1. **State the problem:** We need to find the perimeter of a triangle with side lengths $1 \frac{1}{3}$ miles, $2 \frac{1}{4}$ miles, and $1 \frac{1}{3}$ miles. 2. **Formula for perimeter:** The perimeter $P$ of a triangle is the sum of the lengths of all its sides: $$P = a + b + c$$ where $a$, $b$, and $c$ are the side lengths. 3. **Convert mixed numbers to improper fractions:** - $1 \frac{1}{3} = \frac{4}{3}$ - $2 \frac{1}{4} = \frac{9}{4}$ - $1 \frac{1}{3} = \frac{4}{3}$ 4. **Add the fractions:** $$P = \frac{4}{3} + \frac{9}{4} + \frac{4}{3}$$ 5. **Find a common denominator:** The denominators are 3 and 4, so the least common denominator is 12. 6. **Convert each fraction to denominator 12:** - $\frac{4}{3} = \frac{4 \times 4}{3 \times 4} = \frac{16}{12}$ - $\frac{9}{4} = \frac{9 \times 3}{4 \times 3} = \frac{27}{12}$ - $\frac{4}{3} = \frac{16}{12}$ 7. **Add the fractions:** $$P = \frac{16}{12} + \frac{27}{12} + \frac{16}{12} = \frac{16 + 27 + 16}{12} = \frac{59}{12}$$ 8. **Convert improper fraction to mixed number:** $$\frac{59}{12} = 4 \frac{11}{12}$$ **Final answer:** The perimeter of the triangle is $4 \frac{11}{12}$ miles.