1. **Stating the problem:** We want to understand and solve problems using the reciprocal law, which states that the reciprocal of a number $x$ is $\frac{1}{x}$. 2. **Formula and explanation:** The reciprocal law means if you multiply a number by its reciprocal, the result is always 1: $$x \times \frac{1}{x} = 1$$ This is true for any $x \neq 0$ because division by zero is undefined. 3. **Example 1:** Find the reciprocal of 5. - The reciprocal is $\frac{1}{5}$. - Check: $5 \times \frac{1}{5} = 1$. 4. **Example 2:** Simplify the expression $\frac{1}{\frac{3}{4}}$. - Using the reciprocal law, $\frac{1}{\frac{3}{4}} = \frac{4}{3}$. - Explanation: Dividing by a fraction is the same as multiplying by its reciprocal. 5. **Example 3:** Solve for $x$ if $\frac{1}{x} = 7$. - Multiply both sides by $x$: $1 = 7x$. - Divide both sides by 7: $x = \frac{1}{7}$. 6. **Example 4:** Simplify $\frac{\frac{2}{5}}{\frac{3}{7}}$. - Rewrite as $\frac{2}{5} \times \frac{7}{3}$ (multiply by reciprocal). - Multiply numerators and denominators: $\frac{2 \times 7}{5 \times 3} = \frac{14}{15}$. 7. **Summary:** The reciprocal law helps simplify division of fractions and solve equations involving reciprocals by converting division into multiplication by the reciprocal. Final answers: - Reciprocal of 5 is $\frac{1}{5}$. - $\frac{1}{\frac{3}{4}} = \frac{4}{3}$. - $x = \frac{1}{7}$ when $\frac{1}{x} = 7$. - $\frac{\frac{2}{5}}{\frac{3}{7}} = \frac{14}{15}$.