Comprehensive Math

(-46) - 37 (-22) + (-61) (-25) + (-41) (-61) - 11

(-46) - 37 = -(46 + 37) = -83 (-22) + (-61) = -(22 + 61) = -83 (-25) + (-41) = -(25 + 41) = -66 (-61) - 11 = -(61 + 11) = -72

256 (Place Values)

2 hundreds 5 tens 6 ones 2x100 + 5x10 + 6x1

Simplify: (2x-5)(3x-1)

answer: (6x^2)-17x+5

For positive numbers a and b, let a @ b = (2a^2) + b. What does (1 @ 2) @ 3 equal

a. ... b. 35 (answer)

a. 83 -17 b. 40 - 18 c. 52 - 27 d. 71 - 15

a. 83 -17 = 86 - 20 = 66 b. 40 - 18 = 42 - 20 = 22 c. 52 - 27 = 55 - 30 = 25 d. 71 - 15 = 76 - 20 = 56 (Rule: simplify subtraction by adding the same number to both terms)

Parallel lines

two lines that never intersect, and are always the same distance apart.

congruent

two shapes are congruent if they have the same shape and size. they do not have to have the same orientation.

Solve (no solutions): x-2y= 5 3x-6y

using substitution or elimination, you get two numbers that do not equal each other, so there are no solutions. (Parallel lines on a graph)

complex algebra practice (substitutions): Solve for x: (x^2 + 1)^2 - 15(x^2 + 1) + 50 = 0

a. n^2 - 15n + 50 = 0 b. (n - 10)(n - 5) = 0 c. n = 10 or n = 5 d. 10 = x^2 + 1 or 5 = x^2 +1 e. 9 = x^2 or 4 = x^2 f. x = +/- 3 or x = +/- 2

If N = (2^6)(3^5)(5^2)(7), express √N in simplified form

a. √N= (√2^6)(√3^5)(√5^2)(√7) b. = (√2^3x2^3)(√3^2x3^2x3)(√5^2)(√7) c. = (2^3)(3^2√3)(5)(√7) d. = (8)(9√3)(5)(√7) e. = 360(√3)(√7) f. = 360√21 (answer)

75^2 (Squaring multiples of 5 example)

a.) 75 becomes 7 b.) 7 + 1 = 8 c.) 7*8 = 56 d.) 56 becomes 5625

Simplify the following:(2x^4-8x^2)/(x^2-5x-14)

a.[(2x^2(x^2-4))/... b. [(2x^2(x+2)(x-2))/(x+2)(x-7) c.[(2x^2(x-2))/(x-7)

Rebuilding the dividend formula (dividend = top of fraction)

D= SxQ + r Note: S= divisor; Q= quotient, r= remainder.

Translation practice: Twice A is 100 less than three times B

2A = 3B - 100

84*50

42*100 = 4200

Powers of 4 up to 4^4

4=4 4^2= 16 4^3= 64 4^4= 256

Powers of 5^4

5=5 5^2= 25 5^3= 125 5^4= 625

cubes of 6 to 9

6^3= 216 7^3= 343 8^3= 512 9^3= 729

Prime number facts

a. 1 is not a prime number b. 2 is the only even prime number

a. 1/10 b. 1/100 c. 1/1000 d. 1/20

a. 1/10 = 0.1 b. 1/100= 0.01 c. 1/1000= 0.001 d. 1/20= (1/20)x(5/5)= 5/100= 0.05

a. 1/3 b. 2/3 express as decimals

a. 1/3= 0.33333 b. 2/3= 0.66667

Translation practice: a. A is 50% of B b. A is 50% greater than B

a. A = 0.5B b. A = 1.5B

Finding the slope

a. If you are given the numerical coordinates of two points, try to use the slop triangle first before using algebra. b. Formula: Run is X2 - X1. Rise is Y2 - Y1. Slope = Rise/Run

Solve: kdfj x^2+5= 0

a. Impossible Note: you can't square something and get a negative number.

Right triangles

a. Legs: the sides the meet at the right angles b. hypotenuse: the side opposite the right angle. (always the longest side.)

the number zero

The only number that is neither positive nor negative.

Counting terminology

a. 'and' means multiply b. 'or' means add

Multiplier for a P% Increase

1. 1+(P% as a decimal) Example: multiplier for 46% increase= 1+0.46= 1.46

Finding the percent multiplier for increase or decrease

1. multiplier= new price/old price

Powers of 3 up to 3^4

3=3 3^2 = 9 3^3= 27 3^4= 81

4+3/5(mixed numeral to improper example)

4 + 3/5 = 4x5/5 + 3/5 = 20/5 + 3/5 = 23/5

Area of other quadrilaterals (not square or rectangle)

A = bh Note. that the height has to be perpendicular to the base, so the hight can't be the side of the quadrilateral in some cases.

positive correlation

A correlation where as one variable increases, the other also increases, or as one decreases so does the other. Both variables move in the same direction.

Perpendicular bisector

A line that bisects a segment and is also perpendicular to it.

Integer

All positive & negative WHOLE numbers, including zero

If P is an odd integer and (P^2 + QxR) is an even integer, then which of the following must be true. A. either Q or R is an odd integer B. either Q or R is an even integer C. both Q and R are odd integers D. both Q and R are even integers E. Nothing can be concluded.

Answer. C is tempting, but the answer is E because the problem never specifies whether Q and R are integers. There is not enough info. for us to draw any conclusions. Note: NEVER ASSUME numbers are integers unless specifically specified.

If n is an integer greater than 50, then the expressing (n^2-2n)(n+1)(n-1) MUST be divisible by which of the following? I 8 II 12 III 18 Consecutive Integers.

Answer: I and II only. a. rewrite the expression so it looks like consecutive numbers. n(n-2)(n+1)(n-1)= (n-2)(n-1)n(n+1) b. use basic facts about consecutive numbers coupled with factoring to determine answer.

Inequalities 2 practice: For any number x, which of the following must be greater than x? I. x+2 II. 2x III. x^2

Answer: I only II does not work. If x is negative, the product with be smaller. III does not work. squaring fractions makes them smaller. squaring 1 and 0 produces equalities, not inequalities.

If D = dividend, S= Divisor, Q= quotient, and r= remainder, we can write the following formula: (remainders module)

D/S= Q + (r/S). Note: Q is always an integer. the non-integer part is the remainder or the fraction. Note 2: If the divisor is larger than the dividend, the integer quotient = 0, and the remainder equals the dividend. (the test will ask about this fact).

When isotope QXW radioactively decays, it loses exactly half of its mass in each three-day period. Suppose scientists start with a 96 gram sample of pure isotope QXW on a certain day. What will be the remaining mass in 12 days.

Day 0: 96 Day 3: 48 Day 6: 24 Day 9: 12 Day 12: 6 grams (answer).

What is the sum of al the multiples of 20 from 160 to 840 inclusive?

Determine the number of terms first a. 160 = 8(20), the 8th multiple of 20 b. 840 = 42(20), the 42nd multiple of 20 c. Inclusive counting: 42 - 8 + 1 = 35 d. number of pairs = 17.5 e. Sum of list: 17.5(160+840) = 17.5(1000)= 17,500 (answer)

Standard deviation facts

F1: It can only be positive or zero, never negative. F2: the SD only equals zero if all the numbers on the list are identical to each other. F3: if all the numbers on a list are exactly the same distance from the mean, that distance is the standard deviation. F4: a set with most numbers clustered toward the extremes will have a higher standard deviation than a list with most values equal to or close to the mean. F5: if we add the same number to every number on a list, or subtract the same number from every number on the list, the SD doesn't change. F6: if the spacings between numbers stays the same, multiples sets of numbers can have the same SD. F7: if we multiply every number on a list by positive number K, the standard deviation also gets multiplied by K.

One possible formula for the LCM

Find the LCM of P and R. LCM= (PxR)/GCF Note: Always cancel before multiplying

Finding the number of odd factors (finding factors of large numbers)

Follow the factors of large number procedures, but ignore the factors of 2 1. prime factorization 2. List of exponents of ODD prime factors 3. Add 1 to each: new list 4. Production fo numbers on new list= number of odd factors.

Test for prime numbers less than 100

If any number less than 100 is not divisible by any prime divisor less than 10 (2,3,5,7), then the number has to be prime.

Simplified AND rule

If events A and B are independent, then P(A and B) = P(A) x P(B)

Fundamental Counting Principle

If the first stage can be done in n sub 1 ways, the second in n sub 2 ways, etc., then the complete task can be done in N = (n sub 1)(n sub 2)(s sub 3) ways.

Divisibility rule for 4

If the last two digits form a two-digit number divisible by 4, then the entire number is divisible by 4.

Zero Product Property

If the product of two number is zero, one of the factors MUST BE zero. If ab = 0, then a = 0 OR b = 0

Using pythagorean theorem with two huge measurements.

If the sides given are larger, divide down by the GCF, do the computations in the smaller triangle, then scale back up.

Divisibility Rule for 3

If the sum of the digits is divisible by 3, then the number is divisible by 3. Note: If a number is divisible by 3 & 4, it is also divisible by 12.

Divisibility Rule for 9

If the sum of the digits is divisible by 9, then the number is divisible by 9.

'at least'

If you see this in a probability question, use the complement short cut: P(not A) = 1 - P(A)

The triangle inequality theorem

In any triangle, the sum of any two sides MUST be greater than the third side.

Inequality rules

Inequalities remain the same when... 1. we add or subtract the same thing from both sides 2. we multiply or divide both sides by any positive number. The order of inequalities changes when... 1. we multiply or divide both sides by any negative number (e.g. -x>3 becomes x<-3)

Sam's car was fined for parking when he gave Joe and Peter a ride, so they decided to help Sam pay the fine. Joe paid 3 more than 1/4 of the fine and Peter paid 3 less than 1/3 of the fine, leaving Sam 4 less than 1/2 the fine to complete the payment. How much did Sam pay?

Joe: F/4 + 3 Peter: F/3 -3 Sam: F/2 -4 a. (f/4) + 3 + (f/3) -3 + (f/2) -4 = f b. (f/4) + (f/3) + (f/2) -4 = f c. 3f + 4f + 6f -48 = 12f d. 13f - 48 = 12f e. f - 48 = 0 f. f = 48 g. Sam = (48/2) - 4 h. Sam = $20 (answer)

Divisibility rule for 8

Must be divisible by 2 and 4.

Is 102,334,155 divisible by 9?

No. 24 is not divisible by 9.

If 7k is a positive integer, and if √k > k, then is k and integer?

No. The square root of something can only be greater if what you are working with is not an integer.

Find the positive factors of 36 (method of smaller than 100)

Note: List all the factor pairs. This method works best for numbers smaller than 100 1, 36; 2,8; 3, 12; 4,9; 6,6; 2. 36 has 9 positive factors. You are done when you get to a pair that has two of the same number.

Problem: At the beginning of the year, the price of an item increased 30%. After the increase, an employee purchased it with a 40% discount. The price the employee pid was what percent below the original price? (sequential percent changes)

Note: Never add or subtract percent changes. Use the multipliers: a. 1.30 x 0.6 = 0.78 b. 0.78 is the multiplier for a 22% decrease.

If P and Q are integers, and if (p^2 + PQ) is an odd number, what do we know about P and Q?

Note: We can use the four case method to figure this out, or we can just use logic (see multiplying and adding even and odd integers). answer: It must be true that P is odd and Q is even.

Root properties

Note: everything extends to higher order roots, not just square roots. a. Roots distribute over multiplication and division: √pq = (√p)(√q); √p/q = √p/√q b. Roots DO Not distribute over addition and subtraction.

Practice question: If 7^(2x) = 7^(6-x), then solve for x. Note: a test question will never be this easy.

Note: if the bases are equal, we can set the exponents as equal and solve like a normal equation. a. 2x = 6 -x b. 3x = 6 c. x = 2 (answer)

Dividing evens and odds

Note: there are no absolute rules, but there are patterns. a. E/E could be E or O or not an integer b. O/O could be an O or not an integer. c. E/O could be E or not an integer. d. O/E is never an integer.

Algebraic representations of consecutive numbers (some examples)

Note: these expression all increase by one as you move from left to right. a. {n, n+1, n+2, n+3} b. {n-2, n-1, n, n+1, n+2} c. {n+15, n+16, n+17}

At least one scenario

P(at least one A) = 1 - P(not A)^#of trials

Problem: One cup of butter is enough for 12 of Roger's cookies, and one cup of sugar is enough for 8 of these cookies. If he used five more cups of sugar than butter, how many cookies did he make?

Part 1: Combine ratios B:C = 1:12----> B:C = 2:24 S:C = 1:8----> S:C = 3:24 B:S:C = 2:3:24 Part 2: Use algebra a. Butter= 2n; Sugar= 3n; Cookies= 24 n b. 5= (cups of sugar) - (cups of butter) c. 5= 3n -2n d. 5= n e. 24n= 24(5)= 120 cookies (answer)

A machine working at a constant rate, makes 36 staplers in 28 minutes. How may staplers does it make in 1hr 45 min.

Proportion method quickest a. 36/28 = S/105 b. 9/7 = S/105 c. 9/1 = S/15 d. S = 9 x 15 = 135 (answer)

Procedure for finding GCF. (example 360 and 800)

S1: Find prime factorizations. 360= 2^3x3^2x5; 800= 2^5x5^2 S2: Find the powers of each fact that they have in common. GCF= 2^3x5^1= 40 (answer)

Calculating quartiles

S1: Find the median first S2: Find the median for the upper (Q3) and lower (Q1) lists. Note: make sure to exclude the median of the whole list from the upper and lower list (it should not be part of the calculation).

Counting factors of a large numbers (steps with 8400)

S1: Find the prime factorization. 8400= 84x100= 7x12x100= 7x3x2x2x2x5x2x5= 7x3x2^4x5^2 S2: Make a list for the exponents of the prime factors (4, 1, 2, 1) S3: And 1 to every exponent on the list (5, 2, 3, 2). S4: Multiply numbers in S3 together (60). This means 8400 has 60 different factors.

Integer Properties Strategies

S1: Make sure you are dealing with integers. Words that indicate we are working with integers (integers, even, odd, prime) S2: Remember that integers include 0 and negative. 0 is an even integer. However, Prime numbers must be positive integers. if the question talks about remainders, all numbers involved are positive integers. S3 Remember interchangeable ways of talk about factors (13 is a factor of 78; 13 is a divisor of 78; 78 is divisible by 13; 78 is a multiple of 13; 13 is part of the prime factorization of 78) S4: Memorize all the primes below 60 (1,2,3,5,7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59). 1 is NOT a prime number. 2 is the only even prime number. S5: Prime factorization unlock many secrets. (number with 100 plus) S6: memorize perfect squares through 15 S7: Remember how to get GCF and LCM S8: consecutive number questions often appear in variable form, and you have to recognize that the algebraic expressions represent consecutive numbers. You will likely need to tweak a few. (if it truly is an integer problem it will specify that the variable is an integer).

VICs picking numbers procedures. (Only with variables in the answer choices)

Stage 1- after low hanging fruit 1. Go after low hanging fruit. 2. If there is only one variable in the answer choices and that variable is a percent, then it may be easy to figure out the prompt answer when this variable is either 0 or 100. 3. Do not use 2 if there is more than one variable. Stage 2- after low hanging fruit 1. Avoid 0 or 1 2. pick different numbers for different variables. The number should be different from those given in the problem 3. Don't pick numbers that are multiples of each other; picking different prime numbers is good. 4. Keep the numbers small.

Any even power of x is the square of another power

This means the following x^10 = X^5+5 = (X^5)(X^5) = (X^5)^2

Special right triangles

Triangle 1: Isosceles right triangle Triangle 2: 30-60-90, or the 1-2-√3 triangle (2 being the hypotenuse).

[27(y+5)(2y-2)]/[2(y-1)]= canceling algebraic expressions

[27(y+5)(2y-2)]/[2(y-1)]= [9(y+5)(2y-2)]/[(2y-2]= [9(y+5)x 2(y-1)]/2(y-1)= 9(y+5)

Simplify the following (√2)^48

a = 2^24 Note: and even power of a square root can be written as a power of a whole number

Factorial notation

a Factorial is n times all the positive integers less than n b. Factorial notation: n! (e.g. 5 factorial is 5!) e. Example: 5! = (5)(4)(3)(2)(1)

What is a probability?

a ratio, or fraction b. Probability, P = # of successes/total # of outcomes. c. P is always between 0 and 1: 0 ≤ P ≤ 1

Distributive Property

a(b+c) = ab + ac Or a(b-c) = ab - ac

Probability terminology

a. 'Or' means add b. 'And' means multiply Not: this is generally speaking when it comes to probability.

