Answer

Between two rational numbers, we can find \( \, \boxed{\color{red}{ \, infinitely \, \, many \, }} \) rational number

Answer

Between two rational numbers, we can find as many \( \, \boxed{\color{red}{ \, rational \, \, numbers \, }} \) as we like.

Answer

Between two integers, we can find as many \( \, \boxed{\color{red}{ \, rational \, \, numbers \, }} \) as we like.

Solution

\[ \begin{align*} & = \frac{1}{2} \times (2 + 4) \\ \\ & = \frac{1}{\cancel2_1} \times \cancel6^3 \\ \\ & = 3 \end{align*} \]

Answer \( \boxed{\color{red}3} \)

Solution

\[ \begin{align*} & = \frac{1}{2} \times [-2 + (-6)] \\ \\ & = \frac{1}{2} \times (-2 -6) \\ \\ & = \frac{1}{\cancel2_1} \times (\cancel{-8}^{-4}) \\ \\ & = -4 \end{align*} \]

Answer \( \boxed{\color{red}-4} \)

Solution

\[ \begin{align*} & = \frac{1}{2} \times \left(\frac{1}{4} + \frac{-3}{4} \right) \\ \\ & = \frac{1}{2} \times \left(\frac{1-3}{4} \right) \\ \\ & = \frac{1}{\cancel2_1} \times \left(\frac{-\cancel2^1}{4} \right) \\ \\ & = \frac{-1}{4} \\ \\ \end{align*} \]

Answer \( \boxed{\color{red}\frac{-1}{4}} \)

Solution

\[ \begin{align*} & = \frac{1}{2} \times \left(\frac{-2}{3} + \frac{-7}{3} \right) \\ \\ & = \frac{1}{2} \times \left(\frac{-2-7}{3} \right) \\ \\ & = \frac{1}{2} \times \left(\frac{-\cancel9^3}{\cancel3_1} \right) \\ \\ & = \frac{-3}{2} \\ \\ \end{align*} \]

Answer \( \boxed{\color{red}\frac{-3}{2}} \)

Solution

\[ \begin{align*} & \color{green}\text{Rational number between } \frac{4}{13} \text{ and } \frac{1}{13}\\ \\ & = \frac{1}{2} \times \left(\frac{4}{13} + \frac{1}{13} \right) \\ \\ & = \frac{1}{2} \times \left(\frac{4 + 1}{13} \right) \\ \\ & = \frac{1}{2} \times \frac{5}{13} \\ \\ & = \color{green} \boxed{\frac{5}{26}} \\ \\ & \color{green}\text{Rational number between } \frac{4}{13} \text{ and } \frac{5}{26}\\ \\ & = \frac{1}{2} \times \left(\frac{4}{13} + \frac{5}{26} \right) \\ \\ & = \frac{1}{2} \times \left(\frac{8 + 5}{26} \right) \\ \\ & = \frac{1}{2} \times \frac{\cancel{13}^1}{\cancel{26}_2} \\ \\ & = \color{green} \boxed{\frac{1}{4}}\\ \\ & \color{green}\text{Rational number between } \frac{5}{26} \text{ and } \frac{1}{13}\\ \\ & = \frac{1}{2} \times \left(\frac{5}{26} + \frac{1}{13} \right) \\ \\ & = \frac{1}{2} \times \left(\frac{5 + 2}{26} \right) \\ \\ & = \frac{1}{2} \times \frac{7}{26} \\ \\ & = \color{green} \boxed{\frac{7}{52}} \\ \\ \end{align*} \]

AnswerThree rational numbers between \( \displaystyle \frac{4}{13} \) and \( \displaystyle \frac{1}{13} \) are \( \boxed{\color{red} \frac{5}{26} , \frac{1}{4} , \frac{7}{52}} \)

Solution

\[ \begin{align*} & \color{green}\text{Rational number between } \frac{-7}{10} \text{ and } \frac{11}{10}\\ \\ & = \frac{1}{2} \times \left(\frac{-7}{10} + \frac{11}{10} \right) \\ \\ & = \frac{1}{2} \times \left(\frac{-7 + 11}{10} \right) \\ \\ & = \frac{1}{2} \times \frac{4}{10} \\ \\ & = \frac{\cancel4^1}{\cancel{20}_5} \\ \\ & = \color{green} \boxed{\frac{1}{5}} \\ \\ & \color{green} \text{Rational number between } \frac{-7}{10} \text{ and } \frac{1}{5}\\ \\ & = \frac{1}{2} \times \left(\frac{-7}{10} + \frac{1}{5} \right) \\ \\ & = \frac{1}{2} \times \left(\frac{-7 + 2}{10} \right) \\ \\ & = \frac{1}{2} \times \frac{-\cancel5^1}{\cancel{10}^2} \\ \\ & = \color{green} \boxed{\frac{-1}{4}}\\ \\ & \color{green}\text{Rational number between } \frac{1}{5} \text{ and } \frac{11}{10}\\ \\ & = \frac{1}{2} \times \left(\frac{1}{5} + \frac{11}{10} \right) \\ \\ & = \frac{1}{2} \times \left(\frac{2 + 11}{10} \right) \\ \\ & = \frac{1}{2} \times \frac{13}{10} \\ \\ & = \color{green} \boxed{\frac{13}{20}} \\ \\ \end{align*} \]

