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

A coffee roaster creates a signature blend using only two types of beans: Arabica and Robusta. In a specific batch of this signature blend, what percentage of the total weight of the batch consists of Arabica beans?

(1) If 10 kilograms of Arabica beans were added to the batch, the ratio of the weight of Arabica beans to the weight of Robusta beans in the batch would be 3:23:2.
(2) The total cost of the Arabica beans in the batch is 50%50\% greater than the total cost of the Robusta beans in the batch, and Arabica beans cost 25%25\% more per kilogram than Robusta beans.

  1. A
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.Cevap
  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 (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Statement (2) alone allows us to express the price per kilogram of Arabica beans in terms of Robusta beans (pA=1.25pRp_A = 1.25 p_R) and set up an equation equating total costs (A⋅pA=1.50R⋅pRA \cdot p_A = 1.50 R \cdot p_R). Dividing out the unit price pRp_R gives a direct constant value for the weight ratio AR=65\frac{A}{R} = \frac{6}{5}, which uniquely determines the percentage of Arabica beans in the batch.

Adım Adım Çözüm

1
Define target variables and simplify the question stem.
Let AA be the weight of Arabica beans (in kg) and RR be the weight of Robusta beans (in kg). The target percentage is AA+R×100%\frac{A}{A + R} \times 100\%, which requires finding the ratio AR\frac{A}{R}.
Rephrasing the stem to finding the ratio AR\frac{A}{R} isolates the exact relation needed to answer the question.
2
Evaluate Statement (1) independently.
Statement (1) gives A+10R=32⇒2A+20=3R⇒3R−2A=20\frac{A + 10}{R} = \frac{3}{2} \Rightarrow 2A + 20 = 3R \Rightarrow 3R - 2A = 20. This is one linear equation with two unknowns (AA and RR).
Since the ratio AR\frac{A}{R} varies depending on the specific values of AA and RR (e.g., if A=2,R=8A=2, R=8, AR=14\frac{A}{R}=\frac{1}{4}; if A=14,R=16A=14, R=16, AR=78\frac{A}{R}=\frac{7}{8}), Statement (1) alone is NOT sufficient.
3
Evaluate Statement (2) independently.
Let pAp_A and pRp_R be the price per kg of Arabica and Robusta beans, respectively. We are given pA=1.25pR=54pRp_A = 1.25 p_R = \frac{5}{4} p_R. Total cost of Arabica = A⋅pAA \cdot p_A and total cost of Robusta = R⋅pRR \cdot p_R. Statement (2) states A⋅pA=1.50(R⋅pR)=32R⋅pRA \cdot p_A = 1.50 (R \cdot p_R) = \frac{3}{2} R \cdot p_R. Substituting pA=54pRp_A = \frac{5}{4} p_R gives A(54pR)=32R⋅pRA \left(\frac{5}{4} p_R\right) = \frac{3}{2} R \cdot p_R. Dividing both sides by pRp_R yields 54A=32R⇒AR=3/25/4=65\frac{5}{4} A = \frac{3}{2} R \Rightarrow \frac{A}{R} = \frac{3/2}{5/4} = \frac{6}{5}.
Knowing AR=65\frac{A}{R} = \frac{6}{5} allows us to calculate AA+R=66+5=611\frac{A}{A + R} = \frac{6}{6 + 5} = \frac{6}{11}, giving a unique percentage of 60011%≈54.55%\frac{600}{11}\% \approx 54.55\%. Thus, Statement (2) alone IS sufficient.

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Rates, Ratios, Percentages, and Applied Word Problems

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