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12 Days of Christmas GMAT Competition - Day 4: A 12-liter mixture of

1 2 3 4 5

sushanth21

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D
1:3 ratio of apple and carrot
x + 3x = 12 litre

apple 3 litre
carrot 9 litre

asking us to find the amount of juice to be replaced with apple juice to make the ratio 3:1

which means 9 litre of apple and 3 litre of carrot.

lets assume x litres were removed and x litres of apple is added
then
by the 1:3 ratio, x/4 of the apple will be removed from the mixture before adding the apple juice.

3 - x/4 + x = 9
3 + 3x/4 = 9
x = 8

Bunuel

12 Days of Christmas 2024 - 2025 Competition with $40,000 of Prizes

A 12-liter mixture of apple juice and carrot juice contains apple juice and carrot juice in a 1:3 ratio by volume. What amount of the mixture, by volume, should be replaced with an equal amount of apple juice to change the ratio of apple juice to carrot juice to 3:1 by volume ?

A. 3
B. 4
C. 6
D. 8
E. 10

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A 12-liter mixture of apple juice and carrot juice contains apple juice and carrot juice in a 1:3 ratio by volume. What amount of the mixture, by volume, should be replaced with an equal amount of apple juice to change the ratio of apple juice to carrot juice to 3:1 by volume ?

A. 3
B. 4
C. 6
D. 8
E. 10

Mixture. Apple juice only
25%. 100%
75%
25. 50

25:50=1:2= Mixture: Apple only
Hence 1 part of mixture is mixed with 2 parts of apple juice only hence 2/3*12=8 litres is replaced

Ans D

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19 Dec 2024, 12:24

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Let us assume that X ltr juice is removed and X ltr of apple juice is added. in the initial solution, we have a 3Lt apple juice and 9 ltr of carrot juice,

ATP,

[3-.25X+X][/9-.75X] = [3][/1]

solving the equation results in 8Ltr.

Bunuel

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Total volume = 12L
Apple juice : Carrot juice = 1 : 3

Volume of apple juice = 1/4 * 12 = 3L
Volume of carrot juice = 3/4 * 12 = 9L

Since the ratio is 1 : 3, the juice removed will also be in the same ratio
For ease of calculation, we can assume that the removed juice is a multiple of 4

If 4L is replaced,

In the juice mixture that is removed, 1L Apple juice and 3L Carrot juice will be removed,
Remaining Apple juice = 2L
Remaining Carrot juice = 6L

4L Apple juice is added, making the total 6L
This will make the ratio 1:1

If 8L is replaced,

In the juice mixture that is removed, 2L Apple juice and 6L Carrot juice will be removed,
Remaining Apple juice = 1L
Remaining Carrot juice = 3L

8L Apple juice is added, making the total 9L
This will make the ratio 3:1

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19 Dec 2024, 12:38

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Hi All,

Given that in 12 litres of mixure , Apple juice to carrot juice is 1:3

there no Apple juice in litres= 1/4 x 12= 3

Similarily carrot juice = 3/4 x 12 = 9

We need to take out the mixture and then put apple juice so that the mixture ration for apple and carrot juice changes to 3:1

As we can see if we take out N litres of mixture, to find out how many of apple juice and carrot juice taken out we need to divide it by 4 and then multiply accordingly.

So we first analyze those options that are divisible by 4

Option B) if we take out 4 litres (1 litre AJ and 3 ltrs CJ) , left (2 ltrs AJ and 6 ltrs CJ)

adding 4 litres of AJ in the mixture would make it (6 ltrs of AJ and 6 ltrs of CJ) basically ratio will 1:1. Therefore not our answer

Options D)

if we take out 8 litres (2 litre AJ and 6 ltrs CJ) , left (1 ltrs AJ and 3 ltrs CJ)

adding same 8 litres of AJ in the mixture would make it (9 ltrs of AJ and 3 ltrs of CJ) basically ratio will 3:1.

