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29 Jul 2024, 07:43
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A mixture of orange and carrot juices consists of \(x\) liters of orange juice and \(y\) liters of carrot juice. What percent of the mixture, by volume, is orange juice?
(1) If 2 liters of carrot juice were replaced with 2 liters of orange juice, the percentage of orange juice by volume in the mixture would double.
(2) If half of the carrot juice by volume were replaced with an equal amount of orange juice, the percentage of orange juice by volume in the mixture would double.
(1) If 2 liters of carrot juice were replaced with 2 liters of orange juice, the percentage of orange juice by volume in the mixture would double.
(2) If half of the carrot juice by volume were replaced with an equal amount of orange juice, the percentage of orange juice by volume in the mixture would double.
29 Jul 2024, 07:43
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Official Solution:
A mixture of orange and carrot juices consists of \(x\) liters of orange juice and \(y\) liters of carrot juice. What percent of the mixture, by volume, is orange juice?
Essentially, the question asks to find the value of \(\frac{x}{x + y}*100\).
(1) If 2 liters of carrot juice were replaced with 2 liters of orange juice, the percentage of orange juice by volume in the mixture would double.
This operation increases the amount of orange juice by 2 liters while keeping the total volume the same. This implies that 2 liters is enough to double the original percentage, meaning the original amount of orange juice must also be 2 liters. However, without knowing the value of \(y\), we cannot calculate \(\frac{2}{2 + y} * 100\). Not sufficient.
* For those preferring algebra, here is what this statement means:
\(\frac{x + 2}{x + y} * 100= 2p\).
Since \(\frac{x}{x + y} * 100= p\), then:
\(\frac{x + 2}{x + y} * 100= 2(\frac{x}{x + y} * 100)\)
\(x + 2 = 2x\)
\(x =2\).
(2) If half of the carrot juice by volume were replaced with an equal amount of orange juice, the percentage of orange juice by volume in the mixture would double.
This operation implies that replacing \(\frac{y}{2}\) liters of carrot juice with \(\frac{y}{2}\) liters of orange juice doubles the percentage of orange juice in the mixture. By the same logic as above, we can infer that \(x = \frac{y}{2}\), which gives \(y = 2x\). Therefore, \(\frac{x}{x + y}*100 = \frac{x}{x + 2x} * 100 = 33 \frac{1}{3}\%\). Sufficient.
* For those preferring algebra, here is what this statement means:
\(\frac{x + \frac{y}{2}}{x + y} * 100= 2p\).
Since \(\frac{x}{x + y}*100=p\), then:
\(\frac{x + \frac{y}{2}}{x + y} * 100= 2(\frac{x}{x + y} * 100)\)
\(x +\frac{y}{2}= 2x\)
\(y =2x\).
Answer: B
A mixture of orange and carrot juices consists of \(x\) liters of orange juice and \(y\) liters of carrot juice. What percent of the mixture, by volume, is orange juice?
Essentially, the question asks to find the value of \(\frac{x}{x + y}*100\).
(1) If 2 liters of carrot juice were replaced with 2 liters of orange juice, the percentage of orange juice by volume in the mixture would double.
This operation increases the amount of orange juice by 2 liters while keeping the total volume the same. This implies that 2 liters is enough to double the original percentage, meaning the original amount of orange juice must also be 2 liters. However, without knowing the value of \(y\), we cannot calculate \(\frac{2}{2 + y} * 100\). Not sufficient.
* For those preferring algebra, here is what this statement means:
\(\frac{x + 2}{x + y} * 100= 2p\).
Since \(\frac{x}{x + y} * 100= p\), then:
\(\frac{x + 2}{x + y} * 100= 2(\frac{x}{x + y} * 100)\)
\(x + 2 = 2x\)
\(x =2\).
(2) If half of the carrot juice by volume were replaced with an equal amount of orange juice, the percentage of orange juice by volume in the mixture would double.
This operation implies that replacing \(\frac{y}{2}\) liters of carrot juice with \(\frac{y}{2}\) liters of orange juice doubles the percentage of orange juice in the mixture. By the same logic as above, we can infer that \(x = \frac{y}{2}\), which gives \(y = 2x\). Therefore, \(\frac{x}{x + y}*100 = \frac{x}{x + 2x} * 100 = 33 \frac{1}{3}\%\). Sufficient.
* For those preferring algebra, here is what this statement means:
\(\frac{x + \frac{y}{2}}{x + y} * 100= 2p\).
Since \(\frac{x}{x + y}*100=p\), then:
\(\frac{x + \frac{y}{2}}{x + y} * 100= 2(\frac{x}{x + y} * 100)\)
\(x +\frac{y}{2}= 2x\)
\(y =2x\).
Mustafa10102175
02 Aug 2024, 05:53
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I think this is a high-quality question