Simplify the following (difference of squares: ((0.999856/0.988)-1)

a. ((1-0.000144/1-0.012)-1) b. ((1+012)(1-0.012)/1-0.012)-1) c. 1.012-1 d. 0.012

Simplify the following: [(24x^12)(y^9)]/[(18x^-4)(y^3)]

a. ((4/3)x^16)(y^6) Note: coefficient does not move up with the variable.

a. (1/4)x(8/13)

a. (1/4)x(8/13)= 1x(2/13)= 2/13 Note: Always cancel before you multiply.

Simplify the following: 3^32 - 3^28

a. (3^28)(3^4) - (3^28)(1) b. (3^28)(3^4 - 1) c. 80(3^28)= answer

What is 80% of 200?

a. .80 x 200 b. 160 (answer)

a. 0.68 b. 0.075 c. 2.3 decimals to percents

a. 0.68= 68% b. 0.075= 7.5% c. 2.3= 230%

Proportional reasoning practice In F = (GMP)/R^2, if P triples and R doubles, F is multiplied by what?

a. 1 = [(1)(1)(1)]/1^2 b. F = [(1)(1)(3)]/2^2 c. F = 3/4. Thus F is multiplied by 3/4 when P triples and R doubles.

65^2

a. 65 becomes 6 b. 6 + 1 = 7 c. 6*7 =42 d. 42 becomes 4225

Solve the following problem for a right triangle: If a = 6 and b = 8, c =

a. 6^2 + 8^2 = c^2 b. 100 = c^2 c. 10 = C (answer).

How many even factors does 21,600 have?

a. 72 total factors b. 12 odd factors c. 72-12= 60 even factors (answer).

81^2

a. 81^2 = 80^2 + 80 + 81 b. 6400 + 80 + 81 c. 6561 (answer)

Combination

a. A small group drawn from a large group in which order does not matter. b. Notation: nCr (n choose r). example: 7C2 (there is a groups 7 and we choose 2 randomly from the group. c. Important: nC1 = n d. Important 2: nCr = nC(n-r) Note: only use combination approach when order does not matter.

Simplifying roots

a. Every even power of a prime factor is aperfect square, and can be simplified.

(1/10)- (1/35) LCM module

a. GCF= 5 b. 10= 5x2 c. 35= 5x7 d. LCM= 70 e. (7/70)- (2/70)= 5/70= 1/14 (answer)

a. Negatively sloped line b. Positively sloped line

a. Goes down as we move farther to the right b. Goes up as we move farther to the right.

boxplot

a. Graphical form of a set of numbers that includes minimum, Q1, median, Q3, and max b. 50% of the population is inside the box of the boxplot (between Q1 and Q3). This is the middle 50% of the data.

Binomial Question: djf Three fair coins are flipped. What is the probability of getting exactly two heads?

a. H = 1/2; H = 1/2; T = 1/2. Thus, 1/8 b. However, there are multiple ways this could play out: HHT; HTH; or THH. all of these have a 1/8 chance of happening, so we add them together to get 3/8 (answer)

Divisibility rule for 5

a. If the last digit is a five or a 0, then the number is divisible by five.

Divisibility rule for 2

a. If the one's digit is event, the number is divisible by 2.

Mark is 8 years older than Lisa, and Peter's age is 3 years less than three times Lisa's age. If the sum of their ages is 80, what is Peter's age?

a. M = L + 8 b. P = 3L - 3 c. L + M + P = 80 d. L + (L + 8) + (3L - 3) = 80 e. 5L + 5 = 80 f. 5L = 75 g. L = 15 (plug back into B for peter's age) h. P = 3(15) - 3 i. P= 42

In a certain company, the ratio of programmer to marketers is 3:8, and the ratio of customer service reps to marketers is 2:3. If there are 27 programmers, there are how many CSRs? (strategy 2 for combining ratios)

a. P/M= 3/8=27/m b. 1/8 = 9/m c. m = 72 d. C/M= 2/3= C/72 e. 2/1= C/24 f. C= 48 (answer)

Problem: The price of an item decreases from $250 to 200. What was the percent decrease?

a. P=200/250= 20/25= 80/100= 0.8 b. 0.8 is the multiplier for a 20% decrease (answer). Number Sense: a. the price drops by 50, and 50 is 1/5 of 250. That's a drop of 1/5, or 20%.

Proportional reasoning procedures

a. Pick super easy numbers that satisfy the equation for a start. You can also change constants to 1 b. change whatever values need to be changed, leave the quantity in the question as an unknown, and solve for it.

Work problem categories

a. Proportions b. multiple machines/multiple worker to see how much they get done together.

QA: 449/150 QB: 20/7 (number sense)

a. They are both almost three b. (450/150) - (1/150) vs. (21/7) - (1/7) c. If you subtract something smaller you get something bigger d. 1/150 < 1/7 e. A is bigger

What is 50% of 128?

a. n= (1/2)128 b. n= 64 (answer)

Tangent line

a. passes by a circle and touches it at only one point b. A radius to the point of tangency is a perpendicular line.

Even and Odd integers (special note)

a. zero is even b. evens and odds are both positive and negative c. evens and odds MUST be integers d. the prime factorization of an even number always contains 2. Expresses as 2k e. no odd number is divisible by 2. expressed as (2k+1) or (2k-1). factorization of odd numbers will never contain a factor of 2.

Express the region -3 ≤ x ≤ 11 as an absolute value inequality

a. |x - 4| ≤ 7 answer find the average of the end points to get the answer.

Simplify the following: √(4/50

a. √(2/25) b. √2/√25 c. √2/5

1140/5

1. 1140/10 = 114 2. 114*2 = 228 (answer)

consecutive numbers (basic facts)

1. A set of n consecutive integers will always contain one number divisible by n. 2. if n is odd, then the sum of a set of n consecutive integers will always be divisible by n. (doesn't work if n is even) 3. In a set of 4 consecutive integers you must have two evens and two odds. in a set of three, there will be two evens and 1 odd or vice versa.

Motion questions

Distance Rate: rate or speed Time b. D = RT c. Units fate determines units for distance and time.

Suggested game for developing number sense

S1: Pick four single digit numbers at random (dice, deck of cards, number generator, etc.) S2: Use the four numbers in any combination to get (addition, subtraction, multiplication, division, exponents, etc.) to produce all numbers between 1 and 20.

Strategies for solving two equation, two unknowns

S1: Substitution use this method one at least one of the variables has a coefficient of 1 or -1 S2: Elimination (linear combination) Use this. Use this when no coefficients are equal to +/-1. Eliminate one variable completely from the equations by adding them together.

What is 37% of 700 (Number sense method)

a. 10% of 700= 70 b. 1% of 700= 7 c. Multiply a by 3 and b by 7 d. 30% of 700= 70x3= 210 e. 7% of 700 = 7x7= 49 f. 37% of 700= 210+49= 259(answer)

Find the prime factorization of 9975 (D of S)

a. 10,000-25 b. 100^2-5^2 c. (100-5)(100+5) d. 95x105 e. 19x5x5x3x7

Problem: The price of an item increases from $60 to $102. What was the percent increase?

a. 102= Px60 b. P= 102/60= 17/10= 1.7= 70% increase. Note: The only thing the matters is the number after the decimal. That is the increase. Number Sense: a. 102-60= 42 b. 10% of 60= 6 c. 70% of 60= 42 Therefore, 102 is 70% greater than 60.

Problem: After a 30% increase, the price of something is $78. What was the original price?

a. 78=1.3 x F b. F= 78/1.3= 780/13= $60

Simplify: dkfj a. 7x(x+2x) b. 5x(x^2+6x+12)

a. 7x^2+ 14x^2 b. 5x^3+30x^2+60x

Solve the following (function notation) f(x) =x^2 + 4(x) -21 a. F(0) b. f(3) c. f(-1)

a. = -21 b. = 0 c. = -24

In probability, counting techniques should be used IF

the problem involves selection of several elements from a set, with certain restrictions (e.g. one element is picked, one is not picked, one is next to another, etc.)

Parallel line slopes

these slopes are equal.

Perpendicular line slopes

these slopes have opposite sign (+/-) and they are reciprocals.

Supplementary angles

two angles that add up to 180º

What is 80% of 200? (Number sense method) Watch number sense and percents again to crystalize

1. 10% is 20 2. We want eight of these 3. 8x20= 160

Work Problem formula

1. A = RT 2. A = amount of work done 3. R = Work rate 4. T = time.

Growth and decay questions

1. Very rare 2. Take calculations at 1 change interval at a time.

Simplify: (x^3- 3(x^2)+ 3x) + (x^3+ 3(x^2)-3x)

2(X^3)

Powers of 2 up to 2^9

2=1 2^2= 4 2^3= 8 2^4= 16 2^5= 32 2^6= 64 2^7= 128 2^8= 256 2^9= 512

What is the prime factorization of 96

96= 2x48= 2x6x8= 2x2x3x2x2x2= 2^5x3

FOIL method

F= First (P+q)(R+s) O= Outer (P+q)(r+S) I= Inner (p+Q)(R+s) L= Last (p+Q)(r+S)

Formula for the probability of 'not A'

P(not A) = 1 - P(A)

How many odd factors does 21,600 have?

S1: 2^5x3^3x5^2 S2: 3,2 S3: 4,3 S4: 12 odd factors

Absolute value

The absolute value of a number gives the distance of the number from an origin (written as positive). 0 is the exception. So, a. |x| = the distance of x from the origin. b. |x-5| = the distance of x from +5 c. |x+3| = the distance of x from -3

Vertical angles.

When two lines cross, four angels are formed. The pairs of angles opposite each other, sharing only the vertex in common, are called vertical angles. These angles are ALWAYS congruent. Note: We only need to know on angle measurement to figure out the rest.

Mutually exclusive events

When two or more events are not able to occur at the same time

a. Horizontal line equation b. x-axis equation c. vertical line equation c. y-axis equation

a. y = K b. y = 0 c. x = K d. x = 0

Simplify the following: (54√35)/(18√5)

a. (54/18)(√35/√5) b. = 3√7

Simplify: Y^3+8(y^2)-5(y^2)+5y

Y^3+3(y^2)-3y

260*15

130x30 = 13x3x10x10 = 3900

Rectangle area

A = bh

Multiplying variables with powers

a. Simply add the powers: (x^a)(x^b)= X^(a+b)

Solve v^2 = 2ad for v

a. v = √(2ad)

Simplify: 15xy/3

answer: 5xy

circular sector

slice of a circle.

If P and Q are integers, and (4P+Q) is odd, what must be true?

4P must be even, so the only way the sum will be odd is if Q is odd. Q must be odd. No conclusion can be drawn about P.

Complement

A complement of an event is the absence of that particular event (e.g. the complement of A = 'not A'.

Polygon

Any closed figure with all line-segment sides. Includes everything we've discussed in the cards up to this card and more. Note: The sides cannot cross. The sides have to be straight.

A car and a truck are moving in the same direction on the same highway. The truck is moving at 50 mph, and the car is traveling at a constant speed. at 3:00 pm, the car is 30 miles behind the truck and at 4:30 pm, the care overtakes and passes the truck. What is the speed of the car?

Method 1: a. 30 = R - 50(1.5) b. 30 = 1.5R - 75 c. 105 = 1.5R d. R = 105/1.5= 1050/15 = 70 mph (answer) Method 2 a. R = 30/1.5 = 300/15 = 20 b. add twenty to the slower speed to get the answer: 50 + 20 = 70 mph (answer)

How many factors does 21,600 have?

S1: 2^5x5^2x3^3 S2: 5,2,3 S3: 6, 3, 4 S4: 6x3x4= 72 (answer)

Counting problems with restrictions

S1: Always start with the most restrictive stage, then the next most, etc.

Finding number of even factors

S1: Calculate the total number of factors S2: Calculate the total number of odd factors S3: Subtract the result of S2 from S1

Car X and Y are traveling from A to B on the same route at constant speeds. Car X is initially behind Car Y, but Car X's speed is 1.25 times Car Y's speed. Car X passes Car Y at 1:30 pm. At 3:15 pm, Car X reaches B, and at that moment, Car Y is still 35 miles away from B. What is the speed of Car X

a. 35 = R(7/4) b. R = (4/7)35 = 20 c. X = 1.25y d. x-y= 20 e. 1.25y -y = 20 f. 0.25y = 20 g. y = 80 h 80 + 20 = 100 (X is going 100 mph-- answer)

Inequality practice: 5x+7< 2x-2

a. 3x+7 < -2 b. 3x < -9 c. x < -3

On a certain test, the score had a mean = 300 and standard deviation = 25. If John scored three standard deviations above the mean, what was John's score?

a. 3x25 = 75 b. 300 + 75 = 375 (answer).

Absolute value inequalities practice: Express |x-7| ≤ 3 as an ordinary inequality.

a. 4 ≤ x ≤ 10

Backsolving practice: A chemical supply company has 60 liters of a 40% H solution. How many liters of pure undiluted H must the chemists add so that the resultant solution is a 50% solution. A 12 B 15 C 20 D 24 E 30

a. 40% of 60 is 24, so the current ratio 24/30 b. Starting with answer choice C we get a 44/80. This is more than half, so we have added too much solution (C, D, and E will not work). c. Jumping to answer choice A, we get a 36/72 ratio. This is 50%, so A is the correct answer.

What is 55% of 400? (Number sense method)

a. 50% of 400= 200 b. 5% of 400= 20 c. 55% of 400= 200 + 20 = 220 (answer)

52 is 40% of what number?

a. 52= .4n b. 52/.4= n c. 520/4= n d. n= 130 (answer)

56 is what percent of 800 (method 2)

a. 56=P800 b. 56/800=x c. x= 7/100= .07= d. 7% (answer)

a. 59/100 b. 17/1000 Fractions to percents

a. 59/100= 0.59= 59% b. 17/1000= 0.017= 1.7%

Factoring GCF from a binomial examples a. 5x+45 b. 9x^3 + 12x c. (x^7)(y^4) + (x^5)(y^6)

a. 5x+45 = 5(x+9) b. 9x^3 + 12x= 3x(3x^2 + 4) c. (x^7)(y^4) + (x^5)(y^6)= (x^5)(y^5)(x^2 + y^2)

Solve:kj 5x^2 - 10x-27= 13

a. 5x^2 - 10x- 40 = 0 b. 5(x^2-2x-8)= 0 c. 5(x-4)(x+2)= 0 d. divide by five: (x-4)(x+2)= 0 e. x=4 OR x=-2 (answer)

Problem: The price of stock increased 20% in January, dropped 50% in February, and increased 40% in March. Find the percent change for the three-month period.

a. (1.2)(0.5)(1.4) b. (1.2)(.7) c. 0.84 d. 0.84 is the multiplier for a 16% decrease.

In a certain company, 50% of Ed's salary is 20% of Ruth's salary. If the difference between their salaries is 90,000, what is Ruth's salary?

a. (1/2)E = (1/5)R b. R-E = 90,000 c. E = (2/5)R d. R - (2/5)R = (3/5)R = 90,000 e. (1/5)R = 30,000 f. R = $150,000 (answer)

Three cards are selected from a full deck, each time with replacement. What is the probability of selecting three Spades in a row?

a. (1/4)(1/4)(1/4) = 1/64 (answer)

What is the probability of rolling two sic-sided dice and getting 'snake-eyes'

a. (1/6)(1/6) = 1/36 (answer)

Simplifying complex algebra with substitutions practice: Solve for x: (2x-1)^2 + 5(2x-1) = 24

a. (2x-1)^2 + 5(2x-1) -24 = 0 b. u^2 + 5u -24 = 0 c. (U + 8)(U -3) = 0 d. U = -8 or U = 3 e. -8 = 2x - 1 or 3 = 2x -1 f. -7 = 2x or 4 = 2x g. x = -7/2 or x = 2 (answer)

Simplify: (2x+y)(x+2y)

a. (2x^2)+4xy+ xy+ (2y^2) b. (2x^2)+5xy+(2y^2) --answer

If (4p+3Q-4R)/(P-R)= 19, the Q/(P-R)=?

a. (3Q+4P-4R)/(p-R) b. (3Q/(p-R) + (4P-4R)/(P-R) c....+ 4(P-R)/P-R d....+4=19 e...=15 f. 3Q/(P-R)=15 g. Q/(P-R)= 5 (answer)

If t(he fifth √3)^(3x+7) = 3^2x, find x.

a. (3^(1/5))^(3x+7) = 3^(2x) b. 3^[(3x+7)/5] = 3^2x c. (3x+7)/5 = 2x d. 3x+7 = 10x e. 7x = 7 f. x = 1

Practice question: If 49^x = 7^(6-x), then solve for x.

a. (7^2)^x = 7^(6-x) b. 7^(2x) = 7^(6-x) c. 2x = 6-x b. 3x = 6 c. x = 2 (answer)

Difference of two square factor example(s) a. (9x^2)- 16 b. (25x^2) - (64y^2) c. (x^2)(y^2) - 1

a. (9x^2)- 16= (3x +4)(3x - 4) b. = (5x - 8y)(5x + 8y) c. = (xy + 1)(xy - 1)

point position notation

a. (x, y) b. Thus (5, 4) = 5 over and 4 up on the coordinate plane.

