1. **State the problem:** Roger had 250 more cakes than Fiona. Nicole had $\frac{2}{5}$ as many cakes as Roger. Together, they had 482 cakes. We need to find how many cakes Fiona had. 2. **Define variables:** Let $F$ be the number of cakes Fiona had. 3. **Express Roger's cakes:** Roger had 250 more cakes than Fiona, so $$R = F + 250$$ 4. **Express Nicole's cakes:** Nicole had $\frac{2}{5}$ as many cakes as Roger, so $$N = \frac{2}{5}R = \frac{2}{5}(F + 250)$$ 5. **Write the total cakes equation:** The total cakes are 482, so $$F + R + N = 482$$ Substitute $R$ and $N$: $$F + (F + 250) + \frac{2}{5}(F + 250) = 482$$ 6. **Simplify the equation:** $$F + F + 250 + \frac{2}{5}F + \frac{2}{5} \times 250 = 482$$ $$2F + 250 + \frac{2}{5}F + 100 = 482$$ 7. **Combine like terms:** $$2F + \frac{2}{5}F + 350 = 482$$ 8. **Convert to common denominator and add:** $$2F = \frac{10}{5}F$$ So, $$\frac{10}{5}F + \frac{2}{5}F + 350 = 482$$ $$\frac{12}{5}F + 350 = 482$$ 9. **Isolate $F$:** $$\frac{12}{5}F = 482 - 350$$ $$\frac{12}{5}F = 132$$ 10. **Solve for $F$:** $$F = 132 \times \frac{5}{12}$$ Show cancelation: $$F = 132 \times \cancel{\frac{5}{12}} = 132 \times \frac{5}{12}$$ Actually, simplify $\frac{132}{12} = 11$, so $$F = 11 \times 5 = 55$$ **Final answer:** Fiona had **55** cakes.