1. **Stating the problem:** A family travels a total distance in two days. On the first day, they travel $\frac{2}{5}$ of the total distance. On the second day, they travel $\frac{1}{3}$ of the remaining distance. We need to find: a) The fraction of the total distance traveled on the second day. b) The fraction of the total distance left to travel after the second day. 2. **Step 1: Calculate the remaining distance after the first day.** The family travels $\frac{2}{5}$ of the total distance on the first day, so the remaining distance is: $$1 - \frac{2}{5} = \frac{5}{5} - \frac{2}{5} = \frac{3}{5}$$ 3. **Step 2: Calculate the distance traveled on the second day.** They travel $\frac{1}{3}$ of the remaining distance on the second day, so the fraction of the total distance traveled on the second day is: $$\frac{1}{3} \times \frac{3}{5} = \frac{3}{15} = \frac{1}{5}$$ 4. **Step 3: Calculate the fraction of the total distance left after the second day.** Total distance traveled after two days is: $$\frac{2}{5} + \frac{1}{5} = \frac{3}{5}$$ So, the fraction left to travel is: $$1 - \frac{3}{5} = \frac{2}{5}$$ **Final answers:** a) The fraction of the total distance traveled on the second day is $\frac{1}{5}$. b) The fraction of the total distance left to travel is $\frac{2}{5}$.