Important Math Concepts and Formulas(Review w/Appendix D)
If given a complex quadratic equation like x(x^2 -4x -77) = 0
x = 0 has to be an answer, We also factor to get (x - 11)(x + 7), So x = 0, 11, and -7
Formulas for a Circle
C = 2πr = πd Arc length(fraction of circumference) = (n/360) X 2πr, where n is the angle measurement Area of Circle = πr^2 Area of sector(fraction of area) = (n/360) X πr^2
Graphing Circles
(x - a)^2 + (y - b)^2 = r^2, where a, b, and r are constants
Prime Numbers Under 100
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
How to recognize Multiples of 2, 3, 4, 5, 6, 9, 10, and 12
2: Last digit is even 3: Sum of digits is a multiple of 3 4: Last two digits are a multiple of 4 5: Last digit is 5 or 0 6: Sum of digits is a multiple of 3, and last digit is even 9: Sum of digits is a multiple of 9 10: Last digit is 0 12: Sum of digits is a multiple of 3, and last two digits are a multiple of 4(basically 3 and 4 together)
Special Right Triangles
45-45-90 degree right triangle: both sides opposite of 45 degrees are x and the hypotenuse is x√2 30-60-90 degree right triangle: side opposite of 30 degrees is x, side opposite of 60 degrees is x√3 and the hypotenuse is 2x Other special right triangles are ones with lengths of 3, 4, 5 and 5, 12, 13. These are easier to remember vs. using Pythagorean's theorem.
Area of Triangle
A = 1/2 bh
Area of Paralellogram
A = bh
Combination Formula
A combination question asks you how many unordered subgroups can be formed from a larger group n! / k!(n-k)! Where n is the number of items in the group as a whole and k is the number of items in each subgroup formed Example: A person has 10 books but only space for 3 of them, How many combinations of 3 books in here bag are there? 10! / 3!(10-3)! = 10! / 3!7! (10 X 9 X 8 X 7 X 6 X 5 X 4 X 3 X 2 X 1) / 3 X 2 X 1(7 X 6 X 5 X 4 X 3 X 2 X 1) (10 X 9 X 8) / (3 X 2 X 1) = 720 / 6 = 120 Note: If there are two different groups, such as they have to choose 2 of 4 treats and 1 of 3 drinks, then solve each by themselves and multiply together to get total number of combinations. In harder problems, you may have to find different combinations off of this, so you would then find all of the combinations for each scenario, then add together to get total number of combinations.
A trapezoid
A complex quadrilateral with two parallel sides, Its area is usually determined by breaking it into rectangles and triangles and determining their areas and adding them together. Area of a Trapezoid = (b1 + b2)/2 X h
Isosceles Triangle
A triangle with at least two congruent sides(two equal angles). Make sure you know which side they are talking about, if they only list the length of one side, it may be one of the congruent sides or the base. You'll have to calculate and reason further to figure it out. An equilateral triangle is when all sides are equal(all angles are equal to 60 degrees).
Supplements
Add up to 180 degrees
Complements
Add up to 90 degrees
How to find a Common Factor of two numbers
Break both numbers down to their prime factors to see which they have in common. Then multiply the shared prime factors to find all common factors. Example: What factor greater than 1 do 135 and 225 have in common? Setup: First find the prime factors of 135 and 225; 135 = 3 X 3 X 3 X 5, and 225 = 3 X 3 X 5 X 5. The numbers share 3 X 3 X 5 in common. Thus, aside from 3 and 5, the remaining common factors can be found by multiplying 3, 3, and 5 in every possible combination: 3 X 3 = 9, 3 X 5 = 15, and 3 X 3 X 5 = 45. Therefore, the common factors of 135 and 225 are 3, 5, 9, 15, and 45.
How to find a Common Multiple of two numbers
Example: What is the least common multiple of 28 and 42? Setup: 28 = 2 X 2 X 7 and 42 = 2 X 3 X 7 The LCM can be found by finding the prime factorization of each number, then seeing the greatest number of times each factor is used. Multiply each prime factor the greatest number of time it appears. In 28, 2 is used twice. In 42, 2 is used once. In 28, 7 is used once. In 42, 7 is used once, and 3 is used once. So you multiply each factor the greatest number of times it appears in a prime factorization: LCM = 2 X 2 X 3 X 7 = 84
Converting Radicals into Exponential Terms
Examples: √2 = 2^(1/2) ∛2 = 2^(1/3) Remember radicals can't be added, e.g., √2 + √3 is at its most simplified form, We can simplify √48 into √16√3 = 4√3
When counting a grid with each square unit less than 1
Find the lengths of both sides of the square and multiply them together to get its square units. For example if each side of the square was 0.2, then each square represents 0.2 X 0.2 = 0.04 square units, which is even smaller because it is between two values that are smaller than 1.
For Standard Deviation
Find the mean of the group, then find the difference between each number in the group, how much they deviate away from the mean. Square these numbers and add them together and divide by the total number. You then take the square root of this number and you get the standard deviation of the group. Example: 0, 2, 4, 6, mean = 3 Difference between them and mean: 3, 1, 1, 3 Add those numbers squared: 3^2 + 1^2 + 1^2 + 3^2 = 20 Divide by total number in sets: 20/4 = 5 Square root: √5 is the standard deviation or about 2.236 This number shows how much a number in the set deviates from the mean on average
Probability of Multiple Events
Formula: P(A or B) = P(A) + P(B) - P(A and B) If the events are independent, then P(A and B) = P(A) X P(B), So: *P(A or B) = P(A) + P(B) - P(A)P(B)* Example: A and B are independent, but not mutually exclusive, P(A) = 0.5 and P(A or B) = 0.8, what is P(B)? 0.8 = 0.5 + P(B) - 0.5 P(B) 0.3 = 0.5 P(B) P(B) = 0.6
Dilution or Mixture Problem Example
How many liters of a solution that is 10% alcohol by volume must be added to 2 liters of a solution that is 50% alcohol by volume to create a solution that is 15% alcohol by volume? Setup - The balancing method: Make the weaker and stronger(or cheaper and more expensive, etc.) substances balance. That is, (percent difference between the weaker solution and the desired solution) X (amount of weaker solution) = (percent difference between the stronger solution and desired solution) X (amount of stronger solution). Make n the amount, in liters, of the weaker solution. n(15 - 10) = 2(50 - 15) 5n = 2(35) n = 70/5 = 14 So 14 liters of the 10% solution must be added to the original, stronger solution.
Combined Work Formula
If T is the total time together and a, b, and c are each the time it would take by themselves then, 1/a + 1/b + 1/c = 1/T or T = abc / (ab + bc + ac), the latter is more practical if only between two people
Compound Interest Formula
If interest is compounded, the interest is computed on the principal as well as on any interest earned. To compute compound interest: (Final balance) = (Principal) X (1 + interest rate/c)^[(time)(c)] OR Final Balance = P X (1 + r/c)^[ct] where c = the number of times the interest is compounded annually Example: If $10,000 is invested at 8 percent annual interest, compounded semiannually, what is the balance after 1 year? Setup: Final balance = (10,000) X (1 + 0.08/2)^[(1)(2)] = (10,000) X (1.04)² = $10,816
Use ratios to help you solve for quantities you don't know or to compare them
If there is a small triangle inside a big triangle and you know the bases of both, you can set them as a ratio to solve for other things to make it simpler and cancel units out. These are called Similar Triangles.
How to find the Maximum and Minimum lengths for a Side of a Triangle
If you know the lengths of two sides of a triangle, you know that the third side is somewhere between the positive difference and the sum of the other two sides. Example: The length of one side of a triangle is 7. The length of another side is 3. What is the range of possible lengths for the third side? Setup: The third side is greater than the positive difference (7 - 3 = 4) and less than the sum (7 + 3 = 10) of the other two sides. 4 < x < 10
Know when guessing numbers strategy will work
If you're stuck try it out to eliminate some answer choices, then guess another number to see if you can eliminate more!
Chord of a Circle
Is a line across end to end, which isn't the diameter
The greatest angle of a triangle
Is opposite of its longest side
The largest area of a rhombus with a constant perimeter
Is when it is a square
In questions that say the answer choices must be true
It means they have to be true at all instances, try to work around it and prove it wrong to eliminate it as an answer choice.
When a company recoups
It means when they have enough, or make enough money that they spent, at a zero balance.
Make sure you know PEMDAS on the calculator
Jot some notes down and do simple calculations before entering it into a calculator to avoid simple computational errors.
When it asks you if an expression is equal to another
Just simplify and rearrange it until it equals one of the answer choices.
To find Largest Prime Factor
Keep dividing number in a tree format until you are left with all prime numbers, which are not divisible. The largest number that is left is the largest prime factor. Remember 1 is not a prime number!
When asked the number of distinct positive integer factors of a number
List all of the factors that can be multiplied by the number, starting with the number and 1, then go from here and count how many there are.
When looking at graphs
Make sure you notice where the bar ends, and guess which number it is closest to. Get good at approximating for these question types, they are usually spread out.
Per capita
Means per person
Helpful tips for quantitative comparison
Multiply both sides by the same number to cancel out and simplify one or both parts of the comparison. You should also set variables equal to big and large numbers, positive and negative, fractions, and so on. Don't assume! Also, sometimes you don't have to calculate; just reason through. The multiple choice options are always the same.