Given f(x)= x^2- 2x - 1, find f(x^2+3)

a. (x^2 +3)^2 -2(x^2+3) -1 b. ... c. x^4 +4x^2 +2 (answer)

Simplify: (x^3- 3(x^2)+ 3x) - (x^3+ 3(x^2)-3x)

a. (x^3- 3(x^2)+ 3x) - x^3- 3(x^2)+ 3x b. -6(x^2) + 6x (answer)

Inequality practice: -4 < 5 - 3x ≤ 17

a. -9 < -3x ≤ 12 b. 3 > x ≥ -4 Note: remember that what you do to one of the three expressions, you should do to all of them. When you subtract 5 from the middle expression, subtract five from the two other expressions as well.

Suppose, in Game M, the probability of outcome A is 0.6, the probability of outcome B is 0.7, and the probability of A or B is 0.9. What is the probability of A and B happening at the same time?

a. 0.9 = 0.6 + 0.7 - P(A and B) b. P(A and B) = 0.6 + 0.7 - 0.9 = 0.4 (answer).

Practice problem: For positive numbers p and q, let p @ q = p + (1/q). What does (1 @ 2) @ 3 equal?

a. 1 @ 2= 3/2 b. (3/2) @ 3 = 11/6 answer

Know the first 15 perfect squares. a. 11x11 b.12x12 c. 13x13 d. 14x14 e. 15x15

a. 11x11= 121 b. 12x12= 144 c. 13x13= 169 d. 14x14= 196 e. 15x15= 225

Simplify: a. (3x)(4x^2) b. (7x(^2)y(^2))(6x(y^3) c. (2xy)(3xz)(4yz)

a. 12x^3 b. (42x^3)(y^5) c. 24(X^2)(y^2)(z^2)

Problem: After an item was discounted 80%, the new price is $150. What was the original price?

a. 150= 0.2F b. F= 150/0.2= 1500/2= $750 (answer) Number sense: If 80% is gone, only 20% is left. 20% is 150. Therefore, 10%= 75. 10x75= 750.

Find the prime factorization of 1599

a. 1599= 1600-1 b. 1600-1= 40^2 - 1^2 c. 40^2 - 1^2= (40+1)(40-1) d. (40+1)(40-1)= 41x39 e. 41x39= 41x3x13

Proportional reasoning practice In v^2 = 2ad, if v triples and a doubles, then d is multiplied by what

a. 1^2 = (1)(1)(1) b. 3^2 = (1)(2)d c. 9 = 2d d. d = 9/2 or 4.5. Thus d is multiplied by 9/2 or 4.5 when v triples and a doubles.

Proportional reasoning practice Formula: V^2 = 2ad if v doubles and a stays the same, then d is multiplied by what.

a. 1^2 = (1)(1)d b. 2^2 = (1)(1)d c. 4 = d. Thus, d is multiplied by 4 if v doubles and a remains the same.

Intro to exponents

a. 1^n is always 1 regardless of what n is. b. 0 to any positive exponent is 0 (0^n = 0 if n >0) c. A negative to any even power is positive. A negative to any odd power is negative. d. an equation of the form something squared equals a negative has no solution.

Inequality practice: 2/x ≥ 1/3

a. 2 ≥ x/3 b. 6 ≥ x c. 0 < X ≤ 6 Note we know that x can't be 0 because 0 doesn't work in the denominator. It also can't be a negative number because 2/x is greater than the positive fraction 1/3.

a. Prime numbers less than 20 b. Prime number between 20 and 60 Memorize.

a. 2, 3, 5, 7, 11, 13, 17, 19 b. 23, 29, 31, 37, 41, 43, 47, 53, 59

240 is 30% of what number (method 1)

a. 240=0.3x b. 240/0.3=x c. x= 2400/3 d. x= 800 (answer)

When a car, initially moving at a speed of v decelerates with constant acceleration a, its stopping distance d is related to v & a by the formula: v^2 = 2ad. If v = 30, and a = 10, find stopping distance

a. 30^2 = 2(10)d b. 900 = 2(10)d c. 90 = 2d d. 45 = d (answer).

69^2 (adjacent squares example 2)

a. 69^2 = 70^2 - 70 - (70 -1) b. 4900 - 70 - 69 c. 4830 - 69 d. 4761 (answer)

What is the GCF of 720 and 1200?

a. 720= 2^4x3^2x5 b. 1200= 2^4x3x5^2 c. GCF= 2^4x3x5= 240 (answer)

a. 8/33 b. 11/14 fractions to percent (estimation)

a. 8/33= 24/99> 24/100= 24%+ b. 11/14= 77/98 > 77/100= 77%+

The price of an item increases from $200 to $800. What was the percent increase?

a. 800= Px200 b. P= 800/200= 4; 4-1= 3= 300% increase. Note: Remember when you are looking for the percent increase, always subtract 1 from the initial result.

How many multiples of 8 are there from 200 to 640 inclusive.

a. 8x25 = 200 and 8x80 is 640 b. 200 is the 25th multiple of 8, 640 is the 80th multiple of 8, and both are included. c. number = 80 - 25 + 1 = 56 (answer)

Prime factorization examples

a. 9 = 3x3 b. 10 = 2x5 c. 12 = 2x2x3 d. 15= 2x2x2x3= 2^3x3 e. 100= 2^2x5^2

Under optimal conditions, the V. Bacteria multiplies the size of its population by 5/2 every 4 hours. If there are 24 billion at 9:00 am., and optimal conditions are maintained, how many are there are 5:00 pm of the same day?

a. 9:00 am: 24 b b. 1:00 pm: 24(5/2)= 12(5)= 60 c. 5:00 pm: 60(5/2)= 30(5)= 150 b (answer)

Simplify the following: 2^(3/5)

a. = (2^3)^(1/5 b. = 5√(2^8) c = the fifth root √8

If 27^(2x-2) = 81^(x+1), find the value of x.

a. = (3^3)^(2x -2) = (3^4)^(x+1) b. 3(2x - 2) = 4(x +1) c. 6x - 6 = 4x + 4 d. 2x - 6 = 4 e. 2x = 10 f. x = 5 (answer)

Solve the following problem: 8^4/3

a. = (the cubed root of √8)^4 b. = 2^4 c. = 16 (answer)

Factor the following: a. x^2 + 14x + 24 b. x^2 + 4x - 21 c. x^2 - 2x - 35 d. x^2 -16x + 48

a. = (x+2)(x+12) b. = (x + 7)(x - 3) c. = (x -7)(x+5) d. = (x-12)(x-4); postive product & neg. sum

Simplify the following: 0.999951/0.993

a. = 1-0.000049/1-0.007 b. =1^2- 0.007^2/ 1-0.007 c. (1+0.007)(1-0.007)/1-0.007 d. 1.007 (answer)

Know the rough approximations for the following square roots: a. √2 b. √3 c. √5

a. = 1.4 b. = 1.7 c. = 2.2

Practice with D = RT a. What is the speed of someone who covers 240 miles in 6 hrs at a constant speed? b. How far does someone moving at 8 m/s move in 40 seconds? c. How much time does it take to move 300 feet at a speed of 20 ft/s?

a. = 40 mph b. = 320 meters c. = 15 seconds

a. Is 1296 divisible by 6? b. Is 267,914,296 divisible by 6?

a. Yes. It is divisible by 2 and 3 b. No. It is not divisible by 3

Simplify the following: (y^2+2x-8)/(x-4)

a. [((y^2)/((x-4))-((2x-8)/x-4)} b....- ((2(x-4))/(x-4) c.[((y^2)/((x-4))- 2}

a. Rectangular solid definition b. volume of a rectangular solid c. surface area of a rectangular solid

a. a cube is a rectangular sold + any other box shaped thing. b. V = (h)(w)(d) c. A = 2hw + 2hd + 2wd

Square roots things to know

a. a square root of anything has two answers: negative or positive b. If the question is written with the symbol of the square root and there is not negative sign, assume that you are only working with positive square roots. c. If you, yourself have to include the square root symbol as part of the simplification process, then consider both positive and negative roots. d. Can't take the square root of a negative. e. Test expect you to find square roots of perfect squares. It does not expect you to find square roots of imperfect squares. It does require you to know some approximations. f. √y^2 will always be positive, or the absolute value of y |y|. g. If 0 < B < 1, then positive powers of b get smaller. In particular, if 0<b<1, then b > b^2. Also, in this case √b > b

Degree Facts

a. a straight angle has 180º b. there are 90º in a right angle. c. perpendicular lines meet at right angles. (never assume that two lines are perpendicular unless explicitly told so. d. If tow or more angles lie on a straight line, the sum of their angles is 180º (These are supplementary angles)

For a sequence that has the same rule as the Fibonacci sequence, a sub n = a (sub n -1) + a (sub n -2) for n > 2, but the starting values of a sub 1 = 1 and a sub 2 = 3, find the value of a sub 6

a. a sub 1 = 1 and a sub 2 = 3 b. a sub 3 = 3 + 1 = 4 c. a sub 4= 4 + 3= 7 d. a sub 5 = 7 + 4 = 11 e. a sub 6 = 11 + 7 = 18

14, 23, 32, 41, 50, 59 Find the 41st term of this sequence.

a. a sub 1 = 14; d = 9 b. a sub n = 14 + 9(n-1) c. a sub 41 = 14 + 9(41-1) d. a sub 41 = 14 + 9(40) = 14 + 360 = 374 (answer)

Let S be the set of all positive integers that, when divided by 8, have a remainder of 5. What is the 76th number in this set?

a. a sub 1 = 5; d = 8 b. a sub n = 5 + 8(n-1) c. a sub 76 = 5 + 8(76-1) d. a sub 76 = 5 + 8(75) e. a sub 76 = 5 + 600 = 605 (answer)

For the sequence a sub n = 1/(n+2), find a sub 10 - a sub 6

a. a sub 10 = 1/(10+2) = 1/12 b. a sub 6 = 1/(6+2) = 1/8 c. a sub 10 - a sub 6 = 1/12 - 1/8 = 2/24 - 8/24 = -1/24 (answer)

Recursive sequences

a. a sub n - 1 is the term immediately before a sub n. b. If a sub n is defined in terms of a sub n - 1, this is a recursive definition, and the sequence is recursive. c. You can't figure out random terms in the sequence. You have to work from the first seed number all the way up to the value we want. d. on the test we will get either a sub n -1 or a sub n -2. If a sub n -1, we want to use the term before the term sitting immediately prior to n (two terms back).

symmetrical trapezoid

a. a trapezoid with two equal legs. b. the diagonals also have an equal length.

Cars P & Q are approaching each other on the same highway. Car P is moving at 49 mph and Car Q is moving at 61 mph. At 2:00 pm, the are approaching each other and 121 mi apart. Eventually they pass each other. At what clock time are they moving away from each other and 44 miles apart.

a. add the gaps 121 + 44 = 165 b. add the speeds 61+49 = 110 c. formula: 165 = 110(T) d. T = 165/110 = 15/10 = 3/2 = 1.5 e. 2:00 + 1hr 30 min = 3:30 pm (answer)

How much HCl and how much water must we use to create 5 liters of 30% HCL solutions

a. amount of HCL= 0.3x5= 1.5 liters (answer 1) b. amount of water = 5 -1.5= 3.5 liters (answer 2)

Suppose we start with 5 liters of a 30% HCl solution. How much water must we add to create a 20% solution.

a. amount of HCl: 0.3x5= 1.5 liters b. New solutions: 0.2(x)= 1.5 liters c. x = 1.5/.2 = 15/2 = 7.5 liters d. 7.5 - 5 = 2.5 liters of added water (answer)

Factorial information

a. any factorial n! is divisible by all the integers less than n and all the factorials less than n!. b. n different items can be arranged in n! unique orders.

If b sub n = (b (sub n-1) -1)^2 +3, and b sub 1 = 1, find the value of b sub 4.

a. b sub 1 = 1 b. b sub 2 = (1-1) ^2 + 3= 3 c. b sub 3 = (3-1)^2 + 3 = 7 d. b sub 4 = (7-1)^2 + 3 = 39 (answer).

Fractional exponents

a. b^(1/2) = √b b. b^(1/m) = mth root of √b c. If the denominator of the exponent-fraction is odd, then the base can be negative as well.

Solve the following problem for a right triangle: If a = √7 and b = 2√5, c = ? Note: a and b are the legs.

a. c^2 = √7^2 + (2√5)^2 b. c^2 = 7 + 20 = 27 c. c = √27 = 3√3 (answer)

On a certain high school team, the ratio of sophomores to juniors is 2:3, and the ratio of juniors to seniors is 5:6. Sophomores are what fraction of the whole team. (strategy 1 combining ratios)

a. common terms among ratios are the juniors. b. So:J = 2/3 c. J:Se = 5/6 d. 10:15 & 15:18 e. 10:15:18 f. 10+15+18= 43 g. 10/43 (answer)

algebraic terms

a. constant: a number or symbol such as pi symbol that doesn't change in value. b. term: a product of constants and variables, including powers of variables. terms: 5, x, 6y^2, (x^5)(Y^6) c. coefficient: the constant factor of a term (6 is the coefficient of 6y^2). When no coefficient is written, the coefficient is 1. (See intro to algebra for more terms).

Of the 100 students in a school, 60 are in the band , and 35 are on the baseball team. If 25 students are neither in the band or on the baseball team, how many are in both?

a. create a venn diagram (A, B, C, D areas) b. A + B = 60 c. B+C= 35 d. D =25 e. A+B+C= 75 f. A + 35 = 75 (replacing B+C) g. A = 40 h. 40 + B = 60 i B = 20 students are in both (answer)

Graphing quadratic equations

a. creates a parabola (u-shaped curve). b. parabolas are always symmetrical through the vertex. c. Every point on the parabola must satisfy the equation of the parabola.

circles

a. diameter is two times the radius (d = 2r) b. circumference is π times the diameter (c = πd) or (c = 2πr) c. area is πr^2 (A = πr^2)

If x(2√5 - 3) = 55, find x

a. divide both sides by (2√5 - 3) to isolate x b. x = 55/(2√5 - 3) c. multiply the right side by (2√5 + 3)/(2√5 + 3) d. x = [55(2√5 + 3)]/((2√5)^2 - 9) e. x = [55(2√5 + 3)]/11 f. x = 5(2√5 + 3)

Regular Polygons

a. equilateral (all equal sides) and equiangular Note: regular quadrilateral is a square.

Rhombuses

a. equilateral quadrilaterals (4 equal sides). b. they are parallelograms c. the diagonals are perpendicular, forming right angles.

Multiplying evens and odds

a. even times even is even b. odd times odd is odd (different from addition). c. even times odd is even (different from addition) Note: if we are given an odd product, then we know that none of the variables were even.

Reflection properties

a. every point on a mirror line is equidistant from the original point and the reflected image. b. the reflected line is the perpendicular bisector.

Reflections over the y-axis

a. the original point and the reflected image have the same y-coordinate. the x-coordinate has equal absolute values and opposite signs.

Intercepts

a. the points at which the line crosses the x and y axes. b. horizontal lines have only a y intercept c. vertical lines only have an x-intercept d. Lines that pass through the origin have an x and y intercept of zero

Circle centered at the origin

a. the slope triangle of each radius has a horizontal leg of |x| and vertical leg of |y| b. equation for this circle is x^2 + y^2 = r^2

Quadrilateral facts

a. the sum of the 4 interior angles is 360º b. there are always exactly two diagonals.

