GRE Math Foundations and Formulas 2022
Dilution or Mixture problem
determine the characteristics of a resulting mixture when different substances are combined OR determine how to combine different substances to produce a desired mixture Straightforward Setup EX: If 5 pounds of raisins that cost $1/pound are mixed with 2 pounds of almonds that cost $2.40/pound, what is the cost per pound of the resulting mixture? ($1)(5) + ($2.40)(2) = $9.80 total cost for 7 pounds of the mixture Cost per Pound= $9.80/7 = $1.40 Balancing Method EX: 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? - make the weaker and stronger substances balance (Percent Difference Between the Weaker solution and the Desired solution) x (Amount of weaker solution) = (percent difference between the stronger solution and the desired solution) x (amount of stronger solution) n = amount in liters of weaker solution n(15 - 10) = 2(50 - 15) -> 5n = 2(35) -> n = 70/5 = 14
Combination Formula
nCk = n! / k!(n - k)! n= number in the larger group k= number you're choosing ORDER DOES NOT MATTER EX: How many different ways are there to choose 3 delegates from 8 possible candidates? 8C3 = 8! / 3!(8-3)! = 8! / 3! x 5! = 8x7x6x5x4x3x2x1 / 3x2x1x5x4x3x2x1 = 8 x 7 = 56 possible combinations
Permutation Formula
nPk = n! / (n-k)! n= number in the larger group k= number you're arranging * used for solving questions about permutations, ex, the number of ways to arrange a elements sequentially* ORDER MATTERS EX: Five runners run in a race. The runners who come in first, second, and third place will win gold, silver, and bronze medals. How many possible outcomes for gold, silver, and bronze medal winners are there? 5P3 = 5! / (5-3)! = 5!/2! = 5 x 4 x 3 x 2 x 1 / 2 x 1 = 5 x 4 x 3 = 60
Associative Laws of Addition and Multiplication
regrouping the numbers does not affect the result (a+b) + c = a + (b+c) ; (ab)c = a(bc)
Predicting weather a sum, diff., or product will be EVEN or ODD
save time: DON'T MEMORIZE THE RULES plug 2 in for the "even" #, plug 3 in for the "odd" # and see what the result is EX: If m in even and n is odd, is the product mn even or odd? m=2, n=3; 2 x 3 = 6 -> EVEN
Combined Work
1/r + 1/s = 1/t r & s = number of hours it would take individually t = number of hours it would take together EX: If it takes Joe 4 hours to paint a room and Pete twice as long to paint the same room, how long would it take the two of them, working together, to paint the same room, if each of them works at his respective rate? Joe= 4 hours ; Pete= 2 x 4 = 8 hours 1/4 + 1/8 = 1/t 2/8 + 1/8 = 1/t 3/8 = 1/t t = 1/(3/8) = 8/3 hours OR 2 hours and 40 minutes
Finding the 3rd angle of a Triangle (given two angles)
sum of interior angles is equal to 180º EX: 35º+45º+xº = 180º, solve for x
Rules of Divisibility & How to recognize MULTIPLES of a number
2: (divisible) if its last digit is divisible by 2, (multiple) last digit is even 3: (both) if it's digits add up to a multiple of 3 4: (both) if its last two digits are a multiple of 4 5: (both) if its last digit is 0 or 5 6: (divisible) if it is divisible by both 2 and 3, (multiple) sum of digits is a multiple of 3 and the last digit is an even number 9: (both) if its digits add up to a multiple of 9 10: (multiple) last digit is 10 12: (multiple) sum of digits is a multiple of 3, and last two digits are a multiple of 4
Special RIGHT Triangles
3:4:5 and 5:12:13 -> represent the SIDE lengths 30º-60º-90º and 45º-45º-90º In a 30º-60º-90º the side lengths are multiples of 1, √3, and 2 (hypotenuse), respectively. In a 45º-45º-90º the side lengths are multiples of 1, 1, and √2 (hypotenuse), respectively.
SYMBOLS
= is equal to ≠ is equal to < is less than > is greater than ≤ is less than or equal to ≥ is greater or equal to ÷ divided by π pi (the ratio of the circumference of a circle to the diameter) ± plus or minus √ square root
Area of a Triangle
A = (1/2)bh
Area of Trapezoid
A = (average of parallel sides) x (height)
Area of a Sector
A = (n / 360) x 2π(r^2)
Area of Rectangle
A = LW
Area of Parallelogram
A = bh
Area of Square
A = s^2
Area of Circle
A = πr²
Decimal
A fraction written in decimal system format. 0.6 is a decimal; numerator divided by denominator 5/8=0.625
Operation
A function or process performed on one or more numbers. The four basic arithmetic operations are addition, subtraction, multiplication, and division.
Divisibility
A number is said to be evenly divisible by another number if the result of the division is an integer with NO remainder. A number that is evenly divisible by a second number is also a multiple of the second number.
Cube
A number raised to the 3rd power. 4^3=64; 64 is the cube of 4
Integer
A number without fractional or decimal parts, including positive and negative whole numbers and zero. All integers are multiples of 1 EX: -5, -4, -3, -2, 0, 1, 2, 3, 4, 5
Decimal system
A numbering system base don the powers of 10. ONLY numbering system used on the GRE. 315.246 -> (3)hundreds, (1)tens, (5)ones units. (2)tenths, (4)hundredths, (6)thousandths
Whole
A quantity that is regarded as a complete unit.
Part
A specified number of the equal sections that compose a whole.
Number line
A straight line, extending infinitely in either direction, on which numbers are represented as points. Can contain irrational numbers such as √2 which is between 1 and 2.
Set
A well-defined collection of items, typically numbers, objects, or events. The bracket symbols {} are normally used to define sets of numbers. EX: {2, 4, 6, 8} is a set of numbers
Properties of Zero
Adding or subtracting zero to a number does not change the number. x + 0 = x Subtracting a number from zero changes the number's sign. Subtract 5 from 0: 0 - 5 = -5 The product of zero and any number is zero. 0(z) = 0 Division by zero is undefined. For GRE purposes this translates to " it cannot be done"
Communicative Laws of Addition and Multiplication
Addition and multiplication are both communicative, which means that switching the order of any two numbers added or multiplied does not affect the result. a+b = b+a ; ab = ba Division and subtraction are NOT communicative
Finding an angle formed by a Transversal across Parallel Lines
All acute angles formed are equal All obtuse angles formed are equal Any acute angle plus any obtuse angle equals 180º EX: eº = gº = pº = rº ; fº = hº = qº = sº eº + qº = gº + sº = 180º ......................./ ___________eº/fº__________________ line 1 .............hº/gº ................/ ______pº/qº______________________ line 2 ......sº/rº ......../
The Distributive Law
Allows you to "distribute" a factor over numbers that are added or subtracted; multiply the factor by each number in the group EX: a(b+c) = ab + ac ; (a+b)/c = (a/c) + (b/c) BUT when the sum or difference is in the denominator no distribution is possible! EX: a/(b+c) ≠ (a/b) + (a/c)
Prime Number
An integer greater than 1 that has only two factors: itself and 1 First TEN prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
Sequence
An ordered list of terms. The terms of a sequence are often indicated by a letter with a subscript indicating the position of the number in the sequence. EX: a3 is the third number in a sequence, an is the nth term in a sequence
Finding an Angle in Intersecting Lines
Angles across from each other are EQUAL Angles along a LINE add up to 180º EX: aº = cº ; bº = dº ; aº + bº = 180º aº + bº + cº + dº = 360º ...........\ .........../ .............\ bº / ................ \ / ..........aº / \cº ............./ dº\ .........../ ..........\
Average
Average = Sum of Terms/Number of Terms
Average Rate
Average A per B = Total A / Total B -> convert to totals EX: If the first 500 pages have an average of 150 words per page, and the remaining 100 pages have an average of 450 words per page, what is the average number of words per page for the entire 600 pages? Total pages= 500 + 100 = 600 Total words= (500 x 150) + (100 x 450) = 75,000 + 45,000 = 120,000 Average words per page= 120,000/600 = 200
Average Speed
Average Speed= Total Distance / Total Time EX: Rosa drove 120 miles one way at an average speed of 40 miles per hour and returned by the same 120 mile route at an average speed of 60 miles per hour. What was Rosa's average speed for the entire 240 mile road trip? 120 miles at 40mph takes 3 hours. To return at 60mph takes 2 hours. Total time = 5 hours. AS = 240 miles/5 hours = 48mph
Counting Consecutive Numbers
B - A + 1 = Number of Integers EX: How many integers are there from 73 through 419, inclusive? 419 - 73 + 1 = 347
Linear Equations
y = mx + b m= slope of the line = rise/run b= y-intercept (point where line crosses the y-axis) If a question is asking you to compare a graph and equation, find the y-intercept and another point by using the slope to see what the equation looks like and match it to the graphs
Finding the Common Factor(s) of two numbers
Break both numbers to their prime factors to see what they have in common. Multiply the shared prime factors to find all common factors. EX: what factors greater than 1 do 135 and 225 have in common? 135 = 3 x 3 x 3 x 5; 225 = 3 x 3 x 5 x 5 Common: 3x3x5 ; 3x3= 9, 3x5= 15, 3x3x5= 45 COMMON FACTORS = 3, 5, 9, 15, and 45
Circumference of Circle
C = 2πr C = πd
Distinct
Different from each other. EX: 12 has three prime factors (2, 2, and 3) but only two distinct prime factors (2 and 3)
Cube Roots
EX: (-5) x (-5) x (-5) = -125 -> superscript3√-125 = -5
Scientific Notation
Examples: 123,000,000,000 is 1.23 x 10^11 0.000000003 is 3 x 10 ^-9
Factorials
Factorial is denoted by ! symbol IF n is a number greater than 1 then n factorial is n! n! = the product of all the integers from 1 to n EX: 2! = 2 x 1 = 2 4! = 4 x 3 x 2 x 1 = 24 0! = 1 6! = 6 x 5! = 6 x 5 x 4! EX: 8! / (6! x 2!) = (8 x 7 x 6!) / (6! x 2 x 1) = 28
Compound Interest
Final balance = Principal x (1 + (interest rate/c))^(time)(c) c= the number of times the interest is compounded annually EX: If $10,000 is invested at 8% annual interest, compounded semiannually, what is the balance after 1 year? = 10,000 x (1 + (0.08/2))^(1)(2) = 10,000 x (0.04)^2 = $10,816
Solving a Function problem
Function: algebraic expression of only one variable may be defined as a function, usually symbolized by f or g, of that variable. EX: What is the minimum value of x in the function f(x) = x^2 -1? if x=1 then f(1)= (1)^2 -1 = 0 SO, by inputting 1 the output is f(x) = 0 You're asked to find the minimum value so how can you minimize the expression f(x)=x^2 - 1? Since x^2 cannot be negative, in this case f(x) is minimized by making x = 0; f(0)= 0^2 -1 = -1, so the minimum value of the function is -1.
Overlapping Sets problem involving Both/Neither
Group 1 + Group 2 + Neither - Both = Total EX: Of the 120 students at a certain language school, 65 are studying French, 51 are studying Spanish, and 53 are studying neither language. How many are studying both French and Spanish? 65 + 51 + 53 - Both = 120 169 - Both + 120 Both = 120
Using Actual Numbers to find Rate
Identify the quantities and keep the units straight. EX: Anders typed 9,450 words in 3.5 hours. what was his rate in words per minute? CONVERT: 3.5 hours = 210 minutes 9,450 words / 210 minutes = 45 words per minute
Simplifying Radicals
If the number inside the radical is a multiple of a perfect square, the expression can be simplified by factoring out the perfect square. EX: √72 = (√36)√2 = 6√2
Finding the Maximum and Minimum lengths for a side of a Triangle
If you know the lengths of two side of a triangle, you know that the third side is somewhere between the positive difference and the sum of the other two sides. EX: 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? The third side greater than the positive difference (7 - 3 = 4) and less than the sum (7 + 3 = 10) of the other two sides.
