It takes Neil and Spencer the same amount of time to ride their bikes to school each day, though Nell lives farther away. This Saturday, they plan to study together at the library, which is the same distance from both of their houses. If their average speeds are consistent, who will get to the library in less time?

To solve this problem, we need to analyze the information given about Neil and Spencer's biking speeds and distances. ### Known Conditions: 1. Neil and Spencer take the same amount of time to ride their bikes to school. 2. Neil lives farther away from school than Spencer. 3. The library is the same distance from both of their houses. 4. Their average speeds are consistent. ### Step-by-Step Analysis: 1. **Understanding the relationship between distance, speed, and time:** The relationship can be expressed with the formula: \[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \] 2. **Time taken to ride to school:** Let: - \( d_N \) = distance from Neil's house to school - \( d_S \) = distance from Spencer's house to school - \( v_N \) = average speed of Neil - \( v_S \) = average speed of Spencer Since they take the same time to reach school, we can write: \[ \frac{d_N}{v_N} = \frac{d_S}{v_S} \] 3. **Distance to the library:** Let \( d_L \) be the distance from both Neil's and Spencer's houses to the library. Since the library is the same distance from both houses, we can analyze the time taken by each to reach the library. 4. **Time taken to ride to the library:** - For Neil: \[ \text{Time}_N = \frac{d_L}{v_N} \] - For Spencer: \[ \text{Time}_S = \frac{d_L}{v_S} \] 5. **Comparing the times:** To determine who will get to the library in less time, we need to compare \( \text{Time}_N \) and \( \text{Time}_S \): \[ \text{Time}_N = \frac{d_L}{v_N} \quad \text{and} \quad \text{Time}_S = \frac{d_L}{v_S} \] Since \( d_L \) is the same for both, we can compare the times based on their speeds: - If \( v_N > v_S \), then \( \text{Time}_N < \text{Time}_S \) (Neil gets there faster). - If \( v_N < v_S \), then \( \text{Time}_N > \text{Time}_S \) (Spencer gets there faster). ### Conclusion: Since we know that Neil lives farther away from school but takes the same time as Spencer, it implies that Neil's speed must be less than Spencer's speed (because he covers a greater distance in the same time). Therefore, Spencer must have a higher average speed. Thus, Spencer will get to the library in less time than Neil.