-7}{9}[/tex]
Analogamente, temos:
[tex]\text{B}=0,7-0,\overline{7}[/tex]
[tex]\text{B}=\dfrac{7}{10}-\dfrac{7}{9}[/tex]
[tex]\text{B}=\dfrac{63-70}{9}=\dfrac{-7}{90}[/tex]
Desse modo, temos, [tex]\dfrac{\text{A}}{\text{B}}=\dfrac{\frac{-7}{9}}{\frac{-7}{90}}=\dfrac{-7}{9}\cdot\dfrac{90}{-7}=\dfrac{90}{9}=10[/tex]
c) Observe que:
[tex]2,\overline{5}=2+\dfrac{5}{9}=\dfrac{18+5}{9}=\dfrac{23}{9}[/tex]
[tex]0,\overline{2}=\dfrac{2}{9}[/tex]
O produto é [tex]\dfrac{23}{9}\cdot\dfrac{2}{9}=\dfrac{56}{9}[/tex]