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Difficulty: HardDrawing Inferences and Valid Conclusions

In the mid-twentieth century, manufacturing plants employing automated assembly lines recorded higher worker productivity per hour and lower unit production costs than plants using manual assembly methods. Nevertheless, among plants that operated profitably and sold 100 percent of their output, total operating profits were consistently higher at manual plants producing custom, low-volume goods than at automated plants producing standardized, high-volume goods.

If the statements above are true, which of the following must also be true regarding the manufacturing plants during the specified period?

  1. The average selling price per unit of custom goods produced at manual plants was higher than the average selling price per unit of standardized goods produced at automated plants.Answer
  2. B
    Manual plants generated higher total annual revenues than automated plants did.
  3. C
    Worker wages at automated plants were higher than worker wages at manual plants due to increased productivity per hour.
  4. D
    Automated assembly lines were less cost-effective overall than manual assembly methods for all types of manufactured goods.
  5. E
    If a manual plant producing custom goods adopted automated assembly lines without altering its product line, its total operating profits would decrease.

Answer

The average selling price per unit of custom goods produced at manual plants was higher than the average selling price per unit of standardized goods produced at automated plants.
The correct answer must be true based on the algebraic relationship between total profit, production volume, unit costs, and unit selling prices. Given that manual plants produced a lower volume of goods (QM<QAQ_M < Q_A) with higher unit production costs (CM>CAC_M > C_A) yet earned higher total operating profit (QM⋅MarginM>QA⋅MarginAQ_M \cdot Margin_M > Q_A \cdot Margin_A), the profit margin per unit for custom goods (PriceM−CMPrice_M - C_M) had to be greater than that for standardized goods (PriceA−CAPrice_A - C_A). Combining PriceM−CM>PriceA−CAPrice_M - C_M > Price_A - C_A with CM>CAC_M > C_A logically forces PriceM>PriceAPrice_M > Price_A.

Step-by-Step Solution

1
Analyze the mathematical relationship for total profit.
Total Profit = Quantity Sold × (Unit Selling Price - Unit Cost).
Establishing the core profit formula allows us to compare manual plants producing custom goods (MM) and automated plants producing standardized goods (AA).
2
Identify the given comparative premises.
1) Unit Cost for manual plants is greater than Unit Cost for automated plants (CM>CAC_M > C_A). 2) Quantity sold for manual plants is lower than for automated plants (QM<QAQ_M < Q_A). 3) Total Profit for manual plants is greater than for automated plants (ProfitM>ProfitA>0Profit_M > Profit_A > 0).
Extracting the quantitative constraints provided in the passage sets up the logical inequality.
3
Evaluate the profit per unit constraint.
Since QM⋅(PriceM−CM)>QA⋅(PriceA−CA)Q_M \cdot (Price_M - C_M) > Q_A \cdot (Price_A - C_A) and 0<QM<QA0 < Q_M < Q_A, it must hold that (PriceM−CM)>(PriceA−CA)(Price_M - C_M) > (Price_A - C_A).
For a smaller quantity of goods to yield a larger total profit than a larger quantity of goods, the profit margin per unit for the smaller-quantity goods must be strictly greater.
4
Deduce the relationship between selling prices.
Since (PriceM−CM)>(PriceA−CA)(Price_M - C_M) > (Price_A - C_A) and CM>CAC_M > C_A, adding CMC_M to both sides gives PriceM>PriceA+(CM−CA)>PriceAPrice_M > Price_A + (C_M - C_A) > Price_A. Thus, PriceM>PriceAPrice_M > Price_A.
Because manual unit costs were higher, the unit selling price of custom goods produced at manual plants had to be higher than that of standardized goods to yield both a higher unit margin and higher overall profit.

Key Concept

Quantitative and Logical Deductions from Mathematical Constraints
Estimated Time:2m 0s
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Topics

Drawing Inferences and Valid Conclusions

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