1. Divide
(i) \( \frac{2}{5} \) by \( \frac{-1}{3} \)
Answer
\[ \begin{align*} & = \frac{2}{5} \div \frac{-1}{3} \\ \\ & = \frac{2}{5} \times \frac{-3}{1} \\ \\ & = \frac{-6}{5} \\ \\ \end{align*} \]
(ii) \( \frac{-7}{4} \) by \( \frac{1}{8} \)
Answer
\[ \begin{align*} & = \frac{-7}{4} \div \frac{1}{8} \\ \\ & = \frac{-7}{\cancelto{1}{4}} \times \frac{\cancelto{2}{8}}{1} \\ \\ & = \frac{-14}{1} \\ \\ & = -14 \\ \\ \end{align*} \]
(iii) \( -10 \) by \( \frac{1}{5} \)
Answer
\[ \begin{align*} & = -10 \div \frac{1}{5} \\ \\ & = -10 \times \frac{5}{1} \\ \\ & = -50 \\ \\ \end{align*} \]
(iv) \( \frac{1}{13} \) by \( -2 \)
Answer
\[ \begin{align*} & = \frac{1}{13} \div -2 \\ \\ & = \frac{1}{13} \times \frac{-1}{2} \\ \\ & = \frac{-1}{26} \end{align*} \]
2. By taking \( x =\frac{3}{4} \) and \( y =\frac{-5}{6} \) , verfiy that \( \color{red} x \div y \neq y \div x \)
Answer
\[ \begin{aligned} \text{LHS} & = \color{green}x \div y \\ \\ & = \frac{3}{4} \div \frac{-5}{6}\\ \\ & = \frac{3}{\cancelto{2}{4}} \times \frac{\cancelto{3}{6}}{-5}\\ \\ & = \frac{9}{-10} \\ \\ & = \boxed {\frac{-9}{10}} \\ \\ \text{RHS} & = \color{green} y \div x \\ \\ & = \frac{-5}{6} \div \frac{3}{4} \\ \\ & = \frac{-5}{\cancelto{2}{6}} \times \frac{\cancelto{2}{4}}{3} \\ \\ & = \boxed{\frac{-10}{9}} \\ \\ \end{aligned} \] \[ \begin{aligned} \text{LHS} \neq \text{RHS} \\ \color{red} x \div y \neq y \div x \\ \text{Hence } \text{ Verified} \end{aligned} \]
3. The product of two rational numbers is \( \frac{-3}{7} \). If one of the number is \( \frac{5}{21} \), find the other.
Answer
\[ \begin{aligned} \text{(1st Number)} \times \text{(2nd Number)} &= \frac{-3}{7} \\ \\ \frac{5}{21} \times \text{(2nd Number)} &= \frac{-3}{7} \\ \\ \text{(2nd Number)} &= \frac{-3}{7} \div \frac{5}{21} \\ \\ &= \frac{-3}{\cancelto{1}{7}} \times \frac{\cancelto{3}{21}}{5} \\ \\ \text{Other Number}& = \boxed {\color{red}\frac{-9}{5}} \\ \\ \end{aligned} \]
