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08 Jan 2025, 06:11
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Three ethanol-water solutions are mixed in a 5:3:2 ratio by volume. Is the percentage of ethanol in the resulting mixture by volume greater than 75%?
(1) The percentage of ethanol by volume in the first solution is at most 80%.
(2) The percentage of ethanol by volume in each of the second and third solutions is at most 70%.
(1) The percentage of ethanol by volume in the first solution is at most 80%.
(2) The percentage of ethanol by volume in each of the second and third solutions is at most 70%.
08 Jan 2025, 06:11
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Official Solution:
Three ethanol-water solutions are mixed in a 5:3:2 ratio by volume. Is the percentage of ethanol in the resulting mixture by volume greater than 75%?
For simplicity, let’s assume the three ethanol-water solutions are 50, 30, and 20 liters, respectively. The question asks whether the resulting mixture of 100 liters contains more than 75 liters of ethanol.
(1) The percentage of ethanol by volume in the first solution is at most 80%.
This implies that the first solution contributes at most 50 * 80% = 40 liters of ethanol. If the first solution contains exactly 40 liters of ethanol and the combined second and third solutions contribute more than 35 liters of ethanol, the answer would be YES. However, if the first solution contains exactly 40 liters of ethanol and the combined second and third solutions contribute 35 liters or less, the answer would be NO. Not sufficient.
(2) The percentage of ethanol by volume in each of the second and third solutions is at most 70%.
This implies that the second and third solutions together contribute at most (30 + 20) * 70% = 35 liters of ethanol. Similarly, depending on the contribution of the first solution, we can get both YES and NO answers to the question. Not sufficient.
(1) + (2) With both statements combined, the first solution contributes at most 40 liters of ethanol, and the second and third solutions contribute at most 35 liters of ethanol. This means the final mixture could contain at most 40 + 35 = 75 liters of ethanol, which is not more than 75. Thus, the answer to the question is definitively NO. Sufficient.
Answer: C
Three ethanol-water solutions are mixed in a 5:3:2 ratio by volume. Is the percentage of ethanol in the resulting mixture by volume greater than 75%?
For simplicity, let’s assume the three ethanol-water solutions are 50, 30, and 20 liters, respectively. The question asks whether the resulting mixture of 100 liters contains more than 75 liters of ethanol.
(1) The percentage of ethanol by volume in the first solution is at most 80%.
This implies that the first solution contributes at most 50 * 80% = 40 liters of ethanol. If the first solution contains exactly 40 liters of ethanol and the combined second and third solutions contribute more than 35 liters of ethanol, the answer would be YES. However, if the first solution contains exactly 40 liters of ethanol and the combined second and third solutions contribute 35 liters or less, the answer would be NO. Not sufficient.
(2) The percentage of ethanol by volume in each of the second and third solutions is at most 70%.
This implies that the second and third solutions together contribute at most (30 + 20) * 70% = 35 liters of ethanol. Similarly, depending on the contribution of the first solution, we can get both YES and NO answers to the question. Not sufficient.
(1) + (2) With both statements combined, the first solution contributes at most 40 liters of ethanol, and the second and third solutions contribute at most 35 liters of ethanol. This means the final mixture could contain at most 40 + 35 = 75 liters of ethanol, which is not more than 75. Thus, the answer to the question is definitively NO. Sufficient.
Answer: C
