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

As per the question, Apple juice and carrot juice are in the ratio of one is to 3 by volume. Therefore, in 12 L mixture, the following are the amounts Apple equals to 3 carat equals to 9l
Now we want to change the ratio to 3 is to 1 that is to make Apple equals to 9 L by adding only apple juice
That is, we want six more litres of apple juice
When we remove 1 L of the mixture and add 1 L of apple juice, we add 750 ML of apple juice
So the answer is 6 L divided by 750 ML, which is equals to 8 L

Initial ratio of AJ to CJ=1:3
AJ= 3 liter
CJ= 9 liter

Desired ratio of AJ to CJ in a 12 liter vessel= 3:1, ie 9 liter to 3 liter

we need to remove 6 liters of CJ from the mixture to reduce it to 3 liters. And in order to remove 6 liters of CJ, we need to remove 8 liters of the mixture, because AJ:CJ = 1:3 hence in 8 liters, we get 2 liters of AJ and 6 liters of CJ

IMO Answer (D)

Since the total number of literes is 12 . It is of 3 lits and 9 lits.
Lets assume y lits of mixture are removed and y lits of apple juice is added

(3-y/4+y) /(9-3y/4) =9/3
y=8 lits

12 liter mixture contains apple and carrot juice in ratio 1:3 => 3 liter apple juice and 9 liter carrot juice.

Now when we glance at options there are two easy standouts 4 and 8 - both can be quickly tested as the current ratio of apple:carrot is 1:3 => 1k + 3k => 4k (multiple of 4)

Let us say we remove 4 liter of original solution => this will remove 1 liter of apple and 3 liters of carrot. Now adding 4 liters of apple juice will give us apple:carrot as ((3-1+4):(9-3)) => 6:6 which is not what we're looking for.

Let us say we remove 8 liter of original solution => this will remove 2 liter of apple and 6 liters of carrot. Now adding 8 liters of apple juice will give us apple:carrot as ((3-2+8):(9-6)) => 9:3 => 3:1 which is the answer we're looking for.

Answer D.

Total volume = 12 liters
The ratio of apple: carrot = 1:3
So if we let x be one part: 1x + 3x = 12
4x = 12
x = 3
Therefore: Apple juice = 3L, Carrot juice = 9L

We want a final ratio of 3:1 apple: carrot
Let's say we replace y litres
When we replace y litres of mixture with apple juice:
Original apple juice removed = y × (1/4) = y/4
Original carrot juice removed = y × (3/4) = 3y/4
New apple juice added = y

After replacement:
Apple juice = 3 - y/4 + y = 3 + 3y/4
Carrot juice = 9 - 3y/4

For ratio 3:1:
(3 + 3y/4) = 3(9 - 3y/4)
3 + 3y/4 = 27 - 9y/4
3 + 3y/4 + 9y/4 = 27
3 + 12y/4 = 27
12y/4 = 24
y = 8

Therefore, 8 litres of the mixture should be replaced.

The answer is D. 8 liters.

Lets do it!

Current apple to Carrot ratio = 1:3 -> 1X+3X = 12 so X is 3 and the amount is 3L of apple and 9L of Carrot.
Now the complete mixture is being replaced, so the amount will remain same, which is 12L, and the new equation is basically revered, 9L of apple and 3L of carrot.
So, to achieve this we need to keep 3L of carrot in the end solution since carrot is not being replaced. So remove 6L of Carrot and correspondingly, keeping 1:3 ratio in mind, we will remove 2L of apple juice too, 1L remains.
And 8L of apple juice will be added, and the ratio will be -> 8+1:3 -> 9:3 -> 3:1.

Hence 8L

1:3 means Apple is 3L, carrot is 9L
Say X L is removed and Replaced with X L of apple juice.

3x/12 L of apple juice is removed and 9X/12 L of carrot juice is removed

(3-3x/12+x)/12=3/4=9/12
=> 3+9x/12=9
X= 6.12/9=8L

D)

We are tasked with finding the volume of the initial mixture that must be replaced with apple juice to change the ratio of apple juice to carrot juice from 1:3 to 3:1.

Step 1: Analyze the initial mixture
The total volume of the mixture is 12 liters.
The ratio of apple juice to carrot juice is 1:3.
Thus:
Volume of apple juice = 1/4 ×12=3 liters.
Volume of carrot juice = 3/4 ×12 =9 liters.

Step 2: Replace part of the mixture with apple juice
Let x liters of the mixture be replaced with an equal volume of apple juice.

When x liters are removed, the removed mixture contains:

Apple juice: 1/4 x liters.
Carrot juice: 3/4 x liters.
After replacing x liters with pure apple juice:

Remaining apple juice = 3− 1/4 x+x.
Remaining carrot juice = 9− 3/4 x.
The new ratio of apple juice to carrot juice is 3:1, so:
{3− 1/4 x+x}/ {9− 3/4 x} = 3

Step 3: Solve for x
x=8.

Step 4: Verify the solution
If x=8, then:
Removed apple juice = 1/4 ×8=2 liters.
Removed carrot juice = 3/4 ×8=6 liters.
Remaining apple juice = 3−2+8=9 liters.
Remaining carrot juice = 9−6=3 liters.

The new ratio is:

Apple juice/ Carrot juice = 9/3 =3:1.
This satisfies the condition.
Final Answer: (D) 8

Let x be the amount of replacement.