Reflections over the line y = -x (a line going through the origin with a negative slope).

a. the x and y coordinates are switched and each is give the opposite sign.

negative correlation

as one variable increases, the other decreases

compounding periods smaller than a year

quarterly: n= 4 monthly: n= 12 daily: n= 365 Note: if there are n compounding periods in the year, we divide the annual percent of interest by n to get the percent for each individual compounding period.

12/5x = 8/15 solve for x

12/5x = 8/15; 3/x = 2/3; 9=2x; x=9/2

N = 135 is the lowest number in a set of 41 consecutive multiples of 5. What is the difference between the lowest and highest numbers in the set. A. 165 B. 200 C. 205 D. 335 E. 340

a. 1st = 135 b. 2nd = 135 + 5 c. 3rd= 135x2 d. So, nth = 135 + 5(n-1) e. So, 41st= 135 + 5x40 f. 335-135= 200 answer. Letter B

a. 20% b. 92% c. 0.02% Percents to fractions

a. 20%= 20/100= 1/5 b. 92%= 92/100= 23/25 c. 0.02%= 0.02/100= 2/10000= 1/5000

If (x^2+12x-45)/x+15 = 22, then x=?

a. 22 =((x+15)(x-3))/x+15 b. 22 =x-3 c. x=25 (answer)

a. 24/10 b. 0.02/10 c. 39.85 x 0.1 d. .00072 x 0.1

a. 24/10 = 2.4 b. 0.02/10 = 0.002 c. 39.85 x 0.1 = 3.985 d. .00072 x 0.1 = 0.000072

a. 24x10 b. 2.53x10 c. 6400x10 d. 0.00045x10

a. 24x10 = 240 b. 2.53x10 = 25.3 c. 6400x10 = 64,000 d. 0.00045x10 = 0.0045

Find the prime factorization of 2491 (diff. of squares)

a. 2500-9 b. 50^2 - 3^2 c. (50+3)(50-3) d. 53x47

The area of a circle is 28π. What is the circumference

a. 28π = πr^2 b. 28 = r^2 c. r = √28 = 2√7 d. c = (2)(2√7)π e. c = 4π√7 (answer)

Given the function: f(x)= x^2 + kx +4 find the value of k if f(2) =18

a. 2^2 + 2k + 4 =18 b. 4 +2k +4= 18 c. 2k+8= 18 d. 2k = 10 e. k = 5 (answer).

Simplify the following: (2√3)^4

a. 2^4(√3^4) b. 16(√3x√3x√3x√3) c. 16(3)(3) d. 16(9) e. 144

Absolute Values Solve: |2x+3|= 5

a. 2x+3=5 b. x= 1 OR c. 2x+3= -5 d. 2x=-8 e. x= -4 f. x= 1 or -4 (answer)

Factoring GCF from a trinomial examples a. 2x^4 + 2x^3 + 10x^2

a. 2x^4 + 2x^3 + 10x^2 = 2x^2(x^2 + x + 5)

Approximations of π

a. 3 (roughest) b. 3.14 or 22/7

Pythagorean triplets Note: if we have any of these triplets, we can be assured the triangle is a right triangle even without the right angle symbol.

a. 3, 4, 5 b. 5, 12, 13 c. 8, 15, 17 d. 7, 24, 25 Note: multiplying all members of the triplet by any one number will also yield a right triangle (e.g. 3, 4, 5 multiplied by 2 is 6, 8, 10--another right triangle).

a. 3/8 b. 2/3 Changing fractions to percents

a. 3/8= .375 = 37.5% b. 2/3= 0.6667= 66.67%

complex algebra practice (substitutions): Solve for k: 3/(1 - (8/(7+K))) = 15

a. 3/A = 15 b. 3 = 15A c. A = 1/5 d. 1/5 = 1 - B e. B = 1 - 1/5 f. B = 4/5 g. 4/5 = 8/(7+K) h. 1/5 = 2/(7+K) I 7+K = 10 J K = 3

Simply the following: (3√5)(2√15)

a. = 6√(5x15) b. = 6√(5x5x3) c. = 6(5)(√3) d = 30√3 (answer)

Simplify the following: (2√42)(4√63)

a. = 8√(42x63) b. = 8√(2x3x7x3x3x7) c. = 8√(3x3) x √7x7 x √2x3 d. = (8)(3)(7)(√6) e. = 168√6 (answer.

Solve the following problem: √72 - √32

a. = √(36x2) - √(16x2) b. = 6√2 - 4√2 c. =2√2 (answer) Note: We can combine two terms only if they have the same radical factor.

Solve the following problem: 2 + √(4-3x) = x

a. = √(4-3x) = x - 2 b. = 4-3x = (x-2)^2 c. = 4 - 3x = x^2 - 4x + 4 d. = 0 = x^2 - x e. 0 = (x)(x-1) f. x = 0 or x = 1 g. Plugging both answers back into the equation neither of them work, so there is NO SOLUTION

Factor: dkfj a. 2x^2 - 22x + 48 b. 7x^4 - 56x^3 - 63x (high intermediate) c. -3x^9 + 48xy^4 (advanced: nothing much harder than this)

a. =2(x-8)(x-3) b. = 7x^2(x-9)(x+1) c. = 3x(4y^2 + X^4)(2y+x^2)(2y-x^2)

Alice and Bruce each bought a refrigerator, and the sum of their purchases was 900. If twice of what Alice paid was 75 more than what Bruce paid what did Alice pay for her refrigerator?

a. A + B = 900 b. 2A = B+75 c. B= 2A-75 d. A + 2A -75= 900 e. 3A = 900 + 75 f. A = 300 + 25 g. A = $325 (answer)

Area of a trapezoid

a. A = (average of bases)h d. or divide into a central rectangle and two right triangles. Find the area of all three. (Basically expect to use the pythagorean theorem on anything with a slant). Note if the trapezoid is symmetrical (don't assume unless stated), we really only need to find the area of one triangle and multiply by 2.

Measures of center

a. measures that best represent a set of numbers as a whole: mean, median, and mode.

Problem: A $170 item is discounted 30%. What is the new price?

a. multiplier= 1-0.3= 0.7 b. n= 170 x 0.7= $119 (answer) Number sense: 10% = 17, so 30% is 3x17= 51. the price goes down by $51= 50+1. 170 down by 50 is 120. 120 down by 1 is 119

Rationalize the following: (4 + 2√5)/(3 + √5)

a. multiply by (3 - √5)/(3 - √5) b. [(4 + 2√5)(3 - √5)]/4 c. [2(2 + √5)(3 - √5)]/4 d. [(2 + √5)(3 - √5)]/2 e. (6 -2√5 + 3√5 - 5)/2 f. (1 + √5)/2 (answer)

Rationalize the following: 8/(√7 -√3)

a. multiply by (√7+√3)/(√7+√3) b. [8(√7 + √3)]/ (7 -3) c. [8(√7 + √3)]/4 d. 2(√7 + √3) e. 2√7 + 2√3 (answer).

Solve for x and y using the elimination method: 2x+3y=15 x+2y= 11

a. multiply equation 2 by -2 and add both together to find y b. -2x-4y=11 + 2x+3y=15 c. -y=-7 d. y=7 e. pluggin 7 back into one of the equation, we get x

Solve for x: kdfj [(x/2)+(5/4)]/[(x/3) + (3/2)]

a. multiply numerator and denominator by LCM of all the smaller fractions denominators:12. b. (6x+15)/(4x+18) (answer)

a. Finding distances between vertical lines b. Finding distances between horizontal lines

a. subtract y coordinates b. subtract x coordinates.

Median

a. the middle number on an ordered list. b. on an even numbered list, it is the average of the two numbers.

Reflections over the x-axis

a. the original point and the reflected image have the same x-coordinate. the y-coordinate has equal absolute values and opposite signs.

a. -6 x -7 b. -65/5 c. -30/-12

Answers: a. 42 b. -13 c. 5/2

Answer the following Is (√2)^48 > 2^20?

a. √2^48= 2^24 b. Yes (answer)

To detail a car means to clean it inside and out. When Amelia and Brad detail a car together, 1 car takes 3 hours. How long does it take Brad, working alone, to detail a car?

1. A + B = 1/3 2. A= 1/4 3. 1/4 + B = 1/3 4. B = 1/3 - 1/4 = 4/12 - 3/12 = 1/12 (Brad is one car every 12 hours).

Percent Decrease phrasing

1. Y decreased by P% Or 2. X is P% less than Y

Percent Increase phrasing

1. Y increased by 30% or 2. X is 30% greater than Y (2 ways to say the same thing). Note: in this case the multiplier for a 30% increase is 1 +0.3 = 1.3

Law of exponents 11

1. You can distribute exponents over multiplication: (ab)^n = a^n x b^n 2. You CANNOT distribute exponents over addition and subtraction. 3. We can simplify the sum or difference of powers by factoring out the lower power. 4. If a^m = a^n, then m = n

Taking from pool with groups of identical items

1. You just have to count it out. No formula will work in this case.

Algebraic simplification rule

1.) We can combine like terms (same variable part) by addition or subtraction. e.g. 15y-8y= 7y 2.) We can add or subtract only like terms, not terms with different variables or powers 3.) Multiplication is commutative (ab = ba) 4.) When an addition sign appears in front of the parentheses, we can simply remove the parentheses 5.) When a subtraction sign appears in front of the parentheses, we must change every sign in the parentheses.

Divisibility rule for 12

Must be divisible by 4 and 3.

What is the units digit of 57^123

Note: I only need to worry about the last digit in the base. a. 7^1: 7 b. 7^2: ...9 c. 7^3:...3 d. 7^4:...1 e. 7^5: 7 f. 7^6: ...9 g. 7^7:...3 h. 7^8:...1 I. 7^120:...1 j. 7^121:...7 k. 7^122:...9 l. 7^123:...3 (answer)

a. 1/9 b. 2/9 express as decimals

a. 1/9= 0.1111.... b. 2/9= 0.2222 Note: trend for every fraction with 9

Percent multiplier (Percent increases and decreases)

1. The decimal form of P percent is the multiplier for finding P percent of something. 2. BUT we use different multipliers for a P% increase or P% decrease.

Solve (1/p) + (1/q) = 1/f for q.

a. 1/q = (1/f) - (1/p) b. 1/q = (p/pf) - (f/pf) = (p-f)/pf c. q = pf/(p-f) (answer)

a. P = 0 b. P = 1

a. impossible b. certain. success will always happen

For all the positive integers N such that 80 is less than/equal to N and N is less than/equal to 90, how many numbers are prime.

2 numbers are prime (83, 89) because they are not divisible by any prime numbers less than 10).

What is the LCM (LCD) of 8 and 12?

24 Note: You can make a list to find this, but there is a quicker way to do it. The product of the tow numbers is rarely the LCM.

the prime factorization of 4680 is 2^3x3^2x5x13. Which of the numbers below are or aren't factors of 4680. 25 45 65 85 120 180

25 = 5x5 no 45 = 3x3x5 yes 65= 5x13 yes 85= 5x17 no 120= 2^3xx5 yes 180= 2^2x3^2x5 yes

What is the GCF (Greatest common factor = Greatest common divisor) of 24 and 40?

8 Note: usually takes too long to list, so follow a procedure. We use the GCF to find the LCM.

6/42 (factoring out example)

6/42 = (6x1)/(6x7) = 1/7 Note: canceling has been done here. It is a form of division. Note: on the text always write fractions in the simplest form. (The answer choice will almost always be in simplest form in MC)

Simplify: dffddf A. 6(x^2)- 8x + 10m - 2(x^2) + 3x - 4 B. 3(y^2) + 7xy + 2(x^2) + y^2- 7xy + 2(x^2)

A. 4(x^2)- 5x+ 6 B. 4(y^2)+ 4x2

Problem: Sheila invests $4000 in an account that yields 6% compounding annually for 8 years. What is the total amount after 8 years?

A= 4000(1.06^8)

Inclusive counting

Adding 1 to the answer when the start value and ending value are included in what we are counting.

Problem: Suppose P,Q,R, and S are integers. If P is even and (PxQ + RxS) is an odd integer, then which of the following must be true I. Q is odd II. R is odd III S is odd

Answer: II and III only. (See rules about multiplying and adding evens and odds).

Multiples

If P is a multiple of r, then r is a factor/divisor of P (75 and 1250 are multiples of 5; 5 is a factor/divisor of 75 and 1250). a. Idea 1: Every positive integer is a multiple of the number 1. Every positive integer is a multiple of itself. b. If we need the first five multiples of a number, we simply multiply the original number by 1,2,3,4, & 5. c. If we know P is a multiple of r, then (P-r) and (P+r) are also multiples of r. d. If P and Q are multiples of r, then (P+Q) and (P-Q) must also be multiples of r. e. If P is a multiple of r, then any multiple of P is a multiple or r. f. If P and Q are multiples of r, then the product of PxQ must also be a multiple of r

Calculating standard deviation (advanced)

S1: start with a list of numbers S2: calculate the mean S3: subtract the mean from every number to produce a second list, the list of deviations. S4: Square every deviation to produce a third list, a list of squared deviations. S5: Find the average of this third list, the average squared deviation; this is the VARIANCE. S6: Take the square root of the variance; this is the standard deviation.

If x < 1, and x ≠ 0, is x^7 > X^6? (exponential growth)

The answer is a consistent no.

Multiple workers/machines

The combined work rate of people machines/working together is the sum of the individual work rates

Multiplier for a P% decrease

The multiplier for a P% decrease is 1-(decimal form of P%)= N e.g. Multiplier for a 30% decrease is 1-0.3= 0.7

Divisibility rule for 6

The number must be divisible by 2 and 3.

Independent Events

The outcome of one event does not affect the outcome of the second event Note: The number rolled on one die, will not affect the other die. If you have info. about one event, you still have no info. for the other event.

Test problems that require ginding a diagonal

You typically must find some way of working in the pythagorean theorem to figure these out.

Strange operators strategy

You will be given a symbol that nobody has every used before. Just follow the rules give for that symbol to answer the question.

Isosceles right triangle, 45-45-90 triangle, 1-1-√2 triangle (three titles for the same thing).

a angles must be 45, 45, and 90 b two legs must have equal lengths. c. hypotenuse is larger than the legs, but less than their sum. d. all 45-45-90 triangles are similar. Thus, the sides all have the same proportion. e. the hypotenuse is √2 times leg (s) (sx√2= hypotenuse. f. dividing a square in half along the diagonal yields two of these triangles. Thus, the diagonal of a square it the side times √2

Stating intercepts

a as equations: x-intercept = 5 and y-intercept = -3 b. as points: (5, 0) and (0, -3)

Scatterplot

a graphed cluster of dots, each of which represents the values of two variables

Histogram

a graphical representation of the distribution of numerical data b. the horizontal axis is always the quantitative variable in question. c. The vertical axis is always frequency. That is, the number of people or occurrences.

Rank the following from smallest to biggest I: (1/3)^-8 II: 3^-3 III: (1/3)^5

a iii, ii, i

regression line

a line that describes how a response variable y changes as an explanatory variable x changes

Transversal

a non-parallel line that cuts through two parallel lines.

Prime numbers

a number that has only 2 factors, one and itself. Note: 1 is not a prime number because it has only 1 factor.

rational expression (definition)

a ratio, a fraction, of two algebraic expressions. (x-3)/(x+2)

Finding arclength

a. (arclength/2πr) = angle/360º

18 is what percent of 45?

a. 18= P45 b. 9/45=P c. 2/5= p d. .4=p e. p= 40% (answer)

a. cylinder volume b. cylinder surface area

a. V = (π)(r^2)(h) b. A = 2π(r^2) + 2πrh

If 5x+2y=55 and 2x-y= 19, then find the value of x+y.

answer is twelve. There is a shortcut if you recognize the there is a difference of three for both variables. multiply the second equation by negative one so that when you add them together, you get two coefficients of 3. Dividing by 3 on both sides eliminates the coefficient and give you the answer you need.

Absolute Values: |x|= 5

answer: x= 5 or -5

Rationalize the following: a. 1/√5 b. 2/√3 c. 12/√21

answers: a. √5/5 b. (2√3)/3 c. (2√21)/3

630/5 (Dividing by 5 example 1)

1. (630x2)/10 = 1260/10 =126 Or 2. (630/10)*2 = 63x2 = 126

235/5

1. 235*2 = 470 2. 470/10 = 47 (answer)

To say events A and B are mutually exclusive means...

1. A could happen alone 2. B could happen alone 3. neither A or B happens 4. A and B CANNOT both happen

What is a rate?

1. A rate is a ratio. Thus, we solve them like ratios, using proportions.

What is a ratio?

