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31 Dec 2023, 07:33
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One liter of a Christmas cranberry mimosa cocktail consists of \(x\%\) champagne and \(y\%\) cranberry juice. A liter of champagne costs $12 and a liter of cranberry juice costs $2. Is the ratio of \(x\) to \(y\) greater than \(\frac{3}{2}\)?
(1) If the cocktail consisted of \(y\%\) champagne and \(x\%\) cranberry juice, it would cost more than $6.
(2) The total cost of the cocktail is less than $10.
(1) If the cocktail consisted of \(y\%\) champagne and \(x\%\) cranberry juice, it would cost more than $6.
(2) The total cost of the cocktail is less than $10.
31 Dec 2023, 07:33
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Official Solution:
One liter of a Christmas cranberry mimosa cocktail consists of \(x\%\) champagne and \(y\%\) cranberry juice. A liter of champagne costs $12 and a liter of cranberry juice costs $2. Is the ratio of \(x\) to \(y\) greater than \(\frac{3}{2}\)?
First, note that the cocktail is one liter, so \(x + y = 1\). The question asks whether \(\frac{x}{y}> \frac{3}{2}\), or if \(x\) is more than 60% (since \(\frac{x}{y}> \frac{60}{40}\)).
(1) If the cocktail consisted of \(y\%\) champagne and \(x\%\) cranberry juice, it would cost more than $6.
This implies \(12y + 2x > 6\). Substituting \(y\) with \(1 - x\) gives \(12(1 - x) + 2x > 6\), which simplifies to \(x < 0.6\). Hence, \(x\) is less than 60%. Sufficient.
(2) The total cost of the cocktail is less than $10.
This implies \(12x + 2y < 10\). Substituting \(y\) with \(1 - x\) gives \(12x + 2(1 - x) < 10\), which simplifies to \(x < 0.8\). Hence, \(x\) may or may not be more than 60%. Not sufficient.
Answer: A
One liter of a Christmas cranberry mimosa cocktail consists of \(x\%\) champagne and \(y\%\) cranberry juice. A liter of champagne costs $12 and a liter of cranberry juice costs $2. Is the ratio of \(x\) to \(y\) greater than \(\frac{3}{2}\)?
First, note that the cocktail is one liter, so \(x + y = 1\). The question asks whether \(\frac{x}{y}> \frac{3}{2}\), or if \(x\) is more than 60% (since \(\frac{x}{y}> \frac{60}{40}\)).
(1) If the cocktail consisted of \(y\%\) champagne and \(x\%\) cranberry juice, it would cost more than $6.
This implies \(12y + 2x > 6\). Substituting \(y\) with \(1 - x\) gives \(12(1 - x) + 2x > 6\), which simplifies to \(x < 0.6\). Hence, \(x\) is less than 60%. Sufficient.
(2) The total cost of the cocktail is less than $10.
This implies \(12x + 2y < 10\). Substituting \(y\) with \(1 - x\) gives \(12x + 2(1 - x) < 10\), which simplifies to \(x < 0.8\). Hence, \(x\) may or may not be more than 60%. Not sufficient.
Answer: A
10 Mar 2024, 10:45
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I think this is a high-quality question and I don't agree with the explanation. How did we conclude that cocktail only contains champagne and cranberry juice and assumed x+y=1?
