The ratio, by volume, of soap to alcohol to water in a

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 \(w=2*400=800\).

Answer: E.
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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.

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

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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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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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.

Answer is E.
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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.

Answer = 800

option E

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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

Answer: E

Cheers,
Brent

My method is as follows:-
Step 1 : Present ratio S:A:W = 2 : 50 : 100
Step 2 : Altered Ratio S:A = 4:50 and S:W= 1:100
Step 3 : To make S common multiply S:W x 4 = 4:400
Step 4: The new ratio S:A:W = 4 : 50 : 400
Step 5: Let amount of water be x. Therefore, 50/400=100/x ...which on solving gives x=800

If soap and alcohol is doubled, why it is not 4:100 instead of 4:50

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Initially i also use to get confused with halving or doubling of ratio. Best way to understand is that instead of considering ratio in form S:A use form S/A and than perform the intended operation. In case doubled, S/A*2 =4:100 ....Similarly when halving the ratio S/A*1/2

I hope it is clear now