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Veritas Prep GMAT Instructor D
Joined: 16 Oct 2010
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Re: Eight litres are drawn off from a vessel full of water and s  [#permalink]

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1
guerrero25 wrote:
VeritasPrepKarishma wrote:
nonameee wrote:
Bunuel, thanks a lot. I must have made some miscalculation.

Conceptually the question is interesting. However the numbers are quite "difficult" to work with. Do you think that a real GMAT question would have better numbers?

Actually, the numbers are quite suitable for a very efficient, quick and oral solution. This is what I thought of when I came up with the answer in 20 secs. Mind you, you need to go through the link provided below to understand this theory. Else the 20 sec solution will probably not make sense to you.

We are substituting milk so we should work with water.

Final concentration of water = 9/49
There were two iterations. So, 9/49 = (100%)*(Vi/Vf)^2
Vi/Vf = 3/7
Since we are putting 8 liters of water but difference between Vi and Vf is 4, final volume (which is also equal to volume of the vessel) must be twice too i.e. 7*2 = 14 liters.

http://www.veritasprep.com/blog/2012/01 ... -mixtures/

Hi Karishma , Could you please elaborate the highlighted portion ? I am unable to grasp the last part of the solution . Thanks in advance .

This is the ratios concept in action here even though I haven't used ratio explicitly.
Vi is the volume after you remove 8 lts but before you put it back. Vf is the volume after you put the 8 lts back in.

But you get Vi/Vf = 3/7 i.e. their ratio is 3:7. But the actual difference in their volume is 8lts.
so 7x - 3x = 8 giving you x = 2
Vi = 6 lts, Vf = 14 lts

(All I did above was I saw that the difference between 3 and 7 is 4 (the ratio difference) but actual difference is 8 so the multiplier is 2. Hence the actual volume would be twice of the ratio too. Check out my ratio posts on my blog to understand the multiplier concept)
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Re: Eight litres are drawn off from a vessel full of water and s  [#permalink]

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1
1
Quote:
Eight litres are drawn off from a vessel full of water and substituted by pure milk. Again eight litres of the mixture are drawn off and substituted by pure milk. If the vessel now contains water and milk in the ratio 9:40, find the capacity of the vessel.

A. 21 liters
B. 22 liters
C. 20 liters
D. 14 liters
E. 28 liters

The way the equation is expressed, it seems to me that the number used for Vf should be the volume of the vessel (this is my understanding of what Vf is supposed to represent). However, that is obviously incorrect. Thus, I am having difficulty conceptualizing what Vf actually represents in the solution you have provided and, further, how you were able to intuitively determine that it should be doubled. I understand that Vi of 3 and Vf of 7 must be doubled in order to ensure that Vf-Vi=8. However, I do not understand why this is permissible within the construct of the concentration equation.

Responding to a pm:

You are correct. In this question Vf is the capacity of the vessel.

What is Vf in replacement questions?
Replacement consists of two steps: - 'withdraw from the vessel' and 'put back into the vessel'.
When you withdraw from the vessel, the volume goes down - This is Vi for the next step.
When you put back, the volume comes up again - this is Vf. In this step, since amount of water stays the same (you are putting in milk), CiVi = CfVf

In this question, vessel is FULL of water and you are substituting part of it by milk. So it will be FULL when you put milk in it in step 2. So Vf is the capacity of the vessel.

Also, we are using ratios here.
Say, you know a/b = 1/2. If a = 10, what is b? It is 20, right?
Similarly, you know a/b = 1/2. If b-a = 4, what is a? Note that on the ratio scale, the difference between b and a is 1 (2 - 1). But actually it is 4 so a must be 4 and b must be 8.
Check out my posts on ratios: http://www.veritasprep.com/blog/2011/03 ... of-ratios/

Since Vi/Vf = 3/7 but Vf - Vi = 8 (twice of what it is on the ratio scale), Vi must be 6 and Vf must be 14.
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Re: Eight litres are drawn off from a vessel full of water and s  [#permalink]

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VeritasPrepKarishma wrote:
nonameee wrote:
Bunuel, thanks a lot. I must have made some miscalculation.

Conceptually the question is interesting. However the numbers are quite "difficult" to work with. Do you think that a real GMAT question would have better numbers?

Actually, the numbers are quite suitable for a very efficient, quick and oral solution. This is what I thought of when I came up with the answer in 20 secs. Mind you, you need to go through the link provided below to understand this theory. Else the 20 sec solution will probably not make sense to you.

