SAT Prep Black Book: The Most Effective SAT Strategies Ever Published (26 page)

Example:

If we divide 30 by 4, we see that it doesn’t work out evenly. 4 * 7 = 28, which isn’t enough, and 4 * 8 = 32, which is too much. So if we divide 30 by 4, one way to state the answer is to say that 30 divided by 4 is “7 with a remainde
r of 2,” because 4 * 7 = 28 and
28 + 2 = 30.

The remainder in a division problem must be less than the number we’re dividing by.

Example:

It doesn’t make any sense to say that 30 divided by 4 is “3 with a remainder of 18,” because 18 is bigger th
an 4 and 4 will still go into 18 a few more times.

As a reminder, when you first learned to divide, you were probably taught to use remainders
.

Most calculators don’t give remainders when solving division problems—instead, they give fractions or decimals.

A prime number is a number that has exactly two factors: 1 and itself.

Example:

17 is a prime number because there are no positive integers besides 1 and 17 that can be multiplied by other integers to generate 17. (Try to come up with some—you won’t be able to.)

24 is NOT a prime number because there are a lot of positive integers besides 1 and 24 that can be multiplied by other integers to generate 24. For example, 2, 3, 4, 6, 8, and 12 can all be multiplied by other integers to generate 24.

All prime numbers are positive.

The only even prime number is 2.

1 is NOT a prime number because it has only one factor (itself), while prime numbers must have exactly two factors.

Ratios, proportions, and percentages are all ways to express a relationship between two numbers.

A ratio is written as a pair of numbers with a colon between them.

Example:

If you make 5 dollars for every 1 dollar Bob makes, then the ratio of
your pay
to
Bob’s pay
is
5
:
1
.

A proportion is usually written as a fraction, with a number in the numerator compared to the number in the denominator.

Example:

If you make 5 dollars for every 1 dollar
Bob makes, then your pay can be compared to Bob’s pay with the proportion 5/1. (Or, if we wanted to compare what Bob makes to what you make, that proportion would be 1/5.)

A percentage is a special proportion where one number is compared to 100.

To determine a percentage, first compare two numbers with a proportion, and then divide the top number by the bottom number and multiply the result by 100.

Example:

If Bob makes 1 dollar for every 5 dollars you make, then the proportion that compares Bob’s pay to your pay is 1/5. If we divide 1 by 5 and multiply by 100, we see that Bob makes 20% of what you make.

Ratios can be set equal to each other and “cross-multiplied.”  (If you don’t already know how to do this, don’t worry—it’s just a short cut around regular algebraic techniques.
You don’t have to know how to do it for the SAT.)

If the relationship between two quantities is the kind where increasing one quantity results in an increase in the other quantity, then we say those two quantities “vary directly” or are “directly proportional.”

Example:

If I make 1 dollar for every 5 dollars you make, then when I make 4 dollars you make 20 dollars—increasing my pay to 4 leads to an increase in your pay to 20. That means our two rates of pay are in direct proportion.

If two quantities are related so that increasing one decreases the other, then we say those two quantities “vary indirectly” or are “inversely proportional.”

Example:

If we have two quantities
x
and
y
set up so that
xy
= 20, then
x
and
y
are inversely proportional—every time one increases, the other one decreases, and vice-versa. So if
x
starts out as 10 and
y
starts out as 2, changing
x
to 5 means we have to change
y
to 4—as one decreases, the other increases.

Sequences are strings of numbers
that follow a rule, so that knowing one number in the sequence allows us to figure out another number in the sequence.

Sequence questions on the SAT will rarely operate in exactly the same way that a question about an arithmetic or geometric series would work in a math class, though the College Board often tries to mislead you by making a sequence question look deceptively similar to a traditional question about a series.

SAT sequences can either go on forever or stop at some point, depending on the setup of the question.

There are two common types of SAT sequences, and we can classify them by the rules that are used to figure out which numbers go in the sequence. Let’s look at the different types of SAT sequences:

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:

o
The sum of certain terms in a sequence.

o
The average of certain terms.

o
The 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.)

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.

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.

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 5
x
and 19
x
to get 24
x
, because the 5
x
and 19
x
both involve the same variable expression:
x
. We can subtract 17
xyz
2
from
100
xyz
2
and get 83
xyz
2
because they both involve the variable expression
xyz
2
.
But if we want to add 5
x
to 17
xyz
2
, we can’t combine those two expressions any further because they have different variable expressions. So we would just write “5
x
+ 17
xyz
2
” and leave it at that.

We can multiply any two algebraic expressions by multiplying all the terms in the first expression by all the terms in the second expression.

Example:

5
x
* 7
y
= 35
xy

(5
a
+ 2)(4
b
+ 9) = 20
ab
+ 45
a
+ 8
b
+ 18

We can divide any algebraic expression by another algebraic expression when they share factors. (See the discussion on factoring algebraic expressions.)

Example:

26
xy
/13
x
= 2
y

When multiplying two algebraic expressions on the SAT, we can often use the “FOIL” technique. “FOIL” stands for “First, Outer, Inner, Last,” and refers to the order in which the terms of the two expressions are multiplied by one another.

You have probably used FOIL in your math classes, but if you used some other technique there’s no need to worry.

Example:

To multiply the expressions (5
x
+ 7) and
(3
x
+ 4), we can use FOIL.

The “First” pair in the acronym is the 5
x
and the 3
x
, because they are the first terms in each expression. We multiply these and get 15
x
2
.

The “Outer” pair in the acronym is the 5
x
and the 4. We multiply these and get 20
x
.

The “Inner” pair in the acronym is the 7 and the 3
x
. We multiply these and get 21
x
.

The “Last” pair in the acronym is the 7 and the 4. We multiply these and get 28.

Now we just add up all those terms and we get the expression 15
x
2
+ 20
x
+ 21
x
+ 28, which we can simplify a little bit by combining the two like
x
terms, giving us: