The SAT Prep Black Book(81)
Example:
The sequence 3, 5, 7, 9, 11, 13, . . . follows a very simple rule: to get the next number in the sequence, just add 2 to the number before. So the next number in this sequence would be 15, then 17, and so on.
The sequence 3, 15, 75, 375, . . . also follows a simple rule: to get the next number, multiply the previous number by 5. The next number here would be 1,875.
The Math section MIGHT ask you to figure out:
oThe sum of certain terms in a sequence.
oThe average of certain terms.
oThe value of a specific term.
If you studied sequences in school, they were probably a lot harder in your math class than they will be on the SAT. For example, there’s no sigma notation on the SAT. (If you’ve never heard of sigma notation, don’t worry about it.)
Set theory
Sets are collections of things.
Sets on the SAT are usually groups of numbers.
Example:
The set of factors of 24 is {1, 2, 3, 4, 6, 8, 12, 24}.
On the SAT, the things in a set can be called “members” of that set or “elements” of that set.
The “union ” of two or more sets is what we get when we combine all of the members of those sets into a bigger set.
Example:
The set of factors of 24 is {1, 2, 3, 4, 6, 8, 12, 24} and the set of factors of 36 is {1, 2, 3, 4, 6, 9, 12, 18, 36}. That means the union of those two sets is the set {1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36}.
The “intersection” of two or more sets is the set of members that the two sets have in common.
Example:
Given the sets {1, 2, 3, 4, 6, 8, 12, 24} and {1, 2, 3, 4, 6, 9, 12, 18, 36}, the “intersection” is {1, 2, 4, 6, 12}, because those members are common to both sets.
Counting problems
On the SAT, “counting problems” are problems where you’re asked to give the total number of ways that two or more events might happen.
If you’ve studied these types of problems in math class, you probably called them “permutation and combination” problems.
The general, basic rule of these types of problems is this: when you have two events, and the first event might happen in any one of x ways, and the second event might happen in any one of y ways, then the total number of ways that both events could happen together is given by xy. (That might sound a little complicated—let’s do an example.)
Example:
Imagine there are three roads between your house and your friend’s house, and there are 6 roads between your friend’s house and the library. If you’re driving from your house to your friend’s house and then to the library, how many different ways can you go?
There are 3 ways to get from your house to your friend’s house. So the event of you getting to your friend’s house can happen in any one of 3 ways. Then there are 6 ways to get from your friend’s house to the library, so the event of going to the library from the friend’s house can happen in any one of 6 ways. This means the total number of paths you could travel from your house to your friend’s house and then on to the library is given by 3 * 6, which is 18.
The key to solving these types of problems is making sure you correctly count the number of possible outcomes for each event.
Example:
Imagine that there are 3 roads between your house and your friend’s house. You’re going to visit her and then return home. For some reason, you can’t travel the same road twice. What’s the total number of ways you could go from your house to your friend’s house and back?
Well, the total number of ways to go from your house to your friend’s house is 3, and the total number of ways to come back home is ONLY 2. Why can you only come back from your friend’s house in 2 ways? Because the problem says you’re not allowed to use the same road twice, and when you go back home you will already have used one of the three roads to visit your friend in the first place. So the right way to answer this is to multiply 3 * 2, NOT 3 * 3. That means the answer is 6, NOT 9.
Operations on algebraic expressions
Algebraic expressions are figures that include variables.
Algebraic expressions, just like the regular numbers they represent, can be added, subtracted, multiplied, and divided—but sometimes there are special rules that apply.
We can add or subtract two algebraic expressions when they involve the same variable expressions.
Example:
We can add 5x and 19x to get 24x, because the 5x and 19x both involve the same variable expression: x. We can subtract 17xyz2 from
100 xyz2 and get 83xyz2 because they both involve the variable expression xyz2.
But if we want to add 5x to 17 xyz2, we can’t combine those two expressions any further because they have different variable expressions. So we would just write “5x + 17xyz2” and leave it at that.