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

Question

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?

Accepted Answer

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

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1

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2

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3

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4

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?

Explanation

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