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

I went with the trial and error method by trying out each option:
A. 3
12 - 3 = 9. Hence A and O would be 2.25 and 6.75. Even if I add the 3l of apple juice the ratio of A:O would not be 3:1

B. 4
12 - 4 = 8. Hence A and O would be 2 and 6. Even if I add the 4l of apple juice the ratio of A:O would not be 3:1

C. 6
12 - 6 = 6. Hence A and O would be 1.5 and 4.5. Even if I add the 6l of apple juice the ratio of A:O would not be 3:1

D. 8
12 - 8 = 4. Hence A and O would be 1 and 3. If I add the 8l of apple juice the ratio of A:O would be 3:1 i.e. Apple 9L and Orange 3l
Hence D is the correct answer

12-liter mixture of apple juice and carrot juice
apple juice and carrot juice in a 1:3 ratio by volume.
Current: A : C :: 1 : 3 => A = 3L and C = 9L

Desired ratio : A : C :: 3 : 1 => A = 9L and C = 3L
This means, we need to remove 6L of C and because it will be as mixture, so that will also remove 2L of A
total Mixture is to be replaced = 8L

The initial ratio is A:C = 1:3. The total volume is 12. Means we have 3L of A and 9L of C. If we remove x amount of mixture from the total volume the volume of mixture that will come down will be in the same ratio i.e. 1 part of A and 3 parts of C. But the replaced quantity will only get added to A. This will us in getting the equation that can be further solved to get the answer.

Ans: D. 8

In a 12 liter mixture, Apple juice: Carrot juice = 1:3 i.e., Apple juice = 3 liter and Carrot juice = 9 liter

Now, suppose x liter was removed and replaced with the same amount of apple juice then,
New volume of carrot juice = 9-(3x/4) = (36-3x)/4
New volume of apple juice = 3-(x/4)+x = 3+(3x/4) = (12+3x)/4

In the new mixture, Apple juice: Carrot juice = 3:1
=>((12+3x)/4) / ((36-3x)/4)= 3:1
=> 12+3x = 108-9x
=>12x=96
=>x=8

Therefore, 8 liters of the mixture should be replaced with an equal amount of apple juice

Given: A 12-liter mixture of apple juice and carrot juice contains apple juice and carrot juice in a 1:3 ratio by volume

Hence Volume of Apple Juice (3 L) & Carrot Juice (9L) [1:3 ratio]. To make the value of the mix to be 3:1 ratio, volume of apple juice needs to be 9 L & Carrot Juice must be 3 L.

On analyzing options (E) through (A):

Option (E) is not feasible as replacing 10L of the mixture with 10L apple juice, we would definitely have more than 9L of apple juice in the mixture & fail to get the desired ratio/proportion. Hence option (E) can be eliminated.

Option (D) talks about replacing 8L of the mixture with 8 L apple juice. If apple juice & acrrot juice are mixed in 1:3 ration, the amount of Apple Juice & Carrot Juice replaced would be (x+3x=8),x=2 which gives us 2 L of Apple Juice & 6 L of carrot juice replaced.

Thus the amount of Apple Juice left would be (3-2=1L) & Carrot Juice would be (9-6=3L).On adding 8 L of Apple Juice, we get 9 L of apple juice & the ratio of Apple Juice to Carrot Juice would turn out to be (3:1)

Hence the correct option for this question is (D) 8 L

Apple to carrot: 1:3 → 3 litre / 9 litre. of 12 l

Now if we replace 6 litre of carrot juice with apple juice , the new vol ratio will be 9:3 → 3:1 .Hence Answer C

We have 3 liters of apple and 9 liters of Carrot
We want to remove 6 liters of Carrot that represent 6/9 of the carrot in the mixture
To remove this quantity of carrot, we have to remove 6/9 of the total mixture
6/9*12=8

Apple + Carrot = 12 liters
Apple : Carrot = 1/3
=> By volume, Apple = 1/4 * 12 = 3 liters and Carrot = 12 - 3 = 9 liters

New ratio is the reverse of the current ratio => by volume, Apple will be 9 liters and Carrot will be 3 liters.

