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Difficulty: HardData Sufficiency with Visual and Tabular Contexts

A regional logistics firm operates four distribution centers: Facility Alpha, Facility Beta, Facility Gamma, and Facility Delta. The table below displays the total tonnage of cargo processed and the average processing cost per ton (in dollars) at each facility during the last quarter:

Distribution CenterTotal Tonnage Processed (tons)Average Cost per Ton ($)
Facility Alpha2,40050
Facility Beta1,60040
Facility GammaTT80
Facility Delta3,000CC

Is the overall average processing cost per ton across all four distribution centers combined less than $60?

(1) Facility Gamma processed 2,000 tons of cargo (T=2,000T = 2,000).
(2) Facility Delta's average cost per ton was 55(55 ( C = 55$).

  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 correct answer is the option stating that both statements together are sufficient, but neither alone is sufficient. By setting up the total cost and total tonnage expressions from the table, the question 'Is the overall average cost less than 60?′simplifiestotheinequality′Is60?' simplifies to the inequality 'Is T + 150C < 11,800 ?′.Neitherstatementalonesuppliesbothvariables(?'. Neither statement alone supplies both variables ( T and and C ),buttogethertheyspecify), but together they specify T = 2,000 and and C = 55 ,giving, giving 2,000 + 8,250 = 10,250 < 11,800$, which answers the question with a definitive 'Yes'.

Step-by-Step Solution

1
Set up the algebraic expression for the overall combined average processing cost.
Total Tonnage = 2,400+1,600+T+3,000=7,000+T2,400 + 1,600 + T + 3,000 = 7,000 + T.
Total Cost ()=) = (2,400 \times 50) + (1,600 \times 40) + (T \times 80) + (3,000 \times C) = 120,000 + 64,000 + 80T + 3,000C = 184,000 + 80T + 3,000C$.
Weighted average requires dividing total dollar expenditure across all facilities by total tonnage processed.
2
Rephrase the target question inequality.
Is 184,000+80T+3,000C7,000+T<60\frac{184,000 + 80T + 3,000C}{7,000 + T} < 60?
Multiply by (7,000+T)(7,000 + T):
184,000+80T+3,000C<60(7,000+T)184,000 + 80T + 3,000C < 60(7,000 + T)
184,000+80T+3,000C<420,000+60T184,000 + 80T + 3,000C < 420,000 + 60T
20T+3,000C<236,00020T + 3,000C < 236,000
Divide by 20:
Is T+150C<11,800T + 150C < 11,800?
Simplifying the target question establishes a direct numerical threshold to test against both statements.
3
Evaluate Statement (1) alone.
Given T=2,000T = 2,000, the target inequality becomes: Is 2,000+150C<11,800  ⟹  150C<9,800  ⟹  C<65.332,000 + 150C < 11,800 \implies 150C < 9,800 \implies C < 65.33?
Since CC is unknown, if C=50C = 50, the answer is Yes (9,500<11,8009,500 < 11,800). If C=70C = 70, the answer is No (12,500≮11,80012,500 \not< 11,800).
Statement (1) leaves CC undetermined, allowing both 'Yes' and 'No' outcomes.
4
Evaluate Statement (2) alone.
Given C=55C = 55, the target inequality becomes: Is T+150(55)<11,800  ⟹  T+8,250<11,800  ⟹  T<3,550T + 150(55) < 11,800 \implies T + 8,250 < 11,800 \implies T < 3,550?
Since TT is unknown, if T=2,000T = 2,000, the answer is Yes (10,250<11,80010,250 < 11,800). If T=4,000T = 4,000, the answer is No (12,250≮11,80012,250 \not< 11,800).
Statement (2) leaves TT undetermined, allowing both 'Yes' and 'No' outcomes.
5
Evaluate Statements (1) and (2) together.
Combining T=2,000T = 2,000 and C=55C = 55 into the rephrased inequality:
T+150C=2,000+150(55)=2,000+8,250=10,250T + 150C = 2,000 + 150(55) = 2,000 + 8,250 = 10,250.
Since 10,250<11,80010,250 < 11,800, we obtain a definitive 'Yes'.
Both variables are uniquely determined, resolving the target inequality definitively.

Key Concept

Weighted Average Calculation and Data Sufficiency Inequality Rephrasing
Estimated Time:2m 0s
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Topics

Data Sufficiency with Visual and Tabular Contexts

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