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Manager  Joined: 27 Mar 2008
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The ratio, by volume, of soap to alcohol to water in a  [#permalink]

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Question Stats: 55% (02:07) correct 45% (02:15) wrong based on 2287 sessions

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The ratio, by volume, of soap to alcohol to water in a certain solution is 2:50:100. The solution will be altered so that the ratio of soap to alcohol is doubled while the ratio of soap to water is halved. If the altered solution will contain 100 cubic centimeters of alcohol, how many cubic centimeters of water will it contain?

A. 50
B. 200
C. 400
D. 625
E. 800

Originally posted by mgoblue123 on 16 Aug 2008, 12:57.
Last edited by Bunuel on 26 Apr 2012, 08:25, edited 1 time in total.
Edited the question and added the OA
Math Expert V
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Re: PS: Ratio by volume  [#permalink]

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droopy57 wrote:
The ratio, by volume, of soap to alcohol to water in a certain solution is 2:50:100. The solution will be altered so that the ratio of soap to alcohol is doubled while the ratio of soap to water is halved. If the altered solution will contain 100 cubic centimeters of alcohol, how many cubic centimeters of water will it contain?

(a) 50
(b) 200
(c) 400
(d) 625
(e) 800

Given: $$\frac{s}{{\frac{a}{w}}}=\frac{2}{{\frac{50}{100}}}$$;

The ratio of soap to alcohol is doubled --> $$\frac{s}{a}=2*\frac{2}{50}=\frac{4}{50}$$;

The ratio of soap to water is halved --> $$\frac{s}{w}=\frac{1}{2}*\frac{2}{100}=\frac{1}{100}=\frac{4}{400}$$;

New ratio: $$\frac{s}{{\frac{a}{w}}}=\frac{4}{{\frac{50}{400}}}$$ --> $$\frac{a}{w}=\frac{50}{400}$$ --> if $$a=2*50=100$$ then $$w=2*400=800$$.

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Re: PS: Ratio by volume  [#permalink]

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droopy57 wrote:
The ratio, by volume, of soap to alcohol to water in a certain solution is 2:50:100. The solution will be altered so that the ratio of soap to alcohol is doubled while the ratio of soap to water is halved. If the altered solution will contain 100 cubic centimeters of alcohol, how many cubic centimeters of water will it contain?

(a) 50
(b) 200
(c) 400
(d) 625
(e) 800

soap:alcohol
Intial ratio soap:alcohol: water --> 2:50:100
Inital soap:alcohol = 2/50 =2:50
after doubled soap:alcohol =2* 2/50 = 4:50

Initial soap:water = 2/100=2:100
after halved soap:water: 1/2 * 2/100 = 1/100 = 1:100= 4:400

After soap: alcohol:water --> 4:50:400

given alchol 100 cumeter.
ratio is 8:100:800 (4:50:400)

For 100 cu meter of Alcohol ---800 cu m water is required.

Originally posted by x2suresh on 16 Aug 2008, 13:32.
Last edited by x2suresh on 16 Aug 2008, 13:52, edited 1 time in total.
##### General Discussion
Senior Manager  Joined: 27 May 2008
Posts: 365
Re: PS: Ratio by volume  [#permalink]

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intial soap to alcohol = 2/50
final soap to alcohol = 4/50 = 2/25

initial soap to water = 2/100
final soap to water = 2/200

in fianl solution soap = 2x, alcohol = 25x, and water = 200x

as per question 25x=100, x = 4

water = 200 * 4 = 800 option E
Manager  Joined: 29 Sep 2008
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Re: PS: Ratio by volume  [#permalink]

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1
new values are
S:w=1:100
S:A=2:25
a:s:w=25:2:200

100=25/227*total amnt
total amount=4*227

water content=200/227(4*227)=800
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Re: The ratio, by volume, of soap to alcohol to water in a  [#permalink]

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2
INITIAL-
S:A:W
2:50:100

DOUBLED AND HALVED
S:A:W
4:50:-
1:-:100 OR 4:-:400

so A/W=50/400
100/x=50/400
x=800
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Intern  Joined: 19 Feb 2013
Posts: 3
Re: PS: Ratio by volume  [#permalink]

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Bunuel wrote:
droopy57 wrote:
The ratio, by volume, of soap to alcohol to water in a certain solution is 2:50:100. The solution will be altered so that the ratio of soap to alcohol is doubled while the ratio of soap to water is halved. If the altered solution will contain 100 cubic centimeters of alcohol, how many cubic centimeters of water will it contain?