Sequence problems
In a sequence problem, the nth term in the sequence is generated by performing an operation, which will be defined for you, on either n or on the pervious term in the sequence. Term is expressed a(subscript n). If you are referring to the fourth term in a sequence it is a(subscript 4). EX: What is the positive difference between the fifth and fourth terms in the sequence 0, 4, 18,... whose nth term is n^2(n - 1)? a(sub 5)= 5^2(5 - 1) = 25(4) = 100 a(sub 4)= 4^2(4 - 1) = 16(3) = 48 Positive difference is 100 - 48 = 52
Even
Integers that are evenly divisible by 2
Odd
Integers that are not evenly divisible by 2
Simple Interest
Interest = Principle x Rate x Time EX: if $12,000 is invested at 6% simple annual interest, how much interest is earned after 9 months? 12,000 x 0.06 x (9/12) = 540
Isosceles, Equilateral, and Similar Triangles
Isosceles: have at lease 2 equal sides and angles Equilateral: all sides are equal and all angles are 60º Similar: corresponding angles are equal, corresponding angles are proportional
Length of an Arc
Length of an Arc = (n / 360) x 2πr n= central angle
Simplifying a Radical
Look for factors under the radical that are perfect squares, then find the square root of the perfect squares. Keep simplifying until the term with the square root sign is as simplified as possible, that is, when there are no other perfect square factors (4, 9, 16, 25, 36...) inside the √. Write the perfect squares as separate factors and "unsquare" them. EX: √48 = √16 x √3 = 4√3 √180 = √36 x √5 = 6√5
Finding dimensions or area of an Inscribed or Circumscribed Figure
Look for the connection. Is the diameter the same as the side or a diagonal? EX: If the area of the inscribed square is 36, what is the circumference of the circle? - the circle's diameter is the square's diagonal. This is 6√2 because the diagonal pf the square transforms it into two separate 45º-45º-90º triangles. SO, the diameter is 6√2 a multiple of √2. Circumference = π(Diameter) = π(6√2) = 6π√2
Solving Quadratic Equations
Manipulate the equation so it is equal to 0, factor the left side and break the quadratic into two simple expressions. Then find the value(s) for the variable that make either expression = 0. EX: x^2 + 6 = 0 x^2 - 5x + 6 = 0 (x - 2)(x - 3) = 0 x = 2 or 3
Combined Ratio
Multiply one or both ratios by whatever you need to in order to get the terms they have in common to match. EX: The ratio of a to b is 7:3. The ratio of b to c is 2:5. What is the ratio of a:c? multiply a:b by 2 -> 2(7:3) = 14:6 multiply b:c by 3 -> 3(2:5) = 6:15 a:b:c = 14:6:15 -> a:c = 14:15
Exponent Rules
Multiply: same base: x^a • x^b = x^a+b (2^2)(2^3) = 2^2+3 = 2^5 diff base: (3^2)(5^2) = (3x3)(5x5) = (3x5)(3x5) = (3x5)^2= 15^2 Divide: x^a / x^b = x^a-b 4^5 / 4^2 = 4^5-2 = 4^3 Fraction = smaller number (1/2)^2 = (1/2)(1/2)=1/4 x^1/2 = √x Raise a power to another power: (x^a)^b = x^ab (3^2)^4 = 3 ^(2x4) = 3^8 Neg. exponent: x^-a = 1/x^a 5^-3 = 1/5^3 = 1/125 Exponent = 0 5^0 = 1 ; 161^0 =1 ; (-6)^0 = 1 Base = 0 0^3 = 0^4 = 0^12 = 0 Neg. # to even power = Positive number Neg. # to odd power = Neg. number Even # to even power = Even Number Odd # to ANY integer greater than or equal 0 = Odd number
Properties of 1 and -1
Multiplying or dividing a number by 1 does NOT change the number. Multiplying or dividing a nonzero number by -1 changes the number's sign.