We are substituting milk so we should work with water.

Final concentration of water = 9/49
There were two iterations. So, 9/49 = (100%)*(Vi/Vf)^2
Vi/Vf = 3/7
Since we are putting 8 liters of water but difference between Vi and Vf is 4, final volume (which is also equal to volume of the vessel) must be twice too i.e. 7*2 = 14 liters.

http://www.veritasprep.com/blog/2012/01 ... -mixtures/

Responding to a pm:

Cf*Vf = Ci*Vi
Cf = Ci * (Vi/Vf)

When you add milk for the first time, concentration of water in initial solution is 100%

Cf = 100% (Vi/Vf)

Then you remove some solution which doesn't change its concentration which is 100% (Vi/Vf)
When you add milk again, initial concentration is 100% (Vi/Vf) and the final concentration is given by

Cf2 = 100% (Vi/Vf) * (Vi/Vf) = 100%(Vi/Vf)^2

We know that this Cf2 is given to be 9/49

So Vi/Vf = 3/7

So every time we take out some solution, the volume of the solution reduces to Vi and every time we add it back, it goes up to Vf.
7x - 3x = 8 lts
x = 2 lts

So Vi is 6 lts and Vf is 14 lts. The capacity of the vessel i.e. the volume when it is full (i.e. when we put back the 8 lts of milk) is 14 lts.
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Re: Eight litres are drawn off from a vessel full of water and s  [#permalink]

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oldstudent wrote:
virupaksh2010 wrote:
Eight litres are drawn off from a vessel full of water and substituted by pure milk. Again eight litres of the mixture are drawn off and substituted by pure milk.If the vessel now contains water and milk in the ratio 9:40, find the capacity of the vessel.

A. 21 litrers

B. 22 litres

C. 20 litres

D. 14 litres

E. 28 litres

Using Wine formula:

{ (P-8) / P }^2 = 9/49

P solves to 14 - Ans - D

Hi,

Will this formula holds good for all this types of questions?

Regards,
Rrsnathan.
Senior Manager  Joined: 10 Jul 2013
Posts: 312
Re: Eight litres are drawn off from a vessel full of water and s  [#permalink]

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virupaksh2010 wrote:
Eight litres are drawn off from a vessel full of water and substituted by pure milk. Again eight litres of the mixture are drawn off and substituted by pure milk. If the vessel now contains water and milk in the ratio 9:40, find the capacity of the vessel.

A. 21 liters
B. 22 liters
C. 20 liters
D. 14 liters
E. 28 liters

Milk and water can change in amount but Total capacity will not. So stay with capacity. That's the theme should work on. REMEMBER, STICK WITH CAPACITY.

Capacity = x
1st event:
water : capacity = (x-8) : x

2nd event:
8 liters more solution replaced. but how much water it carried OUT with itself ? So,

EARLIER x liters solution contained (x-8) liter water
or, 1 liter ....................... = (x-8)/x liter water
or, 8 liters ...................... = 8(x-8)/x liter

So now we have water = x-8 - {8(x-8)/x} = (x-8)^2 / x liter

GIVEN, water:milk = 9:40 , So total = 49 . Here, water : capacity = 9:49
Finally,
water : capacity = {(x-8)^2/x} : x = 9:49
or, x = 14 (Answer)
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Re: Eight litres are drawn off from a vessel full of water and s  [#permalink]

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iambroke wrote:
the equation you'd get once 8 lts are drawn the second time is: (w-16)/16 = 9/40 where w is the capacity of the vessel.
Solving for w, w = 19.6. So, its C

I am also getting 19.6
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Re: Eight litres are drawn off from a vessel full of water and s  [#permalink]

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VeritasPrepKarishma wrote:
Quote:
Eight litres are drawn off from a vessel full of water and substituted by pure milk. Again eight litres of the mixture are drawn off and substituted by pure milk. If the vessel now contains water and milk in the ratio 9:40, find the capacity of the vessel.

A. 21 liters
B. 22 liters
C. 20 liters
D. 14 liters
E. 28 liters

The way the equation is expressed, it seems to me that the number used for Vf should be the volume of the vessel (this is my understanding of what Vf is supposed to represent). However, that is obviously incorrect. Thus, I am having difficulty conceptualizing what Vf actually represents in the solution you have provided and, further, how you were able to intuitively determine that it should be doubled. I understand that Vi of 3 and Vf of 7 must be doubled in order to ensure that Vf-Vi=8. However, I do not understand why this is permissible within the construct of the concentration equation.

