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Concentrates P and Q each have a fixed sugar content by weight. Combining P and Q in the weight ratio 3 to 2 yields a blend that is 30 percent sugar by weight, and combining them in the weight ratio 1 to 4 yields a blend that is 20 percent sugar by weight. How many kilograms of P must be used to make 200 kilograms of a blend that is 35 percent sugar by weight?
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Correct answer
E
The concentrate strengths are hidden, so recover them first. Let p and q be the sugar fractions of P and Q. A 3-to-2 blend means 3 parts P and 2 parts Q over 5 parts, so (3p + 2q)÷5 = 0.30, giving 3p + 2q = 1.5. A 1-to-4 blend gives (p + 4q)÷5 = 0.20, so p + 4q = 1.0. Multiply the second equation by 3 to get 3p + 12q = 3.0 and subtract the first: 10q = 1.5, so q = 0.15. Then p = 1.0 − 4(0.15) = 0.40. So P is 40 percent sugar and Q is 15 percent.
Now build the 200-kilogram target. Let x be the kilograms of P, so 200 − x is the kilograms of Q. The blend needs 35 percent of 200 = 70 kilograms of sugar: 0.40x + 0.15(200 − x) = 70. That is 0.40x + 30 − 0.15x = 70, so 0.25x = 40 and x = 160. Check: P gives 0.40 × 160 = 64 and Q gives 0.15 × 40 = 6, total 70, which is 35 percent. So the answer is E.
Note that alligation cannot shortcut this, since the two endpoint strengths are exactly what is concealed, and back-solving the answer choices fails because each is a weight you cannot verify without first reconstructing p and q.
The distractors map to specific errors. 40 (A) swaps which concentrate is stronger, and also catches solvers who report Q's amount. 100 (B) comes from misreading Product Two's ratio as 4 to 1. 120 (C) reuses Product One's 3-to-2 ratio instead of solving the new 35 percent blend. 150 (D) skips recovering the concentrates and treats the product percentages themselves as the stock strengths.
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