1. A ratio is a fraction that may compare part-to-whole or part-to-part 2. The test always gives you ratios in their simplest form. The actual numbers could be much higher.

Solve (infinitely many solutions): 2x-y= 5 2y-4x= -10

-10=-10 (answer has no variable, so there are infinitely many solutions. also, one equation is just a multiple of the other meaning the same thing. (only one line).

Simplify: 5xy + 7yx

12xy Note: order of variables does not matter because we are working with multiplication.

LCM tips

1). If A is a factor of R, then the LCM of A & R must be R (e.g. the LCM of 8 and 24 is 24). 2). If A and B have no factors in common greater than 1, then their LCM would have to be their product, AxB. (The LCM of 7 and 15 must be 7x15). 3.) Develop your number sense whenever possible (choose two random numbers to find LCM and GCF daily).

Working with percents

1. 'is' means equals 2. 'of' means multiply 3. change any percent to the multiplier form (decimal form) 4. replace unknowns with a variable

Translating words to math

1. 'is' means equals 2. 'of' means multiply Example: What is 3/5 of 400 = (3/5)x400

Translating word problems

1. 'is' or 'are' correspond to equals sign. 2. '50 more than B' corresponds to B+50 3. '50 less than B' corresponds to B-50

Arithmetic sequence

1. A sequence in which we add the same constant to get from each term to the next. (5, 12, 26...) the constant is 7. 2. any evenly spaced list is an arithmetic sequence. consecutive multiples of a number and consecutive odds and evens are all arithmetic sequences. Numbers which, when divided by the same divisor have the same remainder are also arithmetic sequences. the remainder is the a sub 1 term. 3. the nth term of an arithmetic sequence with an initial term a sub 1 and a common difference d is: a sub n = a sub 1 + (n-1)xd

Geometry strategies

1. Always draw diagrams to help you answer. Use your scrap paper. 2. It may be helpful to extend lines in diagrams, or add a line that would facilitate calculation. 3. Sometimes it is helpful to assign variables to either the lengths or the angles to use algebra to solve. 4. Look at the big picture even when there are a lot of small details.

Prime factorization important idea (the single most important skill to have for the integer properties section).

1. Any factor of Q must be composed only of prime factors found in Q. 2. If r is a factor of Q, every prime factor of r is included in the prime factorization of Q.

Facts about all triangles (card 2)

1. Area = 1/2 bh (b = base, h = height) 2. Any three side of the triangle works as the base. 3. height is just a line drawn from the base you choose to the opposite vertex. 4. In a right triangle, two of the altitudes are the legs, the sides that meet at the right angle. If one leg is the base, the other is the altitude.

Bar graphs

1. Bars may be vertical or horizontal (merely stylistic) 2. Important data comes from the length of the bar

Rationalizing

1. Basically no answer should have a radical in the denominator.. not kosher. 2. If there's a single radical in the denominator, we rationalize by multiplying by that radical over itself.

Assigning variables

1. Choose a meaningful variable. A letter that begins the name of a person, for example, is helpful. 2. It could be helpful to assign a variable to the smallest value 3. It could be helpful to assign a variable to the target value. 4. If all the variables are related to one quantity, choose that quantity as the variable.

Exponential equations

1. Equations that have variables as exponents. 2. If two powers with the same base are equal, then the exponents must be equal: If b^x = B^y, then x = y. 3. Before we can solve, for variables in the exponents, we must do what we can to make sure the bases are the same.

Infinitely Many solutions

1. If all variables go away in the process of elimination or substitutions and the numbers equal each other, then there is an infinite number of solutions. Note: Also, if one equation is a multiple of another, you have infinite solutions.

Rules for combining inequalities

1. If we have a = b and b = c, we can combine them to get a = c 2. If r < s and s < t, we can say that r < s < t, and so, r < t 3. We cannot draw any conclusion if the same term is greater than both other terms or less then both other terms (e.g. c < f and d < f) 4. We can add inequalities withe same direction. If a > b and c > d, then (a + c) > (b+d). 5. We cannot add inequalities that do not have the same direction. 6. We cannot subtract inequalities with the same direction. 7 We CAN subtract inequalities with the opposite directions. The resultant inequality follows the direction of the first inequality. 8. There are no rules for multiplication and division of inequalities. 9. Remember that any positive number is bigger than any negative number. 10. Adding a positive number always makes the number bigger (x< x + 2/5) 11. Subtracting a positive or adding a negative always makes a number smaller.

Age word problems (strategies)

1. It is usually easier to pick the variable to represent the age right now. (F = Frieda's age right now) 2. Use addition and subtraction to create expressions for ages at other times (e.g. F-5 = Frieda's age five years ago; F+7 = Frieda's age in 7 years.

Simple OR rule

1. Just add Note: If events A and B are mutually exclusive, then P(A or B) = P(A) + P(B)

Sequential Percent Changes Takeaway (mistakes)

1. Mistake 1: an increase and decrease by the same percent do NOT get us back to the original starting point. 2. Mistake 2: In a series of percent changes, Never add and subtract individual percents. 3. For a series of percent changes, multiply the individual multipliers together to get the initial result.

Law of exponents 1

1. Multiplying numbers with the same base: (A^m)(A^n)= A^(m+n) 2. Dividing numbers by the same base: (A^m)/(A^n)= A^(m-n) 3. If a ≠ 0, then a^0= 1 4. Raising a power to a power: (a^m)^n= a^(m x n) Note 1: the laws do not not work if the bases are different. Note: 2: There is no law for the sum or difference of powers. You need to solve these through factoring.

combining ratio strategies

1. S1: Find equivalent fractions for each ratio so the common terms come to be represented by the same number in both ratios. (e.g. 1:2 is the same as 7:14) 2. S2: when given absolute quantities in the prompt, solve one ratio completely for absolute quantities and work with those.

Simple vs. Compound interest

1. Simple interest: same dollar amount is added to the principal each year. 2.Compound: Same percentage is added to the total amount each year. (always outperforms simple interest) 3. Note: figuring out compound interest is nearly impossible to do without a calculator. The answer may be in the form of the final formula used to get the answer. 4. More compounding periods gives us more money.

Factoring quadratics things to remember

1. The primary method for factoring quadratics only works if the quadratic coefficient is one (x^2). If it is something other than one, use the other methods of factoring (Difference of two squares, GCF, etc.).

Sums of sequences

1. The sum of any evenly spaced list that has N items (N is the number of items on the list) has the following formula: [N(a sub 1 + a sub N)]/2

Facts about all triangles

1. The sume of the three angles is always 180º 2. acute angles = less than 90º 3. right angle = 90º exactly 4. obtuse angle = more than 90º 5. At least two angles have to be acute. 6. It is possible to have three acute angles in a triangle. 7. The largest angle is ALWAYS opposite the longest side. 8. The shortest angle is always opposite the shortest side.

Average speed problems (to solve)

1. There is a D = RT for the first leg, there is a D = RT for the second leg, and D = RT for the trip as a whole. 2. average velocity = total distance/total time. 3. you need to find the time for both legs. before you can figure out the average velocity.

Backsolving definition and procedure

1. Use when word problems are mc with numerical answers. Procedure: a. assume one of the answers is correct. b. If the answer is incorrect, we can usually tell if the answer should have been bigger or smaller. c. Always begin with answer choice C

Quadratic Equations Overview

1. Usually have two different solutions. But some have one and some have none 2. Process of solving is different from linear equations. 3. Factor a quadratic to a product of two linear binomials. Make sure things equal 0 3.1. x^2+bx+c= (x-p)(x-q)= 0

GRE Assumptions for diagrams

1. We are allowed to assume that lines that look straight are straight (that's about it). Be suspicious of everything even if your eyes betray you. What looks like a square may not be a square. 2 You CANNOT assume: a. two line lengths are the same, b. two lines are perpendicular or parallel, c. horizontal or vertical, d. right angles. 4. We can trust our deductions based on the rules of geometry. That is it.

240 is 30% of what number (Number sense method)

1. We can divide 240 by 3 to get 10% 2. 240/3= 80 is 10% 3. That's one-tenth of the whole 4. 8x10 = 800 (answer)

Solving ratios on the test

1. We frequently use proportions (fraction = fraction) 2. always use scale factor to solve, especially when sums and differences are involved.

Two travelers moving in the same direction (gaps)

1. We subtract the speed--bigger minus smaller Note: If faster traveler is in front, we are measure the expansion. I the slower traveler is in front we are measure the shrinkage. Note 2: set up a D = RT for the gap to save time.

FOIl Module things to remember

1. You can distribute multiplication across addition and subtraction. You CANNOT distribute exponents across addition and subtraction: (a+b)^2 IS NOT a^2 + b^2 2. So, (a+b)^2 is (a+b)(a+b). Use foil to solve. 3. The sum of a square always takes the following form (a + b)^2= a^2 + 2ab + b^2 4. The square of a difference always takes the following form: (a-b)^2= a^2 -2ab + b^2

Solve for x and y using the elimination method: 7x+3y= 5 2x-3y= 13

1. add equations together: 9x=18 2. x=2; plug two back in to either equation 3. 2(2) - 3y= 13 4. -3y= 9 5. y= -3 6. y=-3 and x=2 (answer)

negative exponents

1. b^-n = 1/(b^n) 2. A base to a negative power is the reciprocal of that same base to the positive power (p/q)^-n = (q/p)^n 3. A negative power in the numerator of a fraction can be moved to the denominator as a positive power, or likewise, from denominator to numerator: (b^5 d^-8)/(h^-4 k^7) = (b^5 h^4)/(d^8 k^7)

Mixture Questions

1. concentration = (amount solute/total amount of solution) x 100 2. Solution: water + solute 3. when amounts of two different solutions are unknown, we have to set up simultaneous equations (most popular question type) 3.1 One equation will always be a "total" equation (mass, volume, weight, etc.). the other equation will be about the amount of solute.

bisector

1. cuts something into two congruent pieces. 2. Divides a line segment into two equal halves

Segmented Bar Graph

1. each bar has two or more segments denoted by different colors or patterns. 2. Each segment breaks down the overall variable into relevant parts

Quadratic equation solving process

1. get everything on one side of the equation, set equal to zero 2. Divide by any GCF (if necessary) 3. Factor 4. Use the zero product property to create two linear equations, and solve.

Trapezoid

1. has only one pair of parallels sides 2. parallel sides are bases. the non parallel sides are legs. 3. the two angles on a leg are supplementary.

Advanced numerical factoring

1. if you can get to the number you want to factor by subtracting a perfect square from another perfect square, factoring becomes easier. 2. You can also use the difference of two squares to simplify decimal problems.

Sequence

1. ordered list of numbers 2 Infinite, following a pattern. 3. the sequence as a whole is represented by a letter and an individual letter by a numerical subscript (e.g. a sub 5 = 28) 4. an algebraic formula is used to represent the repeating pattern 5. a sub n = n is the sequence of all positive integers. 6. a sub n = 2n -1 is the sequence of all positive odd numbers. 7 a sub n = 2n is the sequence of all positive even numbers. 8. a sub n = 7n is the sequence of all positive multiples of 7 (this works for any factor times n) 9. a sub n = n^2 is the sequence of all perfect squares. 10 a sub n = 3^n is the sequence of all powers of 3.

Three equations with three unknowns steps

1. pick tow of the three equations and eliminate one variable. 2. pick another pair of the original equations and eliminate the same variable. 3. Use two-equations-with-twho unknown techniques to solve for the remaining variables. 4. Plug answer into any original equation find the value of the third variable.

Absolute value inequalities

1. remember |x - 3| is the distance of x from positive 3. 2. remember that |x + 3| is the distance of x from negative 3. 3. The middle of a region on the number line is the average of the two endpoints.

Formula question types

1. some questions will give you some numbers, and ask you to solve for others. 2. Some question will ask you to solve algebraically for a variable in the formula. 3. Proportional reasoning questions.

In probability, listing should be used ONLY IF

1. the full list is very short, fewer than 10 possibilities. 2. Note: Listing will almost never be the sole option you should use to solve a problem. It can be used to help you decide what other technique to use (e.g. counting, or algebra)

In probability, algebraic rules should generally be used IF...

1. the problem gives you algebraic expressions: P(A) = 0.5, etc. 2. the items concerned are coins, cards, dice, etc. 3. language used includes 'mutually exclusive', 'independent', or a description directly related to those terms.

Equations with two different variables

1. these equations without powers, can be represented by a straight line. 2. Nobody can ask you to solve a single equation with two variable because it would have infinite solutions. 3. However, if have two equation with the same two unknown variables, we can usually solve for those variables.

Two travelers moving in opposite directions (gaps)

1. we add the speeds of two travelers moving in opposite directions (moving away from or toward each other). Toward: The sum is the speed at which the gap is shrinking. Away from: The speed at which the gap is expanding. Note: set up a D = RT for the gap to save time.

Memorize the cubes up to ten.

1^3 = 3 2^3 = 18 3^3 = 27 4^3 = 64 5^3 = 125 6^3= 216 7^3= 343 8^3= 512 9^3 = 729 10^3= 1000

Factoring Quadratics

Find 2 number for which the sum = the midterm and the product equals the last term

random

Means 2 things: 1) Every individual event is absolutely unpredictable 2) the overall pattern of events is completely predictable (over the long term) Note: think of flipping a coin. With each individual flip, I don't know whether it will be heads or tails. With thousands of flips, I can confidently predict the ration will be 1/2.

Problem: If K, (K+200), (k+350), and 15xK are all multiples of P, then P could equal which of the following? A. 20 B. 25 C. 75 D. 100 E. 150

Note: 15xK is useless information. Differences among the multiples will also be P a. 200 and 350 must be multiples of P b. 350 - 200 = 150 c. 200 - 150 = 50 d. P is a factor of 350, 200, 150, and 50 e. the only number that can be a factor of all of these is 25 (answer option B)

Suppose we roll one fair six-sided die eight times. What is the probability that we will roll at least one six

P( at least one six)= 1 - (5/6)^8

What is the LCM of 24 and 32? (finding the LCM example 1)

S1: Find prime factorizations and GCF. 24= 2^3x3; 32= 2^5; GCF= 8 S2: write each number in the form (GCF) times another factor. 24= 8x3; 32= 8x4. S3: Find LCM by finding the product of the three factors in S2. 8x3x4= 96 (answer)

At a certain school of 200 students, the students can study French, Spanish, both, or neither. Just as many study neither as study both. One quarter of those who study Spanish also study French. The total number who study French is 10 fewer than those who study Spanish only. How many students study French?

a. A = French; B= Both; C= Spanish; D= neither b. B=D c. 1/4(B+C)= B... B+C= 4B... d. C=3B e. A+B = C-10 f. A+B = 3B-10 g. A= 2B-10 h. A+B+C+D= 200 I. (2B-10) + B + 3B + B= 200 j. 7B = 210 k. B = 30 l. A = 2(30)-10 m. A = 50 students study French only (answer).

Exponents of prime factors of squares

a. Always must be even b. if we see an unknown number in prime factorization, and all the exponents are even, it must be a perfect square.

Circle properties

a. Any triangle with two sides that are radii has to be isosceles. b. central angle: An angle with its vertex at the center of a circle. c. the measure of a central angle equals the measure of the ark. d. a diameter is a 180º angle. c. If two different central angles in the same circle have the same measure, then they will intersect arcs of the same size. The opposite is also true. e. equal length chords in the same circle intersect equal length arcs. d. inscribed angles: chords with vertex on the circle. e. the measure of the inscribed angle is half the measure of the arc it intercepts. f. Any inscribed angle that intercepts a semicircle has to be a right angle. g. If two inscribed angles in the same circle intercept the same arc or same chord (on the same side), tehn the two inscribed angles are equal.

On a test in a class of more than 40 students, the scores had mean = median = mode = 81. Two absent students then took the test; they received grades of 83 and 47. What are the new mean and median? A. mean = 81 and median = 81 B. mean < 81 and median = 81 C. mean = 81 and median < 81 D. mean < 81 and median < 81

a. B is the answer Note 1: The median should stay the some because two new score have been add with one being below and one being above the original median. This eliminates C and D. Note 2: The mean should be less because the new lower score is pretty extreme. This eliminates A.