Number Added/Deleted Formulas Using Original Average & New Average to find WHAT was added or deleted
Number Added = New Sum - Original sum Number Deleted = Original Sum - New Sum EX: The average of five numbers is 2. After one number is deleted the new average is -3. What number was deleted? Original Sum= 5 x 2 = 10 New Sum = 4 x (-3) = -12 Number DELETED = 10 - (-12) = 22
Consecutive numbers
Numbers of a certain type, following one another without interruption. Numbers may be consecutive in ascending or descending order. Consecutive integers: (-2, -1, 0, 1 ,2, 3) Consecutive even numbers: (-4, -2, 0, 2, 4, 6) Consecutive multiples of 3: (-3, 0, 3, 6, 9) COnsecutive prime numbers: (2, 3, 5, 7, 11)
Add, Subtract, Multiply, and Divide Roots
ONLY add/subtract when the parts inside the √ are identical EX: √2 + 3√2 = 4√2 ; √2 + √3 cannot be combined √2 - 3√2 = -2√2 Multiplying/dividing roots, deal with what's inside the √ and outside the √ separately EX: (2√3)(7√5) = (2 x 7)(√3x5) = 14√15 10√21 / 5√3 = (10/5)√21/3 = 2√7
Rules for Odds and Evens
Odd + Odd = Even Even + Even = Even Odd + Even = Odd Odd x Odd = Odd Even x Even = Even Odd x Even = Even
Element
One of the members of a set
Digit
One of the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. A number can have several digits 542 -> three digits: 5, 4, and 2 321,321,000 -> nine digits, but four distinct digits: 3, 2, 1, and 0
Addition and Subtraction of Radicals
Only like radicals can be added to or subtracted from one another. EX: 2√3 + 4√2 -√2 - 3√3 = (4√2 -√2) + (2√3 - 3√3) = 3√2 + (-√3) = 3√2 - √3
Overlapping Sets problem involving Either/Or
Organize the info in a grid and fill in the blank boxes from the given information using simple math to find the desired answer
Perimeter of Rectangle
P = 2(L + W)
Perimeter of Square
P = 4s
PEMDAS
Parentheses Exponents Multiplication, Division (simultaneously from left to right) Addition, Subtraction (simultaneously from left to right) "Please Excuse My Dear Aunt Sally" EX: 66(3-2)÷11 = 66(1)÷11 = 66÷11= 6
Percent Formula
Part = Percent x Whole EX- Finding the Part: 12% of 25? Part = (12/100) x 25 -> Part = 300/100 = 3 EX- Finding Percent: 45 is what % of 9? 45 = (Percent/100) x 9 -> 4,500 = Percent x 9 -> 500 = Percent EX- Finding the Whole: 15 is 3/5% of what #? 15 = (3/5)(1/100) x Whole -> 15 = (3/500) x Whole -> Whole = 15(500/3) = (7,500/3) = 2,500
Percent Increase/Decrease Formula
Percent Increase = (Amount of increase/Original Whole) x 100% Percent Decrease = (Amount of decrease/ Original Whole) x 100% EX- Increase: Price goes up from $80 to $100, what's the % increase? (20/80) x 100% = .25 x 100% = 25%
Probability
Probability = (Number of desired outcomes) / (Number of total possible outcomes) EX: What is the probability of throwing a 5 on a fair six-sided die? # of desired: 1 ; # of total: 6 -> Probability= 1/6
Probability: problems when probabilities need to be multiplied
Probability= number of desired outcomes/ number of total possible outcomes EX: If 2 students are chosen at random to run an errand from a class with 5 girls and 5 boys, what is the probability that both students chosen will be girls? 1st student girl: 5/10 = 1/2 2nd girl: 4/9 Both girls: 1/2 x 4/9 = 2/9
Surface Area of a Rectangular Solid
SA = 2(LW) + 2(WH) + 2(LH)
Surface Area of a Cylinder
SA = 2π(r^2) + 2πrh
Using a Ratio to determine an Actual Number
Set up a proportion using the given ratio EX: ratio of boys to girls is 3 to 4. If there are 135 boys, how many girls are there? (3/4) = (135/g) -> 3 x g = 4 x 135 -> 3g = 540 -> g = 180
Slope of a Line
Slope = Rise/Run = (change in y) / (change in x) EX: slope of a line that contains the points (1,2) & (4,-5) Slope = (-5-2) / (4-1) = -7/3
Combined Percent Increase and Decrease with no specified original value
Start with 100 as the starting value EX: A price rises by 10% one year and by 20% the next year. What is the combined percent increase? Year 1: 100 + (10% of 100) = 100 + 10 = 110 Year 2: 110 + (20% of 110) = 110 + 22 = 132 That is a combined 32% increase.
