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

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12 Jun 2010, 14:17

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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

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

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.

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

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05 Sep 2010, 03:00

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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.

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

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14 Jun 2010, 06:49

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bibha wrote:

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

Please explain this part

Step1 :-

Water in vessel = x

8 liters is removed and replaced with milk

so now,

Water = x-8 Milk = 8

Step 2 :-

Now again 8 liters of mixture is removes , so water in that mixture is w = (x-8/x) * 8

So Total water removed = x-8-w = x-8-(x-8/x)*8 = (x^2-8x-8x-64)/x = (x^2-16x-64)/x

Bunuel, could you please pin point to a mistake in my reasoning?

Let T = total capacity of vessel

1.) the concentration of milk after the first round is 8/T 2.) After the second round the concentration of milk is \(40/49\)

\(8/T\) * (T-8) + 8 = \(40/49\) T

Now, I can't solve this equation. And even substituting answer choices won't help.

Where's the mistake?

Thank you very much.

Equation is correct.

\(\frac{8}{t}(t-8)+8=\frac{40}{49}t\) --> \(\frac{t-8}{t}+1=\frac{5t}{49}\). Now, if you substitute t=14 you'll see that it'll hold true.
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Re: Eight litres are drawn off from a vessel full of water and s [#permalink]

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31 Dec 2012, 12:01

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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?

This is not a GMAT like question. Answers in gmat are usually sorted. as for as calculation is concerned it is easy. See below

Following formulae works in such scenario. More about its derivation later: If out of x, y is taken repeatedly after nth trial amount left is x(1-y/x)^n

Here asuming x ltr for water water left after two attempts is : x(1-8/x)^2

=> (x(1-8/x)^2)/x = 9/(9+40) => 1-8/x = 3/7 => 8/x = 4/7 or x = (7/4)*8 = 14ltr

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.

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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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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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.

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.

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

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12 Jun 2010, 19:42

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

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12 Jun 2010, 22:37

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

Ya but second time 8 liters are removed from the mixture of milk and water not only water.

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

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13 Jun 2010, 22:39

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

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

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13 Feb 2012, 14:19

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?

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?

If you and up with an equation shown in my post above then it won't be that difficult to solve it, for your equation it's better to plug numbers of course. So I think that the question, though tough, is still a GMAT type.
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