1. **State the problem:** Julia has run $\frac{5}{6}$ of her weekly goal by Friday and runs 6.5 miles on Saturday. The total distance run this week is 51.5 miles. We need to find her weekly goal distance and how far off she was from the goal after Saturday. 2. **Define variables:** Let $G$ be Julia's weekly goal distance in miles. 3. **Write the equation:** By Friday, Julia ran $\frac{5}{6}G$ miles. On Saturday, she ran 6.5 miles. The total is 51.5 miles. $$\frac{5}{6}G + 6.5 = 51.5$$ 4. **Solve for $G$:** Subtract 6.5 from both sides: $$\frac{5}{6}G + 6.5 - 6.5 = 51.5 - 6.5$$ $$\frac{5}{6}G = 45$$ Divide both sides by $\frac{5}{6}$: $$G = \frac{45}{\frac{5}{6}}$$ Rewrite division as multiplication by reciprocal: $$G = 45 \times \frac{6}{5}$$ Calculate: $$G = 45 \times \frac{6}{5} = \cancel{45}^9 \times \frac{6}{\cancel{5}^1} = 9 \times 6 = 54$$ So, Julia's weekly goal is 54 miles. 5. **Find how far off she was:** She ran 51.5 miles total, goal is 54 miles. Difference: $$54 - 51.5 = 2.5$$ Julia was 2.5 miles short of her weekly goal. **Final answer:** Julia's weekly goal distance is 54 miles, and she was 2.5 miles short of her goal.