1. **State the problem:** We are given that $\frac{2}{5}$ of Mabel's stamps equals $\frac{4}{9}$ of Nancy's stamps. We need to find the ratio of the number of Mabel's stamps to Nancy's stamps. 2. **Write the equation:** Let $M$ be the number of Mabel's stamps and $N$ be the number of Nancy's stamps. The problem states: $$\frac{2}{5}M = \frac{4}{9}N$$ 3. **Solve for the ratio $\frac{M}{N}$:** Multiply both sides by 45 (the least common multiple of 5 and 9) to clear denominators: $$45 \times \frac{2}{5}M = 45 \times \frac{4}{9}N$$ Simplify: $$9 \times 2M = 5 \times 4N$$ $$18M = 20N$$ 4. **Express the ratio:** Divide both sides by $18N$: $$\frac{M}{N} = \frac{20}{18} = \frac{10}{9}$$ 5. **Interpretation:** The ratio of Mabel's stamps to Nancy's stamps is $10:9$. **Final answer:** $10:9$