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Zorluk: OrtaRates, Ratios, Percentages, and Applied Word Problems

A specialty coffee roaster produces a house blend composed entirely of Arabica beans, which cost 12perpound,andRobustabeans,whichcost12 per pound, and Robusta beans, which cost 8 per pound. What was the average cost per pound of the house blend produced last week?

(1) Arabica beans accounted for 60% of the total weight of the house blend produced last week.
(2) The total weight of the house blend produced last week was 500 pounds.

  1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.Cevap
  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. E
    Statements (1) and (2) TOGETHER are NOT sufficient.

Cevap

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
The average cost per pound of a blend depends only on the relative percentage weighting of its components. Since Statement (1) provides that Arabica beans constitute 60% of the blend, the remaining 40% must be Robusta. The weighted average cost per pound can be calculated directly as 0.60($12)+0.40($8)=$10.400.60(\$12) + 0.40(\$8) = \$10.40. Statement (2) gives total weight but no ratio information, leaving the average cost indeterminate. Therefore, Statement (1) alone is sufficient, but Statement (2) alone is not.

Adım Adım Çözüm

1
Formulate the algebraic target for weighted average cost per pound.
Let wAw_A be the proportion of total weight consisting of Arabica beans and wR=1−wAw_R = 1 - w_A be the proportion consisting of Robusta beans. Average cost per pound =12wA+8(1−wA)=4wA+8= 12 w_A + 8 (1 - w_A) = 4 w_A + 8.
Rephrasing the question stem shows that knowing the weight fraction wAw_A alone is sufficient to calculate a unique average cost per pound.
2
Evaluate Statement (1) independently.
Statement (1) specifies wA=0.60w_A = 0.60. Average cost per pound =4(0.60)+8=$10.40= 4(0.60) + 8 = \$10.40.
A single unique numerical value is obtained, making Statement (1) ALONE sufficient.
3
Evaluate Statement (2) independently.
Statement (2) specifies that total weight is 500500 pounds, but the proportion wAw_A can be any value between 00 and 11.
Without knowing the proportion of each bean type, the average cost per pound cannot be uniquely determined, so Statement (2) ALONE is not sufficient.

Anahtar Kavram

Weighted Average Ratios in Data Sufficiency
Tahmini Süre:1m 30s
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Rates, Ratios, Percentages, and Applied Word Problems

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