Distance between Points
Subtract the values that differ. if both values differ make a right triangle and use Pythagorean theorem. EX: (2,3) to (-7, 3) subtract x's as the y's are the same: 2 - (-7) = 9
Finding the SUM using Average
Sum = (Average) x (Number of Terms) EX: 17.5 is the average of 24 numbers, what's the sum? SUM = 17.5 x 24 = 420
SUM of Consecutive Numbers
Sum = (Average) x (Number of terms) EX: What is the sum of the integers from 10 through 50, inclusive? Average: (10 + 50) ÷ 2 = 30; # of integers: 50 - 10 + 1 = 41 SUM= 30 x 41 = 1,230
Finding the sum of all the Angles of a Polygon and one angle measure of a Regular Polygon
Sum of the interior angles in polygon with n sides (n - 2) x 180 *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 EX: What is the measure of one angle of a regular pentagon? Pentagon = 5 sides ; n = 5 (5 - 2) x 180 / 5 = 540/5 = 108
Solving Remainders problems
TIP: Pick a number given the conditions and see what happens EX: When n is divided by 7, the remainder is 5. What is the remainder when 2n is divided by 7? - Find a number that leaves a remainder of 5 when divided by 7. Take any multiple of 7 and add 5 to it. 7 + 5 = 12 -> n=12 -> 2(12) = 24 -> 24 ÷ 7 = 21 r3 ANSWER = 3
Original Whole before percent increase/decrease
TIP: think of a 15% increase over x as 1.15x and set up an equation EX: After decreasing by 5%, the population is now 57,000. What was the original population? 0.95 x (Original Population) = 57,000 Original Population = 57,000/0.95 = 60,000
Average of Consecutive Numbers
The average of the smallest number and the largest number. The average of ALL the integers from 13 to 77 is the same as the average of 13 and 77. EX: (13 + 77) ÷ 2 = (90/2) = 45
Absolute Value
The distance a number is from zero on a number line. ALWAYS POSITIVE |-3| = |+3| = 3 ; -3 and 3 are 3 units from 0 so each have an absolute value of 3. if you are told |x| = 5 then x could be 5 or -5.
Fraction
The division of a part by a whole. Part/Whole = Fraction EX: 3/5
Prime Factorization
The expression of a number as the product of its prime factors (the factors that are prime numbers) There are 2 methods to find this, look in book pg. 189 & 190
Greatest Common Factor (GCF)
The largest factor a group of integers share
Exponent
The number that denotes the power to which another number or variable is raised. 5^3 = (5)(5)(5)
Factors
The positive and negative integers by which a number is evenly divisible AKA: divisors
Range
The positive difference between the highest and lowest values in a distribution EX: Range of 88, 57, 68, 85, 98, 93, 93, 84, and 81 98 - 51 = 41
Square
The product of a number multiplied by itself. A squared number has been raised to the 2nd power. EX: 4^2 = (4)(4) = 16; 16 is the square of 4
Multiple
The product of a specified number and an integer. EX: 3, 12, and 90 are multiples of 3: 3 = (3)(1); 12 = (3)(6); 90 = (3)(30)
Denominator
The quantity in the bottom of a fraction
Numerator
The quantity in the top of a fraction.
