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 ?
- (a) 1
- (b) 2 ✓ UPSC's answer
- (c) 3
- (d) 4
Why the answer is (b)
• Let the length of PQ be 3x and QR be 5x, based on the given ratio PQ : QR = 3 : 5.
• Since P, Q, and R are on a straight line, there are two possible relative positions for the points: Q lies between P and R, or P lies between Q and R (R cannot lie between P and Q because PQ < QR).
• Case 1: If Q is between P and R, the total length PR = PQ + QR = 3x + 5x = 8x. The ratio PQ : PR is 3x : 8x, which simplifies to 3 : 8.
• Case 2: If P is between Q and R, the total length PR = QR - PQ = 5x - 3x = 2x. The ratio PQ : PR is 3x : 2x, which simplifies to 3 : 2.
• These are the only two distinct possible values for the ratio PQ : PR.
• Therefore, the number of possible values, n, is 2.
Why the other options are wrong
- (a) 1
- Option (a) is incorrect because there are two distinct geometric configurations (Q between P and R, or P between Q and R) that yield different ratios.
- (c) 3
- Option (c) is incorrect because the third configuration (R between P and Q) is impossible as it would require PQ > QR, contradicting the 3:5 ratio.
- (d) 4
- Option (d) is incorrect because only two valid linear arrangements exist for three points on a line given the specific segment lengths.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2021, held on 10 October 2021. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.