1. **State the problem:** We are given two speed lines for Ariana and Glenn and need to complete ratio/rate tables, find unit rates, list points, and answer distance/time questions. 2. **Complete Ariana's ratio/rate table:** Given points: (1,8), (2,15), (5,38), (6,45), (8, ?) - To find miles at 8 hours, find the rate or use linear interpolation/extrapolation. 3. **Calculate Ariana's unit rate (pace):** Unit rate = miles per hour = $\frac{\text{miles}}{\text{hours}}$ - Using (1,8), unit rate = 8 miles/hour approximately. - Check with (2,15): $\frac{15}{2} = 7.5$ miles/hour, close to 8. - Average unit rate ~7.6 miles/hour. 4. **Find miles at 8 hours for Ariana:** Using approximate unit rate 7.6 miles/hour, $$8 \times 7.6 = 60.8$$ miles. 5. **Complete Glenn's ratio/rate table:** Given points: (1,5), (2,10), (5,25), (7,35), (8, ?) - Find miles at 8 hours using unit rate. 6. **Calculate Glenn's unit rate:** Using (1,5), unit rate = 5 miles/hour. - Check with (2,10): $\frac{10}{2} = 5$ miles/hour consistent. 7. **Find miles at 8 hours for Glenn:** $$8 \times 5 = 40$$ miles. 8. **List two additional points on Ariana's line:** - Using unit rate 7.6, points at (3, 22.8), (4, 30.4). 9. **List two additional points on Glenn's line:** - Using unit rate 5, points at (3, 15), (4, 20). 10. **After 3 hours, how much farther has Ariana gone than Glenn?** - Ariana: $3 \times 7.6 = 22.8$ miles - Glenn: $3 \times 5 = 15$ miles - Difference: $22.8 - 15 = 7.8$ miles 11. **How much longer will it take Glenn to walk 24 miles than Ariana to run 24 miles?** - Time = distance / rate - Ariana's time: $\frac{24}{7.6} \approx 3.16$ hours - Glenn's time: $\frac{24}{5} = 4.8$ hours - Difference: $4.8 - 3.16 = 1.64$ hours 12. **Solve 13 ÷ 18:** $$\frac{13}{18} \approx 0.722$$ 13. **Solve 28,681 + 9,478:** $$28,681 + 9,478 = 38,159$$ 14. **Solve 78 ) 3,734 (division):** - $78 \times 47 = 3,666$ remainder $68$ - Quotient is 47 remainder 68 **Final answers:** - Ariana's miles at 8 hours: 60.8 - Ariana's unit rate: approx 7.6 miles/hour - Additional points Ariana: (3,22.8), (4,30.4) - Glenn's miles at 8 hours: 40 - Glenn's unit rate: 5 miles/hour - Additional points Glenn: (3,15), (4,20) - After 3 hours, Ariana is 7.8 miles farther - Glenn takes 1.64 hours longer to walk 24 miles - 13 ÷ 18 ≈ 0.722 - 28,681 + 9,478 = 38,159 - 3,734 ÷ 78 = 47 remainder 68