To find the total distance Frankie walked around the rectangular trail, we need to calculate the perimeter of the rectangle.
The formula for the perimeter \( P \) of a rectangle is given by:
\[ P = 2 \times (\text{length} + \text{width}) \]
From the information provided, we can identify the length and width:
- Width of the rectangle (left/right side) = one-third mile = \( \frac{1}{3} \) miles
- Length of the rectangle (top/bottom side) = two-fifths mile = \( \frac{2}{5} \) miles
Now, substituting these values into the perimeter formula:
\[ P = 2 \times \left(\frac{2}{5} + \frac{1}{3}\right) \]
To add the fractions \( \frac{2}{5} \) and \( \frac{1}{3} \), we need a common denominator. The least common multiple of 5 and 3 is 15.
Now convert each fraction:
- For \( \frac{2}{5} \): \[ \frac{2}{5} = \frac{2 \times 3}{5 \times 3} = \frac{6}{15} \]
- For \( \frac{1}{3} \): \[ \frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15} \]
Now we can add the two fractions: \[ \frac{6}{15} + \frac{5}{15} = \frac{11}{15} \]
Now we substitute this sum back into the perimeter formula:
\[ P = 2 \times \frac{11}{15} = \frac{22}{15} \]
Thus, the total distance Frankie walked around the trail is:
\[ \frac{22}{15} \text{ miles} \]
This can also be expressed as a mixed number:
\[ \frac{22}{15} = 1 \frac{7}{15} \]
However, as the problem requests the answer in fraction form, we can conclude that:
Frankie walked \( \frac{22}{15} \) miles.
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