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Math Expert V
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Question Stats: 55% (02:28) correct 45% (02:24) wrong based on 62 sessions

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3L of milk are drawn from a container containing 30L of milk. It is replaced by water and the process is repeated 2 times. What is the ratio of milk to water at the end?

A. $$\frac{729}{271}$$

B. $$\frac{743}{229}$$

C. $$\frac{81}{19}$$

D. $$\frac{2187}{100}$$

E. $$\frac{813}{19}$$

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Re: 3L of milk are drawn from a container containing 30L of milk. It is re  [#permalink]

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We are given that there was a 30L milk in a container. Out of this milk, 3 liters of milk was taken out and replaced with 3L of water. This process of drawing out 3L from the container was repeated twice.
After the first replacement, the amount of milk left$$= 30-3=27L$$ concentration of milk in container = $$0.9$$
After second replacement the amount of milk left$$= 27 - 3*0.9 = 27-2.7 = 24.3L.$$ Concentration is now $$\frac{243}{300} = 0.81$$
After third replacement the amount of milk left $$= 24.3 - 3*0.81 = 24.3-2.43 = 21.87L$$.
Concetration $$=\frac{21.87}{30}=\frac{729}{1000}$$
Hence for every $$729L$$ of milk there is $$(1000-729) = 271L$$ of water.
So ration of milk to water $$= \frac{729}{271}$$

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Amount of Milk after 3 Replacements/ Total Volume of the mixture = $$(1-\frac{3}{30})^3$$

Amount of Milk after 3 Replacements/ Total Volume of the mixture = $$\frac{729}{1000}$$

ratio of milk to water at the end = 729 : (1000-729) = 729 : 271
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Formula in use: Final concentration = Initial concentration * $$(1- \frac{amount replaced}{total amount})^{n}$$

Final concentration of milk = $$100*(1-\frac{3}{30})^3$$ = $$100*(\frac{27}{30})^3$$ = 72.9%
concentration of water = 27.1%
milk:water = $$\frac{729}{271}$$
Ans: A
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Re: 3L of milk are drawn from a container containing 30L of milk. It is re  [#permalink]

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Solution:

• Initial amount of milk in container = 30 L
• Initial amount of water in container = 0 L
• 3 L of milk is removed and replaced with water
o Amount of milk in container = 30 – 3 = 27L
o Amount of water in the container = 3L
 Ratio of milk to water = $$\frac{27}{3} = \frac{9}{1}$$
• Now, again 3L solution is removed from the container.
o In this case only milk is not removed, water is also removed.
o Milk removed from the container = $$\frac{9}{10}*3 = 2.7$$ L
 Milk left in the container = $$27 – 2.7 = 24.3$$ L
o Water removed from the container = $$\frac{1}{10}*3 = 0.3$$
 Water left in the container = $$3 – 0.3 = 2.7$$ L
 Water in the container after the addition of 3 litre water = $$2.7+3 = 5.7$$ L
o Ratio of milk to water = $$\frac{24.3}{5.7}$$.
• Now, 3L solution is replaced with water one more time
o Milk removed = $$\frac{24.3}{30}*3 = 2.43$$
 Milk left in the solution = $$24.3 – 2.43 = 21.87$$ L
o Water removed =$$\frac{5.7}{30}*3=0.57L$$
 Water left = $$5.7 – 0.57 = 5.13$$ L
Water in container after the addition = $$5.13+3=8.13$$
• Ratio of milk to water = $$\frac{21.87}{8.13} = \frac{2187}{813}=\frac{729}{271}$$
Hence, the correct answer is Option A.
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3L of milk are drawn from a container containing 30L of milk. It is replaced by water and the process is repeated 2 times. What is the ratio of milk to water at the end?

A. 729/271

B. 743/229

C. 81/19

D. 2187/100

E. 813/19

After removing 3L of milk 3L of water is added. So the mixture has Milk to Water ratio of 9:1. Milk is 27L and Water is 2.7L. Now the process is repeated again so 3L of the mixture is removed.
Removed mixture has water content of 0.3L and Milk content of 2.7L. Total milk content left = 24.3L. As 3L of water is added total water content is 2.7 + 3 = 5.7L
Ratio of Milk to Water = $$\frac{24.3}{5.7}$$ = $$\frac{81}{19}$$

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Re: 3L of milk are drawn from a container containing 30L of milk. It is re  [#permalink]

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Gotta add up to a whole number no math needed but each time factor out 10% off the next calculation.

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Re: 3L of milk are drawn from a container containing 30L of milk. It is re  [#permalink]

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Quote:
3L of milk are drawn from a container containing 30L of milk. It is replaced by water and the process is repeated 2 times. What is the ratio of milk to water at the end?

con_final=con_initial(vol_intial-pure/vol_initial)^3
x=100%milk(30-3/30-3+3)^3
x=(27/30)^3=(3^3/3*10)^3=3^9/3^3*10^3
x=3^6/10^3=729/1000=milk/total
milk/water=729/(1000-729)=729/271

ans (A)
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Re: 3L of milk are drawn from a container containing 30L of milk. It is re  [#permalink]

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3L of milk are drawn from a container containing 30L of milk. It is replaced by water and the process is repeated 2 times. What is the ratio of milk to water at the end?
After 1st replacement : Milk 27 Water 3
After 2nd replacement, REDUCED milk = 3*27/30 = 2.7 .New ratio M:W = 24.3 : 5.7 = 8.1 : 1.9
After 3rd time, milk reduced = 3 * 81/100 = 2.43. New ratio M:W = (24.3 -2.43) 5.7 +2.43) =21.87 : 8.13 = 7.29 : 2.71
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3L of milk are drawn from a container containing 30L of milk. It is re  [#permalink]

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Worthy question!

