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

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UfuomaOh

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19 Dec 2024, 09:37

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The 12 liters mixture contains apple and carrot juice in the ratio 1:3

It means that apple juice is 3 liters in content

and carrot juice is 9 liters in content

The question is 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 ?

In other words, how many carrot juice should we take out and replace with apple juice to change the ratio to 3:1

Hence lets make apple juice 9litre and Carrot juice 3litre

to bring the carrot juice to 3 litres we would subtract 3litres from 9 litres

In conclusion, the answer is 6 litres

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12l of mixture of Apple and carrot juice, contains Apple Juice and Carrot Juice in the Ratio of 1:3.

SO intial Quantitites of
Apple Juice: 3L
Carrot Juice: 9L

so we need to replace the mixture from the Same quantity of apple juice to make the ratio of APple to carrot in the Mixture: 3:1
Means Final Apple Juice in the mixture: 9L

So, Let x Liters of solution (1:3) apple :Carrot, to be replaced and x liters of Apple juice to be added
3-(x)1/4 + x = 9

Gives us X = 8 Liters

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total juice = 12l
apple juice = (1/4)*12 = 3l
carrot juice = (3/4)*12 = 9l

let x be the amount of total juice removed and apple juice added
since the total has both apple and carrot
new apple juice = 3 - x/4 + x = 3 + 3x/4 ---> equation 1
new carrot juice = 9 - 3x/4 ------> equation 2

equate (eq 1/eq 2) with 3/1

after solving for x we get x = 8 liters

Bunuel

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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

This question was provided by GMAT Club
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Kaxz

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Apple juice/Carrot juice=3:1
Mixture=12l

So, A=3l, C=9l

Let the volume of mixture to be replaced be x litres.

New A:C=1:3

Amount of A removed=x/4
Amount of C removed=3x/4

(3-x/4+x)/(9-3x/4)=3
x=8l

RamaSubramanian

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Apple : Carrot :: 1: 3
total 12L

therefore,
-> apple = 3L
-> Carrot = 9L

Let x litres of apple juice be taken from the mixture and replaced with apple juice:

this x litres would contain 1/4 (x) litres of apple juice and 3/4 (x) litres of carrot juice
after replacing it would be

apple = 3 - x/4 + x = 3-3x/4
carrot = 9 - 3x/4

given that apple : carrot = 3:1

equating 3 - 3x/4 / 9 - 3x/4 to 3/1
we get 8L

aviraj1703

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19 Dec 2024, 10:05

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a = x, c = 3x
a + c = 12
=> a = 3, c = 9

now, x liters are removed from the mixture
=> a' = 3 - x/4
=> c' = 9 - 3x/4

Now, we know that
a'/c' = 3
=> x = 8

Answer: D. 8

Nikhil17bhatt

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19 Dec 2024, 10:08

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IMO D

Using intutive solution

Total = 12ltrs
Ratio of Apple to carrot is 1:3, hence
Apple = 3lts
Carrot =9ltrs

Now new should be Carrot = 3ltrs
and Apple = 9 ltrs because total will be same

Hence Carrot is decreasing and no new is adding so from 9 to 3 is a decrease of 2/3 , hence a simialr decrease will be for the total quanity as well so a 2/3 decrease of 12 means 8. hence 8 is the answer

Fuyu

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19 Dec 2024, 10:14

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Total mixture= Apple juice +Carrot juice=12l
Apple juice: carrot juice= 1:3

we can say, 1x+3x=12l
4x=12; x=3
Therefore, Apple juice=3l
Carrot juice=9l
To flip the ratio to 3:1, we need apple juice to be 9l and carrot juice to be 3l.

9-3=6
6L of carrot juice needs to be replaced.
Hence, C is the answer.

GraCoder

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Bunuel

This question was provided by GMAT Club
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Apple juice = 3L and carrot juice = 9L, Let's assume we remove 12x L from the mixture
So the new volume becomes:
Apple juice = 3 - 3x and carrot juice = 9 - 9x

Now since we are replacing with pure apple juice so adding 12x apple juice:

Apple juice = 3 + 9x and carrot juice = 9 - 9x

Now, 3 + 9x = 3(9 - 9x)
=> 1 + 3x = 9 - 9x
x = 2/3

Therefore 12x L => (D) 8L

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The mixture initially has 3 liters of apple juice and 9 liters of carrot juice. We need to replace some amount x of the mixture with apple juice to get a 3:1 ratio.

After replacing, the amount of apple juice becomes 3 + 3x/4, and the amount of carrot juice becomes 9 - 3x/4.

For a 3:1 ratio:

(3 + 3x/4) / (9 - 3x/4) = 3.

Solving this gives x = 8.

Answer: D. 8

dhaphuong

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The target mixture needs to have 9l of apple juice (3:1, or 75% of 12l). Just plugging in a number can help us get to the right answer.
Say if we remove half of the mixture; we'd be left with 1.5l of apple juice (25% of 6l), and we'd add 6l back. 7.5l is less than the desired quantity.
One step up if we remove 8l, we'd have 1l of apple juice (25% of 4l), and adding 8l, we'd have 9l, which is what we needed.

