# Math Concepts

## Number and Operations

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### The following concepts are covered on the SAT:

- Arithmetic word problems (including percent, ratio, and proportion)
- Properties of integers (even, odd, prime numbers, divisibility, etc.)
- Rational numbers
- Sets (union, intersection, elements)
- Counting techniques
- Sequences and series (including exponential growth)
- Elementary number theory

Integers, Odd and Even Numbers, Prime Numbers, Digits

**Integers**: . . . , -4, -3, -2, -1, 0, 1, 2, 3, 4, . . .

(*Note: zero is neither positive nor negative*.)

**Consecutive Integers**: Integers that follow in sequence; for example, 22, 23, 24, 25. Consecutive integers can be more generally represented by *n*, *n* +1, *n* + 2, *n* + 3, . . .

**Odd Integers**: . . . , -7, -5, -3, -1, 1, 3, 5, 7, . . . , 2*k* + 1, . . . , where *k* is an integer

**Even Integers**: . . . , -6, -4, -2, 0, 2, 4, 6, . . . , 2*k*, . . . , where *k* is an integer

(*Note: zero is an even integer.*)

**Prime Numbers**: 2, 3, 5, 7, 11, 13, 17, 19, . . .

(N*ote: 1 is not a prime and 2 is the only even prime.*)

**Digits**: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

(*Note: the units digit and the ones digit refer to the same digit in a number. For example, in the number 125, the 5 is called the units digit or the ones digit*.)

#### Percent

Percent means hundredths, or number out of 100. For example, 40 percent means

.

**Problem 1**: If the sales tax on a $30.00 item is $1.80, what is the sales tax rate?

**Solution**:

.

#### Percent Increase/Decrease

**Problem 2**: If the price of a computer was decreased from $1,000 to $750, by what percent was the price decreased?

**Solution**: The price decrease is $250. The percent decrease is the value of $n$ in the equation . The value of $n$ is 25, so the price was decreased by 25%.

**Note**:

**Problem**: José traveled for 2 hours at a rate of 70 kilometers per hour and for 5 hours at a rate of 60 kilometers per hour. What was his average speed for the 7-hour period?

**Solution**: In this situation, the average speed is

.

The total distance was

.

The total time was 7 hours. Thus, the average speed was .

**Note**: In this example, the average speed over the 7-hour period is *not* the average of the two given speeds, which would be 65 kilometers per hour.

Two common types of sequences that appear on the SAT are arithmetic and geometric sequences.

An **arithmetic sequence** is a sequence in which successive terms differ by the same constant amount.

For example: 3, 5, 7, 9, . . . is an arithmetic sequence.

A **geometric sequence** is a sequence in which the ratio of successive terms is a constant.

For example: 2, 4, 8, 16, . . . is a geometric sequence.

A sequence may also be defined using previously defined terms. For example, the first term of a sequence is 2, and each successive term is 1 less than twice the preceding term. This sequence would be 2, 3, 5, 9, 17, . . .

On the SAT, explicit rules are given for each sequence. For example, in the sequence above, you would not be expected to know that the 6th term is 33 without being given the fact that each term is 1 less than twice the preceding term. For sequences on the SAT, the first term is *never* referred to as the zeroth term.

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