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Math Facts for the GRE

Algebra

The absolute value of a number is its distance from zero. Absolute values are always positive.

To solve any equation or inequality that has an absolute value, solve it once for the positive version of whats in the absolute value, and solve it once for the negative version of whats in the absolute value.

For example: |x + 3| = 13

• is, are, was, were: =
• of: multiply
• more than, total, sum: add
• percent: /100
• what: the unknown variable, or x
• the ratio of [this] to [that]: this/that
  
• There are 3/5 as many As as Bs: A = 3/5 (B)

For example: What is 8 more than 25% of 6% of of 12?

translates to:

When simplifying an inequality and multiplying or dividing by a negative, flip the greater than or less than sign.

For example: if 3x > 6, then x < 2.

Formula for overlapping sets:

Total = Group 1 + Group 2 Both + Neither

For example:

There are 526 students at Grover Cleveland High School. 156 take Mandarin. 432 take Spanish.

If 231 take both Spanish and Mandarin, how many take neither?

This translates to:

526 = 156 + 432 231 + Neither

526 = 357 + Neither

169 = Neither

Know the factored and foiled forms of these common quadratics:

(x + y)(x + y) = x2 + 2xy + y2

(xy)(xy) = x2 2xy + y2

(x + y)(xy) = x2y2

Distance = Rate Time
Work = Rate Time
Working together? Add the rates.
Working against each other (for example, the faster car trying to catch the slower car)? Subtract the rates.

In the notation an, the n indicates the slot or place in the sequence. So, a1 refers to the first element in the sequence, a2 refers to the second, and so on.

The expression an itself indicates the nth number of the sequence, or any number in the sequence.

The expression an1 means the previous term. The expression an2 means two terms before. The expression an+1 means the following term.

Translate a sequence rule into a set of instructions.

For example: an = 2(an1) 5 is saying For any given term in the sequence, take the previous term, multiply by 2, and then subtract 5.

Arithmetic

Prime numbers = numbers that are divisible only by 1 and themselves
Smallest prime number = 2
Only even prime number = 2
Primes from 1 to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
Factors = numbers that divide evenly into another number

For example: the factors of 12 are 1, 12, 2, 6, 3, and 4.

Multiples = numbers that another number multiplies up to

For example: multiples of 4 include 4, 8, 12, 16, 20...

Prime factors = the prime building blocks that make up another number

For example: the prime factorization of 12 is 2 2 3, or 223.

The greatest common factor of two numbers is the product of the prime factors shared between those two numbers (multiply only the numbers they have in common).

For example, to find the greatest common factor of 16 and 28, start by finding the prime factors of each. 16 is 2 2 2 2, or 24. 28 is 2 2 7, or 227. 24 and 227 have in common two 2s, or 22. The greatest common factor of 16 and 28 is thus 2 2, or 4.

When multiplying two exponents with the same base, add the powers.

For example: 32 35 = 32+5 = 37

When dividing two exponents with the same base, subtract the powers.

For example:

When raising an exponent to another power, multiply the exponents.

For example: (x7)3 = x73 = x21

When the powers match, you can combine bases.

For example: 2x 3x = (2 3)x = 6x

If the bases dont match and the powers dont match, you cant do anything.

For example, 22 35 cannot be simplified further.

If youre adding or subtracting numbers that have the same base, factor:

For example: 25 + 28 = 25 (1 + 23) = 25 (9)

Negative exponents just reciprocate.

For example:

Adding or subtracting fractions? Find a common denominator and add the numerators.

For example:

Multiplying fractions? Multiply across the numerators and the denominators.

For example:

Dividing by a fraction? Multiply by the reciprocal.

For example:

Even +/ Odd = Odd
Even +/ Even = Even
Odd +/ Odd = Even
Even (Even or Odd) = Even
Odd Odd = Odd
Evenany power = Even
Oddany power = Odd
Code for even: multiple of 2; x = 2y
Code for odd: prime greater than 2; x = 2y + 1

Remainders are the integer left over after division.

For example: the remainder when 7 is divided by 5 is 2.

when n is divided by 12, the remainder is 4 = n could be any multiple of 12 + 4

Simplify expressions in the correct order:

Parentheses

Exponents

Multiplication

Division

Addition

Subtraction

Pos Pos = Pos

Pos Neg = Neg

Neg Neg = Pos

Subtracting a negative? Add a positive.

For example, 5 2 = 5 + 2 = 7.

Inequalities as code for positive and negative:

x > 0 means x is positive.

y < 0 means y is negative.

x + y > 0 means at least one is positive.

x + y < 0 means at least one is negative.

xy