19. Which of the following inequalities is consistent with the statement that the time it took train K to travel r miles at a constant rate of s miles per hour is less than the time it took train M to travel y miles at a constant rate of z miles per hour?
Answer: D
sy - rz > 0
The inequality sy - rz > 0 indicates that the time taken by train K, which is represented by sy (distance y divided by speed z), is less than the time taken by train M, represented by rz (distance r divided by speed s). This aligns with the statement that train K's travel time is less than that of train M.
A) yz - rs > 0
This inequality suggests that the product of the speeds and distances of the two trains is being compared, but it does not directly relate to their travel times. It does not confirm that the time for train K is less than the time for train M, making this option incorrect.
B) rs - yz > 0
This inequality implies that the time taken by train M is less than that of train K, which contradicts the statement given in the question. Therefore, this option is also incorrect as it represents the opposite relationship of what is needed.
C) rz - sy > 0
This inequality indicates that the time for train M is less than the time for train K, which is not consistent with the information provided in the question. It incorrectly suggests that train M completes its journey faster than train K, rendering this option incorrect.
D) sy - rz > 0
This inequality accurately reflects the relationship stated in the question. It confirms that the time taken by train K (sy) is less than the time taken by train M (rz), thus validating the given condition.
E) ry - sz > 0
This inequality does not directly relate to the times taken by the trains based on their respective distances and speeds. It involves a different comparison that does not address the travel times, making it an incorrect option.
Conclusion
The correct inequality, sy - rz > 0, clearly demonstrates that the time train K took to travel is less than that of train M, as required by the question's premise. All other options either present incorrect relationships or do not address the travel time comparison, confirming that they are not consistent with the statement provided.