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Difficulty: HardOverlapping Sets, Statistics, and Data Distributions

A survey recorded the monthly electricity costs of 60 small business workshops. Each workshop operated during Shift X, Shift Y, or both shifts. Exactly 35 workshops operated during Shift X, and exactly 40 workshops operated during Shift Y. What was the median monthly electricity cost among all 60 workshops?

(1) For the workshops that operated ONLY during Shift X, the monthly electricity cost was 400perworkshop,andfortheworkshopsthatoperatedONLYduringShiftY,themonthlyelectricitycostwas400 per workshop, and for the workshops that operated ONLY during Shift Y, the monthly electricity cost was 600 per workshop.

(2) For the workshops that operated during BOTH shifts, the median monthly electricity cost was 500,andthetotalmonthlyelectricitycostforall60workshopscombinedwas500, and the total monthly electricity cost for all 60 workshops combined was 30,000.

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

Answer

Statements (1) and (2) TOGETHER are NOT sufficient.
The correct option identifies that even when both statements are combined, the internal distribution of costs in the overlapping group allows for multiple overall medians (such as 500and500 and 600). Therefore, the statements together remain insufficient.

Step-by-Step Solution

1
Determine subgroup counts using the Principle of Inclusion-Exclusion.
Total workshops N=60N = 60. N(X)=35N(X) = 35, N(Y)=40N(Y) = 40. Since every workshop is in XX or YY, N(X∪Y)=60N(X \cup Y) = 60. Thus, N(X∩Y)=35+40−60=15N(X \cap Y) = 35 + 40 - 60 = 15. Workshops in ONLY X=35−15=20X = 35 - 15 = 20. Workshops in ONLY Y=40−15=25Y = 40 - 15 = 25.
Rephrasing the stem establishes the exact count of elements in each of the three disjoint set categories.
2
Evaluate Statement (1) alone.
The 20 ONLY XX workshops cost 400each,andthe25ONLY400 each, and the 25 ONLY Y workshopscost workshops cost 600 each. However, the costs of the 15 BOTH workshops are completely unknown. The overall median (the average of the 30th and 31st values in sorted order) could be 400(ifallBOTHcostsarelow)or400 (if all BOTH costs are low) or 600 (if all BOTH costs are high).
Without data on the overlapping subgroup, the median cannot be uniquely determined.
3
Evaluate Statement (2) alone.
Statement (2) provides the median (500)andsum(500) and sum ( 7,000) for the 15 BOTH workshops, as well as the total sum (30,000),butgivesnocostvaluesfortheONLY30,000), but gives no cost values for the ONLY X orONLY or ONLY Y$ workshops.
Without specific values for the single-shift groups, Statement (2) is insufficient.
4
Evaluate Statements (1) and (2) together.
From Statement (1), the sum of ONLY XX and ONLY YY workshops is 20(400)+25(600)=8,000+15,000=23,00020(400) + 25(600) = 8,000 + 15,000 = 23,000. From Statement (2), total sum is 30,00030,000, so the 15 BOTH workshops sum to 30,000−23,000=7,00030,000 - 23,000 = 7,000, with a median of 500500.
Case A: If 1 BOTH workshop has cost 00 and 14 have cost 500500 (median = 500500, sum = 7,0007,000), the 60 sorted values consist of 1 zero, 20 values of 400400, 14 values of 500500, and 25 values of 600600. The 30th and 31st values are both 500500, so overall median = 500500.
Case B: If 7 BOTH workshops have cost 00, 1 has cost 500500, and 7 have cost 642.85642.85 (median = 500500, sum = 7,0007,000), the sorted values put the 20 values of 400400 in positions 8–27, the single 500500 at position 28, and the 25 values of 600600 in positions 29–53. The 30th and 31st values are both 600600, so overall median = 600600.
Since the overall median can be 500500 or 600600 depending on how the costs within the overlapping group are distributed, both statements combined are insufficient.

Key Concept

Overlapping Sets and Data Sufficiency for Position-Based Measures (Median)
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Topics

Overlapping Sets, Statistics, and Data Distributions

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