Also, please try to remember that the material in the Math Toolbox is pretty dry and technical, and that it’s not the focus of the proper strategic approach to the SAT. It’s just a set of basic ideas that need to be refreshed before we get into the more important stuff.
Properties of integers
An integer is any number that can be expressed without a fraction, decimal, percentage sign, or symbol.
Integers can be negative or positive.
Zero is an integer.
Example:
These numbers are integers: -99, -6, 0, 8, 675
These numbers are NOT integers: pi, 96.7, 3/4
There are even integers and there are odd integers.
Only integers can be odd or even—a fraction or symbolic number is neither odd nor even.
Integers that are even can be divided by 2 without having anything left over.
Integers that are odd have a remainder of 1 when they’re divided by 2.
Example:
These are even integers: -6, 4, 8
These are odd integers: -99, 25, 675
An even number plus an even number gives an even result.
An odd number plus an odd number gives an even result.
An odd number plus an even number gives an odd result.
An even number times an even number gives an even result.
An even number times an odd number gives an even result.
An odd number times an odd number gives an odd result.
Some integers have special properties when it comes to addition and multiplication:
Multiplying any number by 1 leaves the number unchanged.
Dividing any number by 1 leaves the number unchanged.
Multiplying any number by 0 results in the number 0.
Adding 0 to any number leaves the number unchanged.
Subtracting 0 from any number leaves the number unchanged.
It’s impossible, for purposes of SAT Math, to divide any number by 0.
Word problems
SAT word problems are typically simple descriptions of one of the following:
Real-life situations
Abstract concepts
Example:
An SAT word problem about a real-life situation might look like this:
“Joe buys two balloons for three dollars each, and a certain amount of candy. Each piece of the candy costs twenty-five cents. Joe gives the cashier ten dollars and receives twenty-five cents in change. How many pieces of candy did he buy?”
An SAT word problem about an abstract concept might look like this:
“If x is the arithmetic mean of seven consecutive numbers, what is the median of those seven numbers?”
To solve SAT word problems, we have to transform them into math problems. These are the steps we follow to make that transformation:
Note all the numbers given in the problem, and write them down on scratch paper.
Identify key phrases and translate them into mathematical symbols for operations and variables. Use these to connect the numbers you wrote down.
Example:
In the phrase “two balloons for three dollars each,” the each part means we have to multiply the two balloons by the three dollars in order to find out how much total money was spent on the two balloons. 2 * 3 = 6. Six dollars were spent on the two balloons if they cost three dollars each.
After the word problem has been translated into numbers and symbols, solve it like any other SAT Math problem (see the SAT Math Path in this chapter for more on that).
Number lines
A number line is a simple diagram that arranges numbers from least to greatest.
The positions on a number line can be labeled with actual numbers or with variables.
Example:
This number line shows all the integers from -7 to 4:
On the SAT, number lines are drawn to scale and the tick marks are spaced evenly unless the question notes otherwise.
To determine the distance between two numbers on a number line, just subtract the number to the left from the number to the right.
Example:
On the number line above, the distance between 1 and 3 is two units, which is the same thing as saying that 3 – 1 = 2.
On a number line, there is a DIFFERENCE between the distance that separates two numbers and the number of positions between them.
If you’re asked how many positions are BETWEEN two numbers on a number line, remember that you CANNOT answer this question by simply subtracting one number from the other—that’s how you would find the distance. You should actually count the positions—you’ll find the number of positions is one less than the difference you get when you subtract.
Example:
On the number line above, there are NOT two positions between the numbers 2 and 4, even though 4 – 2 = 2. There is only one position between the numbers 2 and 4, which is one less than the difference we get when we subtract the number 2 from the number 4.
On the SAT, the positions on a number line don’t have to represent whole numbers. They might represent groups of five numbers at a time, or hundredths, or any other consistent amount.