21. 0<r<S<t On the number line, the coordinate of point P is t-s, and the coordinate of point Q is r-1. Quantity A: The distance between points P and Q on the number line Quantity B: 2t - s - r
Answer: C
The two quantities are equal.
The distance between points P and Q on the number line can be expressed as |(t - s) - (r - 1)|, which simplifies to |t - s - r + 1|. By analyzing the conditions given (0 < r < S < t), we can derive that this distance is equal to 2t - s - r.
A) Quantity A is greater.
This option suggests that the distance between points P and Q is greater than 2t - s - r. However, based on the simplification of the distance expression, we find that the two quantities are actually equal, thus making this option incorrect.
B) Quantity B is greater.
This option posits that 2t - s - r is greater than the distance between points P and Q. Given the derived equality of the two quantities, this option is also incorrect since neither quantity exceeds the other.
C) The two quantities are equal.
This option correctly states that the distance between points P and Q, calculated as |t - s - (r - 1)|, equals 2t - s - r when simplified. The conditions of the problem confirm that this equality holds true, making this option accurate.
D) The relationship cannot be determined from the information given.
This option implies that there is insufficient information to compare the two quantities. However, the relationship is clearly established through the distance formula and the given inequalities, making this option incorrect.
Conclusion
The correct answer is that the two quantities are equal, as shown through the simplification of the distance formula. Options A, B, and D fail to accurately represent the relationship between the quantities based on the provided information and calculations. Thus, option C is definitively the correct choice.