A 20 litre mixture of milk and water contains milk and water

Question

A 20 litre mixture of milk and water contains milk and water in the ratio 3 : 2. 10 litres of the mixture is removed and replaced with pure milk and the operation is repeated once more. At the end of the two removal and replacement, what is the ratio of milk and water in the resultant mixture?

(1) 17 : 3
(2) 9 : 1
(3) 3 : 17
(4) 5 : 3
(5) 11: 2

A. 17 : 3 
B. 9 : 1 
C. 3 : 17 
D. 5 : 3 
E. 11: 2

xennie · Accepted Answer

2) 9 : 1

From the beginning; 
Initially milk to water is 12l : 8l
remove half 6 - 4
add 10l milk 16 - 4
remove half 8 - 2
add 10l milk 18 - 2 
18:2 = 9:1

KarishmaB · Answer

Use this:

New Concentration of Water = Old concentration of Water * (V1/V2)^n

New Concentration of Water = 2/5 * (1/2)^2 = 1/10

Ratio of milk:water = 9:1

christoph · Answer

x/8=(1-10/20)^2 => x=amount of water left after 2 removements. its 9:1...

xennie · Answer

Chris I dont really understand your reasoning, care to explain a bit where the numbers come from. Thanks.

christoph · Answer

usually i use this formula for these kind of questions...
amount of x after removal/total amount of x=(1-(amount of y that is replaced)/total volume)^number of replacements...not the best but the fastest way unless its trickier...

xennie · Answer

Where did you get it from? Might be useful to memorize a few of those formulas for the actual exam.

mrlofs · Answer

he 20 litre mixture contains milk and water in the ratio of 3 : 2. Therefore, there will be 12 litres of milk in the mixture and 8 litres of water in the mixture.

Step 1.
When 10 litres of the mixture is removed, 6 litres of milk is removed and 4 litres of water is removed. Therefore, there will be 6 litres of milk and 4 litres of water left in the container. It is then replaced with pure milk of 10 litres. Now the container will have 16 litres of milk and 4 litres of water.

Step 2.
When 10 litres of the new mixture is removed, 8 litres of milk and 2 litres of water is removed. The container will have 8 litres of milk and 2 litres of water in it. Now 10 litres of pure milk is added. Therefore, the container will have 18 litres of milk and 2 litres of water in it at the end of the second step. Therefore, the ratio of milk and water is 18 : 2 or 9 : 1.

Shortcut.
We are essentially replacing water in the mixture with pure milk.
Let W_o be the amount of water in the mixture originally = 8 litres.
Let W_r be the amount of water in the mixture after the replacements have taken place.
Then,{W_r}/{W_o}= (1-R/M)^n
where R is the amount of the mixture replaced by milk in each of the steps, M is the total volume of the mixture and n is the number of times the cycle is repeated.

Hence, {W_r}/{W_o} =(1/2)^2 =1/4
Therefore, "W_r ={W_o}/4= 8/4 = 2 litres

Reference: lofoya. com/Aptitude-Questions-And-Answers/Alligation-or-Mixture/l3p1 .htm

Hope it helps

kuttingchai · Answer

I just learned the new method to solve this problem as discussed by Karishma from veritas link, Trying to solve this question using that method

For future reference

total solution : 20 liters
milk : water 3:2

milk = 12 liters
water = 8 liters

Amount = Concentratio * volumn
therefore concentration of water in the solution is

8 = Cw * 20
Cw = 2/5

Step 1: 10 liters from 20 liters solution is removed - in the leftover solution the concentration of water remains the same. i.e. 2/5. Intial volume of solution was 20, new volumn is 10 (after 10 liter is removed)

Step 2: 10 liters of pure milk is added to the solution, therefore new solution is again 20 liters

Initial COncentration (Ci) * Initial Volumn (Vi) = Final Contentration (Cf) * final Volumn (Vf)
(2/5) * 10 = Cf * 20
Cf = 2/5 (1/2) (the amount of water will remain the same when as initial in (10 liter solution) but concentration of water will changes when milk is added to intial 10 liter water)

Step 3 now again from the Step 2 solution we remove 10 liter solution and added 10 liter pure milk

cf = (2/5) (1/2) (1/2)
cf = 1/10 (this is a new concentration of Water in the solution)

there fore concentration of milk is 9/10

therefore ratio of milk : water is 9:1

ankur1901 · Answer

The attachment is written with invisible ink :lol:

i cant see anything..pls can you upload it again thanks

Bunuel · Answer

This attachment is lost.

dcummins · Answer

Great question, but it may be 600 level.

AnirudhaS · Answer

I would not recommend this method, but if you want to save time you can plug in the formula. The solution is given below. Chris gave the original solution, however I think he is lacking a bit of explanation. This is my take on his original solution.

The formula is -

\frac{QuantityOfAleft}{QuantityOfAoriginallyPresent} = (1-\frac{R}{T})^n 
R = Replacing quantity of B
T = Total quantity of mixture
n = number of operations of replacement

We are replacing water in the mixture with pure milk.
Quantity of A originally present = 8 litres (water)
Then,  \frac{Quantity of A left}{8} = (1−\frac{10}{20})^n , where R is the amount of the mixture replaced by milk in each of the steps = 10 litres, T is the total volume of the mixture = 20 litres, and n is the number of times the cycle is repeated.

Therefore, Quantity of A (water) left = 2 litres.
This is the quantity of water in the mixture after the process is done twice.
So, the final quantity of water in the 20-litre mixtures is 2 litres.
Hence, the mixture will have 18 litres of milk and 2 litres of water.

Ratio of milk to water = 18:2 = 9:1

kiranbhasker · Answer

based on the forumula,

Final Volume of Milk /initial volume of milk = (1- quantity replaced/total initial quantity)^n

total 20 ltrs, ratio = 3:2 that is 12 ltrs milk and 8 ltrs water.

x/12 = (1-10/20)^2

x/12 = 1/4

x= 9/1 =  9:1

OllieMartin · Answer

New Concentration of Water = Old concentration of Water * (V1/V2)^n

New Concentration of Water = 2/5 * (1/2)^2 = 1/10

Ratio of milk:water = 9:1

aviddd · Answer

This link is not opening. Can you please check and post the correct link?

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

A 20 litre mixture of milk and water contains milk and water

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

A 20 litre mixture of milk and water contains milk and water

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards