Divya is twice as old as Shruti. What is the difference in their ages? (I)…

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Divya is twice as old as Shruti. What is the difference in their ages?

(I) Five years hence, the ratio of their ages would be 9 : 5.

(II) Ten years back, the ratio of their ages was 3 : 1.

  1. A.

    I alone sufficient while II alone not sufficient to answer

  2. B.

    II alone sufficient while I alone not sufficient to answer

  3. C.

    Either I or II alone sufficient to answer

  4. D.

    Both I and II are not sufficient to answer

Attempted by 1 students.

Show answer & explanation

Correct answer: C

Concept:

In a Data Sufficiency question, a statement is sufficient on its own only if, together with everything already given in the question stem, it yields enough independent equations to compute a UNIQUE value for the exact quantity asked. With two unknown ages, two independent linear relations are needed to fix both values; the stem already supplies one relation, so any statement that supplies just ONE further independent linear relation between the two ages is, by itself, enough.

Application:

Let Divya's present age be D and Shruti's present age be S.

  1. From the stem: Divya is twice as old as Shruti, so D = 2S ....(i)

  2. Statement (I): five years hence the ratio is 9 : 5, so (D + 5)/(S + 5) = 9/5. Cross-multiplying: 5(D + 5) = 9(S + 5), which gives 5D - 9S = 20 ....(ii)

  3. Solving (i) and (ii) together: substitute D = 2S into (ii) -> 10S - 9S = 20, so S = 20 and D = 40. This is a single, unique pair of ages, so Statement (I) alone is sufficient.

  4. Statement (II): ten years back the ratio was 3 : 1, so (D - 10)/(S - 10) = 3/1. Cross-multiplying: D - 10 = 3(S - 10), which gives D - 3S = -20 ....(iii)

  5. Solving (i) and (iii) together: substitute D = 2S into (iii) -> 2S - 3S = -20, so S = 20 and D = 40. Again a single, unique pair of ages, so Statement (II) alone is sufficient.

Cross-check:

Substituting D = 40 and S = 20 back into the original ratios: (40 + 5)/(20 + 5) = 45/25 = 9/5, and (40 - 10)/(20 - 10) = 30/10 = 3/1 — both statements independently confirm the same unique ages, and the age difference works out to 20 years either way.

Since each statement, combined with the stem's D = 2S relation, independently pins down the same unique ages, either statement alone is enough to answer the question — Either I or II alone is sufficient to answer.

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