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An agricultural processing facility produces a liquid concentrate by blending organic apple juice, which contains 12%12\% sugar by volume, and organic grape juice, which contains 18%18\% sugar by volume. If a batch of the concentrate is made by mixing only these two types of juice, is the total volume of grape juice used in the batch greater than the total volume of apple juice used in the batch?

(1) The concentration of sugar in the batch of concentrate is 16%16\% by volume.
(2) The total volume of the batch of concentrate is 150150 liters.

  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.
Evaluating Statement (1) alone shows that a batch with a 16%16\% overall sugar concentration made from 12%12\% and 18%18\% components must contain twice as much grape juice as apple juice (G=2AG = 2A). Because juice volumes are strictly positive, GG must be greater than AA, providing a definitive 'Yes' answer. Statement (2) alone provides only the total volume A+G=150A + G = 150, which allows for infinitely many combinations of AA and GG where G>AG > A could be either true or false. Thus, Statement (1) alone is sufficient and Statement (2) alone is not sufficient.

Adım Adım Çözüm

1
Rephrase the question target using variables
Let AA be the volume of apple juice and GG be the volume of grape juice in liters. The question asks: Is G>AG > A?
Establishing clear algebraic expressions simplifies evaluating sufficiency.
2
Evaluate Statement (1) algebraically
Total sugar volume = 0.12A+0.18G0.12A + 0.18G. Total batch volume = A+GA + G.
Given sugar concentration = 16%16\%, set up equation: 0.12A+0.18GA+G=0.16\frac{0.12A + 0.18G}{A + G} = 0.16.
Multiply out: 0.12A+0.18G=0.16A+0.16G  ⟹  0.02G=0.04A  ⟹  G=2A0.12A + 0.18G = 0.16A + 0.16G \implies 0.02G = 0.04A \implies G = 2A.
Since volumes are positive (A>0A > 0), G=2A>AG = 2A > A is ALWAYS true (Answer is a definitive 'Yes').
Statement (1) uniquely fixes the ratio of grape juice to apple juice at 2:12:1, which fully answers the Yes/No question.
3
Evaluate Statement (2) independently
Given A+G=150A + G = 150.
Case 1: If A=100A = 100 and G=50G = 50, then G>AG > A is False ('No').
Case 2: If A=50A = 50 and G=100G = 100, then G>AG > A is True ('Yes').
Since both 'Yes' and 'No' are possible, Statement (2) is NOT sufficient.
Without concentration or proportion data, knowing total volume alone does not determine which component has the larger volume.

Anahtar Kavram

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Rates, Ratios, Percentages, and Applied Word Problems

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