Two boats P & Q are rowing in two different rivers X & Y respectively. Find…

2022

Two boats P & Q are rowing in two different rivers X & Y respectively. Find the distance covered by boat Q in 15 hours downstream.
(i) Sum of speed of stream in X and Y is 12 km/hr.
(ii) Distance covered by boat Q in Z hours in downstream is 120 km more than the distance covered by boat Q in (Z+2) hours in upstream.
(iii) Boat P can cover 420 km downstream in 8 hours while boat Q can cover 120 km upstream in six hours.

  1. A.

    Both (ii) & (iii)

  2. B.

    Only (iii)

  3. C.

    Both (i) & (ii)

  4. D.

    Both (i) & (iii)

  5. E.

    None

Attempted by 3 students.

Show answer & explanation

Correct answer: E

Concept

In a data-sufficiency problem you must decide whether the given statements pin down the asked quantity to a single value. A statement set is sufficient only if it yields one unique numeric answer; if any free unknown survives, the set is insufficient. Here the target is boat Q's downstream distance in 15 hours, which equals 15 × (Q's downstream speed), where downstream speed = (boat speed of Q) + (stream speed of river Y).

Reading each statement

  • (i) stream(X) + stream(Y) = 12. This alone links the two rivers' currents but says nothing isolating Q's own speed or Y's current.

  • (ii) Q's distance in Z hours downstream is 120 more than in (Z+2) hours upstream: Z·(Qdown) = (Z+2)·(Qup) + 120. This carries a free unknown Z.

  • (iii) P downstream = 420÷8 = 52.5 km/hr and Q upstream = 120÷6 = 20 km/hr. This fixes Q's UPSTREAM speed only, not its downstream speed.

Application — test the strongest combination

  1. From (iii): Qup = 20.

  2. Substitute into (ii): Z·(Qdown) = (Z+2)·20 + 120 = 20Z + 160, so Qdown = 20 + 160÷Z.

  3. Qdown still depends on the unknown Z, so it is not a single number. Statement (i) only relates the two currents (with P's boat speed unknown) and cannot fix Z either.

  4. Hence even all three statements together leave Qdown undetermined.

Cross-check

Even when all three statements hold, two scenarios fit the data yet give different answers. With Z = 8 the streams are valid (current of Y = 10, current of X = 2) and Qdown = 40, i.e. 600 km in 15 h; with Z = 10 (current of Y = 8, current of X = 4) Qdown = 36, i.e. 540 km. Since the value is not fixed, no offered combination — (ii)&(iii), only (iii), (i)&(ii), or (i)&(iii) — is sufficient.

Result

None of the listed combinations can answer the question, so the correct choice is “None”.

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