# Ratios

As always, remember to follow the Knowsys method for math. Read the problem carefully and identify the bottom line (what you are looking for). Then, consider your options. How could you solve it? How should you solve it? Next, attack the problem using the method that you selected. Finally, loop back and verify that your answer matches the bottom line.

In an 8-gram solution of water and alcohol, the ratio by mass of water to alcohol is 3 to 1. If 12 grams of a solution consisting of 2 grams water for each gram of alcohol is added to the 8-gram solution, what fraction by mass of the new solution is alcohol?

This problem is going to test your ability to manipulate ratios. The first thing to remember is that the test makers are always going to try to trick you. When you are given a part-to-part ratio, you will usually need to convert it to a part-to-whole ratio and vice-versa. Before you start, though, make sure that you read the problem carefully and identify the bottom line. You need to know what the final ratio of alcohol to water is when you mix the 8-gram and 12-gram solutions.

Start by finding the amount of alcohol in the 8-gram solution. You know that the ratio of water to alcohol is 3 to 1. That means the ratio of  solution to alcohol is 4 to 1 (notice the conversion from part-to-part to whole-to-part).

$\frac{water}{alchohol}=\frac{3}{1}\therefore \frac{solution}{alchohol}=\frac{4}{1}=\frac{8}{2}$
If the ratio of solution to alcohol is 4 to 1, then there must be a total of 8 grams of solution and 2 grams of alcohol.

Now you need to find out how much alcohol there is in the 12 gram solution. Again, you are given the part-to-part ratio, "2 grams of water for each gram of alcohol." Simply convert it to the whole-to-part ratio.

$\frac{water}{alchohol}=\frac{2}{1}\therefore \frac{solution}{alchohol}=\frac{3}{1}$

You have 3 grams of solution for each gram of alcohol. Since there are 12 grams of solution, there must be 4 grams of alcohol.

Finally, add the two solutions together. You have a total of 20 grams of solution (the 8-gram solution and the 12-gram solution) and you have 6 grams of alcohol (2 grams from the 8-gram solution and 4 grams from the 12-gram solution). That means the ration of alcohol to solution is 6 to 20 or 3 to 10. Now, look at the answers below and select the one that matches your prediction.

(A)
(B)
(C)
(D)
(E)

The correct Answer Choice is (C).

On sat.collegeboard.org 47% of the responses were correct.

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