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A roaster's blends P and Q are each made only from Arabica beans and Robusta beans. A batch of 10 pounds of blend P and 30 pounds of blend Q is 40 percent Arabica by weight, and a batch of 30 pounds of P and 10 pounds of Q is 60 percent Arabica by weight. What percent, by weight, of a batch made from 24 pounds of P and 16 pounds of Q is Arabica?
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Correct answer
C
No per-blend Arabica fractions are given, so treat the Arabica fraction of P and of Q as unknowns and read the two given batches as two equations. Let p be the Arabica fraction of blend P and q the Arabica fraction of blend Q. Batch 1 (10 lb P, 30 lb Q, 40 lb total, 40 percent Arabica): 10p + 30q = 16, so p + 3q = 1.6. Batch 2 (30 lb P, 10 lb Q, 40 lb total, 60 percent Arabica): 30p + 10q = 24, so 3p + q = 2.4. Subtracting the first from the second gives 2p - 2q = 0.8, so p minus q = 0.4. Adding the two original equations gives 4p + 4q = 4, so p + q = 1. Therefore p = 0.7 and q = 0.3: blend P is 70 percent Arabica and blend Q is 30 percent.
For the new batch (24 lb P, 16 lb Q, 40 lb total), the Arabica weight is 24 × 0.7 + 16 × 0.3 = 16.8 + 4.8 = 21.6 pounds, which is 21.6 ÷ 40 = 54 percent. So the answer is C.
Among the traps: B (50 percent) takes the quick p + q = 1 and reports the midpoint, but the midpoint holds only for a 1-to-1 mix; the new batch is 3-to-2, weighting P more, so it must exceed 50 percent. E (60 percent) reuses Batch 2's figure, but the new 3-to-2 ratio matches neither given batch. A (46 percent) assigns each blend the other's Arabica fraction, a variable-swap slip. D (56 percent) sets up correctly but slips in the final addition, reading 16.8 + 4.8 as 22.4 rather than 21.6.
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