Sum
The result of addition.
Product
The result of multiplication.
Difference
The result of subtraction
Least Common Multiple (LCM)
The smallest multiple (other than zero) that two or more integers have in common. How to find it: - Determine the prime factorization of each integer - Write out each prime number the maximum number of times that it appears in any one of the prime factorizations - Multiply those prime numbers together to get the LCM of the original integers EX: LCM of 6 & 8? 6= 2 x 3 8= 2 x 2 x 2 LCM= 2 x 2 x 2 x 3 = 24
Multiplication and Division of Radicals
To multiply or divide one radical by another, multiply or divide the numbers outside the radical signs, then the numbers inside the radical signs. EX: (6√3)2√5 = (6)(2)(√3)(√5) = 12√15 EX: 12√15 ÷ 2√5 = (12/2)(√15/√5) = 6√(15/5) = 6√3
Solving Multiple Equations
Try to combine them so you get closer to the value you are looking for. EX: If 5x - 2y = -9 and 3y - 4x =6, what is the value of x + y? 5x - 2y = -9 + (-4x + 3y = 6) x + y = -3
Solving Digits problem
Use logic and trial and error. EX: If A, B, C, and D represent distinct digits in the addition problem below, what is the value of D? AB +BA = CDC C= 1 as AB and BA create a hundreds number. SO, B + A = 1 ; B + A = 11 because a 1 gets carried therefore A & B are any 2 digits that add up to 11 (3&8, 4&7,...). 47 + 74 = 121 ; 83 + 38 = 121 D = 2
Counting Number of Possibilities
Use multiplication to find the number of possibilities when items can be arranged in various ways. EX: How many three-digit numbers can be formed with the digits 1, 3, and 5 each only use once? hundreds place: 3 possibilities tens place: 2 possibilities ones place: 1 possibility 3 x 2 x 1 = 6 possibilities or numbers can be formed
Finding the Diagonal of a Rectangular Solid
Use the Pythagorean Theorem twice, unless you spot "special" triangles
Finding the New Average when a number is added or deleted
Use the sum of the terms of the old average to help you find the new average. EX: Micheal's average score after four tests is 80. If he scores 100 on the fifth test, what's his new average? Original Sum = 4 x 80 = 320 New Sum = 320 + 100 = 420 New Average = 420/5 = 64
Volume of a Cylinder
V = Area of the base x Height = π(r^2)h
Volume of a Rectangular Solid
V = L x W x H
Weighted Average
WA: (Sum of Weighted terms) / Number of Terms TIP: give each term the appropriate "weight" EX: the girls' average score is 30. Boys' is 24. If there are twice as many boys as girls, what is the overall average? WA= ((1 x 30) + (2 x 24)) / 3 = 78/3 = 26 OR (30 + (2)(24)) / 3 OR (30 + 24 + 24) / 3
Remainder
What is "left over" in a division problem EX: 17 ÷ 3 = 5 with a remainder of 2
Standard Deviation
a measure of how spread out a set of numbers is - greater the spread the higher the SD HOW is it calculated: - find the average of the set - find the differences between the mean and each value in the set - square each of the differences - find the average of the squared differences - take the positive square root of the average
Pythagorean Theorem
a^2 + b^2 = c^2 Used to find the Hypotenuse (c) of a RIGHT triangle
Classic Polynomials
ab + ac = a(b + c) a^2 + 2ab + b^2 = (a + b)^2 a^2 - 2ab + b^2 = (a - b)^2 a^2 - b^2 = (a - b)(a + b)