(a) 50
(b) 200
(c) 400
(d) 625
(e) 800

Given: $$\frac{s}{{\frac{a}{w}}}=\frac{2}{{\frac{50}{100}}}$$;

The ratio of soap to alcohol is doubled --> $$\frac{s}{a}=2*\frac{2}{50}=\frac{4}{50}$$;

The ratio of soap to water is halved --> $$\frac{s}{w}=\frac{1}{2}*\frac{2}{100}=\frac{1}{100}=\frac{4}{400}$$;

New ratio: $$\frac{s}{{\frac{a}{w}}}=\frac{4}{{\frac{50}{400}}}$$ --> $$\frac{a}{w}=\frac{50}{400}$$ --> if $$a=2*50=100$$ then $$w=2*400=800$$.

How you get 1/100=4/400?? Why you multiply by 4??
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Posts: 64957
Re: PS: Ratio by volume  [#permalink]

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Marchikn wrote:
Bunuel wrote:
droopy57 wrote:
The ratio, by volume, of soap to alcohol to water in a certain solution is 2:50:100. The solution will be altered so that the ratio of soap to alcohol is doubled while the ratio of soap to water is halved. If the altered solution will contain 100 cubic centimeters of alcohol, how many cubic centimeters of water will it contain?

(a) 50
(b) 200
(c) 400
(d) 625
(e) 800

Given: $$\frac{s}{{\frac{a}{w}}}=\frac{2}{{\frac{50}{100}}}$$;

The ratio of soap to alcohol is doubled --> $$\frac{s}{a}=2*\frac{2}{50}=\frac{4}{50}$$;

The ratio of soap to water is halved --> $$\frac{s}{w}=\frac{1}{2}*\frac{2}{100}=\frac{1}{100}=\frac{4}{400}$$;

New ratio: $$\frac{s}{{\frac{a}{w}}}=\frac{4}{{\frac{50}{400}}}$$ --> $$\frac{a}{w}=\frac{50}{400}$$ --> if $$a=2*50=100$$ then $$w=2*400=800$$.

How you get 1/100=4/400?? Why you multiply by 4??

We have that s/a=4/50 (s corresponds to 4) and s/w=1/100 (s corresponds to 1). To get s/a/w ratio we need s to correspond to the same number in both ratio, so we multiply 1/100 by 4 to get s/w=4/100.
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Re: The ratio, by volume, of soap to alcohol to water in a  [#permalink]

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S:A:W= 2:50:100

S:A = 4:50

S:W = 2:200 = 4:400

Combining above two,

4:50:400

=> 8:100:800

Hence (E)
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Re: The ratio, by volume, of soap to alcohol to water in a  [#permalink]

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S:A:W = 2:50:100

New solution ratios

S: A = 2*2/50= 2/25

S:W = 2/100 *1/2=. 2/200

So New ratio is

2:25:200
Given that alcohol content is 100cc in the new solution.

Let soap ratio be 2x, alcohol ratio be 25 x and water ratio be 200x in the total solution.

It's already given that alcohol content is 100, so equate 25x = 100. => x= 4

Therefore water content will be 200x = 200*4 = 800.
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Re: The ratio, by volume, of soap to alcohol to water in a  [#permalink]

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I am using the unknown mupltiplier method for 3 quantities for such problems.

