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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% Arabica by weight, and a batch of 30 pounds of P and 10 pounds of Q is 60% 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
The problem never states each blend's Arabica fraction directly, so treat those fractions as unknowns and use the two given batches as two equations. Let p be the Arabica fraction of blend P and q be the Arabica fraction of blend Q.
Batch 1 (10 lb P, 30 lb Q, 40 lb total, 40% Arabica): 10p + 30q = 16, so p + 3q = 1.6. Batch 2 (30 lb P, 10 lb Q, 40 lb total, 60% Arabica): 30p + 10q = 24, so 3p + q = 2.4.
Subtracting the first equation from the second gives 2p - 2q = 0.8, so p - q = 0.4. Adding the two original equations gives 4p + 4q = 4, so p + q = 1. Combining these two results gives p = 0.7 and q = 0.3: blend P is 70% Arabica and blend Q is 30% Arabica.
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%. The answer is C.
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