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Mixture Problem

by jose.mario.amaya » Fri May 31, 2013 6:16 am
Eight liters are drawn off from a vessel full of water and substituted by pure milk. Again eight liters 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

I need help with this problem.
Thanks


D
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by GMATGuruNY » Fri May 31, 2013 8:28 am
jose.mario.amaya wrote:Eight liters are drawn off from a vessel full of water and substituted by pure milk. Again eight liters 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

D
We can plug in the answers, which represent the capacity of the vessel.

Answer choice C: 20 liters
First replacement:
After 8 liters of water are removed, the remaining water = 20-8 = 12 liters.
After 8 liters of milk are added, the amount of milk = 8 liters.

Removal of 8 more liters:
Since 8/20 = 2/5, the amount of water and the amount of milk each decrease by 2/5, implying that 3/5 of each volume remains.
Thus:
Remaining water = (3/5)12 = 36/5 liters.
Remaining milk = (3/5)8 = 24/5 liters.

Second replacement:
After 8 liters of milk are added, the total volume of milk = 24/5 + 8 = 64/5 liters.
Resulting ratio of water to milk:
W:M = (36/5) : (64/5) = 36:64 = 9:16.

Since the required ratio of water to milk = 9:40, the proportion of milk must INCREASE.
Thus, the capacity of the vessel must be LESS than 20 liters, so that the replacement milk constitutes a GREATER proportion of the total volume.

The correct answer is D.

Answer choice D: 14 liters
First replacement:
After 8 liters of water are removed, the remaining water = 14-8 = 6 liters.
After 8 liters of milk are added, the amount of milk = 8 liters.

Removal of 8 more liters:
Since 8/14 = 4/7, the amount of water and the amount of milk each decrease by 4/7, implying that 3/7 of each volume remains.
Thus:
Remaining water = (3/7)6 = 18/7 liters.
Remaining milk = (3/7)8 = 24/7 liters.

Second replacement:
After 8 liters of milk are added, the total volume of milk = 24/7 + 8 = 80/7 liters.
Resulting ratio of water to milk:
W:M = (18/7) : (80/7) = 18:80 = 9:40.
Success!
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by faraz_jeddah » Sat Jun 01, 2013 6:45 am
GMATGuruNY wrote:
jose.mario.amaya wrote:Eight liters are drawn off from a vessel full of water and substituted by pure milk. Again eight liters 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

D
We can plug in the answers, which represent the capacity of the vessel.

Answer choice C: 20 liters


Answer choice D: 14 liters
First replacement:
After 8 liters of water are removed, the remaining water = 14-8 = 6 liters.
After 8 liters of milk are added, the amount of milk = 8 liters.

Removal of 8 more liters:
Since 8/14 = 4/7, the amount of water and the amount of milk each decrease by 4/7, implying that 3/7 of each volume remains.
Thus:
Remaining water = (3/7)6 = 18/7 liters.
Remaining milk = (3/7)8 = 24/7 liters.

Second replacement:
After 8 liters of milk are added, the total volume of milk = 24/7 + 8 = 80/7 liters.
Resulting ratio of water to milk:
W:M = (18/7) : (80/7) = 18:80 = 9:40.
Success!
I dont understand the steps in bold. How do you get that from the question?
Also, is there an algebraic approach to this problem.

Thanks.
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by vivekchandrams » Sat Jun 01, 2013 10:14 am
Hi faraz_jeddah,

Firstly, thanks for posting a good question.
I have got an algebraic solution for this. Here it is

Let's assume the volume of the water in the container after the 1st instance of removal be x.

So the volume of the container after adding 8 liters of milk would be 8+x

Now, from the 8+x liters of mixture, 8 liters is removed and 8 l of milk is added to it.

The volume of the container remains unchanged, i.e., 8+x

Now perform the 2nd iteration of removal.

The condition given is If the vessel now contains water and milk in the ratio 9:40.

So from the variable chosen above, the given condition could be translated like this:

8 + 8 - (64)/(8+x) : x - 8x/(8+x) = 40 : 9

Calm, here's how I arrived at it.

After the first iteration, the ratio of milk in the container is 8:8+x
From this mixture 8 l is removed. So the amount of milk that got removed from the container is 8*8/(8+x)

Similarly, the amount of water that got removed after the 1st iteration : 8*x(8+x)

Hence the equation.

Now let's equate terms from each side.

So it would be like

16 - 64/(8+x) = 40*k

x - 8x/(8+x) = 9*k


k is a constant introduced to remove the fraction.