Hence D is our answer

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19 Dec 2024, 12:43

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Bunuel

This question was provided by GMAT Club
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Let's assume x liters of juice is replace with x liters of apple juice

Volume of apple juice in the resultant mixture = $\frac{12 - x }{ 4} + x $

Volume of carrot juice in the resultant mixture = $\frac{ 3(12 - x) }{ 4} $

We know -

$\frac{\frac{12-x}{4} + x}{\frac{36 - 3x}{4}} = \frac{3}{1}$

$\frac{12 + 3x}{36-3x} = \frac{3}{1}$

12 + 3x = 36*3 - 9x

12x = (36 * 3) - 12

12x = 12(9 - 1)

x = 8

Option D

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Given the initial ratio and the desired one.

1: 3; apple:carrot juice

we can infer that apple juice is 1/4*12 = 3 liters
carrot juice is the remaining 9 litres

And we’d want to alter this ratio and volumes to a 3:1 ratio

thus removing x liters containing 1/4 of apple juice and 3/4 of carrot juice

3/1 = (3 - x/4) / (9 - 3x/4)

Solving for x, we get x = 8 (D)

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Bunuel

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Answer C. 6

A:C is 1:3 or C = 3A
Since 12L A=3 C=9

Convert to A:C 3:1 or A = 3C
Opposite A 3+6=9 and C 9-6 = 3
So Remove 6 C and add 3 A

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Bunuel

This question was provided by GMAT Club
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12L mixture is present - Apple Juice : Carrot Juice is in the ratio 1:3
x + 3x = 12; x = 3

Apple Juice = 3L
Carrot Juice = 9L
Total = 12L

Required - Apple Juice : Carrot Juice in the ratio 3:1

If 8L of the mixture is removed and replaced with 8L Apple juice

Remaining 4L of mixture will contain: 4/4=1
Apple 1L and Carrot 3L

+ 8L of Apple juice = 9L of Apple juice and 3L of Carrot juice

Apple Juice : Carrot Juice = 9:3
Which is 3:1 by volume

[D] is the correct answer.

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19 Dec 2024, 13:25

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Hi everyone

straightforward.

	Apple	Carrot	Total
Ratio	1x	3x	4x
Numbers	3	9	12
	25%	75%	100%
[?]	3x	1x	12-[?]

We are dealing with weighted averages (My approach):

apple	Numbers	Percent
Our old Mix	12-x	25%
New Mix	12	75%
Adding Apple	x	100%

Equation: 0.25(12-x)+x=0.75(12)
3-0.25x+x=9
x=8 Our Answer (D)

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Ratio of apple j to orange j = 1x:3x, we will convert it to fractions to make it easier so this means that apple juice = 1/4 of T and 3/4 of T will be orange juice.

Now given T = 12 L
Apple j = 3
Orange j = 9

Altering the ratio eliminating the mixted juice --> 3:1 apple to orange with a fraction of 3/1
3 - x/4 = new amount of apple juice = 3
9 - 3x/4 = 1

Solving for both equations x = 8 ANSWER (D)

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a=liters of apple juice
c=liters of carrot juice
x=liters we replace

a:c=1:3 -> There are 3 liters of apple juice and 9 liters of carrot juice in the mixture. There are 0.25x liters of apple y x liters of mixture.

a:c=3:1 -> a:(a+c)=3:4

(3-0.25x+x)/12=3/4
3+0.75x=9
0.75x=6
x=8

IMO D

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Bunuel

This question was provided by GMAT Club
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I tested the options. Only Option D gives me the right answer.
Original volume: Apple juice Carrot juice
3 liter 9 liter

After Removal of 8 liter from the mixture

Apple juice Carrot juice
1 liter 3 liter

After adding 8 liter of Apple to the mixture

Apple juice Carrot juice
9 liter 3 liter

Therefore option D (8) is the answer.

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T = 12; A:C = 1:3; X+3X = 12; X = 3; A=3, B = 9

Asked to Make it A:C = 3:1; by replacing some solution with pure apple solution. This means the final carrot solution will remain the same.

So first remove some solution such that carrot solution becomes the desired one.

A:C = 3:1; In a 12-liter solution this means that C would be 3 Liter

Now Replace the solution with pure apple so that the remaining solution will have 3 liters of carrot. Currently, the ratio is A:C = 1:3, which means that if the remaining solution is 4 liters, this ratio can be maintained.

