Is R elder to Q? (i) The present age of P is half of the age of R twelve years…

2022

Is R elder to Q?

(i) The present age of P is half of the age of R twelve years hence.

(ii) The age of P six years hence is equal to the sum of two-thirds of the present age of R and one-fifth of the present age of Q.

(iii) P is 12 years elder to Q’s daughter, and Q is 18 years elder to his daughter.

  1. A.

    Only (i) & (iii) together

  2. B.

    Only (ii) & (iii) together

  3. C.

    Only (iii)

  4. D.

    All of the three together

  5. E.

    Only (ii)

Attempted by 3 students.

Show answer & explanation

Correct answer: B

Concept

In a data-sufficiency question you are not asked to find exact values — only whether the given statements pin down the answer to the asked question (here: is R older than Q?). A statement set is sufficient only when, after translating it into equations, the comparison R versus Q is forced to a single outcome no matter what the still-unknown quantities are. The smallest such set is the intended answer; an extra statement that is not needed makes a larger combination redundant, not 'more correct'.

Translate the statements

  1. (i) Present age of P is half of R's age twelve years hence: P = ½ · (R + 12).

  2. (ii) P's age six years hence equals two-thirds of R's present age plus one-fifth of Q's present age: P + 6 = (2/3)·R + (1/5)·Q.

  3. (iii) P is 12 years older than Q's daughter and Q is 18 years older than that daughter. Eliminating the daughter's age gives Q = P + 6 — but this says nothing about R.

Test each combination

  • Statement (iii) alone fixes only Q in terms of P and never mentions R, so the comparison R versus Q cannot be settled.

  • Combining (i) with (iii): from (i), R = 2P − 12, and from (iii), Q = P + 6, so R − Q = (2P − 12) − (P + 6) = P − 18. The sign of this depends on the unknown value of P, so R could be older, younger, or equal — undecided.

  • Combining (ii) with (iii): substitute Q = P + 6 into (ii): P + 6 = (2/3)·R + (1/5)(P + 6). The left side is exactly (P + 6) = Q, so Q − (1/5)Q = (2/3)·R, i.e. (4/5)·Q = (2/3)·R, giving R = (6/5)·Q. Since ages are positive, R = 1.2·Q is always greater than Q — the question is answered with a definite 'yes'.

Cross-check

Using all three together forces unique values: from (i) R = 2P − 12 and from (ii)+(iii) R = (6/5)Q = (6/5)(P + 6); equating gives P = 24, so Q = 30 and R = 36. Statement (i) is consistent (R = 2·24 − 12 = 36) but was not required, which is why 'all three together' is not the minimal answer. The decision is reached by (ii) and (iii) together, so that pairing is the sufficient set.

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