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Zorluk: ZorOverlapping Sets, Statistics, and Data Distributions

In a commercial survey of 120120 retail stores, each store sells products from Brand P, Brand Q, or both. The average (arithmetic mean) monthly revenue of the stores that sell Brand P is $40,000\$40,000, and the average monthly revenue of the stores that sell Brand Q is $50,000\$50,000. What is the average monthly revenue of all 120120 stores combined?

(1) Exactly 3030 stores sell both Brand P and Brand Q, and their average monthly revenue is $60,000\$60,000.
(2) Exactly 4040 stores sell only Brand P.

  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 correct response identifies that neither statement alone provides enough information to determine both the exact store counts in each category and the overlap revenue sum, but combining both statements gives a unique system of linear equations that yields an exact combined average revenue of $5,000,000120≈$41,666.67\frac{\$5,000,000}{120} \approx \$41,666.67.

Adım Adım Çözüm

1
Formulate the total revenue expression in terms of overlapping sets and subgroup means.
Let n(P)n(P) be the number of stores selling Brand P and n(Q)n(Q) be the number of stores selling Brand Q. Total stores N=n(P only)+n(Q only)+n(P∩Q)=120N = n(P \text{ only}) + n(Q \text{ only}) + n(P \cap Q) = 120. Total revenue Stotal=S(P)+S(Q)−S(P∩Q)=40,000n(P)+50,000n(Q)−S(P∩Q)S_{total} = S(P) + S(Q) - S(P \cap Q) = 40,000 n(P) + 50,000 n(Q) - S(P \cap Q).
Stores selling both brands are counted twice when summing S(P)S(P) and S(Q)S(Q), so their revenue sum S(P∩Q)S(P \cap Q) must be subtracted once.
2
Evaluate Statement (1) alone.
Statement (1) gives n(P∩Q)=30n(P \cap Q) = 30 and average overlap revenue =$60,000  ⟹  S(P∩Q)=30×60,000=1,800,000= \$60,000 \implies S(P \cap Q) = 30 \times 60,000 = 1,800,000. Also n(P)+n(Q)=120+30=150n(P) + n(Q) = 120 + 30 = 150.
Since n(P)n(P) and n(Q)n(Q) can still vary while their sum is 150150 (e.g., n(P)=70,n(Q)=80n(P)=70, n(Q)=80 vs n(P)=90,n(Q)=60n(P)=90, n(Q)=60), Stotal=40,000n(P)+50,000n(Q)−1,800,000S_{total} = 40,000 n(P) + 50,000 n(Q) - 1,800,000 yields multiple possible total values. Statement (1) alone is INSUFFICIENT.
3
Evaluate Statement (2) alone.
Statement (2) gives n(P only)=40n(P \text{ only}) = 40. Thus n(Q)=120−40=80n(Q) = 120 - 40 = 80.
No revenue information or overlap count is given. Statement (2) alone is INSUFFICIENT.
4
Evaluate Statements (1) and (2) together.
From Statement (2), n(P only)=40n(P \text{ only}) = 40. From Statement (1), n(P∩Q)=30n(P \cap Q) = 30. Thus n(P)=40+30=70n(P) = 40 + 30 = 70. Since n(P)+n(Q)=150n(P) + n(Q) = 150, n(Q)=80n(Q) = 80.
We now have exact values: n(P)=70n(P) = 70, n(Q)=80n(Q) = 80, S(P∩Q)=1,800,000S(P \cap Q) = 1,800,000. Stotal=40,000(70)+50,000(80)−1,800,000=2,800,000+4,000,000−1,800,000=5,000,000S_{total} = 40,000(70) + 50,000(80) - 1,800,000 = 2,800,000 + 4,000,000 - 1,800,000 = 5,000,000. Combined mean revenue =5,000,000120=$41,666.67= \frac{5,000,000}{120} = \$41,666.67. Both statements together are SUFFICIENT.

Anahtar Kavram

Weighted averages in overlapping sets requiring exact subgroup counts and sum reconciliations.
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Overlapping Sets, Statistics, and Data Distributions

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