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. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked. C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked. D. EACH statement ALONE is sufficient to answer the question asked. E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

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. Original orange juice volume: x Original carrot juice volume: y New orange juice volume: x+2 New carrot juice volume: y−2 New percentage: \frac{{x + 2}{x + y} Given: {x + 2}{x + y}} =2 × \frac{{x}{x+y} This simplifies to: x+2=2x ⟹ x=y−2 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. Original orange juice volume: x Original carrot juice volume: y New orange juice volume: x+ {y}{2}} New carrot juice volume:\frac{{y}{2} New percentage: x+ {y}{2}}\frac{x+y Given: x+ {y}{2}}\frac{x+y =2 × x x+y} This simplifies to: x+\frac{{y}{2} =2x⟹x= {y}{2}} From Statement (1): x=y−2 From Statement (2): x= {y}{2} Since the two statements provide different equations for x in terms of y, neither statement alone is sufficient, and together they provide conflicting information. Answer: E - Statements (1) and (2) TOGETHER are NOT sufficient to answer the question, and additional data is needed.

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GMAT Club Olympics 2024 (Day 4): A mixture of orange and carrot juices

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Juice mixture contains x liters of orange juice and y liters of carrot juice.
Percent of the mixture, by volume, is orange juice means (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.
After replacing 2 liters of carrot juice with 2 liters of orange juice, new volumes = x+2 liters of orange juice and
y−2 liters of carrot juice.
So, new percentage of orange juice = {x+2/(x+2)+y-2}*100 = (x+2/x+y)*100
As per this statement, new percentage = 2*original percentage,
So, (x+2/x+y)*100 = 2*(x/x+y)×100
upon simplification, x+2 = 2x, x=2.
No info about y. Hence, 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.
If half of the carrot juice y/2 replaced with same volume of orange juice, then new volumes become x+y/2 liters of orange juice and y/2 liters of carrot juice.
New percentage of orange juice = {(x+y/2)/x+y/2+y/2}*100= {(x+y/2)/x+y}*100
As per this statement, new percentage =2* original percentage
{(x+y/2)/x+y}*100 = 2*(x/x+y)*100
x+y/2= 2x
y=2x
Not sufficient as no values of x and y available.

1) +2) we know that x=2 & y=2x, so y=4. So we can substitute both x and y into x/x+y*100 and get unique value.
Hence sufficient.

IMO answer is C.

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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.

Original orange juice volume: x
Original carrot juice volume: y
New orange juice volume: x+2
New carrot juice volume: y−2
New percentage: \frac{{x + 2}{x + y}[}{fraction]
Given:[fraction]{x + 2}{x + y}} =2 × \frac{{x}{x+y}[}{fraction]

This simplifies to: x+2=2x ⟹ x=y−2

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.

Original orange juice volume: x
Original carrot juice volume: y
New orange juice volume: x+[fraction]{y}{2}}
New carrot juice volume:\frac{{y}{2}[}{fraction]
New percentage: x+[fraction]{y}{2}}\frac{x+y[}{fraction]
Given: x+[fraction]{y}{2}}\frac{x+y[}{fraction]=2 × x [fraction]x+y}

This simplifies to: x+\frac{{y}{2}[}{fraction]=2x⟹x=[fraction]{y}{2}}

From Statement (1): x=y−2
From Statement (2): x=[fraction]{y}{2}[/fraction]

Since the two statements provide different equations for x in terms of y, neither statement alone is sufficient, and together they provide conflicting information.

Answer: E - Statements (1) and (2) TOGETHER are NOT sufficient to answer the question, and additional data is needed.

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percentage of orange juice by volume in the mixture = $\frac{X}{X+Y}$

Statement 1
Equations are not needed here.
Since total volume of the mixture is not changing, and adding 2 L of orange juice doubles the percentage, it is clear that orange juice in the original mixture also will be 2L.
However, we don't know the value of Y. Y can be anything. Hence we will not be able to calculate the percentage of range juice by volume in the mixture. Hence not sufficient.

Statement 2
Let us calculate the percentage of range juice by volume in the mixture after the changes in statement 2 are done.
= $\frac{x+\frac{y}{2}}{x+y}$
Notice that the total volume (x+y) is not changing and it is also given that this percentage is equal to double that of original mix.
Hence,
$\frac{x+\frac{y}{2}}{x+y}$ = $2 * \frac{x}{x+y}$

$x+\frac{y}{2} = 2x$
y = 2x
Substituting value of y as 2x we can calculate percentage of range juice by volume in the original mixture. Hence sufficicent.

Correct asnwer B

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We need to find x/(x+y) here.

Statement 1: x+2/x+y = 2x/x+y

x =2. We do not know the value of y and hence the statement is not sufficient

Statement 2: ( x +y/2)/ x+y = 2x/x +y. Simplifying gives x =y/2

Substituting the same in x/x+y, we get 1/3. Statement 2 alone is sufficient.

Therefore B

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Orange juice = x litres
Carrot juice = y litres
x/(x+y) = ?