Although not necessary for this specific question, I will do generalization to such typical MIXTURE PROBLEM so that you can solve similar, but harder question (eg. repeated 10x etc.)

Process #1: (30-3) litre wine left = 30*(1-1/10) = 30*(9/10)^1 litre wine left, total 30L;

#2: {30*(9/10)^1} - {30*(9/10)^1}/30*3L= 30*(9/10)*(1-1/10) = 30*(9/10)^2 litre wine left, total 30L;

... etc

I clearly see now following patterns when process are repeated N times:
Wine left =30*(9/10)^N
Water content =30*{1-(9/10)^N}

N=2
Wine left =30*(9/10)^2 =30*(81/100)
Water content =30*{1-(9/10)^2} = 31*(19/100)
---> Wine to water ratio
= 30*(81/100) / {30*(19/100)}
= 81/19

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Re: 3L of milk are drawn from a container containing 30L of milk. It is re  [#permalink]

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Top Contributor
The ratio of milk: water after two repeats is 24L Milk and 6L water, 24/6 or four times. The closest answer is only C, which is 81/19. However, I am not clear why the option 24/6 is not given as an answer.
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Re: 3L of milk are drawn from a container containing 30L of milk. It is re  [#permalink]

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10% of a 100% solution of milk is replaced with water thrice

Final concentration of milk=100%*(9/10)*(9/10)*(9/10)=72.9%

Final concentration of water=100-72.9=27.1%

Ratio=729/271

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Re: 3L of milk are drawn from a container containing 30L of milk. It is re  [#permalink]

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Ans C
milk proportion intially = 1;
after adding water, mp reduces = 27/30=9/10
when it is done twice = 1*9/10*9/10=81/100
final mp is 81/100
so mp:wp = 81:19
Math Expert V
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Re: 3L of milk are drawn from a container containing 30L of milk. It is re  [#permalink]

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3L of milk are drawn from a container containing 30L of milk. It is replaced by water and the process is repeated 2 times. What is the ratio of milk to water at the end?

Formula for such questions is Ratio of mlk:total =n (1-replaced quantity/total quantity)^(number of times) = $$(1-\frac{3}{30})^3=(\frac{9}{10})^3=729/1000$$.
So water = 1000-729=271
Thus milk:water=$$\frac{729}{271}$$

A
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Re: 3L of milk are drawn from a container containing 30L of milk. It is re  [#permalink]

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30 litre milk - 3litres of milk + 3 litres of water.
27:3= milk : water - litre mixture
27-2.7:3-0.3= 24.3:2.7
+ 3 litre water.
24.3:5.7
- 3 litre of mixture.
24.3-2.43 : 5.7-0.57
21.87 : 5.13
21.87:8.13
2187/813
729/271
OA is A

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Re: 3L of milk are drawn from a container containing 30L of milk. It is re  [#permalink]

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After the first process,
—> $$27L$$ —milk
—> $$\frac{27}{30} = \frac{9}{10}= 0.9L$$ milk in 1L mixture

After 2nd process
—> $$27 —3*0.9= 27–2.7 = 24.3L$$ milk
—> $$\frac{24.3}{30}$$L milk in 1L mixture

After 3rd process,
—> $$24.3 —3*\frac{24.3}{30} = 24.3 —2.43 = 21.87L$$ milk
$$30 —21.87 = 8.13$$L water
—> $$\frac{milk}{water} = \frac{21.87}{8.13} = \frac{2187}{813} = \frac{729}{271}$$

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I really cannot get this..the process is done only twice right, why are we doing it thrice?
Math Expert V
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monikakumar wrote:
I really cannot get this..the process is done only twice right, why are we doing it thrice?

For me "...repeated 2 times" means that something was done once and then repeated 2 more times.
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Bunuel wrote:
monikakumar wrote:
I really cannot get this..the process is done only twice right, why are we doing it thrice?

For me "...repeated 2 times" means that something was done once and then repeated 2 more times.

So this is how we have to view this question in GMAT also?..is this how GMAT also interpret?
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I will do generalization to this typical mixture problem so that we can solve similar, but harder question (eg. repeated 100x etc.)

Process #1: (30-3) litre milk left = 30*(1-1/10) = 30*(9/10)^1 litre milk left, total 30L;

#2: 30*(9/10) - 30*(9/10)*(3L/30) = 30*(9/10)*(9/10) = 30*(9/10)^2 litre milk left, total 30L;

...etc

Now, I clearly see patterns when process are repeated N times:
Milk left =30*(9/10)^N
Water content =30*{1-(9/10)^N}

When N=3,
Milk left =30*(9/10)^3 =30*(729/1000)
Water content =30*{1-(9/10)^3} = 31*(271/1000)
So, milk to water ratio
= 729/271

Posted from my mobile device 3L of milk are drawn from a container containing 30L of milk. It is re   [#permalink] 17 Apr 2020, 01:49

# 3L of milk are drawn from a container containing 30L of milk. It is re  