From a solution containing milk and water in the ratio 2 : 3, 15 litre

Question

From a solution containing milk and water in the ratio 2 : 3, 15 litres of solution is removed and replaced with 15 litres of milk. This operation is done once more. Now the ratio of milk to water is 7 : 5. What was the volume of the initial solution?

A. 60 litres
B. 70 litres
C. 80 litres
D. 90 litres
E. 100 litres

A. 60 litres
B. 70 litres
C. 80 litres
D. 90 litres
E. 100 litres

KarishmaB · Accepted Answer

The logic of the method is discussed here: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2012/0 ... -mixtures/ The formula for multiple replacements is: C2 = C1 * (V1/V2)^n (5/12) = (3/5)* ^2 (V - 15)/V = 5/6 V = 90 Answer (D)

CrackverbalGMAT · Answer

When doing Iterations, we use the equation  \frac{Final }{ Initial}  =  (1 - \frac{b }{ a})^n

Final = The Final amount / Ratio of that substance who's  concentration is being reduced .

Initial = The Initial amount / Ratio of that substance who's  concentration is being reduced .

b = Amount of liquid ( who's concentration is increasing ) is added back

a = Final Volume in the container  after the replacement  of amount b.

n = number of iterations, or the number of times the process is repeated.

In the question Above

Let x L be the initial Volume of solution

Here the concentration of water is reducing and that of milk is increasing

Final Ratio = (5/12) * x

Initial Ratio = (3/5) * x

b = 15

a = x (as 15L are removed and replaced, the volume is unchanged)

n = 2

Substituting in the equation,  \frac{Final }{ Initial}  =  (1 - \frac{b }{ a})^n

\frac{(5/12)x}{(3/5)x}  =  (1 - \frac{15 }{ x})^2

\frac{25}{36}  =  (1 - \frac{15 }{ x})^2

Taking square root on both sides

\frac{5}{6}  =  (1 - \frac{15 }{ x})

\frac{15}{x}  = 1 -  \frac{5}{6}  =  \frac{1}{6}

x = 15 * 6 = 90 L

Option D

Arun Kumar

Archit3110 · Answer

c2 = c1 * (v1/v2)
c1= 2/5
c2= 7/12
7/12 =2/5 * ( v/v+15) ^2
solve for v = 90
IMO D

yashwardhan · Answer

Archit3110  can you please share your working for arriving at the value of v?

yashwardhan · Answer

VeritasKarishma  Ty for your reply. I observe you've used the ratio of water (5/12 and 3/5) due to which you were able to easily take the square root to get 5/6. However, if i had to use the ratio of milk(7/12 and 2/5), I am unable to get the square root of 35/24 which leads to a long quadratic equation. Could you suggest another way of solving for v using the ratio of milk? Note - I have taken vi as "v" and vf as "v+15".

KarishmaB · Answer

You are adding back MILK. So the AMOUNT of milk does not stay same. Hence CiVi = CfVf does not hold for milk.
You have to work with concentration of water only here.

ScottTargetTestPrep · Answer

Solution: Instead of solving this problem algebraically, let’s solve it numerically by checking the given answer choices. Let’s start with choice A using the table below (notice that the volume of the solution is always 60 if we use choice A): milk1.png Now, let’s compute the ratio of milk to water from the last row of the table above: 39.75/20.25 = 159/81 = 53/27 We see that 53/27 is not 7/5. Choice A is not correct. Let’s continue with choice D using the table below (notice that the volume of the solution is always 90 if we use choice D): milk2.png Now, let’s compute the ratio of milk to water from the last row of the table above: 52.5/37.5 = 105/75 = 7/5 We see that choice D is the correct answer. (Note: We check choice A first and then skip to choice D because 15, the amount we are replacing, is 1/4 of 60 and 1/6 of 90. Notice that 1/4 and 1/6 are nice and easy fractions.) Answer: D

priyachandank · Answer

Why are we working out with the Water concentration, when milk is being added here? and if we take 7/12=2/5 (1-15/x)^2, it results to 75 closely. 
but if I assume for 100 and solve the whole problem, it leaves the ratio a little close to 90L. 
Could you add detailed steps, please?

CEdward · Answer

ScottTargetTestPrep VeritasKarishma What if we don't know the initial concentration? Secondly, Karishma, why did you choose water for your fraction as opposed to milk? Does the answer work out to be the same anyway?

Here's a question where we don't know the initial concentration at all.
https://gmatclub.com/forum/8-litres-are ... 41893.html

ScottTargetTestPrep · Answer

Response:
 
Without the initial concentration, it is not possible to determine the volume of the initial solution. If we are not told anything about the initial concentration, the question either has to give us some other information or ask for some other quantity. We would have a different question in that case.

CEdward · Answer

Hi Karishma, Why is this question different from the one you discussed here? https://gmatclub.com/forum/10-of-a-50-a ... 87462.html

Can,Will · Answer

KarishmaB 
here the initial volume is V-15 then why we took V that is 90 as initial volume?

Can,Will · Answer

KarishmaB  
here, C2=C1∗(V1/V2)n V1 is initial volume which we took as V-15 and V2 as final volume which is V.
so Initial volume should be 90-15 ? 
can you explain ? I am not able to understand

KarishmaB · Answer

Replacement has two steps - first removal and then addition. This leads to replacement.

The change in concentration happens at the addition step, not the removal step. So V1 is the volume before addition (which is after removal) and V2 is the volume after addition (when volume comes back to its original quantity). 
Hence if we had V volume to begin with, when we remove 15 litres, our volume becomes V - 15. Now before adding, volume is V - 15 (which is V1) and after adding, the volume is V (which is V2).

sayan640 · Answer

KarishmaB  , Can you please explain the logic behind this formulae here ?  I am struggling with these kind of questions.

sayan640 · Answer

KarishmaB  , I am getting lost while trying to solve this question using the formulae : -
Final concentration of water = initial concentration of water * ( 1 - amount of solution replaced / total amount of the initial solution )^ 2
Can you kindly show as to where I am making mistake ?

sayan640 · Answer

Also , It was not clear to me why you considered the volume of milk here for C1 and C2 . Can you please help ?

KarishmaB · Answer

Explanation of this concept is available on my blog here: https://anaprep.com/arithmetic-replacement-in-mixtures/

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