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Zorluk: OrtaOverlapping Sets, Statistics, and Data Distributions

In a technology firm of 100100 software engineers, each engineer works on at least one of two projects: Project Alpha or Project Beta. If 7070 engineers work on Project Alpha, what is the average (arithmetic mean) monthly salary of all 100100 engineers at the firm?

(1) The average monthly salary of the engineers who work ONLY on Project Alpha is $6,000\$6,000, and the average monthly salary of all engineers who work on Project Beta is $10,800\$10,800.
(2) The average monthly salary of the engineers who work ONLY on Project Beta is $12,000\$12,000, and the average monthly salary of the engineers who work on BOTH Project Alpha and Project Beta is $9,000\$9,000.

  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.Cevap
  4. D
    EACH statement ALONE is sufficient.
  5. E
    Statements (1) and (2) TOGETHER are NOT sufficient.

Cevap

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
The option stating that BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient is correct. Evaluating either statement independently leaves the overlap count or salary undefined. However, combining both statements allows equating two expressions for the total salary of Project Beta, which uniquely yields 2020 engineers working on both projects, leading to a single deterministic overall average salary of $8,400\$8,400.

Adım Adım Çözüm

1
Define variables from the question stem.
Let n1n_1 be the number of engineers working ONLY on Project Alpha, n2n_2 be the number working ONLY on Project Beta, and n3n_3 be the number working on BOTH. Total engineers = n1+n2+n3=100n_1 + n_2 + n_3 = 100. Given 7070 work on Alpha (n1+n3=70n_1 + n_3 = 70), we find n2=100−70=30n_2 = 100 - 70 = 30.
Establishing exact set counts reduces unknowns.
2
Evaluate Statement (1) alone.
Total salary = (n1×6000)+((n2+n3)×10800)=(70−n3)(6000)+(30+n3)(10800)=744,000+4800n3(n_1 \times 6000) + ((n_2 + n_3) \times 10800) = (70 - n_3)(6000) + (30 + n_3)(10800) = 744,000 + 4800 n_3. Since n3n_3 is unknown, overall average cannot be determined.
Statement (1) depends on the unknown overlap n3n_3, so Statement (1) alone is NOT sufficient.
3
Evaluate Statement (2) alone.
We know the average salary for n2=30n_2 = 30 is $12,000\$12,000 and for n3n_3 is $9,000\$9,000, but we have no salary information for the n1n_1 engineers working ONLY on Project Alpha.
Without salary information for Project Alpha only, Statement (2) alone is NOT sufficient.
4
Evaluate Statements (1) and (2) together.
From Statement (2), total salary of Project Beta = (30×12000)+(n3×9000)=360,000+9000n3(30 \times 12000) + (n_3 \times 9000) = 360,000 + 9000 n_3. From Statement (1), total salary of Project Beta = (30+n3)×10800=324,000+10800n3(30 + n_3) \times 10800 = 324,000 + 10800 n_3. Equating both: 360,000+9000n3=324,000+10800n3  ⟹  1800n3=36,000  ⟹  n3=20360,000 + 9000 n_3 = 324,000 + 10800 n_3 \implies 1800 n_3 = 36,000 \implies n_3 = 20.
Finding n3=20n_3 = 20 gives n1=50n_1 = 50, allowing exact calculation of the overall average salary (8,4008,400).

Anahtar Kavram

Weighted Averages and Overlapping Sets in Data Sufficiency
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Konular

Overlapping Sets, Statistics, and Data Distributions

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