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Zorluk: OrtaOverlapping Sets, Statistics, and Data Distributions

An agricultural research station evaluated a sample of 9090 fruit trees. Each tree was treated with Fertilizer X, Fertilizer Y, or both. Exactly 6060 trees were treated with Fertilizer X, and exactly 5050 trees were treated with Fertilizer Y. What was the average (arithmetic mean) yield, in kilograms, of all 9090 trees?

(1) The average yield of the trees treated with Fertilizer X was 4545 kg, and the average yield of the trees treated only with Fertilizer Y was 3535 kg.
(2) The average yield of the trees treated with Fertilizer Y was 4242 kg, and the average yield of the trees treated only with Fertilizer X was 4848 kg.

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

Cevap

Each statement alone is sufficient to answer the question.
Using the principal formula for overlapping sets N(Total)=N(X)+N(Y)−N(X ∩ Y)N(\text{Total}) = N(\text{X}) + N(\text{Y}) - N(\text{X } \cap \text{ Y}), we find that 90=60+50−N(X ∩ Y)90 = 60 + 50 - N(\text{X } \cap \text{ Y}), meaning exactly 2020 trees received both fertilizers. This partitions the 9090 trees into three mutually exclusive groups: 4040 trees receiving Only X, 2020 trees receiving Both, and 3030 trees receiving Only Y.

Statement (1) provides the average for all trees receiving X (which combines 'Only X' and 'Both', totaling 6060 trees) as 4545 kg, giving a subgroup total yield of 60×45=2,70060 \times 45 = 2,700 kg. It also gives the average for the remaining 3030 trees ('Only Y') as 3535 kg, giving 30×35=1,05030 \times 35 = 1,050 kg. Summing these gives the exact total yield of all 9090 trees (3,7503,750 kg), which allows computing a unique overall mean. Hence Statement (1) alone is sufficient.

Statement (2) provides the average for all trees receiving Y (combining 'Only Y' and 'Both', totaling 5050 trees) as 4242 kg, giving a subgroup total yield of 50×42=2,10050 \times 42 = 2,100 kg. It also gives the average for the remaining 4040 trees ('Only X') as 4848 kg, giving 40×48=1,92040 \times 48 = 1,920 kg. Summing these gives the exact total yield of all 9090 trees (4,0204,020 kg), which allows computing a unique overall mean. Hence Statement (2) alone is sufficient.

Since each statement alone is sufficient, the correct option is the one stating that each statement alone is sufficient.

Adım Adım Çözüm

1
Determine the number of trees in each disjoint subset using overlapping set principles.
Number of trees receiving both fertilizers is 2020; 'Only X' is 4040; 'Only Y' is 3030.
By the inclusion-exclusion principle: N(Total)=N(X)+N(Y)−N(Both)N(\text{Total}) = N(\text{X}) + N(\text{Y}) - N(\text{Both}). Thus, 90=60+50−N(Both)90 = 60 + 50 - N(\text{Both}), which yields N(Both)=20N(\text{Both}) = 20. Consequently, N(Only X)=60−20=40N(\text{Only X}) = 60 - 20 = 40 and N(Only Y)=50−20=30N(\text{Only Y}) = 50 - 20 = 30.
2
Evaluate Statement (1) independently.
Total yield =3,750= 3,750 kg, yielding a unique overall average of 3,75090=1253\frac{3,750}{90} = \frac{125}{3} kg.
Statement (1) gives the average yield for all 6060 trees treated with Fertilizer X (4545 kg) and for the 3030 trees treated only with Fertilizer Y (3535 kg). The total yield of all 9090 trees is (60×45)+(30×35)=2,700+1,050=3,750(60 \times 45) + (30 \times 35) = 2,700 + 1,050 = 3,750 kg. Dividing by 9090 gives a single, deterministic value.
3
Evaluate Statement (2) independently.
Total yield =4,020= 4,020 kg, yielding a unique overall average of 4,02090=1343\frac{4,020}{90} = \frac{134}{3} kg.
Statement (2) gives the average yield for all 5050 trees treated with Fertilizer Y (4242 kg) and for the 4040 trees treated only with Fertilizer X (4848 kg). The total yield of all 9090 trees is (50×42)+(40×48)=2,100+1,920=4,020(50 \times 42) + (40 \times 48) = 2,100 + 1,920 = 4,020 kg. Dividing by 9090 gives a single, deterministic value.
4
Synthesize data sufficiency evaluation.
Since each statement independently allows us to calculate the exact average yield, each statement alone is sufficient.
Data Sufficiency requires identifying whether each statement alone yields a unique solution to the question asked.

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