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Difficulty: HardRates, Ratios, Percentages, and Applied Word Problems

A bio-refinery processes two grades of raw biomass, Grade Alpha and Grade Beta, to produce liquid biofuel. Grade Alpha yields 15%15\% biofuel per ton, while Grade Beta yields 35%35\% biofuel per ton. If a single production batch used a combined total of 1,2001,200 tons of Grade Alpha and Grade Beta biomass, resulting in VV total tons of biofuel, is V>300V > 300?

  1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.Answer
  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.

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
The correct answer is the option stating that Statement (1) ALONE is sufficient, but Statement (2) alone is not. Rephrasing the target condition V>300V > 300 yields A<600A < 600. Statement (1) provides A<533.33A < 533.33, which definitively proves A<600A < 600 (a conclusive YES). Statement (2) provides A<700A < 700, which permits values of AA both above and below 600600, rendering it insufficient.

Step-by-Step Solution

1
Set up algebraic equations for total biomass and total biofuel yield.
Let AA be the tons of Grade Alpha biomass and BB be the tons of Grade Beta biomass. Given A+B=1,200A + B = 1,200, we have B=1,200−AB = 1,200 - A. Total biofuel yield is V=0.15A+0.35BV = 0.15A + 0.35B.
Expressing VV in terms of a single variable AA simplifies the target question.
2
Rephrase the target question 'Is V>300V > 300?' in terms of variable AA.
Substitute B=1,200−AB = 1,200 - A into the yield equation: V=0.15A+0.35(1,200−A)=420−0.20AV = 0.15A + 0.35(1,200 - A) = 420 - 0.20A. The condition V>300V > 300 becomes 420−0.20A>300  ⟹  120>0.20A  ⟹  A<600420 - 0.20A > 300 \implies 120 > 0.20A \implies A < 600. Thus, the question asks: 'Is A<600A < 600?'
Rephrasing reduces the complex weighted average inequality to a straightforward threshold check on AA.
3
Evaluate Statement (1): The ratio of Grade Alpha to Grade Beta tonnage was less than 4:54:5.
AB<45  ⟹  5A<4B\frac{A}{B} < \frac{4}{5} \implies 5A < 4B. Substituting B=1,200−AB = 1,200 - A gives 5A<4(1,200−A)  ⟹  9A<4,800  ⟹  A<533.335A < 4(1,200 - A) \implies 9A < 4,800 \implies A < 533.33. Since A<533.33A < 533.33 guarantees that A<600A < 600, the answer to 'Is A<600A < 600?' is a definitive YES. Statement (1) is SUFFICIENT.
Determining if Statement (1) alone guarantees A<600A < 600.
4
Evaluate Statement (2): More than 500500 tons of Grade Beta biomass was used in the batch.
B>500  ⟹  1,200−A>500  ⟹  A<700B > 500 \implies 1,200 - A > 500 \implies A < 700. If A=500A = 500, then A<600A < 600 is TRUE (YES). If A=650A = 650, then A<600A < 600 is FALSE (NO). Because both YES and NO outcomes are possible, Statement (2) is NOT SUFFICIENT.
Checking whether Statement (2) produces a single definitive YES/NO answer.

Key Concept

Data Sufficiency Question Stem Rephrased for Weighted Averages and Mixture Ratios
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Topics

Rates, Ratios, Percentages, and Applied Word Problems

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