Formulas

Question 1

Rates & Work with multiple workers

Answer 1

W = Individual Rate x Number of Workers x Time

Question 2

True or false: if the distances are the same, average speed is always weighted towards the slower speed

Answer 2

True

Question 3

Work formula

Answer 3

W = R x T

Question 4

Distance formula

Answer 4

D = R x T

Question 5

Probability formula

Answer 5

P(Event)= number of favorable outcomes / total number of possible outcomes

Question 6

Independent events (probability)

Answer 6

Two events A and B are independent if the occurrence of one does not affect the occurrence of the other. For independent events, P (A and B) = P (A) x P (B)

Question 7

Mutually exclusive events (probability)

Answer 7

Two events A and B are mutually exclusive if they cannot occur at the same time. For such events, P(A or B) = P(A) + P(B)

Question 8

Combinatorics

Answer 8

Deals with counting and arrangements of objects. The two main concepts here are permutations and combinations.

Question 9

Probability (concept)

Answer 9

Probability measures the likelihood of an event occurring. It’s a value between 0 and 1 inclusive, where 0 means the event is impossible and 1 means the event is certain.

Question 10

Permutations (concept and formula)

Answer 10

The number of ways in which we can arrange r objects out of n distinct objects. Arrangements or sequences. Order is important. Ex: the arrangement of books on a shelf (the sequence matters), the order of runners finishing a race (who finished first, second, etc.), the sequence of letters in a password.

Key words: arrange, sequence, order, rank, in a row, etc.

Question 11

Combinations (concept and formula)

Answer 11

The number of ways to select r objects out of n without considering the order. Ex: a team of players selected from a larger group (it doesn’t matter who was selected first, second, etc.), or a hand of cards (because it doesn’t matter in which order you get the cards).

Key words: select, choose, committee, group, team, pair, etc.

Question 12

Overlapping sets

Answer 12

the Principle of Inclusion-Exclusion is often applied. Let’s say you have two sets, A and B.

If you want to find the total number of elements in sets A and B, you don’t just add the two sets together, because that would count the overlap twice. Instead, you use:

Question 13

Prime numbers (1-100)

Answer 13

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Question 14

Simple interest (how to calculate)

Answer 14

Interest is calculated based on the initial amount every time; the investment earned is not included in future calculations.

Question 15

|x|

Answer 15

Absolute value of a number, which is a mathematical concept used to express the distance of a number from zero on the number line, disregarding its sign.

For a positive number x or zero, |x| is equal to x. For example, |3| = 3 and |0| = 0.

For a negative number -x, |x| is equal to its positive counterpart. For example, |-3| = 3.

In essence, |x| always gives you a non-negative value

Question 16

“Undefined” value

Answer 16

A value is considered “undefined” in mathematics or other contexts when it does not have a meaningful or valid representation within the defined rules or constraints of a particular system or problem.

For example:

Division by zero: In mathematics, dividing a number by zero is undefined because it does not yield a meaningful result. For instance, 5 divided by 0 is undefined.

Square root of a negative number: The square root of a negative number is undefined in the realm of real numbers. For instance, √(-1) is undefined in the real number system, but it can be represented as an imaginary number, denoted as “i” (where √(-1) = i).

Variables without a defined value: In algebra, if a variable doesn’t have an assigned value or expression, it is considered undefined. For example, if you encounter an equation like “x + 2 = ?” without any information about what x represents, the value of x is undefined in that context until it’s defined.

In essence, “undefined” means that there is no valid or meaningful value according to the rules or parameters of a given mathematical or logical system.

Question 17

Difference of Squares

Answer 17

X^2 — y^2 = (x + y)(x — y)

Question 18

Special products: quadratic equations

Answer 18

x^2 — 2xy + y^2 = (x — y)^2

x^2 + 2xy + y^2 = (x + y)^2

Question 19

Multiplying or dividing by a negative value (inequalities and absolute values)

Answer 19

Switch the sign e.g. |< becomes |>

Question 20

A right angle is made up of __ degrees.

Answer 20

90

Question 21

A straight line is made up of __ degrees.

Answer 21

180

Question 22

If two lines intersect, the sum of the resulting four angles equals __.

Answer 22

360

Question 23

Triangle area

Answer 23

Area= 1/2 × bh

Question 24

Isosceles right triangle

Answer 24

45-45-90

Has sides in a ratio of x : x : x√2

Question 25

30-60-90 triangle

Answer 25

Has sides in a ratio of x : x√3 : 2x, with the x side opposite the 30 degree angle.

Question 26

Equilateral triangle

Answer 26

Has 3 equal sides; each angle is ~60 degrees

Question 27

T or F: Any given angle of a triangle corresponds to the length of the opposite side. The larger the degree measure of the angle, the larger the length of the opposite side.

Answer 27

True

Question 28

A right triangle has a right angle (a 90 degree angle); the side opposite the right angle is called the __, and is always the longest side.

Answer 28

Hypotenuse

Question 29

Pythagorean Theorem

Answer 29

For a right triangle with legs a and b and hypotenuse c: 𝑐²=𝑎²+𝑏²

Question 30

Pythagorean triples

Answer 30

* 3-4-5 * 5-12-13 * 8-15-17 * 7-24-25 A multiple of a Pythagorean triple is a Pythagorean triple (e.g., 6-8-10)

Question 31

T or F: The length of the longest side can never be greater than the sum of the two other sides.

Answer 31

True

Question 32

T or F: The length of the shortest side can never be less than the positive difference of the other two sides.

Answer 32

True

Question 33

Circle area

Answer 33

A = πr²

Question 34

Circle circumference

Answer 34

C = 2πr

Question 35

A circle has how many degrees?

