1. **State the problem:** Jada and Diego baked some cookies. They gave away fractions of the cookies, threw some away, and after several steps, 15 cookies remained. We need to find the total number of cookies they baked. 2. **Define the variable:** Let $x$ be the total number of cookies baked. 3. **Step 1:** They gave away $\frac{1}{4}$ of the cookies to their teachers. So, cookies left after this step: $$x - \frac{1}{4}x = \frac{3}{4}x$$ 4. **Step 2:** They threw away 1 burnt cookie. Cookies left: $$\frac{3}{4}x - 1$$ 5. **Step 3:** They delivered $\frac{2}{5}$ of the remaining cookies to a nursing home. The remaining cookies before delivery are $\frac{3}{4}x - 1$, so cookies delivered: $$\frac{2}{5} \left(\frac{3}{4}x - 1\right)$$ Cookies left after delivery: $$\left(\frac{3}{4}x - 1\right) - \frac{2}{5} \left(\frac{3}{4}x - 1\right) = \frac{3}{5} \left(\frac{3}{4}x - 1\right)$$ 6. **Step 4:** They gave 3 cookies to neighborhood kids. Cookies left: $$\frac{3}{5} \left(\frac{3}{4}x - 1\right) - 3$$ 7. **Step 5:** They wrapped up $\frac{2}{3}$ of the remaining cookies to save for friends. So, cookies wrapped: $$\frac{2}{3} \left[ \frac{3}{5} \left(\frac{3}{4}x - 1\right) - 3 \right]$$ Cookies left after wrapping: $$\left[ \frac{3}{5} \left(\frac{3}{4}x - 1\right) - 3 \right] - \frac{2}{3} \left[ \frac{3}{5} \left(\frac{3}{4}x - 1\right) - 3 \right] = \frac{1}{3} \left[ \frac{3}{5} \left(\frac{3}{4}x - 1\right) - 3 \right]$$ 8. **Step 6:** After all this, 15 cookies remained. So, $$\frac{1}{3} \left[ \frac{3}{5} \left(\frac{3}{4}x - 1\right) - 3 \right] = 15$$ 9. **Solve the equation:** Multiply both sides by 3: $$\cancel{\frac{1}{3}} \times 3 \left[ \frac{3}{5} \left(\frac{3}{4}x - 1\right) - 3 \right] = 15 \times 3$$ $$\frac{3}{5} \left(\frac{3}{4}x - 1\right) - 3 = 45$$ Add 3 to both sides: $$\frac{3}{5} \left(\frac{3}{4}x - 1\right) = 48$$ Multiply both sides by $\frac{5}{3}$: $$\cancel{\frac{3}{5}} \times \frac{5}{3} \left(\frac{3}{4}x - 1\right) = 48 \times \frac{5}{3}$$ $$\frac{3}{4}x - 1 = 80$$ Add 1 to both sides: $$\frac{3}{4}x = 81$$ Multiply both sides by $\frac{4}{3}$: $$\cancel{\frac{3}{4}} \times \frac{4}{3} x = 81 \times \frac{4}{3}$$ $$x = 108$$ 10. **Answer:** They baked **108** cookies in total.