Statement 1: Carrot juice = y-2
Orange juice = x + 2
$\frac{(x+2)}{(x+y)} $= $\frac{2x}{(x+y) }$
x=2 , y =?
We still dont know y. We need y to find the percentage.
I is insufficient.

Statement II: Carrot juice = y- (y/2) = y/2
Orange juice = x + (y/2) = x + 0.5y
$\frac{(x+0.5y)}{(x+y)} $= $\frac{2x}{(x+y) }$
x = 0.5y
y=2x
x/(x+y)= x/3x = 1/3
II is sufficient
Ans B

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Choice B

Given: Mixture of orange and carrot juice

Contents: x lts of Orange juice and y lts of carrot juice

Question: Orange juice % by volume ?

% of orange juice = 100 * (volume of orange juice)/ (total volume of juice)

=> 100 * (x/ (x+y))
divide numerator and denominator by x

=> 100 * ( 1 / (1 + y/x)) ---------- (1)

We will know the % of orange, if we know either individual juice quantity or ratio of orange : carrot in the final mixture

Statement 1:
2 lts of carrot juice replaced with 2 lts of orange juice, resulting in double the percentage of orange juice compared to original mixture

y -> y - 2
x -> x + 2

% orange juice = (x + 2)/ (x + y) = 2 times the original % = 2x/ (x+y)
=> x + 2 = 2x
=> x = 2
But what about y?
To get the overall concentration we need y as well here. Hence Insufficient

Statement 2:

Carrot content is halved, and orange content is increased by equal volume, resulting in double the percentage of orange juice compared to original juice mixture

y -> y/2
x -> x + y/2

% orange juice = (x + y/2)/ (x + y) = 2 times the original % = 2x/ (x+y)
=> x + y/2 = 2x
=> y/2 = x
=> y/x = 2
Substitue above ratio in (1) to get the answer

Here we got the ratio needed to calculate the original orange juice %
hence, sufficient

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Orange-X
Carrot-Y
%Organge Juince -X/(X+Y)?

1-
Orange- X+2
Carrot- Y-2

%Organge Juince -X+2/(X+Y)
Given X+2/(X+Y)=2X/(X+Y)
X=2
Ratio- 2/2+Y
Not Sufficient

2-
Orange- X+Y/2
Carrot- Y/2
Ratio-
X+Y/2 /(X+Y)=2X/(X+Y)

Y/2=X
Y=2X
%Organge =
X/(X+2X)
1/3
33.33333%
Sufficient

Ans-B

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Bunuel

This question was provided by GMAT Club
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x=orange y= carrot We have to find Orange percentage in mixture = [x/(x+y)]*100

Statement 1 = (x+2)/x+y = 2x/x+y with this we get x=2 not sufficient as we do not have the value of y.
Statement 2 = [x+(y/2)/x+y] = 2x/x+y with this we get x=y/2. not sufficient

Now by both statement we get x=2 y=4
So Answer is D

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Given that 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 is orange juice?
This would be $\frac{x}{x+y}$ * 100

From statement 1, we know that
$\frac{x+2}{x+y} = 2\frac{x}{x+y}$
x+2 = 2x
x = 2 liters

Without knowing what y is, we will not be able to compute the percent of orange juice in the mixture.
From statement 2, we know that $\frac{x+(half of y)}{x+y} = \frac{2x}{x+y}$
x + y/2 = 2x
x = y/2
We will be able to compute $\frac{x}{x+y}$
Therefore, B alone is sufficient to answer.
The right answer choice is B

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To determine what percent of the mixture, by volume, is orange juice, we need to find the percentage
= x/(x+y)×100%.

Analysis of Statement (1)
Statement (1) states that 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.

The original volume of orange juice is x liters.
The original volume of carrot juice is y liters.
The original percentage of orange juice is x/(x+y)
After replacing 2 liters of carrot juice with 2 liters of orange juice, the volume of orange juice becomes
x+2 liters, and the volume of carrot juice becomes y−2 liters.

According to Statement (1), this new percentage is double the original percentage:
(x+2)/(x+y) = 2*[x/(x+y)]

x+2=2x
So, we have x=2.

Analysis of Statement (2)
Statement (2) states that 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 original volume of orange juice is x liters.
The original volume of carrot juice is y liters.
The original percentage of orange juice is x/(x+y)
Half of the carrot juice is y/2.

The new percentage of orange juice is
[x+(y/2)]/ (x+y) = 2* [x/(x+y)]

According to Statement (2), this new percentage is double the original percentage:

Simplify the equation :
[x+(y/2)] / (x+y)= 2x/ (x+y)

So, we have
y=2x.

Combining Statements (1) and (2)
From Statement (1), we have
x=2. From Statement (2), we have
y=2x.
Substituting x=2 into the equation from Statement (2), we get
y=4.

Now, we can find the percentage of orange juice in the mixture:
x/(x+y) =2/(2+4)=2/6=1/3

So, the percentage of orange juice is:
(1/3)×100%=33.33%

Conclusion
Both statements together are necessary to determine the percentage of orange juice. Therefore, the correct answer is:
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.

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Let orange be x, carrot be y

x/(x+y) =?

