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

An industrial bakery uses two automated mixing machines, Machine X and Machine Y, operating simultaneously at their respective constant rates, to prepare batches of dough. How many hours would it take Machine X, working alone at its constant rate, to complete a production order of 1,2001,200 batches of dough?

(1) Working together at their respective constant rates, Machine X and Machine Y can complete the production order of 1,2001,200 batches of dough in 88 hours.
(2) Machine Y's constant production rate is 50%50\% greater than Machine X's constant production rate.

  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. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.Answer
  4. D
    EACH statement ALONE is sufficient.
  5. E
    Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

Both statements together are sufficient, but neither statement alone is sufficient.
The question asks for the time Machine X needs to produce 1,2001,200 batches alone, which requires finding Machine X's individual rate, rXr_X. Statement (1) gives the combined rate rX+rY=150r_X + r_Y = 150 batches/hour, which is insufficient on its own because rYr_Y is unknown. Statement (2) gives the relative relationship rY=1.5rXr_Y = 1.5 r_X, which is insufficient on its own because no absolute rate is given. Combining both statements yields rX+1.5rX=150r_X + 1.5 r_X = 150, giving a unique value rX=60r_X = 60 batches/hour and a single solution TX=20T_X = 20 hours. Therefore, both statements together are sufficient, but neither statement alone is sufficient.

Step-by-Step Solution

1
Define the target variable and rephrase the question stem.
Let rXr_X be Machine X's rate (batches per hour) and rYr_Y be Machine Y's rate. The required time for Machine X working alone is TX=1,200rXT_X = \frac{1,200}{r_X}. The target is to determine the exact value of rXr_X.
Rephrasing simplifies the Data Sufficiency goal from finding time to finding a unique numerical value for rXr_X.
2
Evaluate Statement (1) independently.
Statement (1) states (rX+rY)×8=1,200(r_X + r_Y) \times 8 = 1,200, which simplifies to rX+rY=150r_X + r_Y = 150 batches per hour.
Since there are two unknown rates (rXr_X and rYr_Y) and only one equation, rXr_X cannot be uniquely determined. Statement (1) alone is NOT sufficient.
3
Evaluate Statement (2) independently.
Statement (2) states rY=rX+0.50rX=1.5rX=32rXr_Y = r_X + 0.50 r_X = 1.5 r_X = \frac{3}{2} r_X.
This establishes a proportional relationship between the two rates but gives no absolute values. rXr_X can take infinitely many positive values, so Statement (2) alone is NOT sufficient.
4
Evaluate Statement (1) and Statement (2) together.
Substitute rY=1.5rXr_Y = 1.5 r_X into rX+rY=150r_X + r_Y = 150:
rX+1.5rX=150  ⟹  2.5rX=150  ⟹  rX=60r_X + 1.5 r_X = 150 \implies 2.5 r_X = 150 \implies r_X = 60 batches per hour.
Then TX=1,20060=20T_X = \frac{1,200}{60} = 20 hours.
Combining the statements yields a single linear equation with one variable, giving a unique value for rXr_X and thus for TXT_X. Both statements together are sufficient.

Key Concept

Data Sufficiency evaluation for combined work rate problems with linear algebraic systems
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Topics

Rates, Ratios, Percentages, and Applied Word Problems

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