Math Flashcards
Multiplying by 5
When we need to divide a NUMBER by 5 we:
- double the NUMBER and
- divide by 10
Doubling and Halving
Divide one number by 2 and multiply the other number by 2.
ex. 1635 = 870 = 560
you can do this multiple times to the numbers as wellex.
1635 = 870 = 4140 = 2280
Picking Numbers
Don’t pick numbers that are 0 or 1.
Don’t pick numbers appear in the answers.
Try and pick numbers are realistic to facilitate checking.
Plugging in for Integer Properties
Smart Numbers:
-2, -1, -1/2, 0, 1/2, 1, 2
Plugging in for Percents
Smart Numbers:
100
Word Problem Backsolving Strategies
pick answer (C) and then see if you’re too high or too low
1/6 in decimal form
0.16667
5/6 decimal form
0.83333
1/8 decimal form
0.125
3/8 in decimal form
0.375
5/8 in decimal form
0.625
7/8 in decimal form
0.875
1/9 in decimal form
0.1111 (any change in the numerator/9 will just be a multiple of 0.1111 ex. 2/9 = 0.2222)
Comparison using cross multiplication
a/d ?? b/c (ac ?? bd look at this to see which is larger)
proportion and fractions
fraction = fraction (cannot do diagonal cancellation in a fraction = fraction format; you can do diagonal cancellation when multiplying two fractions)
the word “is” “are” means
(word problems with fractions)
equals “=”
the word “of” means
(word problems with fractions)
multiply
Multiplyer for percent increase
= 1 + (P% as a decimal)
Multiplyer for a percent decrease
= 1 - (P% as a decimal)
Percent change formula
(new - old)/old
Divisibility Rule for 3
Add all the digits of the number and if the sum is divisible by 3 then the number is divisible by 3.
ex. 135 = sum 9 so the number 135 is divisible by 3
Divisiblity rule for 2
Look at the last digit in the number. If last digit is divisible by 2 then the number is divisible by 2.
Divisibility Rule for 5
If the last digit is a 5 or 0 then the number is divisible by 5.
Divisibilty rule for 4
If the last two digits form a two-digit number divisible by 4 then the number is divisible by 4.
Divisibility rule for 9
If sum of the digits is divisible by 9 then the number is divisible by 9.
Divisibility rule for 6
The number must be divisible by 2 and 3.
First 17 Prime Numbers
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59
Fundamental Theorem of Arithmetic
Every integer greater than 1 that’s not a prime can be expressed as a product of primes. This product is called the prime factorization of the number.
Greatest Common Factor GCF or Greatest Common Divisor
ex. GCF for 360 and 800.
360 = 6610 = 2^33^25
800 = 81010 = 2^55^2
they both have 3 factors of 2 and 1 factor of 5
2225 = 40 = GCF
Least Common Multiple or Least Common Denominator
For any two integers P and Q the LCM = P*Q/GCF
ex. what is the LCM of 12 and 75
Find greatest common factor = 3
12 = GCF 3 * 4
75 = GCF 3 * 25
LCM = 3425
FOIL
First
Outer
Inner
Last
a^2 - b^2 =
(a+b)(a-b)
(a+b)^2 =
a^2 + 2ab + b^2
(a-b)^2 =
a^2 - 2ab + b^2
inequality rules
- multiply or divide by a negative switches the direction of the inequality
- can multiply and divide inequalities by positive number and the inequality stays the same
- can add and subtract with inequalities, exactly as we do with equations and the inequality stays the same
absolute value
- (absolute value) = - value or + value
- need to check extraneous solutions
Age Problems
Choose variables to represent the ages now and use addition and subtraction to create expessions for ages at other times.
ex. Right now, Steve’s age is half of Tom’s age. In eight years, twice Tom’s age will be five mor than three times Steve’s age. How old is Tom right now?
T = 2S
2(T+8) = 3(S+8) + 5
Motion Questions
- Distance, Rate (speed), and Time
- D = RT
- ## Keep track of units
Average Speeds
- need to perform a D=RT for both legs
Multiple Traveler Questions
Need to do D = RT for each traveler, each trip, and/or each leg
Shrinking and Expanding Gaps
- Going in opposite directions – add the speeds
- Going in the same direction – subtract the speeds
- Think about the directions to determine shriking vs expanding gaps
- Solving a D=RT for the gap itself can enormously simplify such problems
Work Questions (the work equation)
A = RT
A = amout of work done
R = the work rate
T = time
Arithmetic Sequence Formula
An = A1 + (n - 1)*d
An = value of the nth term (e.g., value of the 41st term)
A1 = starting term
n = desired term (e.g., 41st term in sequence)
d = common difference between two adjacent terms
Recursive sequences
In a recursive sequence, the nth term An is defined in terms of fthe previous term (or previous terms).
The test would have to give us a formula for An in terms of An -1, it would also have to give us a numerical seed value, typically the value of A1.
There are no shortcuts you have to go number by number through the sequence.
Inclusive counting (sets and sequences)
We use inclusive counting whenever the situation demands that both endpoints, the lowest value and the highest value, are part of what we are counting.
We perform the ordinary subtraction of high minus low, and then add one for the included lower endpoint.
Sums of sequences
Sum = N(A1+An)/2
N = number of pairs
COME BACK TO THIS ONE
(a^n)(a^m) =
a^(n+m)
(a^n)*(a^m) =
a^(n*m)
(a^m)/(a^n) =
a^(m-n)
a^0 =
1
(a^m)^n =
a^(m*n)
(P/Q)^-n =
(Q/P)^n
(ab)^n =
(a/b)^n =
(a^n)(b^n)
ex. 18^8 = (23^2)^8 = (2^8)(3^2)^8 = (2^8)*(3^16)
(a^n)/(b^n)
Exponents and Law of Distribution
Expontential Equations
Need to get equal bases on both sides and then we can solve for X
Rationalize the radical
Use the conjugate if necessary. Otherwise multiply by the radical to elminate the radical in the denominator.
What is a bisector?
a bisector divides something into two equal halves
Triangle inequality
The sum of any two sides of a triangle has to be greater than the other side of the triangle.
Pythagorean Theorem
Only works for right triangles (90 degrees)
Pythagorean Triplets
can multiple any triplet
(3, 4, 5)
(5, 12, 13)
(8, 15, 17)
45-45-90 Triangle (isosceles)
30-60-90 triangle
Area of an equilateral triangle
Quadrilateral Facts
- the sum of the four interior angles is 360 degrees
Parallelogram (special quadrilateral)
Area of a trapezoid
Polygon Properties - Sum of the angles
(n-2)*180
Inscribed Angle that intercepts a semi-circle
How do you find an arc length and area of a sector?
Perpendicular lines and slopes
Perpendicular lines have slopes that are opposite signed reciprocals of each other.
Slope intercept equation
y = mx + b
(m = slope)
(b = y-intercept; where it intercepts the y-axis)
Definition of Mean
Ordinary average
Definition of Median
middle number on a list (put list has to be in order first)
- if there’s a even number in the list then average the two middle numbers
Defintion of Mode
the mode frequent number in the list
- there can be no mode, 1 mode, or multiple modes
Definition of Range
range is the max - min (highest number - lowest number)
Normal Distribution
- 34% between mean and 1 standard deviation
- 13.5% between 1 and 2 standard deviations
For positive fractions you can
cross multiply to determine which one is larger
Inequality operations
- Can always multiply or divide both quantities by any positive number.
- Can always add any number to both quantities or substract any number from both quantities