Responding to a pm:

You are correct. In this question Vf is the capacity of the vessel.

What is Vf in replacement questions?
Replacement consists of two steps: - 'withdraw from the vessel' and 'put back into the vessel'.
When you withdraw from the vessel, the volume goes down - This is Vi for the next step.
When you put back, the volume comes up again - this is Vf. In this step, since amount of water stays the same (you are putting in milk), CiVi = CfVf

In this question, vessel is FULL of water and you are substituting part of it by milk. So it will be FULL when you put milk in it in step 2. So Vf is the capacity of the vessel.

Also, we are using ratios here.
Say, you know a/b = 1/2. If a = 10, what is b? It is 20, right?
Similarly, you know a/b = 1/2. If b-a = 4, what is a? Note that on the ratio scale, the difference between b and a is 1 (2 - 1). But actually it is 4 so a must be 4 and b must be 8.
Check out my posts on ratios: http://www.veritasprep.com/blog/2011/03 ... of-ratios/

Since Vi/Vf = 3/7 but Vf - Vi = 8 (twice of what it is on the ratio scale), Vi must be 6 and Vf must be 14.

What is the wine Formula, your solutions are always amazing.

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Re: Eight litres are drawn off from a vessel full of water and s  [#permalink]

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VeritasPrepKarishma wrote:
VeritasPrepKarishma wrote:
nonameee wrote:
Bunuel, thanks a lot. I must have made some miscalculation.

Conceptually the question is interesting. However the numbers are quite "difficult" to work with. Do you think that a real GMAT question would have better numbers?

Actually, the numbers are quite suitable for a very efficient, quick and oral solution. This is what I thought of when I came up with the answer in 20 secs. Mind you, you need to go through the link provided below to understand this theory. Else the 20 sec solution will probably not make sense to you.

We are substituting milk so we should work with water.

Final concentration of water = 9/49
There were two iterations. So, 9/49 = (100%)*(Vi/Vf)^2
Vi/Vf = 3/7
Since we are putting 8 liters of water but difference between Vi and Vf is 4, final volume (which is also equal to volume of the vessel) must be twice too i.e. 7*2 = 14 liters.

http://www.veritasprep.com/blog/2012/01 ... -mixtures/

Responding to a pm:

Cf*Vf = Ci*Vi
Cf = Ci * (Vi/Vf)

When you add milk for the first time, concentration of water in initial solution is 100%

Cf = 100% (Vi/Vf)

Then you remove some solution which doesn't change its concentration which is 100% (Vi/Vf)
When you add milk again, initial concentration is 100% (Vi/Vf) and the final concentration is given by

Cf2 = 100% (Vi/Vf) * (Vi/Vf) = 100%(Vi/Vf)^2

We know that this Cf2 is given to be 9/49

So Vi/Vf = 3/7

So every time we take out some solution, the volume of the solution reduces to Vi and every time we add it back, it goes up to Vf.
7x - 3x = 8 lts
x = 2 lts

So Vi is 6 lts and Vf is 14 lts. The capacity of the vessel i.e. the volume when it is full (i.e. when we put back the 8 lts of milk) is 14 lts.

What is Ci and Cf here.

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Re: Eight litres are drawn off from a vessel full of water and s  [#permalink]

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trafficspinners wrote:

What is Ci and Cf here.

I am not sure what you mean by "wine formula". Here is the link which discussed Ci and Cf i.e. the replacement concept: http://www.veritasprep.com/blog/2012/01 ... -mixtures/
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Re: Eight litres are drawn off from a vessel full of water and s  [#permalink]

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Here is the general formula

water/capacity =(1-(quantity taken out/total volume))^n
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Re: Eight litres are drawn off from a vessel full of water and s  [#permalink]

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Bunuel wrote:
virupaksh2010 wrote:
Eight litres are drawn off from a vessel full of water and substituted by pure milk. Again eight litres of the mixture are drawn off and substituted by pure milk.If the vessel now contains water and milk in the ratio 9:40, find the capacity of the vessel.

A. 21 liters
B. 22 liters
C. 20 liters
D. 14 liters
E. 28 liters

Let the capacity of the vessel be $$x$$.

