Comprehensive Study Flashcards for Math Foundations: Key Concepts and Formulas

# of Integers in an Interval

Last number - first number + 1

Sum of integers in an interval

n(a+l)/2 where n=# of terms, a=first term and l=last term

# of multiples in an interval

(Last multiple-first multiple/multiple in question) + 1

EX: # of multiples of 3 from 29-112

(111-30/3) + 1 = 28

Sum of multiples in an interval

Find the median and multiply the number of multiples in the interval

Prime numbers up to 50

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Non-factors of factorials

For an integer n, prime numbers greater than n cannot be factors of n!. The same applies to multiples of these prime numbers.

EX: 11, 22, and 33 cannot be factors of 8!

Prime number trick (# of factors)

• first prime factories the integer (ex: 3000 = 2^3 x 3 x 5^3)

• add 1 to each exponent if the prime factorization (4, 2, 4)

• multiply the result (4 x 2 x 4 = 32)

Prime number trick (odd factors)

• Prime factor the number

• take ONLY the odd factors in the integer

• find the number of factors amongst the odd factors

Prime number trick (GCF)

Prime factorize both integers and take what's in common with both

Prime number trick (LCM)

Prime factorizas both then take the GREATEST power of each prime and multiply.

Prime number trick (even factors)

Total # of factors - number of odd factors = number of even factors

Exponent unit digit pattern

0: always 0

1: always 1

2: 2,4,8,6,2,4,8,6...

3: 3,9,7,1,3,9,7,1...

4: 4,6,4,6....

5: always 5

6: always 6

7: 7,9,3,1,7,9,3,1...

8: 8,4,2,6....

9: 9,1,9,1...

Converting Terminating Decimals to Fractions

Just write it as a fraction with the denominator a power of 10 and simplify:

0.125 = 125/1000 = 1/8

Converting Repeating Decimals to Fractions

Put the repeating part over the correct number of 9s:

0.3... = 3/9

0.32... = 32/99

0.111.... = 111/999

Exponent Rules 1

• If x^p=x^q and x doesn't = 0, then p = q

• a^-n = 1/a^n

• x^a • x^b = x^a+b

• x^a/x^b = x^a-b

• a^0 = 1 when a doesn't = 0

Exponent Rules 2

• a^m • b^m = (ab)^m

• (a/b)^m = a^m/b^m

• (a^m)^n = a^mn

• in general, a(^p^q) doesn't = (a^p)^q

Roots as Exponents

Operation on Roots 1

Operation on Roots 2

"Pulling Out" Perfect Squares

Rationalizing Denominators 1

Rationalizing Denominators 2

Real Number Properties 1

Real number properties 2

Real Number Properties 3

Real Number Properties 4

Percent Increase

(Difference)/(smaller number)

Percent Decrease

(Difference)/(larger number)

% of versus % more

• b is 200% of a means that b=2a

• b is 200% MORE than a means that b=3a

Quadratic Formula

x = -b ± √(b² - 4ac)/2a

Quadratic Equations

a function in the form: f(x) = ax² + bx + c = 0, a≠0.

Discriminant

D= b²-4ac

• D>0 = 2 solutions

• D=0 = one solution

• D

Linear and Quadratic Polynomials

Linear = degree 1

Quadratic = degree 2

Polynomial Degree

Algebraic Identities

Solving Absolute Value Equations

1. Isolate absolute value

2. set up two equations

3. solve each equation

Factoring a Quadratic

Completing the Square

Absolute Value Quadratics

Distance Formula

d=rt

Relative Speed

• Toward each other? Add the speeds.

• Opposite directions? Add the speeds.

• Catching up? Subtract the speeds.

Solving inequalities

when you multiply or divide by a negative flip the sign

Words to Algebra

• +: addition, plus, sum, more, added to

• -: minus, difference, decrease, less than, under, remaining

• x: times, product, twice, quadruple, into, increase by a factor of

• \: divided by, over, ratio, slices, halve, quarter, decreased by a factor of

Words to Algebra 2

Simple Interest Formula

I=PRT (Interest = Principal x Rate x Time)

Mixture problems

Compound Interest

Work/Rate Problems

w = rt

Slope

(y₂- y₁) / (x₂- x₁)

Quadratics Graphed

slope-intercept form

y = mx + b

•x and y are the x and y intercepts of the line

•m is the slope

•b is the y intercept

Distance formula on a coordinate plane

d = √[( x₂ - x₁)² + (y₂ - y₁)²]

Discriminants and the plane

sum of exterior angles of any polygon

360

Sum of interior angles

180(n-2) n=number of sides

angles of regular polygon

(n-2)180/n

Angle versus side length

Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Area of a triangle

A= 1/2 bh

Pythagorean Theorem

a²+b²=c²

Area of a rectangle

A=lw

Area of a trapezoid

A=1/2(b1+b2)h

Area of a regular hexagon

Let the side of the hexagon be x. Area is just finding the area of the 6 equilateral triangles of side length x

isosceles trapezoid

Pythagorean Triplets

3,4,5

5,12,13

8,15,17

7,24,25

30-60-90 Triangle

x, x√3, 2x

45-45-90 triangle

x, x, x√2

Area of an equilateral triangle

A=(s²√3)/4 s=side

Parallelogram

Area of a regular polygon

Divide the polygon with n sides into n congruent triangles. Find the area of one and multiply by the number of sides

Central Angle Theorem

Circumference of a circle

2πr

length of an arc of a circle

ϴ/360 × circumference of full circle let ϴ = the central angle

Area of a circle

A=πr² or A=πd

Area of a sector

Area of a Sector = (central angle/360) x πr²

Volume of a cube

V=s³

Surface area of a cuboid

2(bh+bl+hl)

Surface area of a cube

6s²

Volume of a cuboid

length x width x height

Volume of a right cylinder

V=πr²h

Surface area and volume relation

The surface area of a 3D figure is usually greater than it's volume when the figure is small but the opposite when it's large

Diagonal of a cuboid

√l² + b² + h²

Surface Area of a Right Cylinder

SA=2πr²+2πrh

Diagonal of a cube

Diagonal = side√3

D=s√3

When mean = median

•evenly spaced set

•symmetrical set

Interquartile Range (IQR)

Q3-Q1

Three overlapping sets

Total = a + b + c - exactly two -2(all three)

overlapping sets formula

Total = Group 1 + Group 2 - Both + Neither

Inclusion/Exclusion Formula

Total = a + b - both

Yes or no

Converting to Letters/Words

Permutations with Repeats

Permutation Formula

nPr = n!/(n-r)!

Subtracting Out

Permutations vs. Combinations

Questions to Ask:

1. Is Order Important? Yes = Permutation

2. Does it matter which object is chosent first, second, third, etc.? Yes = Permutation

3. Can they be chosen in any order? Yes = Combination

The "Choice" method

Permutations in a Circle

(n-1)!

Multiple Groups

Find the number of combinations for each group separately and then multiply together

Combination Formula

nCr = n!/r!(n-r)!

Conditional Probability Formula

P(B|A) = P(A and B) / P(A)