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OG Problem #192

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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OG Problem #192

by GHong14 » Sun Nov 28, 2010 2:40 pm
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?

Right answer is 800.

Anyone care to explain the steps to solve?!!?!?
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by tomada » Sun Nov 28, 2010 4:33 pm
In initial solution, we can state the following relationships based on what's given:

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

The initial ratio of soap to alcohol = 2x/50x = 2/50
If we double the ratio of soap to alcohol, this ratio becomes 2*(2/50) = 4/50 = 4x/50x

The initial ration of soap to water = 2x/100x = 2/100
If we double the ratio of soap to water, this ratio becomes 1/2*(2/100) = 2/200 = 4/400 = 4x/400x

The revised proportions are:

4x = amount of soap
50x = amount of alcohol
400x = amount of water

We're given that in the final solution, there are 100 cubic centimeters of alcohol, so we can set 50x = 100; x=2.
When x=2, there are 400(2) = 800 cubic centimeters of water.



GHong14 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?

Right answer is 800.

Anyone care to explain the steps to solve?!!?!?
I'm really old, but I'll never be too old to become more educated.
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by GHong14 » Sun Nov 28, 2010 5:49 pm
Would the answer be the same if I initially simplified the initial ratio by 2 so the result is 1:25:100?
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by tomada » Sun Nov 28, 2010 6:14 pm
You have an incorrect ratio, so you'll get an incorrect answer.

Keep in mind that that ratio of soap to alcohol is doubled. You've merely simplified the ratio of soap to water from 2/50 to 1/25, so that ratio has not changed. Your ratio of soap to water, however, is correct. If you want to use the proportions of soap and water as you have stated below, the proportion of alcohol would need to be 25/2, or 12.5. Then, the ratio of soap to alcohol will be doubled, and the ratio of soap to water will be halved.

Using 1:12.5:100 and the statement that you have 100 cubic centimeters of alcohol, the multiplier for all proportions would be 8, from 12.5x = 100, and solving for x.

Then, you'd have 8 cubic centimeters of soap and 800 cubic centimeters of water.
I'm really old, but I'll never be too old to become more educated.
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