Answer 35

360

Question 36

Circle arc

Answer 36

Portion of the circumference of a circle in x degrees of the circle.

Question 37

Arc length

Answer 37

Arc Length = 𝑥 ÷ 360 × 2πr

Question 38

Area of sector

Answer 38

A(sector) = 𝑥 ÷ 360 × π𝑟²

Question 39

A fraction of the circumference of a circle is called an ___.

Answer 39

Arc

Question 40

To find the degree measure of an arc, look at the ___.

Answer 40

Central angle

Question 41

Chord

Answer 41

Line segment between two points on a circle

Question 42

A chord that passes through the middle of the circle is a ___

Answer 42

Diameter

Question 43

T or F: If two inscribed angles hold the same chord, the two inscribed angles are unequal.

Answer 43

F. They are equal.

Question 44

T or F: An inscribed angle holding the diameter is a right angle (90 degrees).

Answer 44

T

Question 45

T or F: Inscribed angles holding chords/arcs of equal length are equal.

Answer 45

T

Question 46

Area of parallelogram

Answer 46

Base × Height (the base always forms a right angle with the height)

Question 47

T or F: The diagonals of a rhombus bisect one another, forming four 60 degree angles.

Answer 47

F. The diagonals form four 90 degree angles.

Question 48

Perimeter of a rectangle

Answer 48

Sum of its sides (2𝑙+2𝑤)

Question 49

Area of a rectangle

Answer 49

l x w

Question 50

T or F: The diagonals of a square bisect one another, forming four 90 degree angles.

Answer 50

T

Question 51

Area of a square

Answer 51

A = s²

Question 52

Perimeter of a square

Answer 52

P = 4s

Question 53

Polygon

Answer 53

Any figure with three or more sides (e.g., triangles, squares, octagons, etc.).

Question 54

Total degrees of a polygon

Answer 54

= 180(𝑛–2)

Question 55

Average degrees per side or degree measure of congruent polygon

Answer 55

= 180(𝑛−2)/𝑛

Question 56

Cube volume

Answer 56

V = s³

Question 57

Cube surface area

Answer 57

= 6s²

Question 58

The volume of a cube and the surface area of a cube are equal when...

Answer 58

s = 6

Question 59

Volumes of rectangular solids (including cubes)

Answer 59

V = height × depth × width

Question 60

Surface area of rectangular solids (including cubes)

Answer 60

V = 2 × height × width + 2 × depth × width + 2 × depth × height

Question 61

Volume of cylinders

Answer 61

V = π𝑟2h

Question 62

Surface area of cylinders

Answer 62

2π𝑟² + 2π𝑟ℎ = 2π𝑟(𝑟+ℎ)

Question 63

In any triangle, the sum of the three angles is...

Answer 63

180 degrees

Question 64

Obtuse angle

Answer 64

> 90 degrees

Question 65

Acute angle

Answer 65

< 90 degrees

Question 66

Diameter of a circle =

Answer 66

2r

Question 67

Circumference of a circle =

Answer 67

πd or 2πr

Question 68

The only geometry rule that applies to all quadrilateral?

Answer 68

The sum of the four angles in any quadrilateral is 360°

Question 69

Quadrilateral

Answer 69

A shape with four line segment sides

Question 70

Parallelogram (definition and BIG FOUR properties)

Answer 70

Quadrilateral with two pairs of parallel sides Four properties: 1. Opposites sides are parallel 2. Opposites sides are equal 3. Opposite angles are equal 4. Diagonals bisect each other

Question 71

Three categories of special parallelograms

Answer 71

Rhombuses, rectangles, and squares The BIG FOUR parallelogram properties above apply to all of them

Question 72

Rhombus

Answer 72

EEquilateral quadrilateral, that is, a quadrilateral with four equal sides

Question 73

T or F: The diagonals of a rhombus are always perpendicular.

Answer 73

T. Any rhombus can be subdivided into four congruent right triangles.

Question 74

Multiplying or dividing an inequality by a negative does what to the sign?

Answer 74

Flips the sign! So if >, now <

Question 75

When x is divided by 10, the quotient is y with a remainder of 4. If x and y are both positive integers, what is the remainder when x is divided by 5?

Answer 75

Answer: 4 If the remainder is less than each of two different divisors, and all multiples of the first number are also multiples of the second number, then the remainder for each will be the same

Question 76

What is the remainder when 3^17 + 17^13 is divided by 10?

Answer 76

Answer: 0 The remainder when dividing an integer by 10 always equals the units digit (e.g., units digit for 14 = 4). So you can ignore all but the units digit and re-write this question as: What is the units digit of 3^17 + 7^13? The pattern for the units digit of 3 is [3,9,7,1]. This is because 3^2=3; 3^3=9; 3^4=27 (and you ignore all but the units digit, in this case 7); 3^5=81. Every fourth term is the same. The 17th power is 1 past the end of the repeat: 17-16=1. Thus, 3^17 must end in 3. The pattern for units digit of 7 is [7,9,3,1]. The 13th term is 1 past the end of the repeat: 13-12=1. Thus, 7^13 must end in 7. The sum of these units digits is 3 + 7 = 10. Thus, the units digit is 0.

Question 77

Divisibility rules: Divisibility by 2

Answer 77

Divisibility by 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8).

Question 78

Divisibility rules: divisibility by 3

Answer 78

Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.

Question 79

Divisibility rules: divisibility by 5

Answer 79

Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5.

Question 80

T or F: any prime tree for 10, 100, 1000, etc. will only contain prime factors 2 and 5, occurring in pairs

Answer 80

True