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Formulas Flashcards

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1
Q

Rates & Work with multiple workers

A

W = Individual Rate x Number of Workers x Time

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2
Q

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

A

True

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3
Q

Work formula

A

W = R x T

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4
Q

Distance formula

A

D = R x T

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5
Q

Probability formula

A

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

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6
Q

Independent events (probability)

A

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)

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7
Q

Mutually exclusive events (probability)

A

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)

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8
Q

Combinatorics

A

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

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9
Q

Probability (concept)

A

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.

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10
Q

Permutations (concept and formula)

A

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.

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11
Q

Combinations (concept and formula)

A

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.

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12
Q

Overlapping sets

A

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:

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13
Q

Prime numbers (1-100)

A

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

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14
Q

Simple interest (how to calculate)

A

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

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15
Q

|x|

A

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

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16
Q

“Undefined” value

A

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.

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17
Q

Difference of Squares

A

X^2 — y^2 = (x + y)(x — y)

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18
Q

Special products: quadratic equations

A

x^2 — 2xy + y^2 = (x — y)^2

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

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19
Q

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

A

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

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20
Q

A right angle is made up of __ degrees.

A

90

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21
Q

A straight line is made up of __ degrees.

A

180

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22
Q

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

A

360

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23
Q

Triangle area

A

Area= 1/2 × bh

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24
Q

Isosceles right triangle

A

45-45-90

Has sides in a ratio of x : x : x√2

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25
Q

30-60-90 triangle

A

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

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26
Q

Equilateral triangle

A

Has 3 equal sides; each angle is ~60 degrees

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27
Q

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.

A

True

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28
Q

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.

A

Hypotenuse

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29
Q

Pythagorean Theorem

A

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

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30
Q

Pythagorean triples

A
  • 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)

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31
Q

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

A

True

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32
Q

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

A

True

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33
Q

Circle area

A

A = πr²

34
Q

Circle circumference

A

C = 2πr

35
Q

A circle has how many degrees?

A

360

36
Q

Circle arc

A

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

37
Q

Arc length

A

Arc Length = 𝑥 ÷ 360 × 2πr

38
Q

Area of sector

A

A(sector) = 𝑥 ÷ 360 × π𝑟²

39
Q

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

A

Arc

40
Q

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

A

Central angle

41
Q

Chord

A

Line segment between two points on a circle

42
Q

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

A

Diameter

43
Q

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

A

F. They are equal.

44
Q

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

A

T

45
Q

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

A

T

46
Q

Area of parallelogram

A

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

47
Q

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

A

F. The diagonals form four 90 degree angles.

48
Q

Perimeter of a rectangle

A

Sum of its sides (2𝑙+2𝑤)

49
Q

Area of a rectangle

A

l x w

50
Q

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

A

T

51
Q

Area of a square

A

A = s²

52
Q

Perimeter of a square

A

P = 4s

53
Q

Polygon

A

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

54
Q

Total degrees of a polygon

A

= 180(𝑛–2)

55
Q

Average degrees per side or degree measure of congruent polygon

A

= 180(𝑛−2)/𝑛

56
Q

Cube volume

A

V = s³

57
Q

Cube surface area

A

= 6s²

58
Q

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

A

s = 6

59
Q

Volumes of rectangular solids (including cubes)

A

V = height × depth × width

60
Q

Surface area of rectangular solids (including cubes)

A

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

61
Q

Volume of cylinders

A

V = π𝑟2h

62
Q

Surface area of cylinders

A

2π𝑟² + 2π𝑟ℎ = 2π𝑟(𝑟+ℎ)

63
Q

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

A

180 degrees

64
Q

Obtuse angle

A

> 90 degrees

65
Q

Acute angle

A

< 90 degrees

66
Q

Diameter of a circle =

A

2r

67
Q

Circumference of a circle =

A

πd or 2πr

68
Q

The only geometry rule that applies to all quadrilateral?

A

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

69
Q

Quadrilateral

A

A shape with four line segment sides

70
Q

Parallelogram (definition and BIG FOUR properties)

A

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

71
Q

Three categories of special parallelograms

A

Rhombuses, rectangles, and squares

The BIG FOUR parallelogram properties above apply to all of them

72
Q

Rhombus

A

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

73
Q

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

A

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

74
Q

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

A

Flips the sign! So if >, now <

75
Q

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?

A

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

76
Q

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

A

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.

77
Q

Divisibility rules: Divisibility by 2

A

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

78
Q

Divisibility rules: divisibility by 3

A

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

79
Q

Divisibility rules: divisibility by 5

A

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

80
Q

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

A

True

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