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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% sugar by weight, and combining them in the weight ratio 1 to 4 yields a blend that is 20% sugar by weight. How many kilograms of P must be used to make 200 kilograms of a blend that is 35% 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 out of 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% sugar and Q is 15% sugar.
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% of 200, which is 70 kilograms of sugar: 0.40x + 0.15(200 − x) = 70. That gives 0.40x + 30 − 0.15x = 70, so 0.25x = 40 and x = 160. Check: P contributes 0.40 × 160 = 64 kilograms of sugar and Q contributes 0.15 × 40 = 6 kilograms, for a total of 70 kilograms out of 200, which is exactly 35%. So the answer is E.
Note that alligation cannot shortcut this problem, since the two endpoint strengths are exactly what is hidden at the start, and you cannot verify any answer choice by plugging in without first reconstructing p and q.
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