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Difficulty: MediumRates, Ratios, Percentages, and Applied Word Problems

Two automated cargo vessels, Vessel X and Vessel Y, transported grain along a straight route of 12001{}200 nautical miles from Port A to Port B at constant speeds of vxv_x and vyv_y knots, respectively. What was the average speed vxv_x, in knots, of Vessel X for the trip?

(1) Vessel X completed the trip in 20%20\% less time than Vessel Y.
(2) If Vessel X had traveled at a speed 1010 knots faster, it would have completed the trip in 1010 hours less time.

  1. A
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.Answer
  3. C
    BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
  4. D
    EACH statement ALONE is sufficient.
  5. E
    Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
The choice stating that Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient is correct. Statement (1) provides only a relative ratio between the speeds of Vessel X and Vessel Y (vy=0.8vxv_y = 0.8v_x), leaving infinitely many possible values for vxv_x. Statement (2) sets up the equation 1200vx−1200vx+10=10\frac{1{}200}{v_x} - \frac{1{}200}{v_x + 10} = 10, which reduces to the quadratic equation vx2+10vx−1200=0v_x^2 + 10v_x - 1{}200 = 0. Because speed must be positive, vx=30v_x = 30 is the unique solution.

Step-by-Step Solution

1
Rephrase the question stem target
The target is to find the exact numerical value of vx=1200Txv_x = \frac{1{}200}{T_x}, where TxT_x is the time in hours taken by Vessel X.
Establishing the relationship between distance, rate, and time (D=v⋅tD = v \cdot t) clarifies what data is necessary.
2
Evaluate Statement (1) independently
Statement (1) states Tx=0.80TyT_x = 0.80 T_y, which implies 1200vx=0.80(1200vy)\frac{1{}200}{v_x} = 0.80 \left(\frac{1{}200}{v_y}\right), or vy=0.80vxv_y = 0.80 v_x.
This relationship gives the ratio of the two speeds but provides no specific numerical value for vxv_x or TxT_x. Thus, statement (1) ALONE is not sufficient.
3
Evaluate Statement (2) independently
Statement (2) gives 1200vx−1200vx+10=10\frac{1{}200}{v_x} - \frac{1{}200}{v_x + 10} = 10. Dividing by 1010 yields 120vx−120vx+10=1\frac{120}{v_x} - \frac{120}{v_x + 10} = 1, which simplifies to 120(10)=vx(vx+10)120(10) = v_x(v_x + 10), or vx2+10vx−1200=0v_x^2 + 10 v_x - 1{}200 = 0. Factoring gives (vx+40)(vx−30)=0(v_x + 40)(v_x - 30) = 0. Since speed must be positive, vx=30v_x = 30 knots.
Statement (2) yields a single valid positive value for vxv_x. Thus, statement (2) ALONE is sufficient.

Key Concept

Data Sufficiency evaluation for rate problems involving algebraic equations of degree two where physical constraints eliminate non-positive roots.

Alternative Method

For statement (2), test clean factors of 12001{}200 for vx(vx+10)=1200v_x(v_x + 10) = 1{}200. Since 30×40=120030 \times 40 = 1{}200, vx=30v_x = 30 can be quickly verified without full quadratic expansion.
Estimated Time:1m 45s
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Topics

Rates, Ratios, Percentages, and Applied Word Problems

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