Probability
Number of Desired Outcomes / Number of Possible Outcomes Example: What is the probability of getting 4 independent coin flips and being exactly 1 head and 3 tails? The Number of Desired Outcomes is 4(HTTT, THTT, TTHT, and TTTH). If using two die, take both into consideration, so 6 and 1 and 1 and 6 would make two desired outcomes, because each die be a 6 and a 1. The Number of Possible Outcomes is 16(2 X 2 X 2 X 2) Probability = 4/16 = 1/4 Note: It may sometimes be easier to find the probability of not getting something and substract this from 1 to calculate the probability of getting it.
Overlapping Sets
Overlapping sets have a "∩" or "over" shape and if they are not overlapping have a "U" or "under" shape. To get the number that aren't overlapping add each separate part and subtract the overlapping parts to get the number of parts that aren't overlapping, e.g., A + B - A ∩ B = A U B A + B + A U B - A ∩ B = Total If either or, May have to make a grid and solve to make easier.
Permutations
Permutations differ from combinations in that permutations are ordered. By definition, each combination larger than 1 has multiple permutations. Example: If there are 7 participants and 4 awards, how many permutations of 4 awards are there(first, second, third, and fourth)? There are 7 people and 4 spots, so: 7 X 6 X 5 X 4 = 840 Formula: n! / (n-k)!
Tips for data interpretation
Pick an easy number when using percentages, such as 100, so the problem becomes much easier. Because it lists a fact in one of the questions, don't assume this is true for other questions. Be very careful in seeing what the question is asking, as it may be very specific, so interpret the graphs well.
Tips for factoring a complex equation, get better at reverse FOIL
Rearrange the equation into standard quadratic form so that it can be factored, i.e., 6x² - 11x - 10 = 0 Identify the factors of the last term and those of the coefficient of the first term, i.e., The factors of -10 are (-10 and 1), (10 and -1), (-5 and 2), and (5 and -2). The factors of 6 are (1 and 6), (-1 and -6), (2 and 3), and (-2 and -3). Find the pairs that will result in an algebraic sum of -11, which is the coefficient of the middle term. Note that 3 X (-5) = -15 and 2 X 2 = 4; -15 + 4 = -11, so (2x - 5)(3x + 2) are the correct factors. Then set each factor equal to zero to find the possible values of x, i.e., 2x - 5 = 0 and 3x + 2 = 0 giving us x = 5/4 and -2/3
Volume and Surface Area Formulas
Rectangular solid: V = lwh SA = 2lw + 2lh + 2hw Cylinder: V = πr²h Lateral SA = 2πrh Total SA = 2πr² + 2πrh
If you have a right triangle and don't know angles, but know side lengths
Remember the side lengths for the special right triangles. Using this information, if the short leg is less than half the hypotenuse, which is x vs. 2x for a 30-60-90 triangle, then it is flatter, meaning the smallest angle will be less than 30 degrees. Use similar reasoning for other scenarios where angels aren't given.
Slope
Rise over run
Another Example of Combination
Say we have a set of 6 Letters: A, A, A, B, B, C, How many different arrangements are there? We have 6 letters so 6 combinations, 6! For A there are 3!, For B there are 2!, and for C there are 1! 6! / 3!2!1! = (6 X 5 X 4) / 2 = 60 different arrangements
To see how many times parabolas intersect
Set them equal to each other and solve
When comparing two quantities with variables
Simplify both expressions to make it easier, such as dividing both sides, subtracting a number from both sides, etc., or guess numbers and solve. If you guess high and low for it and they change which is bigger, than the relationship cannot be determined.
How to find the Sum of Consecutive Numbers
Sum = (Average) X (Number of terms) Example: What is the sum of integers from 10 through 50, inclusive? Setup: Average = (10 + 50) / 2 = 30 Number of terms = 50 - 10 + 1 = 41 Sum = 30 X 41 = 1,230
How to find the sum of all the Angles of a Polygon and one angle measure of a Regular Polygon
Sum of the interior angles in a polygon with n sides: (n - 2) X 180 The term regular means all angles in the polygon are of equal measure. Degree measure of one angle in a regular polygon with n sides: [(n - 2) X 180] / n Example: What is the measure of one angle of a regular polygon? Setup: Since a pentagon is a five-sided figure, plug n = 5 into the formula: Degree measure of one angle: [(5 - 2) X 180] / 5 = 540/5 = 108
Sums of Angles
Sum of triangle's interior angles is 180 degrees, Exterior angles add up to 360 degrees Sum of quadrilateral's interior angles, or all angles centered around a point is 360 degrees
Helpful tips for problem solving
The answer choices are in order from lowest to highest so check A or D and work from there. Pick a small number to represent a variable all the rest depend on instead of solving with unknown variables.
