1. **State the problem:** We have a cookie dough mix made up of butter, sugar, and flour. 2. **Given:** Butter is $\frac{2}{5}$ of the mix, sugar is $\frac{1}{3}$ of the mix. 3. **Goal:** Find the fraction of the mix that is flour. 4. **Formula:** The total parts of the mix add up to 1 (whole). 5. **Calculate the sum of butter and sugar fractions:** $$\frac{2}{5} + \frac{1}{3} = \frac{2 \times 3}{5 \times 3} + \frac{1 \times 5}{3 \times 5} = \frac{6}{15} + \frac{5}{15} = \frac{11}{15}$$ 6. **Find the remaining fraction for flour:** $$1 - \frac{11}{15} = \frac{15}{15} - \frac{11}{15} = \frac{4}{15}$$ 7. **Answer:** The fraction of the mix that is flour is $\frac{4}{15}$. This means flour makes up $\frac{4}{15}$ of the total cookie dough mix.