AnswerThree rational numbers between \( \displaystyle \frac{-7}{10} \) and \( \displaystyle \frac{11}{10} \) are \( \boxed{\color{red} \frac{1}{5} , \frac{-1}{4} , \frac{13}{20}} \)

Solution

\[ \begin{align*} & \color{green}\text{Rational number between } \frac{-4}{3} \text{ and } \frac{-19}{3}\\ \\ & = \frac{1}{2} \times \left(\frac{-4}{3} + \frac{-19}{3} \right) \\ \\ & = \frac{1}{2} \times \left(\frac{-4 - 19}{3} \right) \\ \\ & = \frac{1}{2} \times \frac{-23}{3} \\ \\ & = \color{green} \boxed{\frac{-23}{6}} \\ \\ & \color{green}\text{Rational number between } \frac{-4}{3} \text{ and } \frac{-23}{6}\\ \\ & = \frac{1}{2} \times \left(\frac{-4}{3} + \frac{-23}{6} \right) \\ \\ & = \frac{1}{2} \times \left(\frac{-8 - 23}{6} \right) \\ \\ & = \frac{1}{2} \times \frac{-31}{6} \\ \\ & = \color{green} \boxed{\frac{-31}{12}}\\ \\ & \color{green}\text{Rational number between } \frac{-23}{6} \text{ and } \frac{-19}{3}\\ \\ & = \frac{1}{2} \times \left(\frac{-23}{6} + \frac{-19}{3} \right) \\ \\ & = \frac{1}{2} \times \left(\frac{-23 - 38}{6} \right) \\ \\ & = \frac{1}{2} \times \frac{-61}{6} \\ \\ & = \color{green} \boxed{\frac{-61}{12}} \\ \\ \end{align*} \]

AnswerThree rational numbers between \( \displaystyle \frac{-4}{3} \) and \( \displaystyle \frac{-19}{3} \) are \( \boxed{\color{red} \frac{-23}{6} , \frac{-31}{12} , \frac{-61}{12}} \)

Solution

\[ \begin{align*} & \color{green}\text{Rational number between } \frac{1}{8} \text{ and } \frac{-15}{8}\\ \\ & = \frac{1}{2} \times \left(\frac{1}{8} + \frac{-15}{8} \right) \\ \\ & = \frac{1}{2} \times \left(\frac{1 - 15}{8} \right) \\ \\ & = \frac{1}{\cancel2_1} \times \frac{-\cancel{14}^7}{8} \\ \\ & = \color{green} \boxed{\frac{-7}{8}} \\ \\ & \color{green}\text{Rational number between } \frac{1}{8} \text{ and } \frac{-7}{8}\\ \\ & = \frac{1}{2} \times \left(\frac{1}{8} + \frac{-7}{8} \right) \\ \\ & = \frac{1}{2} \times \left(\frac{1 - 7}{8} \right) \\ \\ & = \frac{1}{\cancel2_1} \times \frac{-\cancel6^3}{8} \\ \\ & = \color{green} \boxed{\frac{-3}{8}}\\ \\ & \color{green}\text{Rational number between } \frac{-7}{8} \text{ and } \frac{-15}{8}\\ \\ & = \frac{1}{2} \times \left(\frac{-7}{8} + \frac{-15}{8} \right) \\ \\ & = \frac{1}{2} \times \left(\frac{-7 - 15}{8} \right) \\ \\ & = \frac{1}{\cancel2_1} \times \frac{-\cancel{22}^{11}}{8} \\ \\ & = \color{green} \boxed{\frac{-11}{8}} \\ \\ \end{align*} \]

AnswerThree rational numbers between \( \displaystyle \frac{1}{8} \) and \( \displaystyle \frac{-15}{8} \) are \( \boxed{\color{red} \frac{-7}{8} , \frac{-3}{8} , \frac{-11}{8}} \)

