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11 Jul 2024, 21:27
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From the task, our question is:
\(R = \frac{x }{ x+y} = 1?\)
\(\frac{x+2 }{ x+y} = 2R = \frac{2x }{ x+y}\)
Therefore, \(x+2 = 2x\) and \(x = 2.\)
This is useful but doesn't give us anything about y. This is not sufficient.
\(\frac{x + 0.5y }{ x+y} = 2R = \frac{2x }{ x+y}\)
Therefore, \(x + 0.5y = 2x\) and \(x = 0.5y\)
The ratio R will look: \(R = \frac{x }{ x+y} =\frac{ 0.5y }{ 0.5y + y} = \frac{0.5y }{ 1.5y }=\frac{ 1}{3}\), and this is litreally what we are asked. This is sufficient.
The right answer is B.
\(R = \frac{x }{ x+y} = 1?\)
- (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.
\(\frac{x+2 }{ x+y} = 2R = \frac{2x }{ x+y}\)
Therefore, \(x+2 = 2x\) and \(x = 2.\)
This is useful but doesn't give us anything about y. This is not sufficient.
- (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.
\(\frac{x + 0.5y }{ x+y} = 2R = \frac{2x }{ x+y}\)
Therefore, \(x + 0.5y = 2x\) and \(x = 0.5y\)
The ratio R will look: \(R = \frac{x }{ x+y} =\frac{ 0.5y }{ 0.5y + y} = \frac{0.5y }{ 1.5y }=\frac{ 1}{3}\), and this is litreally what we are asked. This is sufficient.
The right answer is B.
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11 Jul 2024, 21:37
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To determine the percentage of the mixture that is orange juice, we need to find the value of \(\frac{x}{x+y} \times 100\).
### Statement (1):
If 2 liters of carrot juice are replaced with 2 liters of orange juice, the percentage of orange juice by volume in the mixture would double.
Initially, the percentage of orange juice in the mixture is:
\[
\frac{x}{x + y} \times 100
\]
After replacing 2 liters of carrot juice with 2 liters of orange juice, the new volumes of orange juice and carrot juice are \( x + 2 \) and \( y - 2 \), respectively. The new percentage of orange juice is:
\[
\frac{x + 2}{(x + 2) + (y - 2)} \times 100 = \frac{x + 2}{x + y} \times 100
\]
According to the statement, this new percentage is double the initial percentage:
\[
\frac{x + 2}{x + y} = 2 \times \frac{x}{x + y}
\]
Simplifying, we get:
\[
\frac{x + 2}{x + y} = \frac{2x}{x + y}
\]
Cross-multiplying, we get:
\[
x + 2 = 2x \implies x = 2
\]
So, we now have \( x = 2 \), but we need \( y \) to determine the percentage. Statement (1) alone is not sufficient to determine \( y \).
### Statement (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.
Initially, the percentage of orange juice in the mixture is:
\[
\frac{x}{x + y} \times 100
\]
After replacing half of the carrot juice with an equal amount of orange juice, the new volumes of orange juice and carrot juice are \( x + \frac{y}{2} \) and \( \frac{y}{2} \), respectively. The new percentage of orange juice is:
\[
\frac{x + \frac{y}{2}}{x + \frac{y}{2} + \frac{y}{2}} \times 100 = \frac{x + \frac{y}{2}}{x + y} \times 100
\]
According to the statement, this new percentage is double the initial percentage:
\[
\frac{x + \frac{y}{2}}{x + y} = 2 \times \frac{x}{x + y}
\]
Simplifying, we get:
\[
x + \frac{y}{2} = 2x \implies \frac{y}{2} = x \implies y = 2x
\]
From this, we know \( y = 2x \), and thus:
\[
\frac{x}{x + y} = \frac{x}{x + 2x} = \frac{x}{3x} = \frac{1}{3}
\]
So, the percentage of orange juice by volume is:
\[
\frac{1}{3} \times 100 = 33.33\%
\]
Therefore, Statement (2) alone is sufficient to determine the percentage of orange juice by volume.
### Statement (1):
If 2 liters of carrot juice are replaced with 2 liters of orange juice, the percentage of orange juice by volume in the mixture would double.
Initially, the percentage of orange juice in the mixture is:
\[
\frac{x}{x + y} \times 100
\]
After replacing 2 liters of carrot juice with 2 liters of orange juice, the new volumes of orange juice and carrot juice are \( x + 2 \) and \( y - 2 \), respectively. The new percentage of orange juice is:
\[
\frac{x + 2}{(x + 2) + (y - 2)} \times 100 = \frac{x + 2}{x + y} \times 100
\]
According to the statement, this new percentage is double the initial percentage:
\[
\frac{x + 2}{x + y} = 2 \times \frac{x}{x + y}
\]
Simplifying, we get:
\[
\frac{x + 2}{x + y} = \frac{2x}{x + y}
\]
Cross-multiplying, we get:
\[
x + 2 = 2x \implies x = 2
\]
So, we now have \( x = 2 \), but we need \( y \) to determine the percentage. Statement (1) alone is not sufficient to determine \( y \).
### Statement (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.
Initially, the percentage of orange juice in the mixture is:
\[
\frac{x}{x + y} \times 100
\]
After replacing half of the carrot juice with an equal amount of orange juice, the new volumes of orange juice and carrot juice are \( x + \frac{y}{2} \) and \( \frac{y}{2} \), respectively. The new percentage of orange juice is:
\[
\frac{x + \frac{y}{2}}{x + \frac{y}{2} + \frac{y}{2}} \times 100 = \frac{x + \frac{y}{2}}{x + y} \times 100
\]
According to the statement, this new percentage is double the initial percentage:
\[
\frac{x + \frac{y}{2}}{x + y} = 2 \times \frac{x}{x + y}
\]
Simplifying, we get:
\[
x + \frac{y}{2} = 2x \implies \frac{y}{2} = x \implies y = 2x
\]
From this, we know \( y = 2x \), and thus:
\[
\frac{x}{x + y} = \frac{x}{x + 2x} = \frac{x}{3x} = \frac{1}{3}
\]
So, the percentage of orange juice by volume is:
\[
\frac{1}{3} \times 100 = 33.33\%
\]
Therefore, Statement (2) alone is sufficient to determine the percentage of orange juice by volume.
11 Jul 2024, 22:05
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Bunuel
According to me, both the statements are sufficient since they do provide what we are looking for.
we need to find x/x+y*100 and we can do that through both the statements
D. EACH statement ALONE is sufficient to answer the question asked.