Initial ratios:
S:A:W
2:50:100

From stem:
(2/50)2=4/50= 2/25
(2/100)/2=2/200=1/100

Then we use the above mentioned approach like this:

S___A___W
2__25___W --> This is the new ratio for S and A
1__A___100--> This is the new ration for S and W

What we need to do now is create a common denominator for S, as we have 2 different values: LCM of 1 and 2 is 2, so we multiply the first row by 1 (since it already is 2) and the second row by 2 (since it is 1):

S___A___W--->S___A___W
2__25___W --->2__25___W (*1)
1__A___100--->2__25__200 (*2). A just goes down because there was no value there are we already have this ratio.

Now normally you would have the amount for the whole such as this:

S___A___W__
2x+25x+200x= the given amount.

Now we have that 25x=100-->x=4
Then 200x = 200*4=800. ANS E
Manager  Joined: 28 Dec 2013
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Re: The ratio, by volume, of soap to alcohol to water in a  [#permalink]

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x2suresh wrote:
droopy57 wrote:
The ratio, by volume, of soap to alcohol to water in a certain solution is 2:50:100. The solution will be altered so that the ratio of soap to alcohol is doubled while the ratio of soap to water is halved. If the altered solution will contain 100 cubic centimeters of alcohol, how many cubic centimeters of water will it contain?

(a) 50
(b) 200
(c) 400
(d) 625
(e) 800

soap:alcohol
Intial ratio soap:alcohol: water --> 2:50:100
Inital soap:alcohol = 2/50 =2:50
after doubled soap:alcohol =2* 2/50 = 4:50

Initial soap:water = 2/100=2:100
after halved soap:water: 1/2 * 2/100 = 1/100 = 1:100= 4:400

After soap: alcohol:water --> 4:50:400

given alchol 100 cumeter.
ratio is 8:100:800 (4:50:400)

For 100 cu meter of Alcohol ---800 cu m water is required.

How do we know only to multiply by soap ratio for the first few steps?
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Re: The ratio, by volume, of soap to alcohol to water in a  [#permalink]

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droopy57 wrote:
The ratio, by volume, of soap to alcohol to water in a certain solution is 2:50:100. The solution will be altered so that the ratio of soap to alcohol is doubled while the ratio of soap to water is halved. If the altered solution will contain 100 cubic centimeters of alcohol, how many cubic centimeters of water will it contain?

A. 50
B. 200
C. 400
D. 625
E. 800

MATH REVOLUTION VIDEO SOLUTION:

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Re: The ratio, by volume, of soap to alcohol to water in a  [#permalink]

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droopy57 wrote:
The ratio, by volume, of soap to alcohol to water in a certain solution is 2:50:100. The solution will be altered so that the ratio of soap to alcohol is doubled while the ratio of soap to water is halved. If the altered solution will contain 100 cubic centimeters of alcohol, how many cubic centimeters of water will it contain?

A. 50
B. 200
C. 400
D. 625
E. 800

The first thing we want to do is set up the ratio of the original solution using variable multipliers.

soap : alcohol : water = 2x : 50x : 100x

We are given that the ratio of soap to alcohol is doubled. Our original ratio of soap to alcohol is:

soap/alcohol = 2x/50x = x/25x

(Notice that we only canceled the 2 and 50; we didn’t cancel the x because it’s our ratio multiplier.)

If we double the ratio we multiply the entire ratio by 2, so we have:

2(x/25x) = 2x/25x

So now our new ratio is:

soap/alcohol = 2x/25x

Next we are given that the ratio of soap to water is halved. Our original ratio of soap to water is:

soap/water = 2x/100x = x/50x

If we halve the ratio, we multiply the entire ratio by 1/2, so we have:

(1/2)(x/50x) = x/100x

So now our new ratio is:

soap/water = x/100x

Next we must notice that the amount of soap in the two new ratios is not the same value. In the first new ratio, soap = 2x and in the second new ratio, soap = x. Thus, we need to make these values equal before continuing to the answer. To do this, we multiply the second ratio by 2/2, so we have:

soap/water = (2/2)(x/100x) = 2x/200x

Now we can set up the ratio of the altered solution.

soap : alcohol : water = 2x : 25x : 200x

Lastly, we are given that the altered solution will contain 100 cubic centimeters of alcohol.