1. (x+2)/x+y = 2x/(x+y)
x=2... no value of y, Insufficient

2. 3x/(3x+y) = 2x/(x+y)
y=3x

so , x/(x+y) = x/(x+3x) = 1/4 ....Sufficient

Answer B

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Quote:

we need to find,
$\frac{100x }{ x + y}$

Statement 1:
$\frac{100(x + 2) }{ x + y} = \frac{100*2x }{ x + y}$
$x = 2$
We dont know y. Insufficient.

Statement 2:
$\frac{100(x + 0.5y)}{ x + y} = \frac{100*2x }{ x + y}$
$x = \frac{y}{2}$

our original fraction becomes
$\frac{100*0.5y}{ 0.5y + y}$

y cancels out and we can calculate the expression.
Sufficient

B

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11 Jul 2024, 11:19

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Statement I will give x=2, not sufficient alone
II will give y=2x which is sufficient to calculate the ratio of orange juice, so B is the answer

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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.

Percentage of orange juice = 100 * x/(x+y)

Let that percentage be P

P = 100*x/(x+y)

P/100 = x/(x+y)

Cross multiply

P(x+y) = 100x

Px +Py = 100x

(P-100)x +Py = 0 - (I)

Statement (1) -

100* (x+2)/(x+2+y-2) = 2P

100* (x+2)/(x+y) = 2P

Divide both sides by 2

50* (x+2)/(x+y) = P

P = 50* (x+2)/(x+y)

P/50 = (x+2)/(x+y)

Px + Py = 50x +100

(P-50)x + Py =100 (II)

Since we have 3 unknown quantities, 2 simultaneous equations are not enough to find P

Statement (2) -

Carrot juice volume = y/2

Orange juice volume = x +y/2

Total volume = x +y/2 +y/2 = x +y

2P = 100*(x+y/2)/ (x+y)

2P = (100x + 50y)/ (x+y)

2Px + 2Py = 100x + 50y

(2P-100)x + (2P-50)y = 0 (III)

Statement (2) alone is also not enough to find P

Using Statement (1) and (2) together,

- Method I - solving equations I, II, III simultaneously

- Method II

The 2 statements suggest that half of the carrot juice is 2 litres

OR 1/2 of y = 2

y = 4 ltrs

(x+2)/(x+4) = 2P (from statement 2)

x(x+4) = P

2x(x+4) = 2P

2x^2 +8x = 2P

2x^2 +8x = (x+2)/(x+4)

And after solving for x, we can find P

Therefore, the correct answer is C.

BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.

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Orange: x
Carrot : y

Ask is x/(x+y) = ?

S1: (x+2)/ (x+y) = 2x/(x+y) ; total volume of juice is not changing
=> x=2
But no idea about y. So not sufficient

s2: (x+y/2)/(x+y) = 2x/(x+y)
=> x= y/2

hence x/(x+y) = x/3x = 1/3 , i.e., 33%

Hence S2 is sufficient

Opton B

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Bunuel

This question was provided by GMAT Club
for the GMAT Club Olympics Competition

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In a mixture of two liquid, if you are going to replace volume of 1 liquid with another the total volume won't change only the percentage of both liquids in total mixture will change

Let orange juice be x ltrs & carrot juice be y ltrs
% of orange juice is represented by x/(x+y) & their ratios be a/b

Equation 1 be x/(x+y) = a/b

Statement 1: (x+2)/(x+y) = 2a/b

Divide this by eqn 1
X= 2
This doesn't says anything about y
So insufficient

Statement 2: (x+y/2)/(x+y) = 2a/b

Divide this by eqn 1

Y/x = 2
Add 1 to both sides
(Y+x)/X = 3

X/(y+x) = 1/3

Sufficient
Hence, B is the correct option

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11 Jul 2024, 12:31

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After replacing 2 liters of carrot juice with 2 liters of orange juice, the new mixture has (x + 2) liters of orange juice and (y - 2) liters of carrot juice.
The new percentage of orange juice is $\frac{(x + 2)}{(x + y)}$ * 100

It is given that this new percentage is double the original percentage. This means,
$\frac{(x + 2)}{(x + y)}$ * 100 = 2 * $\frac{x}{(x+y)}$ * 100
x + 2 = 2x
Thus, x = 2
But we didn't get any value of y. Hence, it is not possible to determine the desired percentage.
INSUFFICIENT

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.

After replacing half of the carrot juice with an equal amount of orange juice, the new mixture has x + $\frac{y}{2}$ liters of orange juice and $\frac{y}{2}$ liters of carrot juice.
The new percentage of orange juice is $(x + \frac{y}{2})/(x + y)$ * 100

It is given that this new percentage is double the original percentage. This means,
$(x + \frac{y}{2})/(x + y)$ * 100 = 2 * $\frac{x}{(x+y)}$ * 100

x + $\frac{y}{2}$ = 2x

x = $\frac{y}{2}$
or y = 2x
The percent of the mixture by volume that is orange juice = $\frac{x}{(x+y)}$ * 100
= $\frac{x}{(x+2x)}$ * 100
= $\frac{x}{3x}$ * 100
= $\frac{100}{3}$
= 33.33%
SUFFICIENT

The correct answer is B