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

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

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

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

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)

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

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.

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)

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)

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

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.

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.

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

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.

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

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

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.

1. 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.
2. 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.
3. 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

Answer: D. 8