So now 12-4=8 Liter be replaced with pure apple solution.

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A 12-liter mixture of apple juice and carrot juice contains apple juice and carrot juice in a 1:3 ratio by volume. That means we have 3 liters of apple and 9 liters of carrot.

To achieve the ratio of apple juice to carrot juice to 3:1 by volume, we need to have 9 liters of apple and 3 liters of carrot.

That means we need to first remove an amount of juice so that carrot goes from 9 to 3, which is 6 liters of carrot. Since the original ratio of apple to carrot is 1:3, that would be equivalent to 2 liters of apple. So in total we need to remove 6 carrot + 2 apple = 8 liters.

Answer: 8

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Bunuel

This question was provided by GMAT Club
for the 12 Days of Christmas Competition

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The total mixture is 12 liters with apple juice:carrot juice in a 1:3 ratio. So, there are 3 liters of apple juice and 9 liters of carrot juice.

If x liters of the mixture are replaced with pure apple juice. When x liters are removed, the removed mixture contains (1/4)x liters of apple juice and (3/4)x liters of carrot juice.
After replacing x liters with pure apple juice:

Apple juice becomes 3 - (1/4)x + x
Carrot juice becomes 9 - (3/4)x

To achieve a 3:1 ratio:
{3 - (1/4)x + x}/{ 9 - (3/4)x} = 3/1
x = 8
Thus, 8 liters should be replaced.

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12L mix: 3L apple juice + 9L carrot juice (1:3 ratio)
let the amount of mix removed be x
hence x is also the amount of apple juice added.
Since the final ratio is 3:1, lets put all this in equation
(3- x/3 + x)/(9- 4x/3) = $\frac{3}{1}$
Solving the equation we get the answer 8

Bunuel

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19 Dec 2024, 22:26

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Total mixture = 12L
Ratio = 1:3

Let the quantities of Apple and Carrot juice be 1x and 3x.

So,
1x + 3x = 12
4x = 12
x=3

Therefore, Apple juice =3L
Carrot Juice = 9L

Now,

Let the quantity removed be Y
As per the question, if we add Y litres of Apple juice in place of Y litres of mixture, we are basically adding Y litres of apple juice and subtracting this Y from Carrot juice to make the ratio 3:1.
So,

(3+Y)/(9-Y) = 3/1

1*(3+Y) = 3*(9-Y)
3+Y = 27-3Y
4Y = 24
Y = 6

Therefore,
If we remove 6 Litres of mixture and replace it with 6 litres of apple juice, our ratio will become 3:1.

FINAL ANSWER - Option C

Bunuel

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Answer is D. 8

In the 12 liter mixture there are currently 3 liters of apple and 9 liters of carrot. To invert the ratio we need 9 liters of apple and 3 liters of carrot. However, we cannot simply take out 6 liters of the mixture and replace it all with apple because it's not all carrot. If we take out 8 liters of the mixture, then we are taking out 2 liters of apple and 6 liters of carrot. There are 4 liters remaining, consisting of 1 liter of apple and 3 liters of carrot. Then, we can replace the 8 liters with all apple juice, giving us 9 liters of apple and 3 liters of carrot.

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Initial Quantities:
- Total mixture: 12 liters.
- The ratio of apple juice to carrot juice: is 1:3.
- Apple juice: 1/4 × 12 = 3 liters.
- Carrot juice: 3/4 × 12 = 9 liters.
Replacement Process:
- Let x liters of the mixture be replaced with x liters of apple juice.
- When x liters of the mixture are removed:
  - Apple juice removed: 1/4x liters.
  - Carrot juice removed: 3/4x liters.
- After replacement:
  - Apple juice: 3 − 1/4x + x = 3 + 3/4x liters.
  - Carrot juice: 9 − 3/4x liters.
Desired Ratio:
- The new ratio of apple juice to carrot juice should be 3:1.
- Set up the equation:
  (3 + 3/4x)/(9 − 3/4x) = 3/1
- Solving for x:
  3 + 3/4x = 27 − 9/4x
  12/4x = 24
  3x = 24
  x=8