4. With what number should we multiply \( \frac{-36}{35} \), so that the product be \( \frac{-6}{5} \)?
Answer
\(\text{Let the other number be } x \) \[ \begin{aligned} x \times \frac{-36}{35} &= \frac{-6}{5} \\ \\ x &= \frac{-6}{5} \div \frac{-36}{35} \\ \\ x &= \frac{\cancelto{1}{-6}}{\cancelto{1}{5}} \times \frac{\cancelto{7}{35}}{\cancelto{6}{-36}} \\ \\ x &= \frac{7}{6} \\ \\ \text{Other Number}& = \boxed {\color{red}\frac{7}{6}} \\ \\ \end{aligned} \]
5. By taking \( x = \frac{-5}{3}, y = \frac{2}{7}, z = \frac{1}{-4} \)
(i) \( x \div (y + z) \neq x \div y + x \div z \)
Answer
\[ \begin{aligned} \text{LHS} & = \color{green} x \div (y + z) \\ \\ & = \frac{-5}{3} \div \left( \frac{2}{7} + \frac{1}{-4} \right)\\ \\ & = \frac{-5}{3} \div \left( \frac{2}{7} - \frac{1}{4} \right)\\ \\ & = \frac{-5}{3} \div \left( \frac{8-7}{28} \right)\\ \\ & = \frac{-5}{3} \div \frac{1}{28} \\ \\ & = \frac{-5}{3} \times \frac{28}{1} \\ \\ & = \boxed {\frac{-140}{3}} \\ \\ \text{RHS} & = \color{green} x \div y + x \div z \\ \\ & = \left( \frac{-5}{3} \div \frac{2}{7} \right) + \left( \frac{-5}{3} \div \frac{1}{-4} \right)\\ \\ & = \left( \frac{-5}{3} \times \frac{7}{2} \right) + \left( \frac{-5}{3} \times \frac{-4}{1} \right)\\ \\ & = \frac{-35}{6} + \frac{20}{3} \\ \\ & = \frac{-35 + 40}{6} \\ \\ & = \boxed{\frac{5}{6}} \\ \\ \end{aligned} \] \[ \begin{aligned} \text{LHS} & \neq \text{RHS} \\ \color{red} x \div (y + z) & \color{red}\neq x \div y + x \div z \\ \text{Hence } &\text{ Verified} \end{aligned} \]
(ii) \( x \div (y - z) \neq x \div y - x \div z \)
Answer
\[ \begin{aligned} \text{LHS} & = \color{green} x \div (y - z) \\ \\ & = \frac{-5}{3} \div \left( \frac{2}{7} - \frac{-1}{4} \right)\\ \\ & = \frac{-5}{3} \div \left( \frac{2}{7} + \frac{1}{4} \right)\\ \\ & = \frac{-5}{3} \div \left( \frac{8 + 7}{28} \right)\\ \\ & = \frac{-5}{3} \div \frac{15}{28} \\ \\ & = \frac{-\cancelto{1}{5}}{3} \times \frac{28}{\cancelto{3}{15}} \\ \\ & = \boxed {\frac{-28}{9}} \\ \\ \text{RHS} & = \color{green} x \div y - x \div z \\ \\ & = \left( \frac{-5}{3} \div \frac{2}{7} \right) - \left( \frac{-5}{3} \div \frac{-1}{4} \right)\\ \\ & = \left( \frac{-5}{3} \times \frac{7}{2} \right) - \left( \frac{-5}{3} \times \frac{-4}{1} \right)\\ \\ & = \frac{-35}{6} - \frac{20}{3} \\ \\ & = \frac{-35 - 40}{6} \\ \\ & = \boxed{\frac{-75}{6}} \\ \\ \end{aligned} \] \[ \begin{aligned} \text{LHS} & \neq \text{RHS} \\ \color{red} x \div (y - z) & \color{red}\neq x \div y - x \div z \\ \text{Hence } &\text{ Verified} \end{aligned} \]
(iii) \( (x + y) \div z = x \div z + y \div z \)
Answer
\[ \begin{aligned} \text{LHS} & = \color{green}(x + y) \div z \\ \\ & = \left( \frac{-5}{3} + \frac{2}{7} \right) \div \frac{1}{-4} \\ \\ & = \left( \frac{-35 + 6}{21} \right) \div \frac{1}{-4} \\ \\ & = \frac{-29}{21} \div \frac{1}{-4} \\ \\ & = \frac{-29}{21} \times \frac{-4}{1} \\ \\ & = \boxed {\frac{116}{21}} \\ \\ \text{RHS} & = \color{green} x \div z + y \div z \\ \\ & = \left( \frac{-5}{3} \div \frac{1}{-4} \right) + \left( \frac{2}{7} \div \frac{1}{-4} \right) \\ \\ & = \left( \frac{-5}{3} \times \frac{-4}{1} \right) + \left( \frac{2}{7} \times \frac{-4}{1} \right) \\ \\ & = \frac{20}{3} + \frac{-8}{7} \\ \\ & = \frac{140 -24}{21} \\ \\ & = \frac{116}{21} \\ \\ & = \boxed{\frac{116}{21}} \\ \\ \end{aligned} \] \[ \begin{aligned} \text{LHS} & = \text{RHS} \\ \color{red} (x + y) \div z & \color{red}= x \div z + y \div z \\ \text{Hence } &\text{ Verified} \end{aligned} \]
6. From a rope of the length 40 metres, a man cuts some equal sized pieces. How many pieces can be cut if each piece is of \( \frac{4}{9} \) metres length?
Solution
\[ \begin{aligned} \text{Length of the rope} & = 40 \, m \\ \\ \text{Length of each piece} & = \frac{4}{9} \, m \\ \\ \text{Total number of pieces} & = 40 \div \frac{4}{9} \\ \\ & = \cancelto{10}{40} \times \frac{9}{\cancelto{1}{4}} \\ \\ & = 90 \text{ pieces} \\ \\ \end{aligned} \]
Answer\( \boxed{\color{red}90} \) pieces can be cut.