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How many liters of pure alcohol must be added to a 100-liter

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How many liters of pure alcohol must be added to a 100-liter [#permalink]

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How many liters of pure alcohol must be added to a 100-liter solution that is 20 percent alcohol in order to produce a solution that is 25 percent alcohol?

(A) 7/2
(B) 5
(C) 20/3
(D) 8
(E) 39/4
[Reveal] Spoiler: OA

Last edited by Bunuel on 10 Dec 2013, 04:11, edited 1 time in total.
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Re: PS 1000 Section6 Question 19 [#permalink]

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New post 31 Mar 2011, 10:32
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Acer86 wrote:
How many liters of pure alcohol must be added to a 100-liter solution that is 20 percent alcohol in order to produce a solution that is 25 percent alcohol?

(A) 7/2
(B) 5
(C) 20/3
(D) 8
(E) 39/4

Hi guys just a bit lost in this question...would appreciate if someone could help me out :)


20% Alcohol solution means; in the 100 liter solution, 20 liters of solution is alcohol and 80 liters other solvents.

If we add "x" liters of alcohol to the solution, the solution becomes "100+x" liters and alcohol, which was 20 liters, becomes 20+x liters.

According to the statement;
20+x = 25% of (100+x)
OR
20+x=(100+x)/4
80+4x=100+x
3x=20
x=20/3

Ans: "C"
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Re: PS 1000 Section6 Question 19 [#permalink]

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New post 31 Mar 2011, 13:28
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Acer86 wrote:
How many liters of pure alcohol must be added to a 100-liter solution that is 20 percent alcohol in order to produce a solution that is 25 percent alcohol?

(A) 7/2
(B) 5
(C) 20/3
(D) 8
(E) 39/4

Hi guys just a bit lost in this question...would appreciate if someone could help me out :)


Here is my crack at it:

Given:
  • 20% of solution is alcohol
  • 80% of solution is other
  • Total volume = 100 liters

Derived:
  • 20% * 100 liters = 20 liters

Now, if we add additional alcohol to the solution, we are altering the volume, so it's no longer going to be just 100 liters, it's going to be 100 plus some arbitrary amount. This gives us 100 + x liters of solution after adding more alcohol. In addition, when we added more alcohol, there is not just 20 liters of alcohol anymore (we just added some more), there is going to be 20 + x liters of alcohol.

What we're wanting to figure out is: how much alcohol must be added to result in there being 25% alcohol.

As mentioned before, we started out with:

20% * 100 = 20 liters

But, now we have:

25% * (100 + x) = (20 + x)

Now, let's just solve for x:

Convert the percentage and multiply:
\({
\frac{25}{100} * (100 + x) = (20 + x)
}\)

\({
\frac{(2500 + 25x)}{100} = (20 + x)
}\)

Simply the left side:
\({
\frac{25(100 + x)}{100} = (20 + x)
}\)

Cancel out terms:
\({
\frac{(100 + x)}{4} = (20 + x)
}\)

Multiple both sides by 4 to get rid of the 4 in the denominator on the left:
\({
4 \bigg( \frac{(100 + x)}{4} \bigg) = 4(20 + x)
}\)

\({
(100 + x) = (80 + 4x)
}\)

Subtract the x from the left:
\({
(100) = (80 + 3x)
}\)

Subtract the 80 from the right:
\({
20 = 3x
}\)

Divide....success!
\({
x = \frac{20}{3}
}\)

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Re: PS 1000 Section6 Question 19 [#permalink]

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New post 31 Mar 2011, 21:24
Equating total volume of alcohol :

(x + 100) * 25/100 = x + 20

=> x/4 + 25 = x + 20

=> 3x/4 = 5

=> x = 20/3

Answer - C
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Re: PS 1000 Section6 Question 19 [#permalink]

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New post 02 Apr 2011, 14:07
x+20 = 25/100(100+x)

x= 20/3

Answer is C

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Re: PS 1000 Section6 Question 19 [#permalink]

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Using Alligations:

We have 20% alcohol and 80% water. As per the question in the final mixture alcohol should be 25% and water should be 75%. Hence we can say that 80% water will be mixed with 0% water to yield a solution that will contain 75% water. So using alligations:

80%---------0%

75

75-----------5
75:5 = 15:1

Say we added x units of alcohol to the original mixture. These x units will have 0% water. And from above we see 80% water solution is mixed with 0% water solution in 15:1.

so, 15/1 = 100/x
x= 100/15
x=20/3 is the answer

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Re: How many liters of pure alcohol must be added to a 100-liter [#permalink]

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New post 02 Dec 2014, 23:53
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Alcohol .......... Non-Alcohol .................... Total

20 ..................... 80 ................................. 100

20+x ................... 80 .............................. 100+x

Setting up the equation

\(\frac{75}{100} (100+x) = 80\) OR

\(\frac{25}{100}(100+x) = 20+x\)

\(x = \frac{20}{3}\)

Answer = C
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How many liters of pure alcohol must be added to a 100-liter sol [#permalink]

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New post 03 Jun 2015, 03:05
We can solve such weighted average problem by understanding the underlying concept. You need to look at the difference between the alcohol level of the 20% solution and the pure alcohol.
_ The 20% solution has a alcohol level that is 5% lower than the level of alcohol of the final mixture, let's say -5 differential.
_ The pure (100%) alcohol has a alcohol level that is 75% higher than the level of alcohol of the final mixture, let's say +75 differential.
You need to make these differentials cancel out, so just multiply both differentials by respective volumes so that the positive will cancel out with the negative: 75x - 5.100 = 0 -> x = 500/75 = 20/3

Hope it's clear.