Wendy left the house with D dollars in cash. She made the following cash purchases: gas, for 3 less than 1/3 of D; a book, fro 1/6 of D; stationary for six dollars more than 1/6 of D; and groceries, for 1/4 of D. After these four purchases, she had 4 left. What is the value of D?

a. D = (1/3)D - 3 + (1/6)D + (1/6)D + 6 + (1/4)D + 4 b. D = (1/3)D + (1/3)D + (1/4)D + 7 e. D= (4/12)D +(4/12)D + (3/12)D +7 f. D = 11/12D + 7 g. 7 = D - 11/12D h. 7 = 1/12D i. D = 84 (answer)

Cassandra drive from A to B at a constant 60 mph speed. Shen then returned, on the same route, from B to A, at a constant speed of 20 mph. What was her average speed?

a. D = distance from a to b b. First leg: T = D/60 c. Second leg: T = D/20 d. Total time: T = D/20 + D/60 = 3D/20+D/60= 4D/60 = D/15 e. Total distance: 2D f. average speed= (2D)/(d/15) = (15/1D)(2D/1) = 30 mph (final answer)

Slope

a. Def. 1: A measure of how steep a line is. b. Def. 2: Rise over run. run is the horizontal separation. rise is the vertical separation.

Facts about Polygons

a. Diagonal: any segment that connects two non-adjacent vertices. b. Pentagons: have five diagonals. c. Hexagons: have nine diagonals. d. sum of angles in a pentagon is 540º e. sum of angles in a hexagon is 720º

Comparing mean and median

a. Don't always need to calculate. b. Look to the outliers. The mean will follow the outliers. high-value outliers cause the mean to be higher than the median. Low-value outliers cause the mean to be lower than the median.

Adding and subtracting evens and odds

a. E+E=E; E-E=E b. O+O=E; O-O=E c. O+E=O: O-E= O Note: Only when we add or subtract unlikes, do we get an odd result.

Graphing lines big ideas

a. Every line in the x-y plane has its own equation. b. For any line, all the points on the line have x and y coordinates that satisfy the line's equation. In other words, any point on the line has to satisfy the line's equation b. Any linear equation that relates x^1 and y^1, with no multiplication or division of variables, must be the equation of some line on the x-y plane. (e.g. 3y - 4x = 12

Frank has 13 more dollars than Glenda does, and together they have 81. How much does Frank have.

a. F = 13 + G b. G + F = 81 c. G = F-13 d. F + F - 13 = 81 f. 2F = 94 g. F = 47 (answer)

Problem (ratios and rates): A bumblebee's wing flaps 1440 times in 8 seconds. How many times does it flap in a minute?

a. F/S= 1440/8= 720/4= 360/2= 180/1 b. 180 flaps/second c. 1 minute= 60 seconds d. F= 180 x 60 e. F= 10,800 (answer)

Solve for x: dkfj a. 7x/6 + 2/3 = 13/2

a. Find LCM for fractions on both sides: 6 b. 7x+4=39 c. 7x=35 d. x=5 (answer).

Probability question that must be solved with counting: A committee of three will be selected from a group of eight employees, including Alice and Bob. What is the probability that the chosen committee of three includes Alice and not Bob?

a. Find the denominator first (the combination for the entire group): 8C3 = 56 b. Find the numerator second (the combination for the remaining 6 employees after Alice and bob are out of the picture: 6C2 = 15 c. Answer: 15/56

Factoring Quadratics (example): x^2 + 8x +15

a. Find two numbers whose sum = 8 and whose product equals 15: 3 and 5 b. Factor therefore is the following: (x+3)(x+5)

Suppose events A and B are independent. If P(A) = 0.6 and P(B) = 0.8, what does P(A or B) equal?

a. First figure out P(A and B). P(A and B) = (0.6)(0.8) = 0.48 b. use the generalized OR formula to figure out the rest: P(A or B) = 0.6 + 0.8 - 0.48 = 0.92 (answer).

An airplane has a 3600 mile trip. It covers the first 1800 miles of a trip at 400 mph. Which of the following is the closest to the constant speed the plane would have to follow in the last 1800 miles so that the average speed of the whole trip is 450 mph. A. 450 B. 455 C. 500 D. 514 E. 600

a. First leg: 1800 = 400(t) b. First leg T = 4.5 c. Whole trip: 3600 = 450(t) d. whole trip: T = 8 e. Time for second leg: 8 - 4.5 = 3.5 hrs. f. Second leg 1800 = 3.5R g. Second leg: R = 1800/3.5 = 3600/7 = (3500/7) + (100/7) = 500 + approx 14 f. 514 mph = answer.

Bob drove 120 miles at 60 mph, then another 120 at 40 mph. What was he average speed for the total trip.

a. First leg: T = 120/60 = 2hrs. b. Second leg: T = 120/40 = 3hrs. c. Vavg = 240miles/5hrs = 48 mph (answer)

In a certain game, in Phase 1 you flip one coin as many as three times. If you flip three tails, you lose. As soon as you get your first head, you advance to Phase 2. In Phase 2, you roll a sic-sided die once. If you roll a 6, you win. For any other roll, you lose. What is the probability of winning?

a. First, calculate the probability of getting 'at least one tail' in three trials: a1. P(at least one T) = 1 - (1/2)^3 a2. 1 - (1/8) = 7/8 b. Next, calculate the probability of getting a six in one roll: 1/6 c. To win, you have to accomplish the tasks in both phase 1 AND (and rule) phase 2, so multiply the answers for a and b together. c1. (7/8)(1/6) = 7/48 (final answer)

Suppose we know that two sides of a triangle are 8 and 13, and we want to know the possible lengths of the third side x.

a. First, it must be true that x > 13 - 8 b. Second, it must also be true that x < 13 + 8 c. Thus, 5 < x < 21 d. Generalized takeaway: If we know the lengths of two of the three sides of a triangle, P and Q then... P - Q < third sides < P + Q

If a line goes through (2, -1) and has a slope of m = 5/3, find all the points (a,b) on the line where a and b are integers whose absolute values are less than or equal to 10.

a. First, move right: that is , add 3 to x and 5 to y--(2, -1), (5,4) (8, 9) (answers) b. Now move left: subtract 3 from x and 5 from y--(2, -1) (-1, -6) (answer)

Exponential Growth

a. For a positive base greater than 1, the powers continually get larger, at a fast rate. b. for a base less than one. The exponents produce smaller powers (there is an exponential decrease): (1/2)^2 = 1/4 (smaller value). c. negative bases less than 1 (negative numbers): The absolute values continue to get bigger, but the signs alternate. d. negative base between 0 and 1 (fractions): absolute values get smaller, but the signs are alternating.

Frank and Georgia started traveling from A to B at the same time. Georgia's constant speed was 1.5 times Frank's constant speed. When Georgia arrived at B, she turned around immediately and returned by the same route. She crossed paths with Frank, who was coming toward B, when they were 60 miles away from B. How far away are A and B?

a. Frank: D-60 = RT b. Georgia: D+60 =1.5RT c. Combined: D + 60 = 1.5(D -60) d. D + 60 = 1.5D - 90 e. D + 150 = 1.5D f. 150 = 0.5D g. D = 150/0.5 = 1500/5 = 300 (answer)

Problem (ratios and rates): A block of ice in a warm room is melting at a rate of 8 grams/hour. If at noon there are 30 grams of ice, when will the block first be entirely melted?

a. G/H = 8/1 b. 8/1 = 30/H c. 4/1 = 15/H d. 4H = 15 hours e. H = 15/4 hours = 3+ (3/4) hrs. f. Melts at 3:45 pm (answer)

Find the LCM of 48 and 75

a. GCF = 3 b. (48x75)/3 c. 48x25= 24x50= 12x100= 1200 (answer)

What is the LCM of 12 and 75

a. GCF= 3 b. 12=3x4 c. 75= 3x25 d. LCM= 3x4x25= 300

Similar triangles (figures in general)

a. Have the same shape but different sizes b. have equal angles. c. If a segment across a triangle is parallel to one of the sides, it automatically creates a smaller similar triangle. d. If just two angles are equal in two triangles, that's enough to prove they are similar. e. sides in similar figures are proportional. ration of any two sides in one triangle is equal to the ratio of the corresponding sides in the other triangle. f. Scale factor (k): when the sides of the bigger triangle are written in the numerator. g. to find the area of a similar triangle, square the scale factor (k) and multiply the result by the area of the first triangle.

Which of the following could be true of at least some of the terms of the sequence defined by b sub n = (2n-1)(2n+3) I. Divisible by 2 II. Divisible by 3 III. Divisible by 4

a. II and III only b. Never I because it will always work out to be odd time odd, which always has an odd product. Odd numbers are never divisible by 2.

When are the mean and the median the same

a. If the list consists of evenly spaced numbers, then the mean equals the median. (e.g. consecutive integers and consecutive multiples of the same number are evenly spaced lists). b. They are also equal when the list is symmetrical.

System of equations (no solutions)

a. If the result has not variable and two numbers that do not equal each other, then there are no solutions. (parallel lines) b. If there are fewer equations than variables, you will not be able to solve fore the individual values of variables. c. Pay attention to wording. You may not need to solve everything.

Special triangle lines continued

a. If we get any information that a line segment from a vertex to the opposite side of a triangle is playing more than one role (median, perpendicular bisector, angle bisector, altitude), that's enough to prove that a triangle is isosceles.

Equations with square roots

a. It is not always true that √k^2 = k. This is only true fore positive numbers and 0. b. only square both sides when the radical is isolated on one side.

Problem: If y = 5+x and y = 12 - x, and if y^2 = x^2 + K, the K equals which of the following? a. 17 b. 25 c. 60 d. 119

a. K = y^2 - x^2 b. K = (y - x)(y + x) c. y-x = 5 d. y + x = 12 e. K = (5)(12) f: K = 60, letter C (answer)

The equation of Line M is Kx + 3Ky = 17, for some number K. if the line M passes through the point (2,1), then find the value of K.

a. K(2) + 3(K)(1) = 17 b. 2K + 3K = 17 c. 5K = 17 d. K = 17/5 (answer).

Other roots

a. K^3 is 3√k; K^4 is 4√k; etc. b. x^3 = positive has only one positive solution; x^3 = negative has only one negative solution. We can take a cubed root of a negative number: 3√-8 = -2 c. extend b to any odd roots. d. 19 > √19 >3√19.... e. If 0 < b <1, and if n > m, then 0 < b < m√b <n√b; 2/5 < √2/5 < 3√2/5...

Let T be a sequence of the from a sub n = a sub 1 + d(n-1). If a sub 3 = 17 and a sub 19 = 65, find a sub 10.

a. Let the initial term, a sub 1 = b, and let the common difference equal d. b. a sub 3 = b + 2d = 17 c. a sub 19 = b + 18d = 65 d. subtracting b from c we get 16d = 48; d = 3 e. plug 3 back into b and we get b = 11 f. Thus, a sub 10= 11 + 3(9) = 11 + 27 = 38 (answer)

Contract negotiations opened on the morning of March 20th, continued every day without a break, and ended late in the eventing of May 10th. For how many calendar days were contract negotiations in session.

a. March 20th to the 31s: 31 -20 + 1 = 12 days b. April = 30 c. In may, 10 days. e. 12 + 30 + 10 = 52 days.

Martha and Paul started traveling from A to B at the same time. Martha traveled at a constant speed of 60 mph, and Paul at a constant speed of 40 mph. When Martha arrived at B, Paul was still 50 miles away. What is the distance between A and B?

a. Martha: D = 60(T) b. Paul: D-50 = 40(T) c. combined: 60T-50= 40T d. 60T= 40T+50 e. 20T= 50 f. T = 50/20= 5/2 (plug back into a for distance) g. D = 60(5/2) h. D= 30(5)= 150

special triangle lines

a. Median: goes from a vertex to the midpoint of the opposite side. divides the side in half, but not the angle of origin. b. perpendicular bisector of a side: does not typically pass through the opposite vertex. c. angle bisector: divides the angle of origin in half, but not the opposite side. d: altitude: Goes through a vertex and is perpendicular to the opposite side. Note: The line down the middle of an isosceles triangle from the vertex to the midpoint of the base, plays all four of these roles at once.

A librarian has seven books to arrange: four different novels, and three identical copies of the same dictionary. In how many different orders could these seven books be put on the shelf?

a. N = 7!/3! b. N = [(7)(6)(5)(4)(3)(2)(1)]/[(3)(2)(1)] c. N = (7)(6)(5)(4) d N = 840 (answer)

Counting Identical items

a. N = total factorial/number of identical items factorial (N = n!/b!)

When positive integer N is divided by positive integer P, the quotient is 18, with a remainder of 7. When N is divided by (P+2), the quotient is 15 and the remainder is 1. What is the value of N? (use rebuilding the dividend formula)

a. N= 18P+7 b. N= 15(P+2) + 1= 15P+31 c. 18P+7 = 15P +31 d. 3P = 24 e. P= 8 f. N= 15(8+2) +1= 151 (answer).

Problem: An item originally cost $800. The price increased by 20%. What is the new price.

a. N=800x1.2 b. N= 960.00 (answer) Number sense: 10% is $80, so 20% is $160. That the increase. 800 + 160= $960

Normal distribution properties

a. On any normal distribution, 34% of the population is between the mean and one standard deviation (M+S), and 13.5% is between (M+S) and (M+2S)

Options for writing quotients of two numbers that don't divide evenly

a. Option 1: have an integer quotient and an integer remainder b. option 2: express quotient as a fraction or decimal 12/8= 3/2= 1+1/2= 1.5

From a standard shuffled deck of 52 cards, what's the probability of picking three hearts on the first three cards drawn, if the cards are selected without replacement?

a. P(1 = H) = 1/4 b. P(2 = H| 1= H) = 12/51 = 4/17 c. P(3 = H| 1=H and 2=H) = 11/50 d. P(first 3 H) = (1/4)(4/17)(11/50) e. (1/17)(11/50) = 11/850 (answer)

A box had 5 green balls and 7 red balls. Assume that all balls in the box are equally likely, and that the balls are picked without replacement. What is the probability that the first two balls picked are both green?

a. P(1st = G) = 5/12 b. P(2nd = G | 1st = G) = 4/11 d. (5/12)(4/11) = 5/33 (answer) .

Notation: P(A)

a. P(A) = the probability of event A

Conditional probability notation

a. P(A|B). This means what is the probability of A given B?

Practice with VICS: At the store, Sam bought a shirt and a toaster. There was an 8% sales tax on each item, and with tax, Sam paid a total of Q. If the price of the toaster before tax was T, what, in terms of Q and T, is the price of the shirt. A: 0.92(Q-T) B: 0.92Q-T C: 0.92(Q-1.08T) D: (Q-T)/1.08 E: (Q/1.08) - T

a. Q = 1.08(S+T) b. Q/1.08 = S+T c. (Q/1.08) -T = S (answer choice E) NOTE: VICS is a difficult concept. I may need to review the videos.

Problem: If Susan invests $1000 in an account that yields 5% annually, compounding quarterly then how much does she have after 6 years?

a. Quarterly percent= 5/4= 1.25% b. multiplier= 1.25/100= 0.0125 + 1= 1.0125 c. The amount in the account experiences that percent increase 4 times each year, or 24 times in six years. d. A= 1000(1.0125^24) (answer) Note: use simple interest to estimate. 5% is 50 dollars a year. 50 x 6 is 300. 1000 + 300 is 1300. The compounded interest will be slightly more than 1300

A lichen advances 4 cm each year across a rock slab. If this rate remains constant over time, how many years will it take to cross 30 meters? (1 m = 100 cm)

a. R = 4 cm/yr b. D = 30 m = 3000 cm c. 3000 cm = 4cm/yr (T) d. T = 3000 cm/ (4cm/yr) e. T = 750 yrs.

A car moving at 72 km/hr moves how many meters in one second? (1 km = 1000)

a. R = D/T b. 1hr = 60 x 60 = 3600 s c. 72 km/hr = 72,000 m/3600 s = 20 m/s

Counting Factors in a perfect square

a. Remember it will always have an odd number of factors. (This only happens with perfect squares.

Right now, Steve's age is half of Tom's age. In eight years, twice Tom's age will be five more than three times Steve's age. How old is Tom right now?

a. S = (1/2)T b. T = 2S (this avoids fractions) c. 2(T+8) = 3(S+8) + 5 d. 2(2S+8) = 3(S+8) + 5 e. 4S + 16 = 3S + 29 f. S + 16 = 29 g. S = 13 (plug back in to b) h. T = 26

A sequence is defined by S sub n = (S(sub n -1) -1)(S (sub n-2) for n >2, and it has the starting values of S sub 1 = 2 and S sub 2 = 3. Find the value of S sub 6.

a. S sub 1 = 1 b. S sub 2 = 3 c. S sub 3 = (2-1)(2)= 4 d. S sub 4 = (4-1)(3)= 9 e. S sub 5 = (9-1)(4)= 32 f. S sub 6 = (32-1)(9) = 279

Proportional reasoning practice In T^2 = KR^3, if T is multiplied by 5, R is multiplied by what?

a. Set everything equal to 1 in the equation and adapt according to the directions. b. 5^2 = (1)R^3 c. 25 = R^3 d. R = cubed √25 = cubed √(5^2) = 5^(2/3). Any three of these could potentially be the possible answer.

