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Zorluk: Çok zorRates, Ratios, Percentages, and Applied Word Problems

A manufacturing plant produces two types of electronic components: Component A and Component B. In Year 1, the unit production cost of Component A was 20%20\% greater than the unit production cost of Component B. In Year 2, the unit production cost of Component A increased by 10%10\%, while the unit production cost of Component B increased by 25%25\%. Was the overall average unit production cost across all components produced by the plant higher in Year 2 than in Year 1?

(1) In Year 2, the plant produced 50%50\% more units of Component B than it did in Year 1.
(2) In Year 1, Component A accounted for 60%60\% of the total number of components produced by the plant.

  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 with a definitive 'Yes'.
The correct answer is the option stating that EACH statement ALONE is sufficient. Analyzing the range of possible weighted averages directly from the stem reveals that the minimum possible average unit cost in Year 2 (1.25c1.25c) is strictly greater than the maximum possible average unit cost in Year 1 (1.20c1.20c). Thus, the average unit cost in Year 2 must be greater than in Year 1 regardless of the proportion of Component A and Component B produced in either year. Since the stem alone proves a definitive 'Yes', each statement alone is sufficient.

Adım Adım Çözüm

1
Define variables for Year 1 unit costs and establish the range of Year 1 overall average cost.
Let the Year 1 unit cost of Component B be cc, where c>0c > 0. Then the Year 1 unit cost of Component A is 1.20c1.20c. Any weighted average of these two costs in Year 1, Avg1\text{Avg}_1, must satisfy c≤Avg1≤1.20cc \le \text{Avg}_1 \le 1.20c. In particular, Avg1≤1.20c\text{Avg}_1 \le 1.20c.
The overall weighted average of a set of values cannot exceed the maximum individual value in that set.
2
Calculate Year 2 unit costs and establish the range of Year 2 overall average cost.
In Year 2, the unit cost of Component A becomes 1.20c×1.10=1.32c1.20c \times 1.10 = 1.32c. The unit cost of Component B becomes c×1.25=1.25cc \times 1.25 = 1.25c. Any weighted average in Year 2, Avg2\text{Avg}_2, must satisfy 1.25c≤Avg2≤1.32c1.25c \le \text{Avg}_2 \le 1.32c. In particular, Avg2≥1.25c\text{Avg}_2 \ge 1.25c.
The overall weighted average of a set of values cannot be less than the minimum individual value in that set.
3
Compare the upper bound of Year 1 average cost with the lower bound of Year 2 average cost.
Since Avg2≥1.25c>1.20c≥Avg1\text{Avg}_2 \ge 1.25c > 1.20c \ge \text{Avg}_1, it follows that Avg2>Avg1\text{Avg}_2 > \text{Avg}_1 holds unconditionally for all possible production volumes and ratios in both years.
The minimum possible average cost in Year 2 (1.25c1.25c) is strictly greater than the maximum possible average cost in Year 1 (1.20c1.20c).
4
Evaluate Data Sufficiency statements based on the stem analysis.
Because the stem alone yields a definitive 'Yes' answer, Statement (1) alone is sufficient and Statement (2) alone is sufficient.
When the question stem itself contains sufficient information to answer the target question definitively, each statement independently yields a definitive answer.

Anahtar Kavram

Weighted Average Extreme Value Bounds in Data Sufficiency
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Rates, Ratios, Percentages, and Applied Word Problems

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