How to find the Average of Consecutive Numbers
The average of evenly spaced numbers is simply the average of the smallest number and the largest number. The average of all the integers from 13 to 77, for example, is the same as the average of 13 and 77: (13 + 77) / 2 = 90 / 2 = 45
Multiplying exponents with the same base
The base stays the same, the exponents are added together. Example: 4^(3/6) X 4^(2/6) = 4^(5/6)
How to Count Consecutive Numbers
The number of integers from A to B inclusive is B - A + 1. Example: How many integers are there from 73 through 419, inclusive? Setup: 419 - 73 + 1 = 347
When Using Ratios of x to y
They are similar to fractions. If x:y = 1:3, we can say x/y = 1/3. We can then cross multiply and get 3x = y, which we can then substitute in another equation for y and solve them.
A set is comprised of all positive integers x, such that x^3 is a multiple of both 72 and 216. Which of the following integers are factors of every member of set J?
This is basically saying that x^3 has to be a multiple of both 72 and 216, so we need to find the least common multiple between them, which happens to be 216, if not, we would just do a prime factorization tree for both. We now set x^3 = 216, which is 6. This means that the set of J is all positive integers that are multiples of 6. To finish this problem, you would find the factors of 6, which are 1, 2, 3, and 6, and see which are listed for the possible answer choices.
How to find how many odd (prime) factors are in a big number
To a prime factorization tree until you are left with all prime numbers, then see how many odd there are. Don't forget about 1 not being prime! This is how many odd factors it has.
When working with inequalities and Absolute Values
Treat it like a normal equation, just remember to flip the sign if you are multiplying or dividing by a negative number Remember that absolute values need to be solved twice, for the positive and negative values of the quantities inside the symbols. Ex: |b + 4| ≤ 8 b + 4 ≤ 8, so b ≤ 4 b + 4 ≥ -8, so b ≥ -12
How to find the Diagonal of a Rectangular Solid
Use the Pythagorean theorem twice, unless you spot special right triangles Get the diagonal measurement of the top of the surface, then use that and the length of one of the sides and find the measure of the hypotenuse using Pythagorean's theorem or the knowledge of special right triangles in certain cases.
Another Probability Problem Example
What are the odds of rolling three dice and getting only one even number? We have two outcomes each roll, odd or even, so 2 X 2 X 2 for the 3 rolls = 8 possible outcomes, or EEE, EEO, EOE, OEE, EOO, OEO, OOE, OOO. We have only 3 desired outcomes, which are EOO, OEO, and OOE, therefore probability = desired outcomes/possible outcomes = 3/8 or 37.5% chance
Exponent help
When comparing exponents, it is easier to find equal bases. Remember negative exponents are just reciprocals, when common bases are multiplied their exponents are added, and when exponents are raised to another power, they are multiplied. Also know that a square root is a number to the 1/2 power and so on.
Simple Interest Formula
With simple interest, the interest is computed on the principal only and is given by: Interest = Principal X rt, OR Interest = Prt In this formula, r is defined as the interest rate per payment period and t is defined as the number of payment periods.
Factoring out from an Exponential Term
You can break apart the base, not the exponents. Example: 20^(1/2) = 5^(1/2) X 4^(1/2)
X = (86)(47)(94)(123)(64)(56)(72), What is the units digit of X?
You don't need to multiply this all the way out, we just need to pay attention to the last digit. We will multiply all of the last digits and drop the tens if necessary. Take the 6 and 7 from 86 and 74: 6 X 7 = 42 Then take the 2 from 42 and 4 from 94: 2 X 4 = 8 Then take the 8 and 3 from 123: 8 X 3 = 24 Then take the 4 from 24 and 4 from 64: 4 X 4 = 16 Then take the 6 from 16 and 6 from 56: 6 X 6 = 36 Then take the 6 from 36 and 2 from 72: 6 X 2 = 12 Thus, the last digit is 2 from the final 12.
Difference of Squares
a^2 - b^2 = (a+b)(a-b) Example: 4x^2 - 49 = (2x - 7) (2x +7) You would then solve for x by setting both equal to zero, 2x - 7 = 0, so x = 7/2, 2x + 7 = 0, so x = -7/2, x = 7/2 or -7/2
Learn to spot these classic polynomial equations
ab + ac = a(b + c) a² + 2ab + b² = (a + b)² a² - 2ab + b² = (a - b)² a² - b² = (a + b)(a - b), this is a difference of squares scenario
Percent Change Formula
difference / original X 100%
Graphing Quadratic Functions
f(x) = ax^2 + bx + c If a is positive it faces up, if negative it faces down If f(x - 7), it moves to the right by 7 If 2f(x), it is stretched vertically by a factor of 2, it will be shrunken if multiplied by a fraction If f(x) + 7, it moves up 7 Tips to graph: set f(x) and x to 0 to get intercepts, then find roots, plug in other numbers, etc.