Now, solving the equations (omitting the solving here) we arrive at

5x^2 - 18x - 72 = 0

The solutions for the above equation are x = 6 and x = -12.
As -12 cannot be volume of water, eliminate it

Hence, x = 6

Hence, volume of the container is 8 + x = 14 liters!!!!!!

Ah! At last! The answer!

So, I would advise you not to go by the conventional approach. Rather, go by Mitch's approach.

Hope it helps.

Pls hit Thank if you find my post useful
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by arpshriv » Sat Jun 01, 2013 11:16 am
GMATGuruNY wrote:
jose.mario.amaya wrote:Eight liters are drawn off from a vessel full of water and substituted by pure milk. Again eight liters 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

D
We can plug in the answers, which represent the capacity of the vessel.

Answer choice C: 20 liters
First replacement:
After 8 liters of water are removed, the remaining water = 20-8 = 12 liters.
After 8 liters of milk are added, the amount of milk = 8 liters.

Removal of 8 more liters:
Since 8/20 = 2/5, the amount of water and the amount of milk each decrease by 2/5, implying that 3/5 of each volume remains.
Thus:
Remaining water = (3/5)12 = 36/5 liters.
Remaining milk = (3/5)8 = 24/5 liters.

Second replacement:
After 8 liters of milk are added, the total volume of milk = 24/5 + 8 = 64/5 liters.
Resulting ratio of water to milk:
W:M = (36/5) : (64/5) = 36:64 = 9:16.

Since the required ratio of water to milk = 9:40, the proportion of milk must INCREASE.
Thus, the capacity of the vessel must be LESS than 20 liters, so that the replacement milk constitutes a GREATER proportion of the total volume.

The correct answer is D.

Answer choice D: 14 liters
First replacement:
After 8 liters of water are removed, the remaining water = 14-8 = 6 liters.
After 8 liters of milk are added, the amount of milk = 8 liters.

Removal of 8 more liters:
Since 8/14 = 4/7, the amount of water and the amount of milk each decrease by 4/7, implying that 3/7 of each volume remains.
Thus:
Remaining water = (3/7)6 = 18/7 liters.
Remaining milk = (3/7)8 = 24/7 liters.

Second replacement:
After 8 liters of milk are added, the total volume of milk = 24/7 + 8 = 80/7 liters.
Resulting ratio of water to milk:
W:M = (18/7) : (80/7) = 18:80 = 9:40.
Success!
Could you please also show algebraic approach.
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by GMATGuruNY » Sat Jun 01, 2013 1:51 pm
jose.mario.amaya wrote:Eight liters are drawn off from a vessel full of water and substituted by pure milk. Again eight liters 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

D
Let the capacity of the container = x.
Every time 8 liters are removed, the volume of liquid inside the container decreases by 8/x.
To illustrate:
If x=16, removing 8 liters from the container -- 1/2 of the volume -- is equivalent to reducing the volume by 8/x = 8/16 = 1/2.
If x=32, removing 8 liters from the container -- 1/4 of the volume -- is equivalent to reducing the volume by 8/x = 8/32 = 1/4.
Thus, every time 8 liters are removed, the volume of water inside the container decreases by 8/x.

At the start, the container is full of water.
Thus, the original amount of water = x.
When 8 liters are removed, the decrease in the water = (8/x)x = 8.
Remaining amount of water = x-8.
When 8 more liters are removed, the decrease in the water = (8/x)(x-8).
Remaining amount of water = (x-8) - (8/x)(x-8) = (x-8)*(1 - 8/x) = (x-8)*(x-8)/x.

At the end of the process, water:milk = 9:40.
Since 9+40 = 49, water constitutes 9 of every 49 liters inside the container, implying that the amount of water = (9/49)x.

Since both expressions in red above represent the amount of water at the end of the process, we get:
(x-8)*(x-8)/x = (9/49)x

(x-8)² = (9/49)x²

x-8 = (3/7)x

7x - 56 = 3x

4x = 56

x = 14.

The correct answer is D.
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by GMATGuruNY » Sat Jun 01, 2013 2:58 pm
faraz_jeddah wrote:
jose.mario.amaya wrote:Eight liters are drawn off from a vessel full of water and substituted by pure milk. Again eight liters 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

D
is there an algebraic approach to this problem.

Thanks.
Check my post above for an algebraic solution.
Below is a slightly different way to solve with the answer choices.

The resulting ratio of water to milk = 9:40.
Since 9+40 = 49, the resulting amount of water must constitute 9/49 of the final solution.
We can plug in the answers, which represent the original amount of water in the vessel.

Answer choice C: 20 liters
After 8 liters are removed, the remaining amount of water = 20-8 = 12.