The 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 would be 9 - 3 = 6 liters

IMO: C

Initial Condition: A=3L, C=9L
let x liters be removed
Apple = 3-x/4+x (as x liters replaced by apple juice)
carrot=9-3x/4

apple/carrot =3/1= (3-x/4+x)/(9-3x/4)
solving for x will give x= 8

Initial ratio Apple : Carrot = 1:3

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

Final ratio of Apple : Carrot = 3:1

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

Lets say 'x' amount of mixture is replaced with apple juice

Apple juice in the mixture when x is replaced = 3 - x * 1/4 + x = 9 liters

3x/4 = 6 liters
x = 8 liters (OPTION D)

for it to be in the ratio of 3;1 there needs to be 9 liters of Apple juice. hence option a b & c is eliminated as they will not be enough. test case for D and E.
remove 8
2lts from apple 6lts from carrot.
we now have 1lt and 3 lt
adding 8lts of Apple will give 9:3 i.e
3:1

Let 'x' be the liters of mixture which is removed & amount of apple juice added to the the mixture
Currently, apple juice = 12* 1/4 = 3 & carrot juice = 12 - 3 = 9

For x getting removed & added -
apple juice = 3 + x - (x/4) = 3 + 3x/4
carrot juice = 9 - 3x/4

So, (3 + 3x/4) / (9 - 3x/4) = 3
Which gives us x = 8

IMO D

Total parts in the 1:3 ratio = 1 + 3 = 4 parts
One part = 12 liters / 4 parts = 3 liters
Initial apple juice: 1 part * 3 liters/part = 3 liters
Initial carrot juice: 3 parts * 3 liters/part = 9 liters

Let 'x' be the amount of mixture to be replaced.

After replacing 'x' liters of the mixture with 'x' liters of apple juice, we have:

Apple juice: 3 - (x/4) + x = 3 + (3x/4)
(We subtract x/4 because the removed mixture contains 1/4 apple juice. We add x because we're adding pure apple juice.)
Carrot juice: 9 - (3x/4) (We subtract 3x/4 because the removed mixture contains 3/4 carrot juice.)
The new ratio of apple juice to carrot juice is 3:1, so:

[3 + (3x/4)] / [9 - (3x/4)] = 3/1

Simplify and solve for 'x':

3 + (3x/4) = 27 - (9x/4)
3x + 9x = 108 - 12
12x = 96
x = 8

The ratio is in 1:3 format which means apple juice = 3 lit and carrot juice = 9 lit

Now to make it 1:3 we will add more apple juice and subtract carrot juice.

So apple juice = 3 -1/4x +x (new mixture added)

carrot juice = 9-3/4x ( no added x as we are reducing the proportion for carrot juice)

Solve the ratios for = 3/1 and you will get x=8

12-liter mixture with 1:3 ratio means 3 liters of apple juice and 9 liters of carrot juice.
In M liters there are M/4 liters of apple juice.

3:1 means that in the final mixture there will be 9 liters of apple juice.

3-M/4+M=9
3M/4=6
M=8

Answer D

Final % of apple juice in the mixture after replacement= 75%
Final % of carrot juice in the mixture= 25%

Replacement formula- Final% of carrot juice= [(Initial% of carrot juice) x(V-R)/V].... here, V= initial vol. and R=
Therefore, 25%= 75x(12-R)/12... solving for R we get R= 8.

ans= 8

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 ?

So, in the 12 litre mixture there are 3 litre apple and 9 litre carrot. Now, if we take 4 litre from the mixture, there will 2 litre apple and 6 litre carrot. Adding 4 litre apple in the mixture, new ratio would be 6:6 meaning 1:1. If we take 8 litre mixture, there will be 1litre apple and 3 litre carrots, adding 8 litre apple, ew mixture would be 9 litre apple and 3 litre carrot and the ratio would be 3:1.

So, Answer: D. 8