There are three points P, Q and R on a straight line such that PQ : QR = 3 : 5. If n is the number of possible values of PQ : PR, then what is n equal to?

Let the lengths of the segments be PQ = 3k and QR = 5k, where k is a positive constant. Since P, Q, and R are on a straight line, there are three possible orderings of these points. 1. **Q is between P and R (P - Q - R):** In this arrangement, the total length PR is the sum of PQ and QR. PR = PQ + QR = 3k + 5k = 8k. The ratio PQ : PR = 3k : 8k = 3 : 8. 2. **P is between R and Q (R - P - Q):** In this arrangement, the length QR is the sum of RP (which is PR) and PQ. QR = RP + PQ 5k = PR + 3k Solving for PR, we get PR = 5k - 3k = 2k. The ratio PQ : PR = 3k : 2k = 3 : 2. 3. **R is between P and Q (P - R - Q):** In this arrangement, the length PQ is the sum of PR and RQ (which is QR). PQ = PR + RQ 3k = PR + 5k Solving for PR, we get PR = 3k - 5k = -2k. Since length cannot be negative, this arrangement is not possible. Thus, there are two possible valid arrangements of the points, leading to two distinct values for the ratio PQ : PR: Value 1: 3 : 8 Value 2: 3 : 2 The number of possible values of PQ : PR, denoted by n, is 2. The final answer is B