With this we can set up the following equation and determine x:

25x = 100

x = 4 (This means that the ratio multiplier is 4.)

Thus, the altered solution will contain (200)(4) = 800 cubic centimeters of water.

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Re: The ratio, by volume, of soap to alcohol to water in a  [#permalink]

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s:a:w=2:50:100
step 1
ratio of soap to alcohol is doubled = s1/a1=2*s/a=4/50

step 2
Ratio of soap to water is halved = s2/w2=1/2(2/100)=2/200

replace the values back to original

final ratio = s:a:w=4:50:400

final solution = 100 cc of alcohol ,which is 2*50
therefore water = 2*40=800
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Re: The ratio, by volume, of soap to alcohol to water in a  [#permalink]

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Top Contributor
droopy57 wrote:
The ratio, by volume, of soap to alcohol to water in a certain solution is 2:50:100. The solution will be altered so that the ratio of soap to alcohol is doubled while the ratio of soap to water is halved. If the altered solution will contain 100 cubic centimeters of alcohol, how many cubic centimeters of water will it contain?

A. 50
B. 200
C. 400
D. 625
E. 800

soap:alcohol is doubled implies the ratio is 4:50:100
Also, soap :water is halved implies the ratio is 4:50:400
So when alcohol is 100 cubic cm, water is 800 cubic cm.
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Re: The ratio, by volume, of soap to alcohol to water in a  [#permalink]

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I think this problem can be solved with accuracy by setting up a table. Please see the attached table. The table is self explanatory. Let know if an explanation needed.

option E
Attachments OG13_Problem 200.png [ 24.13 KiB | Viewed 27320 times ]

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Re: The ratio, by volume, of soap to alcohol to water in a  [#permalink]

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soap=2x
alcohol=50x
water=100x

new soap=4x
new alcohol=50x
new water=400x

50x=100
so 400x=8(50x)=8(100)=800
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Re: The ratio, by volume, of soap to alcohol to water in a  [#permalink]

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droopy57 wrote:
The ratio, by volume, of soap to alcohol to water in a certain solution is 2:50:100. The solution will be altered so that the ratio of soap to alcohol is doubled while the ratio of soap to water is halved. If the altered solution will contain 100 cubic centimeters of alcohol, how many cubic centimeters of water will it contain?

A. 50
B. 200
C. 400
D. 625
E. 800

S:A:W=2:50:100
=>S/A=2x/50x --- new ratio =4x/50x
=>S/W=2x/100x -- new ratio =x/100x

So, A/S=50x/4x
and, W/S=100x/x or 400x/4x

Therefore, new S:A:W=4:50:400
Now A=100 =>50x=100 =>x=2

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The ratio, by volume, of soap to alcohol to water in a  [#permalink]

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Top Contributor
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droopy57 wrote:
The ratio, by volume, of soap to alcohol to water in a certain solution is 2:50:100. The solution will be altered so that the ratio of soap to alcohol is doubled while the ratio of soap to water is halved. If the altered solution will contain 100 cubic centimeters of alcohol, how many cubic centimeters of water will it contain?

A. 50
B. 200
C. 400
D. 625
E. 800

S : A : W = 2 : 50 : 100

The solution will be altered so that the ratio of soap to alcohol is doubled
The ORIGINAL ratio is S : A = 2 : 50.
NEW ratio is S : A = 4 : 50

the ratio of soap to water is halved
The ORIGINAL ratio is S : W = 2 : 100
NEW ratio is S : W = 1 : 100

If the altered solution will contain 100 cubic centimeters of alcohol, how many cubic centimeters of water will it contain?
We need to COMBINE our two ratios.
Take S : W = 1 : 100 and create the EQUIVALENT ratio S : W = 4 : 400

We can now combine both ratios to get: S : A : W = 4 : 50 : 400
We want water = 100
So, take S : A : W = 4 : 50 : 400 and rewrite as S : A : W = 8 : 100 : 800

Cheers,
Brent
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