Solution

\[ \begin{align*} & \frac{-4}{7} \text{ and } \frac{4}{7}\\ \\ & \color{green}\text{Rational numbers between } \frac{-4}{7} \text{ and } \frac{4}{7}\\ \\ & = \frac{1}{2} \times \left(\frac{-4}{7} + \frac{4}{7} \right) \\ \\ & = \frac{1}{2} \times \left(\frac{-4 + 4}{7} \right) \\ \\ & = \frac{1}{2} \times \frac{0}{7} \\ \\ & = \boxed{0} \\ \\ & \color{green}\text{Rational number between } \frac{-4}{7} \text{ and } 0\\ \\ & = \frac{1}{2} \times \left( \frac{-4}{7} + 0 \right) \\ \\ & = \frac{1}{\cancel2_1} \times \frac{-\cancel4^2}{7} \\ \\ & = \boxed{\frac{-2}{7}}\\ \\ & \color{green}\text{Rational number between } 0 \text{ and } \frac{4}{7}\\ \\ & = \frac{1}{2} \times \left( 0 + \frac{4}{7} \right) \\ \\ & = \frac{1}{\cancel2_1} \times \frac{\cancel4^2}{7} \\ \\ & = \boxed{\frac{2}{7}}\\ \\ & \color{green}\text{Rational number between } \frac{-2}{7} \text{ and } 0\\ \\ & = \frac{1}{2} \times \left( \frac{-2}{7} + 0 \right) \\ \\ & = \frac{1}{\cancel2_1} \times \frac{-\cancel2^1}{7} \\ \\ & = \boxed{\frac{-1}{7}}\\ \\ & \color{green}\text{Rational number between } 0 \text{ and } \frac{2}{7}\\ \\ & = \frac{1}{2} \times \left( 0 + \frac{2}{7} \right) \\ \\ & = \frac{1}{\cancel2_1} \times \frac{\cancel2^1}{7} \\ \\ & = \boxed{\frac{1}{7}}\\ \\ \end{align*} \]

AnswerFive rational numbers between \( \displaystyle \frac{-4}{7} \) and \( \displaystyle \left| \frac{-4}{7} \right| \) are \( \boxed{\color{red} 0 , \frac{-2}{7} , \frac{2}{7} , \frac{-1}{7} , \frac{1}{7} } \)

Solution

\[ \begin{align*} & \frac{-8}{3} \text{ and } \frac{8}{3}\\ \\ & \color{green}\text{Rational numbers between } \frac{-8}{3} \text{ and } \frac{8}{3}\\ \\ & = \frac{1}{2} \times \left(\frac{-8}{3} + \frac{8}{3} \right) \\ \\ & = \frac{1}{2} \times \left(\frac{-8 + 8}{3} \right) \\ \\ & = \frac{1}{2} \times \frac{0}{3} \\ \\ & = \boxed{0} \\ \\ & \color{green}\text{Rational number between } \frac{-8}{3} \text{ and } 0\\ \\ & = \frac{1}{2} \times \left( \frac{-8}{3} + 0 \right) \\ \\ & = \frac{1}{\cancel2_1} \times \frac{-\cancel8^4}{3} \\ \\ & = \boxed{\frac{-4}{3}}\\ \\ & \color{green}\text{Rational number between } 0 \text{ and } \frac{8}{3}\\ \\ & = \frac{1}{2} \times \left( 0 + \frac{8}{3} \right) \\ \\ & = \frac{1}{\cancel2_1} \times \frac{\cancel8^4}{3} \\ \\ & = \boxed{\frac{4}{3}}\\ \\ & \color{green}\text{Rational number between } \frac{-4}{3} \text{ and } 0\\ \\ & = \frac{1}{2} \times \left( \frac{-4}{3} + 0 \right) \\ \\ & = \frac{1}{\cancel2_1} \times \frac{-\cancel4^2}{3} \\ \\ & = \boxed{\frac{-2}{3}}\\ \\ & \color{green}\text{Rational number between } 0 \text{ and } \frac{4}{3}\\ \\ & = \frac{1}{2} \times \left( 0 + \frac{4}{3} \right) \\ \\ & = \frac{1}{\cancel2_1} \times \frac{\cancel4^2}{3} \\ \\ & = \boxed{\frac{2}{3}}\\ \\ \end{align*} \]

AnswerFive rational numbers between \( \displaystyle \frac{-8}{3} \) and \( \displaystyle \left| \frac{-8}{3} \right| \) are \( \boxed{\color{red} 0 , \frac{-4}{3} , \frac{4}{3} , \frac{-2}{3} , \frac{2}{3} } \)