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Re: How many liters of pure alcohol must be added to a 100-liter [#permalink]

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New post 04 Jun 2015, 00:02
Hi All,

Since this question is built around the concept of a "weighted average", there are several different ways that you can go about solving it. Since the question asks how much alcohol must be added, and the answer choices are NUMBERS, we can TEST THE ANSWERS and do some basic arithmetic to get to the solution.

We're told that a 100-liter solution is 20% alcohol. This means that 20 liters are alcohol and 80 liters are not alcohol. We're asked how many liters must be added to this 100-liter solution to get a solution that is 25% alcohol.

Let's start with ANSWER B: 5 liters
IF....we add 5 liters of alcohol to this existing mixture....
Total alcohol = 20+5 = 25 liters
Total liquid = 100+5 = 105 liters
25/105 is LESS than 25% alcohol (since 25/100 = 25%, 25/105 is LESS than 25%), thus Answer B CANNOT be the answer. We need MORE alcohol to raise the percentage.
Eliminate Answers A and B.

Next, let's TEST ANSWER D: 8 liters
IF....we add 8 liters of alcohol to this existing mixture....
Total alcohol = 20+8 = 28 liters
Total liquid = 100+8 = 108 liters
28/108 is MORE than 25% alcohol (since 28/112 = 25%, 28/108 is MORE than 25%), thus Answer D CANNOT be the answer. We need LESS alcohol to raise the percentage.
Eliminate Answers D and E.

There's only one answer left...

Final Answer:
[Reveal] Spoiler:
C


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Re: How many liters of pure alcohol must be added to a 100-liter [#permalink]

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New post 01 Aug 2015, 03:49
yenh wrote:
We can solve such weighted average problem by understanding the underlying concept. You need to look at the difference between the alcohol level of the 20% solution and the pure alcohol.
_ The 20% solution has a alcohol level that is 5% lower than the level of alcohol of the final mixture, let's say -5 differential.
_ The pure (100%) alcohol has a alcohol level that is 75% higher than the level of alcohol of the final mixture, let's say +75 differential.
You need to make these differentials cancel out, so just multiply both differentials by respective volumes so that the positive will cancel out with the negative:75x - 5.100 = 0 -> x = 500/75 = 20/3

Hope it's clear.


Hi, could you please explain this concept in a simple form? or in graphical form? and Why the positive will cancel out with the negative?

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Re: How many liters of pure alcohol must be added to a 100-liter [#permalink]

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New post 17 Mar 2016, 05:15
How many liters of pure alcohol must be added to a 100-liter solution that is 20 percent alcohol in order to produce a solution that is 25 percent alcohol?

(A) 7/2
(B) 5
(C) 20/3
(D) 8
(E) 39/4

Alcohol-------------------- Total solution--------------% of alcohol
20 -------------------------100 --------------------------20%
20+x-----------------------100+x ----------------------25%
(20+x)/(100+x) =25/100
x=20/3

Hence C
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Re: How many liters of pure alcohol must be added to a 100-liter [#permalink]

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Re: How many liters of pure alcohol must be added to a 100-liter [#permalink]

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New post 01 Apr 2017, 20:23
Acer86 wrote:
How many liters of pure alcohol must be added to a 100-liter solution that is 20 percent alcohol in order to produce a solution that is 25 percent alcohol?

(A) 7/2
(B) 5
(C) 20/3
(D) 8
(E) 39/4


let x=liters of alcohol to be added
.2*100+x=.25(100+x)
x=20/3
C

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Re: How many liters of pure alcohol must be added to a 100-liter [#permalink]

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Acer86 wrote:
How many liters of pure alcohol must be added to a 100-liter solution that is 20 percent alcohol in order to produce a solution that is 25 percent alcohol?

(A) 7/2
(B) 5
(C) 20/3
(D) 8
(E) 39/4


\(\frac{Alcohol}{Solution} = \frac{20+x}{100+x} = \frac{1}{4}\)

\(\frac{Alcohol}{Solution} => 80 + 4x = 100 + x\)

\(\frac{Alcohol}{Solution} => 3x = 20\)

Or, \(x = \frac{20}{3}\)

Hence, answer must be (C) \(\frac{20}{3}\)
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Re: How many liters of pure alcohol must be added to a 100-liter [#permalink]

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New post 03 Aug 2017, 16:52
This question can be solved in around 40s using x method.

20........100
25

75.........5

20 - is orginal alcohol%
100 - as new solution contains only alcohol
25 - new value of alcohol

Final value is subatraction

Now divide 15/5 to bring it to lower fraction. So the ratio will come 15:1

15x= 40, x= 20/3

1x=? 20/3. So that's our answer

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Re: How many liters of pure alcohol must be added to a 100-liter   [#permalink] 03 Aug 2017, 16:52
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