Based on the apple juice and carrot juice 1:3 ratio, the apple juice is 3 liter while carrot juice is 9 liter.

Replacement means deduct carrot juice by x liter and add apple juice by x liter:
(3+x)/(9-x) = 3/1
3+x = 27-3x
4x = 24
x = 6

Therefore the answer is C

Concept
If from a jar of juice we take out one glass of juice, then the glass and the remaining juice inside the jar has the same concentration has the original juice.

Now given, We have total 12lt of mixture and A is 25% by volume = 3lt
(as A:O = 1:3 So A:total = 1:4)
we remove 'x' lt of mixture --> this 'x' lt also has 25% Apple juice.--> Apple juice removed = x/4 lt.
then we add 'x' lt of apple juice.

So finally Total is still = 12lt and Apple juice is = {3 - (x/4) + x}
And given the new concentration of apple is 75% by volume

75% of 12 = {3 - (x/4) + x}
on solving we get x=8lt

Initial state:
Total volume: 12 liters
Apple juice: 1/4 * 12 = 3 liters
Carrot juice: 3/4 * 12 = 9 liters
Final state:
Total volume: 12 liters (remains constant)
Apple juice: 3/4 * 12 = 9 liters
Carrot juice: 1/4 * 12 = 3 liters
Analyzing the replacement:
remove 'x' liters of the mixture, which contains 1/4 * x liters of apple juice and 3/4 * x liters of carrot juice.We add 'x' liters of pure apple juice.
Setting up the equation:After replacement, the apple juice volume becomes: 3 - 1/4 * x + x = 3 + 3/4 * x
want this to be 9 liters.
So, we solve the equation: 3 + 3/4 * x = 9 3/4 * x = 6 x = 8

IMO D

The total volume of the mixture is 12 liters, and the initial ratio of apple juice to carrot juice is 1:3
Let:

• A = volume of apple juice
• C = volume of carrot juice

From the ratio A:C=1:3.

So A=3 l, C =9 l, We are just replacing, the total volume has to be same. The final volume of the mixter is in 3:1 SO A=9 l , C =3L.

3/4(X) = 6L (When we remove the mixture, it will also be in the same ratio. As the Carrot juice removed is 6L, X is the volume replaced.

From this, we can get X= 8L.


We can also crosscheck if 8L is removed, Then A=2L, B=6L will be removed and filled with 8L of A.

Remainging A is 1L + New A 8L = 9L.

IMO D

Lets approach this step by step:
1) Let's first determine the initial volumes of apple juice and carrot juice - The ratio is 1:3 so out of 4 parts, 1 parts is apple juice and 3 parts are carrot juice - total volume is 12 liters - Apple juice: 12*(1/4) = 3 liters - carrot juice : 12*(3/4)= 9 liters

2) let x be the amount of mixture replaced with apple juice. This means :- x liters of the original mixture (containing bothe juices) is removed - x liters of pure apple juice is added

3) after the replacement we want the ratio to be 3:1 this means - New apple juice volume :
3-(x/4)+x=3+3x/4
new carrot juice volume = 9-(3x/4)

4) set up the equation based on the new 3:1 ratio: (3+3x/4)/(9-3x/4) = 3/1

5) solve the equation : 3+3x/4=3(9-3x/4)3+3x/4 = 27 -9x/4 12x/4 =24.
x = 8
therefore 8 liters of the original mixture should be replaced with 8 liters of pure apple juice

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 ?

Initial Mixture: -
Apple Juice = 12*1/4 = 3 liters
Carrot Juice = 12*3/4 = 9 liters
Total = 12 liters

Let the amount of mixture to be replaced with apple juice be 4x liters

Desired mixture: -
Apple Juice = 3 - x + 4x = 3 + 3x
Carrot Juice = 9 - 3x
Total = 12 liters

(3+3x)/12 = 3/4
3 + 3x = 9
x = 2 liters

Desired mixture: -
Apple Juice = 3 - x + 4x = 3 + 3x = 9
Carrot Juice = 9 - 3x = 3
Total = 12 liters

The amount of mixture to be replaced with apple juice = 4x = 8 liters

IMO D

With the info given, as per initial ratio Apple Juice = 3 liters, and carrot juice = 9 liters. Check with multiples of 4(as the given ratio is 1:3 --- 1+3=4) instead of going with too much calculations. Pick 8 liters of Apple Juice--> option D,

Removing 8 liters of mixture will leave 4 liters of which, 1 liter is Apple Juice and 3 liters is carrot juice. Adding pure apple juice to this mixture will make it
9 liters apple juice and 3 liters carrot juice-- 9/3= 3:1 Option D is your answer!!!

You initial convert the 12 liter mixture to a ratio of 1:3 which equals a 3:9 ratio of apple juice to carrot juice.

Convert 12 liter mixture to the desired ratio 3:1 which equals 9:3 ratio of apple juice to carrot juice

Take that ratio convert to a fraction and add your replacement apple juice variable which will be the exact same as your carrot juice removal variable

3+X / 9 - X which is equal to the desired fraction of 9 / 3

Solve for x and you see that x equals 6

The answer is 8.

4x=12 ; x=3.

Apple juice is 3liters and carrot juice is 9liters. To make it to 3:1,

3-x/4+x (3 is the initial volume and x is the new amount of juice to be added) = 3+3x/4

9-3x/4--> carrot juice

3+3x/4 / 9-3x/4 = 3/1

Solving this we get 8.