Question

Find three rational numbers between $\frac{2}{3}\mathrm{and}\frac{3}{4}.$

Answer 1

$$\begin{gathered} \mathrm{Rational}\mathrm{number}\mathrm{between}\frac{2}{3}\mathrm{and}\frac{3}{4}: \\ \frac{1}{2}(\frac{2}{3}+\frac{3}{4}) \\ =\frac{1}{2}\left(\frac{8+9}{12}\right) \\ =\frac{17}{24} \\ \mathrm{We}\mathrm{know}: \\ \frac{2}{3}<\frac{17}{24}<\frac{3}{4} \\ \mathrm{Rational}\mathrm{number}\mathrm{between}\frac{2}{3}\mathrm{and}\frac{17}{24}: \\ \frac{1}{2}(\frac{2}{3}+\frac{17}{24}) \\ =\frac{1}{2}\left(\frac{16+17}{24}\right) \\ =\frac{1}{2}\left(\frac{33}{24}\right) \\ =\frac{33}{48}=\frac{33÷3}{48÷3}=\frac{11}{16} \\ \mathrm{Rational}\mathrm{number}\mathrm{between}\frac{17}{24}\mathrm{and}\frac{3}{4}: \\ \frac{1}{2}(\frac{17}{24}+\frac{3}{4}) \\ =\frac{1}{2}\left(\frac{17+18}{24}\right) \\ =\frac{1}{2}\left(\frac{35}{24}\right) \\ =\frac{35}{48} \\ \mathrm{We}\mathrm{know}: \\ \frac{2}{3}<\frac{11}{16}<\frac{17}{24}<\frac{35}{48}<\frac{3}{4} \\ \mathrm{Thus},\mathrm{the}\mathrm{three}\mathrm{rational}\mathrm{numbers}\mathrm{are}\frac{11}{16},\frac{17}{24}\mathrm{and}\frac{35}{48}. \end{gathered}$$