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Question

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

A regional freight company operates two transport fleets: Fleet X and Fleet Y. Last quarter, what was the ratio of the total operating expense of Fleet X to the total operating expense of Fleet Y?

(1) Fleet X's fuel expense was 40%40\% of its total operating expense, and Fleet Y's fuel expense was 25%25\% of its total operating expense.
(2) The combined fuel expense of both fleets was 30%30\% of their combined total operating expenses.

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

Answer

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
The statement expressing that both statements together are sufficient while neither alone is sufficient is correct because Statement (1) gives the component percentages (40%40\% and 25%25\%) and Statement (2) gives the overall weighted percentage (30%30\%). Equating the component sum 0.40EX+0.25EY0.40E_X + 0.25E_Y to the weighted total 0.30(EX+EY)0.30(E_X + E_Y) yields a single linear equation that solves directly for the ratio EX/EY=1/2E_X / E_Y = 1/2.

Step-by-Step Solution

1
Define variables and target ratio
Let EXE_X be the total operating expense of Fleet X and EYE_Y be the total operating expense of Fleet Y. The target is to determine the ratio EXEY\frac{E_X}{E_Y}.
Rephrasing the question stem into a formal mathematical expression highlights what relationship needs to be determined.
2
Evaluate Statement (1) alone
Fuel expense of Fleet X =0.40EX= 0.40 E_X and Fuel expense of Fleet Y =0.25EY= 0.25 E_Y.
Statement (1) provides individual percentages, but contains no information relating EXE_X to EYE_Y. Hence, Statement (1) alone is NOT sufficient.
3
Evaluate Statement (2) alone
Total combined fuel expense =0.30(EX+EY)= 0.30 (E_X + E_Y).
Without knowing the individual fuel percentages for Fleet X and Fleet Y, we cannot express the total combined fuel expense in terms of EXE_X and EYE_Y. Hence, Statement (2) alone is NOT sufficient.
4
Evaluate Statements (1) and (2) together
Equate the sum of individual fuel expenses to the total combined fuel expense:
0.40EX+0.25EY=0.30(EX+EY)0.40 E_X + 0.25 E_Y = 0.30 (E_X + E_Y)
0.40EX+0.25EY=0.30EX+0.30EY0.40 E_X + 0.25 E_Y = 0.30 E_X + 0.30 E_Y
0.10EX=0.05EY0.10 E_X = 0.05 E_Y
EXEY=0.050.10=12\frac{E_X}{E_Y} = \frac{0.05}{0.10} = \frac{1}{2}
Setting the two expressions for total fuel expense equal creates a linear homogeneous equation that uniquely determines the exact ratio of EXE_X to EYE_Y.

Key Concept

Weighted Average Ratios in Data Sufficiency

Practice More

Practice Data Sufficiency questions involving weighted average mixtures where component percentages lie on opposite sides of the combined percentage.

Alternative Method

Alligation / Weighted Average Formula: The distance between Fleet X's percentage (40%40\%) and the combined percentage (30%30\%) is 10%10\%. The distance between Fleet Y's percentage (25%25\%) and the combined percentage (30%30\%) is 5%5\%. The ratio of Fleet X to Fleet Y is inversely proportional to these distances: EX/EY=5/10=1/2E_X / E_Y = 5 / 10 = 1 / 2.
Estimated Time:1m 30s
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Topics

Rates, Ratios, Percentages, and Applied Word Problems

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