1. **State the problem:** Madison plans to cover a total distance of $1 \frac{4}{5}$ miles on a treadmill. She has already walked $\frac{2}{5}$ miles and wants to find out how far she needs to run, represented by $r$. 2. **Write the equation:** The total distance is the sum of the distance walked and the distance run: $$\frac{2}{5} + r = 1 \frac{4}{5}$$ 3. **Convert the mixed number to an improper fraction:** $$1 \frac{4}{5} = \frac{5 \times 1 + 4}{5} = \frac{9}{5}$$ So the equation becomes: $$\frac{2}{5} + r = \frac{9}{5}$$ 4. **Solve for $r$ by subtracting $\frac{2}{5}$ from both sides:** $$r = \frac{9}{5} - \frac{2}{5}$$ 5. **Subtract the fractions:** Since denominators are the same, $$r = \frac{9 - 2}{5} = \frac{7}{5}$$ 6. **Convert the improper fraction back to a mixed number:** $$\frac{7}{5} = 1 \frac{2}{5}$$ **Answer:** Madison needs to run $1 \frac{2}{5}$ miles. This means she will run a little more than one mile to complete her total planned distance.