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

A biopharmaceutical facility operates two automated synthesis lines, Line X and Line Y, to produce a therapeutic compound. On a specific shift, Line X operated for hXh_X hours at a constant production rate of rXr_X liters per hour, and Line Y operated for hYh_Y hours at a constant production rate of rYr_Y liters per hour. Was the overall average hourly production rate for both lines combined across the total operating hours greater than 45 liters per hour?

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

Answer

Each statement alone is sufficient.
To determine whether the combined average hourly rate Ravg=hXrX+hYrYhX+hYR_{avg} = \frac{h_X r_X + h_Y r_Y}{h_X + h_Y} is greater than 45 liters per hour, we evaluate each statement independently. Statement (1) provides rX=40r_X = 40, rY=60r_Y = 60, and hX=2hYh_X = 2 h_Y. Substituting hX=2hYh_X = 2 h_Y gives Ravg=2hY(40)+hY(60)3hY=1403=4623R_{avg} = \frac{2 h_Y (40) + h_Y (60)}{3 h_Y} = \frac{140}{3} = 46\frac{2}{3} liters per hour. Since 4623>4546\frac{2}{3} > 45, Statement (1) alone is sufficient. Statement (2) provides rX=50r_X = 50 and rY=55r_Y = 55. Because any weighted average of two positive values must lie strictly between the smaller and larger value (50<Ravg<5550 < R_{avg} < 55), the combined rate must be strictly greater than 50, which guarantees it is greater than 45. Thus, Statement (2) alone is also sufficient. Since each statement independently yields a definitive 'Yes', the correct choice is that each statement alone is sufficient.

Step-by-Step Solution

1
Formulate the algebraic expression for the target condition.
The combined average hourly rate is Ravg=hXrX+hYrYhX+hYR_{avg} = \frac{h_X r_X + h_Y r_Y}{h_X + h_Y}. We need to determine if Ravg>45R_{avg} > 45.
Establishing the target formula is necessary to analyze statement inputs.
2
Evaluate Statement (1): rX=40r_X = 40, rY=60r_Y = 60, and hX=2hYh_X = 2 h_Y.
Substitute hX=2hYh_X = 2 h_Y into RavgR_{avg}: Ravg=2hY(40)+hY(60)2hY+hY=140hY3hY=1403=46.67R_{avg} = \frac{2 h_Y (40) + h_Y (60)}{2 h_Y + h_Y} = \frac{140 h_Y}{3 h_Y} = \frac{140}{3} = 46.67 liters per hour. Since 46.67>4546.67 > 45, Statement (1) yields a definitive 'Yes'.
Knowing the ratio of hours allows the variable hYh_Y to cancel out, providing an exact rate.
3
Evaluate Statement (2): rX=50r_X = 50 and rY=55r_Y = 55.
Since both individual rates rXr_X and rYr_Y are strictly greater than 45 liters per hour, any weighted average of these two positive quantities must lie between 50 and 55 liters per hour (50<Ravg<5550 < R_{avg} < 55). Thus, RavgR_{avg} is guaranteed to be greater than 45, yielding a definitive 'Yes'.
A weighted average of two values is strictly bounded by the minimum and maximum of those values.
4
Conclude Data Sufficiency decision.
Since Statement (1) alone yields a definitive 'Yes' and Statement (2) alone yields a definitive 'Yes', each statement alone is sufficient.
Each statement independently answers the question.

Key Concept

Weighted Average Bounds in Data Sufficiency
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Topics

Rates, Ratios, Percentages, and Applied Word Problems

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