1. **Problem a (Hikers between 8 and 18 miles):** We want to find the minimum number of hikers who walked between 8 and 18 miles. 2. **Understanding the intervals:** - The intervals given are 0 ≤ x < 5, 5 ≤ x < 10, 10 ≤ x < 15, 15 ≤ x < 20, 20 ≤ x < 25. - Frequencies are 3, 4, 7, 6, 1 respectively. 3. **Identify relevant intervals for 8 to 18 miles:** - 8 to 18 miles overlaps parts of 5 ≤ x < 10, 10 ≤ x < 15, and 15 ≤ x < 20. 4. **Minimum number of hikers between 8 and 18 miles:** - Minimum assumes only those definitely in the range. - From 5 ≤ x < 10 (frequency 4), only those from 8 to 10 count. Minimum is 0 because some could be between 5 and 8. - From 10 ≤ x < 15 (frequency 7), all are between 8 and 18, so count all 7. - From 15 ≤ x < 20 (frequency 6), only those from 15 to 18 count. Minimum is 0 because some could be between 18 and 20. - So minimum = 7. 5. **Maximum number of hikers between 8 and 18 miles:** - Maximum assumes all possible hikers in overlapping intervals are in the range. - From 5 ≤ x < 10 (frequency 4), assume all 4 are between 8 and 10. - From 10 ≤ x < 15 (frequency 7), all 7 count. - From 15 ≤ x < 20 (frequency 6), assume all 6 are between 15 and 18. - So maximum = 4 + 7 + 6 = 17. 6. **Problem b (Dogs with mass more than 27 kg):** - Mass intervals: 0 ≤ x < 10 (3), 10 ≤ x < 20 (9), 20 ≤ x < 30 (13), 30 ≤ x < 40 (4). 7. **Minimum number of dogs with mass > 27 kg:** - Only dogs in 30 ≤ x < 40 (4) definitely have mass > 27. - Dogs in 20 ≤ x < 30 (13) may or may not be > 27. - Minimum assumes none of the 20 ≤ x < 30 are > 27. - So minimum = 4. 8. **Maximum number of dogs with mass > 27 kg:** - Maximum assumes all dogs in 20 ≤ x < 30 are > 27. - So maximum = 13 + 4 = 17. **Final answers:** - Minimum hikers between 8 and 18 miles: $7$ - Maximum hikers between 8 and 18 miles: $17$ - Minimum dogs with mass > 27 kg: $4$ - Maximum dogs with mass > 27 kg: $17$