Math: a1fd2304

ID: a1fd2304

(SAT Suite Question Bank > Find Questions > Assessment: SAT + Test: Math + Domain: Algebra)

Comment: Grow comfortable with mixtures.

Method 1: I have discussed what is known as allegation in an earlier post, here. The short version is that you can use ratios to determine the weighted average of different components in a mixture, and with a little practice and an understanding of ratios, the method proves incredibly efficient. Start by counting the gaps in the percentages given. I will use a dash, -, to count by 5.

25--15-10

Note that 25 is 10 away from 15, while 10 is 5 away from 15. In terms of dashes, that is 2 to 1. The correct mixture must fit this 2:1 ratio, and since 15, the end product, is closer to 10 than it is to 25, it stands to reason that the average is pulled down to be closer to the 10. Hence, there will be 2 parts at 10 percent for every 1 part at 25 percent. If 3 liters (at 10%) represents 2 parts, then half that amount must represent 1 part in the ratio.

3L/2 parts = 1.5L/part

1 part at 25% will be 1.5L.

The correct answer is 1.5.

As I wrote in my earlier post, this method takes practice to get used to, but once you have it down, this type of question might take 10-30 seconds, and you can be certain of the answer.

Method 2: A calculator might best be put to use once you have set up the algebraic framework. You can jot down one piece of information at a time to create an equation. In general form,

percent of x * amount of x + percent of y * amount of y = combined percent of x and y * combined amount of x and y

This equation seems to make sense to the majority of students I work with, and there is nothing wrong with putting it to use here. Rather than converting percentages to decimals, you can use integers to make the calculations easier (not that that would matter to a calculator).

25(x) + 10(3) = 15(x + 3)

25x + 30 = 15x + 45

10x = 15

x = 1.5

The correct answer is 1.5.

Method 3: Certainly not the best way, but one method you could put to use is guessing and checking. If 25 percent and 10 percent solutions were added in equal proportion, the percentage saline (in this problem) of the resulting solution would be easy to work out.

(25 + 10)/2

35/2

17.5

17.5 percent is not too far from 15 percent, the target, and if 3 liters of 10 percent solution cannot be negotiated, then there must be less than 3 liters of 25 percent solution to get to 15. You might try 2 for starters.

3L @ 10% = 0.3L salt (saline)

2L @ 25% = 0.5L salt

3L + 2L = 5L, and 0.3L + 0.5L = 0.8L salt

0.8/5 = 0.16 (16%)

This value is close, but not quite there. You would have to go lower. I would like to point out, though, that with the above work in place, tweaking an input value or two will not take too much time to work out. How about 1?

3L @ 10% = 0.3L salt (saline)

1L @ 25% = 0.25L salt

3L + 1L = 4L, and 0.3L + 0.25L = 0.55L salt

0.55/4 = 0.138 (13.8%)

Clearly, 1 is too low and 2 is too high, so a reasonable guess might be to split the difference at 1.5.

3L @ 10% = 0.3L salt (saline)

1.5L @ 25% = 0.375L salt

3L + 1.5L = 4.5L, and 0.3L + 0.375L = 0.675L

0.675/4.5 = 0.15 (15%)

The correct answer is 1.5.

Again, this is not the recommended route with two far easier methods outlined above, but if you cannot think of a way to navigate a mixture question, then guessing an answer and using logic is passable.