Picking numbers practice: Before January, the price of a dress was D and the price of a matching pair of shoes was H. In January, the price of the dress increased by 40% and the price of the shoes increased by 50%, and didn't change again in the following months. In March, Roberta bought both items with a 30% discount. If D = 5H, which of the following represents the amount that Roberta paid? A. D+40 B. D+40 -1 C. D + 2H D. 5.95H E. 1.21D

a. Suppose D = 200 and H = 40 b. After increases D = 280 and H= 60 c. 280 + 60 = 340 d. 30% decrease of 340 is 238 e. plugging the amounts for D and H before increase into each potential answer should yield 238 for the correct answer only. f. D is the only one that works (6-0.05)(40) = 240 - 2 = 238 (answer) Note: VICs is still very difficult for me. This will take some practice.

Quartiles

a. The median is the 50% mark, it is not included in the list of quartiles. b. Quartile 1 is the medial of the lower list and Quartile 3 is the median of the upper list.

Mode

a. The most frequently appearing number on a list. There can be more than one mode. There can also be no mode at all. (this measure is not as important as median and mean).

Middle 50% in Boxplot (aka: interquartile range--IQR)

a. The population that falls in the box b. There are no outliers in this set. It is the group of people closest to the median.

generalized OR rule

a. This occurs when two events are not mutually exclusive b. P(A or B) = P(A) + P(B) - P(A and B). The last step eliminates repetition.

Three criteria venn diagrams

a. Three groups with binary options b. there are 8 variables. c. start from the central region and work outward.

divisor

a. To say A is a DIVISOR of C is to say that we A divides evenly into C (the quotient is an integer). Note: Factor and divisor are essentially the same thing, with divisor being used in the context of division. 1. 8 is a factor of 24 2. 8 is a divisor of 24 3. 24 is divisible by 8 numbers 1-3 state exactly the same thing.

factor

a. To say A is a FACTOR of C is to say that we can multiply A by some integer and the product will be C (e.g. 3 is a factor of 6, 1 is a factor of every positive integer. ) b. every integer is a factor of itself. So, every integer greater than one has at least two factors

There are a total of 400 students at a school, which offers a chorus, baseball, and Italian. This year, 120 students are in the chorus, 40 students in both chorus and Italian, 45 students in both chorus & baseball, and 15 students do all three activities. If 220 students are in either Italian or baseball, then how many students are in none of the three activities.

a. Use a 3 circle venn diagram to solve. Depending on letter assignments, the answer will look something like the following b. T=400 c. B+C+E+F= 120 d. B+C= 40 e. C+F = 45 f. C=15 g. A+ B+ C+ D+ F+ G =220 h. B = 25; F= 30; E= 50 I. (A+ B+ C+ D+ F+ G)+ E= 270 j. H = 400 -270= 130 students who do none (answer)

Double Matrix problem In a company of 300 employees, 120 are females. a total of 200 employees have advanced degrees and the rest have a college degree only. If 80 employees are males with college degree only, how many are females with advanced degrees?

a. Use a box matrix with one more column and and one more row then there are categories to solve. b. 100 is the answer.

Double Matrix Problem: In a certain school, there are 80 freshmen, 100 sophomores, and 220 upperclassmen, drawn from three cities: A, B, and C. Sixty percent of students are from A, 30% from B, and the rest from C. All the students from C are freshmen. Half the students from B are upperclassman and the rest are split evenly between the other two grades. How many sophomores are from A?

a. Use a box matrix with one more column and and one more row then there are categories to solve. b. 70 students are sophomores from A (answer)

Think about 10C4. We have a pool of 10 different items, and we want to select a set of 4--how many different sets of four could we pick?

a. Use the FCP: N = (10)(9)(8)(7) b. Since order of the 4 selected does NOT matter, we need to divide a by 4! to eliminate repetition: [(10)(9)(8)(7)]/[(4)(3)(2)(1)] c. (10)(3)(7) = 210 (answer) = 10C4 = 10C6

a. Cube volume b. Cube surface area

a. V = s^3 b. A = 6s^2

Distance between points big idea 2

a. We can always use the slope triangle and pythagorean theorem to find the the distance between points on a sloped line.

Adding and subtracting radicals

a. We cannot add or subtract through the radical signs b. We have to simplify each radical separately, and then we can add the ones that have the same radical factor.

56 is what percent of 800? (Number sense method)

a. We know 10% of 800= 80 b. We know 1% of 800= 8 c. We need 7 times the last piece d. 7% (answer) of 800= 56

Parabola features based on the quadratic standard equation: y = a(x^2) + bx + c

a. When a > 0, the parabola opens upward; when a < 0, the parabola opens downward. b. when |a| > 1 the parabola is skinny; when the |a| < 1, the parabola is wide. c. When y = 0, these are the intercepts. These are the solutions to the quadratic equation.

Generalized AND rule

a. When events A and B are not independent. In other words, whether one happens changes the probability of whether the other will happen. This is a CONDITIONAL PROBABILITY. b. Formulas: P(A and B) = P(B) x P(A|B); P(A and B) = P(A) x P(B|A)

Solve the following problem: √(x+3) = x -3

a. X + 3= (x -3)^2 b. X + 3 = X^2 - 6X + 9 c. 0 = X^2 -7x +6 d. 0 = (x -6)(x- 1) e. x = {1, 6} f. pluggin both answer back into the question, the only one that works is 6, so x=6 (answer). Note: if both answer do not work, then there is no solution.

Pum X takes 28 hours to fill a pool. Pump Y takes 21 hours to fill the same pool. How long does it take them to fill the same pool if they are working simultaneously?

a. X = 1/28 b. Y = 1/21 c. 1/21 + 1/28 = 4/84 + 3/84 = 7/84 = 1/12 (12 hours to fill the pool).

What is 60% of 60?

a. X= .6x60 b. X= 36 (answer)

rectangular solide diagonals

a. face diagonal: a diagonal of only one face (find with pythagorean theorem. b. space diagonal: passes through the interior of the solid to the opposite vertex (AD)^2 = (AB)^2 + (BC)^2 + (CD)^2 c. The space diagonal of a cube is √3 times the edge length.

Solve h = [(1/2)gt]^2 + vt for g.

a. g = (2h - 2vt)/t^2 or g = (2h/t^2) - 2v/t (both are the same answer).

Slope intercept form

a. getting y on one side by itself. b. y = mx + b Note: m is the slope and b is the y-intercept Note 2: horizontal lines will have a slope of 0. (e.g. y = 0x +4) Note 3: vertical lines have an undefined slope (e.g. something/0). Can't really use slope intercept form to express vertical lines.

30-60-90 or 1-2-√3 triangle

a. half of an equilateral triangle. b. all 30-60-90 triangles are similar, and the sides have the same proportion. c. hypotenuse is always twice the length of the short leg. the long leg is √3x the short leg.

Cube facts

a. has 6 faces (6 congruent squares). b. 8 vertices. 3 mutually perpendicular edges meet at each vertex. c. 12 edges

Rectangles

a. have four 90º angles. b. they are parallelograms c. Diagonals are equal to each other in length (congruent).

Distances between points big idea 1

a. horizontal and vertical distances are very easy to find. If two points are on a horizontal line just subtract the x-coordinates (smaller from bigger). If two points are on the same vertical line we subtract the y coordinates

Geometry strategies 2

a. look big and look small. Train your eye to zoom in and out. b. look for similar triangles when there are more than two c. Any time there is a right triangle, ask yourself if the p. theorem applies. (finding the length of an altitude are diagonal are commonly p. theorem problems)

Units Digit Questions Strategy

a. look for a repeating pattern b. Figure out where the pattern will be at the desired power c. More often than not, it will be a pattern of 4 different numbers. Figure up to the power of 8. The number at 4 and 8 will establish the pattern. Note: the units digit will be influenced only by the units digits of the two factors.

Range

a. measure of spread. highest number - lowest number. It only tells us the difference between extremes, not much else.

Simplifying a complex fraction

a. multiply the numerator and denominator of the big fraction by the LCM of all the denominators of the little fraction

Problem: In class, the ratio of boys to girls is 3:7. If there are 32 more girls than boys, how many boys are there.

a. number of boys = 3n, and the number of girls = 7n. b. The difference, 7n-3n= 4n= 32 c. n= 8 d. number of boys= 3n= 3x8= 24.

standard deviation

a. numbers below the mean have a negative deviation, and numbers above, a positive deviation. b. measures how far away from the mean are the individual points.

In a class, 18 students took a test and had an average of 70. Alicia and Burt then took the test and the average of all 20 students was 71. If Alicia got a 77, then what was Burt's grade?

a. old sum = 18(70) = 1260 b. new sum = 20(71) = 1420 c. difference = 1420 - 1260 = 160 f. 160 - 77 = 83 (Answer--Burt's grade).

Parallelograms

a. opposite sides are parallel b. opposite sides are equal c. opposite angles are equal d. diagonals bisect each other. If any of the 4 properties above are true, then they are all true. If any are not true, then the rest are not true of a particular quadrialteral.

Presenting ratio information

a. p to q form: The ratio of boys to girls is 3 to 4. b. fraction form: The ratio of boys to girls is 3/4 c. colon form: the ration of boys to girls is 3:4 d. idiom form: for every 3 boys, there are 4 girls. (order is important)

4 quadrants

a. quadrants go counterclockwise from the upper right b. Q1: x > 0 and y > 0 c. Q2: x < 0 and y > 0 d. Q3: x < 0 and y < 0 e. Q4: x > 0 and y < 0

Problem: General purpose concrete is created using a 1:2:3 ratio of cement to sand to gravel. If we have 150 kgs of sand available, how many kgs of concrete can we make? (assume we have more than enough cement and gravel)

a. sand:concrete= 2:6= 1:3 b. 1/3= 150/w c. w= 150 x 3= 450 (answer)

Square area

a. side = s b. A = s^2

Mean

a. simple average. b. N(mean) = sum of entries. (very useful in certain circumstances.

Multiplying and dividing radical expressions

a. simply multiply whole numbers by whole numbers, and radicals by radicals b. sometimes what is left under the radical can be simplified even further.

Presenting rate informations

a. so-many (units) per (units) 330 meters/second $4/ gallon 4 centimeters/year $7.25/hour 60 minute/hour (unit conversion) 360 degrees/revolution (unit conversion)

Suppose we start with 8 liters of 60% H solution. We add 4 liters of C% H solution, and the result is 12 liters of 50% H solution. What is C?

a. solute in first solution: 8(0.6)= 4.8 liters b. solute in result: 12(0.5)= 6 liters c. Thus, solute added = 6-4.8 = 1.2 liters d. C = 1.2/4 = .3 = 30% (answer)

What is the slope of the line with the equation 3x + 5y = 8?

a. solve for y b. 5y = -3x + 8 c. y = (-3/5)x + 8/5 d. m = -3/5 (answer)

Solve the following problem: √(2x-2) = √(x-4)

a. square both sides and solve: x = -2 b. Plugging -2 back into the equation √-6 = √-6 c. There is not solution since it is impossible to take the square root of a negative number.

Backsolving practice: IN a certain state, schools by 2% tax on food and 8% on stationery. A school placed a combined order of $500 on food and stationery, and paid 19 on tax on the order. How much of that money was spend on food A: 120 B: 250 C: 300 D: 350 E: 400

a. start with C: Food = 300; stationery =200 b. food tax = 300(0.02) = 6 c. stationery tax = 200(0.08) = 16 d. Total tax = 22 (too much). We can reason food needs to cost more and stationery needs to cost less. e. answer choice D: Food = 350; Stationery = 150 f. Food tax 350(0.02)= 7 g. Stationery tax= 150(0.08) = 12 h. 7 + 12 = 19 (answer choice D is correct)

What happens if a line has a slope of 1 (m = 1)

a. then rise = run, and the slope triangle is a 45-45-90 triangle b. lines with slopes of m = 1 or m = -1 make 45º with the axes. If greater than 1, the slope is greater the 45º. If less than one the slop is smaller than 45º c. if m > 1 or m < -1 are steep

Squares

a. they are rectangles, rhombuses, and parallelograms. Thus, they have the same properties. b. Never assume a shape is a square even if it look it.

Weighted averages

a. think in terms of sums for multiple groups. you CANNOT add averages together and divid by N and expect to get the correct answer. b. You can also think in terms of proportions (P). So, if you have three groups, you would multiply each proportion= n/whole by the groups average. Do that that for each group and and them together to get the the average of the whole. (e.g. (A1)(P1) + (A2)(P2) + (A3)(P3).

Equilateral triangle characteristics

a. three equal sides and three equal angles. b. each angle must be 60º c. an equilateral triangle is technically an isosceles triangle. Thus, every fact about isosceles triangles applies to equilateral triangles.

Rationalize the following: (4 - √6)/(2√3)

a. times by √3/√3 b. [4√3 - (√6)(√3)]/[2(3)] c. [4√3 - 3√3]/6 (answer)

Finding x and y intercepts.

a. to find x-intercept plug 0 in for y. to find y-intercept, plug 0 in for x in any line equation.

Solving algebraic equations

a. to solve for x follow the order of operations (GEMDAS) backwards.

Isosceles triangle characteristics

a. two or more sides are equal b. the angles opposite the equal sides are also equal c. if we know the equal angels and one opposite side, we can deduce the other side. If we have two sides that are equal, we can deduce one opposite angle if the other is present. d. they can have a right angle or an obtuse angle.

Reflections over the line y = x (a line going through the origin with a positive slope)

a. when points are reflected over this line, the x and y coordinates switch. b. all reflected points are equidistant from the mirror line

Most popular mixture question type: Suppose we start with unlimited supplies of 20% H solution and of 50% H solution. We combine X liters of the first with Y liters of the second to produce 7 liters of 40% H solution. What does X equal?

a. x + y = 7 b. Total solute: 7(0.4)= 2.8 liters c. 0.2X + 0.5Y= 2.8 liters d. go back to a (y= 7-x) e. 0.2X + 0.5(7-x)= 2.8 f. 0.2x + 3.5 - 0.5x = 2.8 g. times by 10 to get rid of decimals: 2x + 35 -5x = 28 h. 35-3x = 28 i. -3x = -7 j x= 7/3 liters (answer)

Rationalize the following: x = 12/√3

a. x = (12/√3)(√3/√3) b. x = (12√3)/3 c. x = 4√3

If (x-3)^2 = 16, solve for x.

a. x-3 = √16 b. x-3 = +/- 4 c. x = 7 or x =-1 Note: this sort of equation is extremely quick if you're working with a perfect square to begin with.

coordinate plane

a. x-axis: horizontal number line b. y-axis: vertical number line c. origin: where y and x axes cross (zero).

What is 75% of 280 (method 3)

a. x= (3/4)280 b. x = 3x70 c. x = 210 Note: use this method only if the percent is a very easy fraction. We already know 75%= 3/4

Absolute values solve: |1+2x| = 4-x

a. x= 1 or x=-5 b. Plugging both back into the original equation, both numbers work so both answer are viable. Note: Both solutions will not always work when plugging them back in.

Solve the for x: dkfj a. 3x/5= 2/7

a. x= 10/21

Solve for x and y using the substitution method: 1. x+2y= 11 2. 2x+3y= 15

a. x= 11-2y b. 2(11-2y) + 3y= 15 c.22-4y+3y=15 d. 22- y= 15 e. 22= 15+y f. y = 7 g. x +2(7)= 11 h. x= -3 and y= 7

Absolute values solve |2x+5| = x+1

a. x=-4 or x=-2 b. Plug answers back into the right side of the original equation first c. -4+1= -3 (no because no absolute values can be negative. d. -2+1= -1 (no because no absolute values can be negative) e. there are no solutions.