After the first removal there would be $$x-8$$ liters of water left in the vessel. Note that the share of the water would be $$\frac{x-8}{x}$$;

After the second removal, the removed mixture of 8 liters will contain $$8*\frac{x-8}{x}$$ liters of water, so there will be $$x-8-8*\frac{x-8}{x}=\frac{(x-8)^2}{x}$$ liters of water left.

As the ratio of water to milk after that is $$\frac{9}{40}$$, then the ratio of water to the capacity of the vessel would be $$\frac{9}{40+9}=\frac{9}{49}$$.

So $$\frac{\frac{(x-8)^2}{x}}{x}=\frac{9}{49}$$ --> $$\frac{(x-8)^2}{x^2}=\frac{9}{49}$$ --> $$\frac{x-8}{x}=\frac{3}{7}$$ --> $$x=14$$.

I tried it like this:

water remaining after 2 times = (x-8)^2 / x
Milk rem = 8x-64 / x

thus, (x-8)^2 / 8x-64 = 9/20
this gives the ans x = 11.6
can u tell me what I'm missing here?
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Joined: 02 Sep 2009
Posts: 53721
Re: Eight litres are drawn off from a vessel full of water and s  [#permalink]

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Cartman2013 wrote:
Bunuel wrote:
virupaksh2010 wrote:
Eight litres are drawn off from a vessel full of water and substituted by pure milk. Again eight litres of the mixture are drawn off and substituted by pure milk.If the vessel now contains water and milk in the ratio 9:40, find the capacity of the vessel.

A. 21 liters
B. 22 liters
C. 20 liters
D. 14 liters
E. 28 liters

Let the capacity of the vessel be $$x$$.

After the first removal there would be $$x-8$$ liters of water left in the vessel. Note that the share of the water would be $$\frac{x-8}{x}$$;

After the second removal, the removed mixture of 8 liters will contain $$8*\frac{x-8}{x}$$ liters of water, so there will be $$x-8-8*\frac{x-8}{x}=\frac{(x-8)^2}{x}$$ liters of water left.

As the ratio of water to milk after that is $$\frac{9}{40}$$, then the ratio of water to the capacity of the vessel would be $$\frac{9}{40+9}=\frac{9}{49}$$.

So $$\frac{\frac{(x-8)^2}{x}}{x}=\frac{9}{49}$$ --> $$\frac{(x-8)^2}{x^2}=\frac{9}{49}$$ --> $$\frac{x-8}{x}=\frac{3}{7}$$ --> $$x=14$$.

I tried it like this:

water remaining after 2 times = (x-8)^2 / x
Milk rem = 8x-64 / x

thus, (x-8)^2 / 8x-64 = 9/20
this gives the ans x = 11.6
can u tell me what I'm missing here?

Milk = x - (x-8)^2/x = 16(x-4)/x not 8x-64/x.

Also, the ratio is 9:40 not 9:20.

Hope it helps.
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Re: Eight litres are drawn off from a vessel full of water and s  [#permalink]

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Let the capacity of the vessel be $$x$$.

After the first removal there would be $$x-8$$ liters of water left in the vessel. Note that the share of the water would be $$\frac{x-8}{x}$$;

After the second removal, the removed mixture of 8 liters will contain $$8*\frac{x-8}{x}$$ liters of water, so there will be $$x-8-8*\frac{x-8}{x}=\frac{(x-8)^2}{x}$$ liters of water left.

As the ratio of water to milk after that is $$\frac{9}{40}$$, then the ratio of water to the capacity of the vessel would be $$\frac{9}{40+9}=\frac{9}{49}$$.

So $$\frac{\frac{(x-8)^2}{x}}{x}=\frac{9}{49}$$ --> $$\frac{(x-8)^2}{x^2}=\frac{9}{49}$$ --> $$\frac{x-8}{x}=\frac{3}{7}$$ --> $$x=14$$.

I tried it like this:

water remaining after 2 times = (x-8)^2 / x
Milk rem = 8x-64 / x

thus, (x-8)^2 / 8x-64 = 9/20
this gives the ans x = 11.6
can u tell me what I'm missing here?[/quote]

Milk = x - (x-8)^2/x = 16(x-4)/x not 8x-64/x.

Also, the ratio is 9:40 not 9:20.

Hope it helps.[/quote]

Thanks...that is very helpful
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Re: Eight litres are drawn off from a vessel full of water and s  [#permalink]

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Dear Brunel ,

Can we use this formula?

