5. A student athlete runs \( 3 \frac{1}{3} \) miles in 30 minutes. A professional runner can run \( 1 \frac{1}{4} \) times as far in 30 minutes. How far can the professional runner run in 30 minutes?

Alright, I need to figure out how far the professional runner can run in 30 minutes based on the information provided. Let's break down the problem step by step. First, the student athlete runs \( 3 \frac{1}{3} \) miles in 30 minutes. To make calculations easier, I'll convert the mixed number to an improper fraction. \( 3 \frac{1}{3} \) is the same as \( \frac{10}{3} \) miles. Next, the professional runner can run \( 1 \frac{1}{4} \) times as far as the student athlete in the same amount of time. Again, converting \( 1 \frac{1}{4} \) to an improper fraction gives me \( \frac{5}{4} \). Now, to find out how far the professional runner can run, I'll multiply the student athlete's distance by \( \frac{5}{4} \): \[ \text{Professional Runner's Distance} = \frac{10}{3} \times \frac{5}{4} \] Multiplying these fractions: \[ \frac{10}{3} \times \frac{5}{4} = \frac{10 \times 5}{3 \times 4} = \frac{50}{12} \] Simplifying \( \frac{50}{12} \) by dividing both numerator and denominator by 2: \[ \frac{50 \div 2}{12 \div 2} = \frac{25}{6} \] So, the professional runner can run \( \frac{25}{6} \) miles in 30 minutes. To express this as a mixed number: \[ \frac{25}{6} = 4 \frac{1}{6} \text{ miles} \] But wait, let me double-check my calculations to ensure accuracy. Starting from the beginning: - Student athlete: \( 3 \frac{1}{3} = \frac{10}{3} \) miles - Professional runner: \( 1 \frac{1}{4} = \frac{5}{4} \) times the student's distance - Calculation: \( \frac{10}{3} \times \frac{5}{4} = \frac{50}{12} = \frac{25}{6} = 4 \frac{1}{6} \) miles Everything checks out. Therefore, the professional runner can run \( 4 \frac{1}{6} \) miles in 30 minutes.