Problem Solving

niketdoshi123 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 will be doubled while the ratio of soap to water is halved. If the altered solution 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

Solution:

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.

The answer is E