(x-y/x)^n = Qualtity after n operations / Initial qty in vessel

(x-8/x)^2 = 9/49
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Re: Eight litres are drawn off from a vessel full of water and s  [#permalink]

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Brunel ,

Please explain this concept in little detail ...

Let the capacity of the vessel be x.

After the first removal there would be x-8 liters of water left in the vessel. Note that the share of the water would be \frac{x-8}{x};

After the second removal, the removed mixture of 8 liters will contain 8*\frac{x-8}{x} liters of water, so there will be x-8-8*\frac{x-8}{x}=\frac{(x-8)^2}{x} liters of water left.
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Re: Eight litres are drawn off from a vessel full of water and s  [#permalink]

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when you state that (x-8)-8*[(x-8)/x] = (x-8)^2/x , what happens to the -8 since you have two? would that not then be (x-16)*[(x-8)/x]?
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Re: Eight litres are drawn off from a vessel full of water and s  [#permalink]

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mmcneilly wrote:
when you state that (x-8)-8*[(x-8)/x] = (x-8)^2/x , what happens to the -8 since you have two? would that not then be (x-16)*[(x-8)/x]?

$$(x-8)-\frac{8*(x-8)}{x} =\frac{x(x-8)-8*(x-8)}{x}=\frac{(x-8)(x-8)}{x}=\frac{(x-8)^2}{x}$$.

Hope it's clear.
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Eight litres are drawn off from a vessel full of water and s  [#permalink]

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virupaksh2010 wrote:
Eight litres are drawn off from a vessel full of water and substituted by pure milk. Again eight litres of the mixture are drawn off and substituted by pure milk. If the vessel now contains water and milk in the ratio 9:40, find the capacity of the vessel.

A. 21 liters
B. 22 liters
C. 20 liters
D. 14 liters
E. 28 liters

(1-8/x)^2=(3/7)^2 ( So square - square is out)
=x-8/x =3/7
=7x-56=3x
=7x-3x=56
x=56/4
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Re: Eight litres are drawn off from a vessel full of water and s  [#permalink]

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1
guerrero25 wrote:
VeritasPrepKarishma wrote:
nonameee wrote:
Bunuel, thanks a lot. I must have made some miscalculation.

Conceptually the question is interesting. However the numbers are quite "difficult" to work with. Do you think that a real GMAT question would have better numbers?

Actually, the numbers are quite suitable for a very efficient, quick and oral solution. This is what I thought of when I came up with the answer in 20 secs. Mind you, you need to go through the link provided below to understand this theory. Else the 20 sec solution will probably not make sense to you.

We are substituting milk so we should work with water.

Final concentration of water = 9/49
There were two iterations. So, 9/49 = (100%)*(Vi/Vf)^2
Vi/Vf = 3/7
Since we are putting 8 liters of water but difference between Vi and Vf is 4, final volume (which is also equal to volume of the vessel) must be twice too i.e. 7*2 = 14 liters.

http://www.veritasprep.com/blog/2012/01 ... -mixtures/

Hi Karishma , Could you please elaborate the highlighted portion ? I am unable to grasp the last part of the solution . Thanks in advance .

This is a ratios concept.
Vi = 3x, Vf = 7x
7x - 3x = 8 (because 8 lts of water was put)
x = 2

So Vi = 3*2 = 6
Vf = 7*2 = 14

You can do the same thing orally like this:
We get Vi : Vf = 3 : 7
The difference between Vi and Vf on the ratio scale is 4 (7 - 3) but actually it is 8. This means the multiplier is 2. So actual values of Vi and Vf must be 3*2 = 6 and 7*2 = 14.

Check this post for more on ratios: http://www.veritasprep.com/blog/2011/03 ... of-ratios/
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Eight litres are drawn off from a vessel full of water and s  [#permalink]

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If we look at it logically.. its quite a simple problem.... Lets go step by step...
We know that 8 liters of milk was added twice and some Milk would have come out in the second take out.. So we know that
8 liters < Milk in the mixture < 16 liters
now we already know the final ratio.. lets apply that ..
8 * 49/40 liters < Total mixture < 16 * 49/40
9.8 liters < total mixture < 19.6 liters

Only solution in that range is 14.. hence our answer.. However if the answer choices had been close we might have issue.. but then we can reassuringly rely on GMAT to not require us to make all these calculations .. Eight litres are drawn off from a vessel full of water and s   [#permalink] 05 Jun 2015, 20:59

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# Eight litres are drawn off from a vessel full of water and s

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