Bunuel

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Total mixture -> 12 L ; Apple juice : Orange juice -> 1 : 3 ; therefore Apple juice (in L) & Orange juice (in L) -> 3L & 9L Respectively; if we remove 8 L from the mixture -> Then Apple juice (in L) & Orange juice (in L) -> 1L & 3L Respectively ; Now 8L of apple juice is added -> 9L of Apple Juice and 3L of Orange Juice => 3:1

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19 Dec 2024, 10:51

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A: apple; C: Carrot
A= 0.25*12=3 lt
C=.75*12=9lt

let x lt of mix removed. Then,
A=3-0.25X
C=9-0.75X

only x lt A added:
A=3-0.25x+x=3+0.75x

3+0.75x/9-0.75x = 3

x=8 lt

Ans D

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The total juice is 12 liters, with 3 liters of apple juice and 9 liters of carrot juice. Let x be the amount of juice removed and replaced by apple juice. The new amounts are:

Apple juice: 3+3x/4
Carrot juice: 9−3x/4

Setting the ratio of apple to carrot juice to 3:1 and solving gives x=8liters.

Bunuel

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aanamahmed12

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19 Dec 2024, 11:06

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Suppose part of the old mixture remaining has x apple juice and 3x carrot juice (in the old ratio of 1:3)
Adding k apple juice makes the mixture (x+k)/3x = 3/1 which is the required ratio (3:1)
On solving k = 8x
Hence adding 8 litres of apple juice will give us the answer.

Bunuel

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nishantswaft

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We can solve this step by step using algebra, first let's understand the question.

The total mixture volume is 12 liters with a ratio of 1:3 (apple juice : carrot juice). This means:

Apple juice volume = $\frac{1}{4}*12=3$
Carrot juice volume = $\frac{3}{4}*12=9$

Let x be the amount (in liters) of the mixture removed and replaced with pure apple juice. When x liters are removed:

Apple juice removed = $\frac{x}{4}$
Carrot juice removed = $\frac{3x}{4}$

The new volumes of apple juice and carrot juice after removal are:

Apple juice = 3- $\frac{x}{4}$
Carrot juice = 9- $\frac{3x}{4}$

When x liters of pure apple juice are added, the new apple juice volume becomes:
New apple juice volume= 3- $\frac{x}{4}$ +x =3+$\frac{3x}{4}$

The carrot juice volume remains:
New carrot juice volume= 9- $\frac{3x}{4}$

We want the new ratio of apple juice to carrot juice to be 3:1. This means:

(3+$\frac{3x}{4}$)/( 9- $\frac{3x}{4}$) =$\frac{3}{1}$

Cross multiplying we have;
3+$\frac{3x}{4}$ = 3* ( 9- $\frac{3x}{4}$)
3+$\frac{3x}{4}$ = 27- $\frac{9x}{4}$
$\frac{3x}{4}$ + $\frac{9x}{4}$ = 27-3
$\frac{12x}{4}$ = 24

x=8

Hence answer is D

skry209

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Given that, 12 litres of mixture,
Initial
A(apple juice) = 3
C(Carrot juice)= 9
Final, Apple juice = 9, Carrot Juice = 3
Let x be amount removed from initial mixture, final ratios after mixing
(3 - (1/4)x + x) / (9 - (3/4)x) = 3/1
Solving the above equation gives x = 8

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Bunuel

This question was provided by GMAT Club
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Let's say we replace x litres of the mixture with apple juice.
Volume of apple juice = [(12-x)/4] + x
Volume of carrot juice = 3(12-x)/4
Ratio = Volume of apple juice/Volume of carrot juice
Simplifying we get,
Ratio = (12+3x)/(36-3x) = (4+x)/(12-x)
We need ratio = 3:1
(4+x)/(12-x) = 3
=> 4+x = 36-3x
=> 4x = 32
=> x = 8

BatrickPatemann

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Step one is easiest:

First ratio is apple juice: carrot juice 1x:3x = 4x Total
Thus apple juice = 1/4*12 litres =3 litres
carrot juice = 3/4*12=9 litres

After replacing volumes, the new ratio of apple: carrot is 3x:1x = 4x Total

Supposing x litres of the mixture is replaced with x litres of apple juice:

Since apple juice is 1/4*x of the original mixture, this would be the removed amount.
remaining apple juice would be 3-(x/4) litres

For carrot juice, x litres are removed, since carrot juice is 3/4 of the original mixture, the amount removed would be 3/4*x = 3*x/4 litres
remaining carrot juice would be 9-(3x/4) litres

Since the new ratio of apple juice to carrot juice is 3:1

3/1 = 3-(x/4) / 9-(3x/4) -> rearranging -> 3*(9-3x/4/ = 3+3x/4

3x = 24 -> x = 8 liters, answer (D)

prashantthawrani

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19 Dec 2024, 11:26

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we start with a 12-liter mixture of apple juice and carrot juice in a 1:3 ratio.

It means there are 3 liters of apple juice and 9 liters of carrot juice.

If 4x litres mixture is removed, x litres of apple juice and 3x litres of carrot juice is removed.

Then, we add same quantity 4x litres of apple juice.

Finally, we end up with a 12-liter mixture of apple juice and carrot juice in a 3:1 ratio.
It means there are 9 liters of apple juice and 3 liters of carrot juice.

A/C=(3-x+4x)/(9-3x)
3/1=(3+3x)/(9-3x)
3=(1+x)/(3-x)
9-3x=1+x
4x=8 litres

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