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11 Jul 2024, 12:59
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Bunuel
The mixture has \(\frac{x}{x+y}*100\) liters of orange juice.
Statement 1
Two liters of carrot juice were replaced so now we have:
\(\frac{x+2}{x+y}*100\) (note that the total volume remains the same, but there is just more orange juice)
We are told the percentage of orange juice would double so we know that:
\(\frac{x+2}{x+y}*100 = 2*(\frac{x}{x+y}*100)\)
\(\frac{x+2}{x+y} = 2*(\frac{x}{x+y})\)
\(x+2=2x\)
\(x=2\) but we don't know y so we cannot find the percentage of orange juice.
-> Statement 1 alone is not sufficient.
Statement 2
Half the carrot juice was replaced with orange juice so:
\(\frac{x+0.5y}{x+y}*100\) (note that the total volume remains the same, but there is just more orange juice)
And we know that the percentage of orange juice would double:
\(\frac{x+0.5y}{x+y}*100 = 2*(\frac{x}{x+y}*100)\) -> we can do a lot of cancelling out here on both sides
\(x+0.5y=2x\)
\(y = 2x\)
Therefore,
\(\frac{x}{x+y}*100\) becomes
\(\frac{x}{x+2x}*100\)
\(\frac{1}{3}*100 = 33.3% \)
-> Statement 2 alone is sufficient.
The answer is B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
ThatIndianGuy
11 Jul 2024, 13:29
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What is required: x/(x+y) = p%
S1:
Given
2p% = (x+2)/(x+y)
not sufficient
S2:
2p%= (3x/2)/(3x/2+y/2)
not sufficient.
Together will also not give value of p%.
E
S1:
Given
2p% = (x+2)/(x+y)
not sufficient
S2:
2p%= (3x/2)/(3x/2+y/2)
not sufficient.
Together will also not give value of p%.
E
11 Jul 2024, 13:52
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Statement (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.
Let x be the initial volume of orange juice and y be the initial volume of carrot juice. The initial percentage of orange juice in the mixture is:
x/(x+y)
According to the statement, replacing 2 liters of carrot juice with 2 liters of orange juice results in x+2 liters of orange juice and y−2 liters of carrot juice. The new percentage of orange juice is:
x+2/(x+2+y−2)=x+2/(x+y)
The statement tells us that this new percentage is double the original percentage:
x+2/x+y=2⋅(x/x+y)
Simplifying this, we get:
x+2/x+y=2x/x+y
Equating the numerators since the denominators are the same:
x+2=2x
2=x
Therefore, x=2
To find the percentage of orange juice in the original mixture:
x/x+y=2/2+y
We don't have the value of y yet, so this alone is insufficient to find the percentage.
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.
The initial percentage of orange juice in the mixture is still:
x/(x+y)
Replacing half of the carrot juice with an equal amount of orange juice results in:
x+y/2
and the remaining carrot juice is:
y/2
The new percentage of orange juice in the mixture is:
(x+y/2)/(x+y/2+y/2)=(x+y/2)/(x+y)
According to the statement, this new percentage is double the original percentage:
(x+y/2)/(x+y)=2⋅x/(x+y)
Simplifying this, we get:
x+y/2/x+y=2x/x+y
Equating the numerators:
x+y/2=2x
y/2=x
y=2x
Now we know y=2x
Substituting y in terms of x into the initial percentage:
x/x+2x=x/3x=1/3
The percentage of orange juice by volume in the original mixture is 1/3 or approximately 33.33%
Therefore, Statement (2) alone is sufficient to answer the question.
The correct answer is:
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
Let x be the initial volume of orange juice and y be the initial volume of carrot juice. The initial percentage of orange juice in the mixture is:
x/(x+y)
According to the statement, replacing 2 liters of carrot juice with 2 liters of orange juice results in x+2 liters of orange juice and y−2 liters of carrot juice. The new percentage of orange juice is:
x+2/(x+2+y−2)=x+2/(x+y)
The statement tells us that this new percentage is double the original percentage:
x+2/x+y=2⋅(x/x+y)
Simplifying this, we get:
x+2/x+y=2x/x+y
Equating the numerators since the denominators are the same:
x+2=2x
2=x
Therefore, x=2
To find the percentage of orange juice in the original mixture:
x/x+y=2/2+y
We don't have the value of y yet, so this alone is insufficient to find the percentage.
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.
The initial percentage of orange juice in the mixture is still:
x/(x+y)
Replacing half of the carrot juice with an equal amount of orange juice results in:
x+y/2
and the remaining carrot juice is:
y/2
The new percentage of orange juice in the mixture is:
(x+y/2)/(x+y/2+y/2)=(x+y/2)/(x+y)
According to the statement, this new percentage is double the original percentage:
(x+y/2)/(x+y)=2⋅x/(x+y)
Simplifying this, we get:
x+y/2/x+y=2x/x+y
Equating the numerators:
x+y/2=2x
y/2=x
y=2x
Now we know y=2x
Substituting y in terms of x into the initial percentage:
x/x+2x=x/3x=1/3
The percentage of orange juice by volume in the original mixture is 1/3 or approximately 33.33%
Therefore, Statement (2) alone is sufficient to answer the question.
The correct answer is:
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