When 8 more liters are removed from the 20-liter container, the volume of liquid inside the container will decrease by 8/20 = 2/5.
This means that the amount of water inside the container also will decrease by 2/5:
12 - (2/5)12 = 12 - 24/5 = 36/5.

Proportion of water in the final solution:
(resulting amount of water)/(total volume) = (36/5) / 20 = 36/100 = 9/25.

Since the resulting amount of water must constitute a SMALLER fraction of the total volume -- 9/49 -- the original amount of water must be LESS than 20.

The correct answer is D.
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by sy323 » Sat Jun 22, 2013 9:31 am
I can't wrap my head around doing this question within 2.5 minutes on the GMAT... I think it would take me 1 minute to think of what this question is asking and then I wouldn't have time to test the choices.
Would this be a 700 range question?
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by guerrero » Sat Jun 22, 2013 9:35 am
That's right . Mitch is an Expert( he can solve whatever comes his way) .. An ordinary guy like me can't think of doing it in ~2mins . Good question (700+ type ,most likely )
Last edited by guerrero on Sat Jun 22, 2013 9:50 am, edited 2 times in total.
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by guerrero » Sat Jun 22, 2013 9:45 am
jose.mario.amaya wrote:Eight liters are drawn off from a vessel full of water and substituted by pure milk. Again eight liters 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

I need help with this problem.
Thanks


D

1st step -8 liters of pure milk added
2nd step - Another 8 litre milk added.
so total quantity of milk will be less than 16 (b'cause some of milk is removed)
since final ratio of water to Milk is 9:40

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

the only option that fits is 14
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by sy323 » Sun Jun 23, 2013 8:12 am
thanks, this is dumbed down enough for me
guerrero wrote:
jose.mario.amaya wrote:Eight liters are drawn off from a vessel full of water and substituted by pure milk. Again eight liters 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

I need help with this problem.
Thanks


D

1st step -8 liters of pure milk added
2nd step - Another 8 litre milk added.
so total quantity of milk will be less than 16 (b'cause some of milk is removed)
since final ratio of water to Milk is 9:40

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

the only option that fits is 14
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by sy323 » Sun Jun 23, 2013 8:27 am
Can you explain this part? i'm not sure I understand the logic behind 16*9/40
guerrero wrote:
jose.mario.amaya wrote: water shud be less than 16*9/40 i.e less than 3
so total must be less than 19

the only option that fits is 14
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by vipulgoyal » Wed Jun 26, 2013 12:10 am
alternate approach

8 liters of x (water) is replaced by 8 liters of milk
milk ratio in 1 liter of mixture = 8/x
in 8 liters 8*8/x = 64/x
now 8 liters of mixture which contain 64/x milk is replaced by 8 leter of milk
remaining milk after removing 8 liter of mixture
8 - 64/x
now 8 liters of milk added
total milk in mixture = 8 - 64/x +8 = 16 - 64 / x^2
Equation becomes

16 - 64 / x^2 = 40/49 OR
5x^2 -98x + 392 = 0
solving further we get
x= (98 +- 42)/ 10
x= 56/10 = 5.6 Not an option
x= 140/10 = 14 BINGO
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by GMATGuruNY » Thu Jun 27, 2013 2:22 am
sy323 wrote:Can you explain this part? i'm not sure I understand the logic behind 16*9/40
Here's the reasoning:
jose.mario.amaya wrote:Eight liters are drawn off from a vessel full of water and substituted by pure milk. Again eight liters 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
8 liters of milk are poured into the container.
When liquid from the container is drawn off, the amount of milk in the container decreases to LESS then 8 liters.
Then 8 more liters of milk are poured into the container.
Resulting amount of milk = LESS than 16 liters.

After the two substitutions, W/M = 9/40.
If there are 16 liters of milk, the following amount of water is implied:
9/40 = W/16
W = (16*9)/40 = 3.6 liters.
In this case, M+W = 16 + 3.6 = 19.6 liters.
Since the ACTUAL amount of milk is LESS than 16 liters, the capacity of the container must be LESS than 19.6 liters.

The correct answer is D.
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by hemant_rajput » Thu Jun 27, 2013 9:38 am
For people, who needed quick solution, mug this formula

milk/ (milk + water) = (1 - b/a)^n

here,
b is the amount of milk taken out in iteration.

a is the total capacity of the vessel containing solution

n is the no. of times the process of taking milk out and replacing it with water is repeated.

so, for current problem : -

9/(40 + 9) = ( 1 - (8/a))^2

taking root on both sides.

3 / 7 = 1 - 8/a

8/a = 4/7

a = 14

Hope this helps.
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