Math SATs- Sets, Ratios and Modules

Posted Wed, 07 May 2008 15:04:06 -0000 by Oren Lahav

Welcome to a Math SAT Preparation Lesson. In this lesson we will take a look at a few important concepts that have to do with sets, ratios and modules.

SAT tests, and many other standardized tests, usually have a few of these type of questions thrown in, since they don't only test your math skills, but also your logic and critical-thinking skills. Therefore, it's important to prepare for these sort of questions.

Let's start with sets.

A set is a collection of objects. The big difference between a set and a sequence is that a sequence is a list, and it has order. Sets have no order for their items. Sets can also be empty, but in that case there's not much you can do. The thing mathematicians enjoy doing best is counting the number of objects in a set. For example, the set of even numbers between 0 and 10 contain 6 objects (the 0 counts).

Counting sets is simple, but there's an extra thing involved- subsets.

As you can imagine, a subset is a set which contain elements that all belong to a greater set. Our earlier example is really a subset of all the numbers between 0 and 10, which is a subset of all natural numbers. The greatest subset of a set is always the set itself, by the definition, and the smallest subset is always the empty set.

Now let's introduce the tricky part- multiple subsets.

Multiple subsets of a common set give rise to two more concepts- first, the intersection, which represents the set of elements in both subset A and subset B. Then comes the union, which is the set of all elements which are either in subset A, B, or both. For example, if A is our earlier set and B is the set of all numbers between 0 and 10 divisible by 3, the union would be {0,2,3,4,6,8,9,10} while the intersection is {6}. Now comes the formula for counting subsets: $N_{union}=N_A+N_B-N_{intersection}$ . This works in our example, since, as we have 6 even numbers and 3 numbers divisible by 3, and 1 that's both, we have that 8=6+3-1 . This formula should help you count the number of elements in different sets and subsets.

Wasn't that fun? Now let's look at ratios.

Ratios are pretty simple when you think about them as fractions. Let's say for example that I have 10 shirts, 3 of which are black and 7 are blue. It's simple to see that 3/10 of my shirts are black and 7/10 are blue. The ratio of black to blue shirts is then 3:7. Note that the ratio of blue to black shirts is 7:3, so order matters. We can apply ratios to sets to find actual numerical values. For example, say I have 40 shoes that are either black and brown, and the ratio of black to brown shoes is 3:1. Then, if I apply the fraction of $\frac{3}{3+1}$ to the value of 40, I get 30 black shoes, and similarly I have 10 brown shoes.

Finally, on a slightly unrelated note, let's look at modules.

We've seen in the Basic Algebra lesson that when dividing a number a by some other number b, we can always write a=qb+r for a quotient q and a remainder r. This is called the Division Algorithm for Real Numbers. We can say the same thing if we write $a \bmod b \equiv r$ . All this does is say that the remainder of a when divided by b is r, without caring about the quotient or anything else. For example, $18 \equiv 4 \bmod 7$ . Note that 25, 32, 39 and 11 are also 4 in mod 7. How do we work with modules? It's easier than it sounds. For example, say $a \equiv 2 \bmod 5$ while $b \equiv 4 \bmod 5$ . Then what about >math>a+b? We don't need to know anything about the actual values of a or b, we can just add the modules: $a+b \equiv 2+4 \equiv 1 \bmod 5$ . The same thing works for multiplication: $a * b \equiv 2 * 4 \equiv 8 \equiv 3 \bmod 5$ . Modules are great as they simplify remainder calculations with big numbers and help us work with unknown variables.

So, we've covered a lot of material in seemingly unrelated topic, but remember that all mathematical concepts (and some non-mathematical ones too) are related, and when writing your SATs you may get a question that combines sets and modules, or ratios with geometry.

Scary, isn't it? Here are a few tricks to help you relax:

1. Remember that calculators are allowed- be comfortable with your calculator and use it as much as you can.

2. If you're stuck, try to figure out what the question is asking. Is this a ratio question? A trigonometry one? Maybe it's combined? This will give you a hint towards arriving at a solution.

3. Word problems always seem scarier than they actually are. Try to ignore the different-colored beans, the cookies and the students and focus on the numbers.

Hopefully, you're now comfortable with a variety of question types and you're ready to try on a sets, modules, ratios and other subjects test.

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