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Let's solve this step-by-step.

Kelly traveled from City C to City D in three days. Here's the breakdown:

1. **First day:** She traveled \(\frac{2}{5}\) of the total distance.
2. **Second day:** She traveled \(\frac{2}{3}\) of the remaining distance.

Let’s calculate the total distance Kelly traveled over the first two days, and then determine how much distance is left for the third day.

### Step 1: Fraction of distance remaining after the first day
On the first day, she traveled \(\frac{2}{5}\) of the distance. So, the remaining distance after the first day is:
\[
1 - \frac{2}{5} = \frac{5}{5} - \frac{2}{5} = \frac{3}{5}
\]
So, \(\frac{3}{5}\) of the total distance remains.

### Step 2: Distance traveled on the second day
On the second day, she traveled \(\frac{2}{3}\) of the remaining \(\frac{3}{5}\). The distance traveled on the second day is:
\[
\frac{2}{3} \times \frac{3}{5} = \frac{6}{15} = \frac{2}{5}
\]
So, she traveled another \(\frac{2}{5}\) of the total distance on the second day.

### Step 3: Fraction of distance remaining for the third day
After traveling \(\frac{2}{5}\) on the first day and \(\frac{2}{5}\) on the second day, the remaining distance for the third day is:
\[
1 - \left( \frac{2}{5} + \frac{2}{5} \right) = 1 - \frac{4}{5} = \frac{1}{5}
\]

### Step 4: Answer
The fraction of the distance that Kelly traveled on the third day is \(\frac{1}{5}\).

Now, comparing this result to the provided options, we are looking for the expression that simplifies to \(\frac{1}{5}\). The correct option is the second one:
\[
1 - \frac{2}{5} - \frac{2}{3}\left(1 - \frac{2}{5}\right)
\]
This matches our calculations.

Would you like further clarification or details on any step? Here are some related questions:

1. How do you calculate the remaining distance in fraction form?
2. What is the general method for solving fractional word problems?
3. How does multiplying fractions work in multi-step problems?
4. What other techniques can be used for checking your work in such problems?
5. Can a similar problem be solved using algebraic equations instead of fractions?

**Tip:** When dealing with fractional parts of a whole, always ensure that your remaining parts sum up to 1 (the whole) to verify your results.

Kelly took three days to travel from City C to City D by automobile. On the first day, Kelly traveled 2/5 of the distance from City C to City D; and on the second day, she traveled 2/3 of the remaining distance. Which of the following is equivalent to the fraction of the distance from City C to City D that Kelly traveled on the third day?

This problem involves calculating the distance Kelly traveled on the third day after traveling specific fractions of the total distance over the first two days. Learn how to solve it using fraction operations, suitable for Grades 6-8.

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