1. **State the problem:** We want to find how many times the goal distance of 4 \frac{1}{2} miles fits into the actual distance biked, which is 6 miles. 2. **Convert mixed number to improper fraction:** \[4 \frac{1}{2} = \frac{(4 \times 2) + 1}{2} = \frac{9}{2}\] 3. **Set up the division to find how many times the goal fits into the actual distance:** \[\text{times their goal} = \frac{6}{4 \frac{1}{2}} = \frac{6}{\frac{9}{2}}\] 4. **Divide by a fraction by multiplying by its reciprocal:** \[\frac{6}{\frac{9}{2}} = 6 \times \frac{2}{9}\] 5. **Multiply and simplify:** \[6 \times \frac{2}{9} = \frac{6 \times 2}{9} = \frac{12}{9}\] 6. **Simplify the fraction by dividing numerator and denominator by 3:** \[\frac{\cancel{12}^{4}}{\cancel{9}^{3}} = \frac{4}{3}\] 7. **Convert to mixed number:** \[\frac{4}{3} = 1 \frac{1}{3}\] **Final answer:** The actual distance biked is \textbf{1 \frac{1}{3}} times their goal.