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

one more way:

Lets start with option:

Vessel capacity be 20litre:

8litre water removed and milk added: W=12litre, M=8litre

8litre mixture removed, (removed proportionately) : W= 12-4.8 = 7.2litre, M= 8-3.2 = 4.8 litre

8litre milk added ; W=7.2 litre, M=12.8 litre

Ratio will be 72/128 = 18/32 = 9/16

As we can see that milk is less so we have to select the ratio even less than 20,

Vessel capcity be 14 litre

8litre water removed and milk added: W=6 litre, M=8 litre

8litre mixture removed, (removed proportionately) : W= 6-3.4 = 2.6 litre, M= 8-4.6 = 3.4 litre

8litre milk added ; W=2.6 litre, M=11.4 litre

Ratio will be 26/114 = 8.66/38 (approx) very close to 9/40

So correct option is D

I solved it 2:28 mins in reverse way.

First 8 l milks added.
Then another 8 litre milk added.
so total is milk < 16 (as some milk is removed)
since final ratio of water is 9:40

water shud be < 16*9/40 i.e < 3....
so total must be less than 19...

in the given options it is clear that 14 is the answer.

Using Wine formula:

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

P solves to 14 - Ans - D

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.

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?

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 ratios video to understand the concept of multiplier: https://youtu.be/5ODENGG5dvc

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: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/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.

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.

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)

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.

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.

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: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/03 ... of-ratios/

1. We know a total of 16 L milk were added. Choice C which is 20 L capacity, implies that finally, at least 4L of water in excess was there after drawing .
2. So water is (4+x) and we know the milk added was 16L giving a ratio of (4+x)/16 which is greater than 9/40. On same reasoning all choices except D can be ruled out.