Given the function f(x)= x^2+4x-21 Find the value(s) of x that would satisfy f(x) =24

a. x^2+4x-21= 24 b. x^2 + 4x - 45 = 0 c. (x+9)(x-5) = 0 d. x= -9 or x= 5 e. f(5) or f(-9)= 24 (answer)

Solve the following: x^2-3x-43=11

a. x^2-3x-54=0 b. (x-9)(x+6)=0 c. x-9= 0 d. x= 9 e. x+6= 0 f. x= -6 g. x= 9 or x= -6 (answer)

simplify the following: [(x^2 y^3)^4]/(X^5 Y^-5)

a. x^3 y^17

More examples of an even power of x is the square of another power a. x^6 b. x^8 c. x^6 - 16 d. x^7 - 4x^5 e. x^4 - 81 f. x^9 - x

a. x^6 = (X^3)^2 b. x^8 = (X^4)^2 c. x^6 - 16 = (x^3 - 4)(x^3 +4) d. x^7 - 4x^5 = x^5(x^2 -4)= X^5(x - 2)(x + 2) e. = (x^2 - 9)(x^2 + 9) = (x-3)(x+3)(x^2 + 9) f. = x(x^4+1)(x-1)(x+1)(x^2+1) Note for e and f: There is no way to factor a sum of squares, just a difference of squares.

Solve (three equations three unknowns): W-2x+3y= 13 2w+x-4y=-14 3w-x+2y=8

a. y=3 b. w=0 c. x= -2

Answer the following: a. Is the √2 > 1 b. Let K^2 = 2. Is K > 1

a. yes b. Cannot be determined Note: since the square root symbol is not included in b, there are both a positive and negative answer.

Absolute values solve: |3x+2| + 1= 5

a. |3x+2|= 4 b. 3x+2= 4 or 3x+2= -4 c. x=2/3 or x=-2 (answer)

Simplify the following: (√12)(√27)

a. √(12)(27) b. √(12)(3)(9) c. √(36)(9) d. (√36)(√9) = 6x3 = 18 (answer)

Simplify the following: √2800

a. √(28 x 100) b. 10√28 c. 10√(4x7) d 10(2√7) e. 20√7

Pythagorean Theorem (right triangles only)

a^2 + b^2 = c^2 Note: c is the hypotenuse. Note 2: We can work backwards with the formula to determine whether the triangle is a right triangle if we don't already know. If the formula works, the triangle is a right triangle.

Difference of two squares factor

a^2 - b^2 = (a+b)(a-c). example: x^2 - 49 = (x+7)(x-7)

-(1/5): (different looks)

-1/5 = 1/-5 = -(1/5) Note: the negative sign can move about

0.0013/0.025

0.0013/0.025 = 0.013/0.25 = (0.013x4)/(0.25x4) = 0.052/1 = 0.052 Note: If you can easily times the denominator by something to make it 1, do it as soon ass you can (make sure to times numerator and denominator by the same number).

QC Strategies: Matching operations

1. Add the same number to both quantities 2. Subtract the same number from both quantities 3. Multiply or Divide the same POSITIVE number from both quantities

Multiplication of Decimals Procedures

1. Count the number of digits to the right of the decimal point: 6.25x0.048 2. The first factor has 2 dec. places and the second has three. Add those two: the product will have 2+3 = 5 decimal places. 3. Ignore the decimal, and find the product of the two positive integers: 625x48

Real Number

A real number is any number on the number line. This includes round numbers as well as fractions, decimals, and negative numbers.. NOTE: "number" always means "real number" on the test.

a. (7/10)/(7/15)= b. (24/35)/(25/36)= c. (8/9)/6=

a. (7/10)/(7/15)= 3/2 b. (24/35)/(25/36)= 10/21 c. (8/9)/6= 4/27 Note: Always cancel before you multiply

a. 0.1 b. 0.01 c. 0.001

a. 0.1 = one tenth = 1/10 = 10^-1 b. 0.01 = one hundredth = 1/100 = 10^-2 c. 0.001 = one thousandth = 1/1000 = 10^-3

Note: Memorize every decimal form of a fraction with a single digit denominator--there are gaps because some fractions reduce down: a. 1/2 b. 1/4 c. 3/4

a. 1/2 = 0.5 b. 1/4 = 0.25 c. 3/4 = 0.75 q. 1/9= 0.11111... (trend for every fraction with 9) r. 2/9= 0.2222...

(54x56x72)/(64x45x60)=

(6x56x72)/(64x5x60)= (6x56x9)/(8x5x60)= (1x56x9)/(8x5x10)= (7x9)/(5x10)= 63/50 (answer) Note: Always cancel before your multiply.

Multiplying and Dividing positive and negative numbers procedures

1. Determine the sign of the product/quotient 2. Treat both factors as positive, and perform the mult. or division. 3. Give the result the appropriate sign.

AxB/CxD (different combinations. Remember Division and Multiplication is at the same level)

1. = (B/CxD)xA 2. = (A/D)x(B/C) 3 =[(A/C)xB]/D Note: ALWAYS choose to cancel before you multiply

QC Strategies Estimation Module

1. Don't do long detailed calculations 2. Estimation is super common on QC 3. Try part-wise comparisons and comparisons of round integers 4. Always look for the simplest/quickest way to solve

GEMDAS (Order of operations)

1. Grouping symbols 2. Exponents 3. Multiplication & Division (same level) 4. Addition & Subtraction (same level) Note: Always work from the inside out if there are multiple layers of parentheses/grouping symbols.

Doubling & Halving

1. Half one factor & double the other factor 2. Best to use when one factor ends in 5 or 50 3. Apply the procedure twice when one factor is 25 or a multiple of 25

"Comparing Fractions II (Advanced)" Takeaway (Review the lesson again to crystalize).

1.. If we start with a proper fraction and add the same number to both the numerator and the denominator, that resultant fraction is closer to 1. 2. If we start with a fraction, and add p to the numerator and q to the denominator, that resultant fraction is closer to p/q

24*75

12x150 = 6x300 = 1800

Rounding module takeaway

Only look immediately to the right of the place value you are rounding. Look no further (e.g.3.14159 rounded to the nearest thousandths is 3.142 & 59,049 rounded to the nearest hundreds place is 59,000). DO NOT double round.

Developing Number Sense

Suggestions: 1. Play with patterns on paper & with a calculator 2. Curiosity and Playing around are key 3. Make games out of calculating

Reciprocal of a fraction

The reciprocal of a fraction, a/b, is the flipped over fraction, b/a (a not 0, and b not 0) 1. the product of any fraction with its reciprocal is 1 ((4/17)x(17/4)= 1). 2. The reciprocal of a positive integer is one divided by that integer (6 is 1/6) 3. One divided by any fraction equals the reciprocal of that fraction (1/(3/7)= 7/3 4. If a number is bigger than 1, then its reciprocal is smaller, between 0 and 1. If a number is between 0 and 1, its reciprocal is larger than 1

a. 11 - 78 b. 47 -65 c. 28 - 43 d. 62 - 74

a. 11 - 78 = -(78 - 11) = -67 b. 47 -65 = -(65 - 47) = -18 c. 28 - 43 = -(43 - 28) = -15 d. 62 - 74 = -(74 - 62) = -12

115^2

a. 115 becomes 11 b. 11 + 1 = 12 c. 11 * 12 = 132 d. 132 becomes 13225

a. 1235/100 b. 0.064x10^-2 c. 37.5/10000 d. 64,000x0.0001 e. 5.4x 10^-5 f. 20.25/10^-6

a. 1235/100 = 12.35 b. 0.064x10^-2 = 0.00064 c. 37.5/10000 = 0.00375 d. 64,000x0.0001 = 6.4 e. 5.4x 10^-5 = 0.000054 f. 20.25/10^-6 = 0.00002025

a. 24/10 b. 0.02/10 c. 39.85 X 0.1 d. 0.00072 x 0.1

a. 24/10 = 2.4 b. 0.02/10 = 0.002 c. 39.85 X 0.1 = 3.985 d. 0.00072 x 0.1 = 0.000072 Note: When we divide any number by ten, or multiply by .1, we move the decimal point one place to the left.

a. 350x100 b. 0.01728x1000 c. 8.3 x 10^6

a. 350x100 = 35,000 b. 0.01728x1000 = 17.28 c. 8.3 x 10^6 = 8,300,000

39^2

a. 39^2 = 40^2 - 40 - 39 b. 1600 - 40 -39 c. 1521 (answer)

|x-1| > 4 (think number line)

the distance between x and +1 is grater than +4 Or X < -3 OR x > 5 = |x-1| > 4

40^2 (squaring multiples of 10 example 1)

1. 4^2 = 16 2. add two 0's: 1600 (answer)

Fractions (properties 1)

1. If a>b, then (a/c) > (b/c) (4/13 > 3/13) 2. Bigger denominators with same numerators make smaller fractions (2/5 > 2/7) 3. If the numerator gets bigger and the denominator gets smaller, the fraction gets bigger (3/8 < 4/7) 4. Cross multiply to decide if two random fractions are bigger.

Possible forms of a fraction that is greater than one.

1. Improper fraction: numerator > denominator 2. Mixed numeral: integer part + fraction part Note: the mixed numeral represents an addition relationship, NOT a multiplication relationship Note 2: Usually better to use improper fractions on the test, but not always.

Percent

1. Percent means divided by 100 (37% means 37/100 or the decimal .37)

Squaring Shortcuts

1. To square a multiple of 10, square the digit(s) without the zero, then tack on two zeros at the end 2. Squaring a number ending in 5 a.) remove the 5 b.) add one to remaining digit(s) c.) multiply the numbers in (a) & (b) d.) put this product in front of 25 3. If we know the value of n^2 (e.g. if n is a multiple of 10 or 5), then we can get the next square up, (n + 1)^2, by adding n and (n+1) = n^2 + n + (n+1) a.) square down: n^2 - n - (n-1)

When to use mixed numeral or improper fractions.

1. Used mixed numerals to locate a number on the number line. 2. For adding and subtracting (doesn't really matter) 3. For multiplication, division, and exponents ALWAYS use improper fractions.

Fraction Properties II (takeaway)

1. We CANNOT separate a fraction into two fractions by addition or subtraction in the denominator (e.g. a/(b+C) IS NOT a/b + a/c) 2. 1 does not hold true for addition and subtraction in the numerator (e.g. (a+b)/c IS (a/c + b/c) 3. If we have addition and subtraction in both the numerator and denominator, the numerator can be split up, but the denominator must stay the same--(a+b)/(c+d)= [a/(c+d)]+[b/(c+d)].

Operations with proportions (takeaway) Note: proportions have an equal sign in the middle

1. We can get rid of proportions through cross multiplication (5/7 = 3/x; 5x=21; x=21/5) 2. For proportions with larger numbers we should try to cancel first. However, the rules for cancelation differ, so be careful. diagonal cancellation IS ILLEGAL...DON'T DO IT. Not the same as multiplication. 3. Horizontal and Vertical Cancelations are okay.

Dividing by 5

1. double N 2. Divide result by 10 = answer Or 1. Divide N by 10 2. Double result = answer

Positive/Negative sign rules for multiplication

1. positive x positive = positive 2. negative x negative = positive 3. postive x negative = negative

Postive/Negative sign rules for division

1. positive/positive = positive 2. negative/negative = positive 3. positive/negative = negative (any order)

16*35 (doubling and halving example 1)

16x35 = (8x2)x35 = 8x(2x35) = 8x70 = 560

56*25

28x50 = 14x100 = 1400

39.0625 (Place Values)

3 in the tens place (10) 9 in the ones place (10) 0 in the tenths place (1/10) 6 in the hundredths place (1/100) 2 in the thousandths place (1/1,000) 5 in the ten thousandths place (1/10,000)

QA 3/7 + 2/5 Or QB 13/27 + 41/97 (Sample "Part Estimation" QC Strategies)

A. 13/27 > 13/28 > 12/28 = 3/7 B. 41/97 > 41/100 > 40/100 = 2/5 Quantity B is larger

Example of number sense game

A. Original set: [2, 3, 4, 5] 1. (2x3) + 4 + 5 = 6 +9 = 15 2. (2x4) + 3 + 5 = 8+8 = 16 3. 2^3 + 4 + 5 = 8+9 = 17 4. 3^2 + 4 + 5 = 9+9 = 18

Absolute Value sample problem: Consider the positive integers from 1-100. If n is a number in that set, then for how many numbers n is it true that |n-30| > 20?

Answer is 59

Practice Problem QA: 147/200 QB: 150/203

Answer: Quantity B is bigger Principle: If you add the same number to both the num. and dem. of a proper fraction, the result is bigger. However, if you add the same number to an improper fraction, the result is smaller. (Very easy for comparison in some situations).

Equivalent fractions

Fractions that have the same numerical values, but may differ with respect to their numerators and denominators. 2/3 = 10/15

0.56/0.0007 (dividing by decimals examples)

Note: slide decimal until denominator is an integer a. 0.56/0.0007 = 5.6/0.007 = 56/0.07 = 560/0.7 = 5600/7 = 800

Numerator vs Denominator: 3/16

Numerator: top 3 Denominator: bottom 16

QA: 32.8% of 5929 QB: 41.6% of 5041 (Sample rounding estimation QC Strategies)

QA: 32.8% of %929 = (33.3%-) of (6000-)=2000- QB: 41.6% of 5041 = (40+)% of (5000+)=2000+ Answer: QB is larger

a. (1/4)+(2/3) b. (3/5)-(1/10 c. (5/6)+(1/4)

a. (1/4)+(2/3)= (3/12)+(8/12)= 11/12 b. (3/5)-(1/10)= (6/10)-(1/10)= 5/10= 1/2 c. (5/6)+(1/4)= (10/12)+(3/12)= 13/12

(0.03)^3 (decimal multiplication example)

a. 0.03^3 = .03x.03x.03 b. (2 + 2 + 2 = 6 decimal places) c. 3^3 = 27, so the 7 must land six places to the right of the decimal d. 0.000027 (answer)

a. 1/40 b. 1/600 (write in decimal form)

a. 1/40= (1/4)x(1/10)= (0.25)(0.1)= 0.025 b. 1/600= (1/6)x(1/100)= (0.16666..)(0.01)= 0.00166..) Note: extrapolate for all multiples of 10

a. 1/5 b. 2/5 c. 3/5 d. 4/5 express as decimals

a. 1/5= 0.2 b. 2/5= 0.4 c. 3/5= 0.6 d. 4/5= 0.8

a. 1/6 b. 5/6 c. 1/7 d. 2/7 e. 3/7 f. 4/7 g. 5/7 h. 6/7 express as decimals

a. 1/6= 0.1667 b. 5/6= 0.8333... c. 1/7= 0.143 d. .285 e. 0.428 f. 0.571 g. 0.714 h.0.857

a. 1/8 b. 3/8 c. 5/8 d. 7/8 express as decimals

a. 1/8= 0.125 b. 3/8= 0.375 c. 5/8= 0.625 d. 7/8= 0.875

25^2

a. 25 becomes 2 b. 2 + 1 +3 c. 2 * 3 = 6 d. 6 becomes 625

47 + 36

a. 40 + 30 = 70 b. 7 + 6 = 13 c. 70 + 13 = 83 (Rule: You can simplify addition of two digit numbers by treating the digits separately).

47 + 36 (mental addition 2 digit example)

a. 40 + 30 = 70 b. 7 + 6 = 13 c. 70 + 13 = 83 (Rule: You can treat the digits separately in addition if two digits)

41^2 (adjacent squares example 1)

a. 41^2 = 40^2 + 40 + (40+1) b. 1600 + 40 + 41 c. 1681 (answer)

a. 42.5% b. 4% c. 0.25% Percents to decimals

a. 42.5%= 0.425 b. 4%= 0.04 c. 0.25%= 0.0025

56^2

a. 56^2 = 55^2 + 55 + 56 b. 3025 + 55 + 56 c. 3136 (answer)

Reciprocal practice problem: The reciprocal of a positive number times the cube of the same number equals 5. What is the number?

a. 5=(1/x)(x)(x)(x) b. 5 = (1)(x)(x)--one x has canceled out c. 5 = x^2 d. square root of 5 = x (answer)

Practice Problem QA: 6/200 QB: 7/235

a. 6/200= 3/100= 1/33.3 b. we have to add 1/35 to reach quantity B c. 1/35 is smaller than 1/33.3, so adding 1 to the num. and 35 to the denom. will decrease the ratio. Thus, quantity A is larger. Note: when the ratio decreases the original fraction is the largest. When the ration increases, the other option is the largest.

84^2

a. 84^2 = 85^2 - 85 - 84 b. 7225 - 85 -84 c. 7140-84 d. 7056 (answer)

9/20 ?? 4/9 (use cross multiplication)

a. 9x9 = 81 ?? 4x10=80 b. 81 > 80 c. 9/20 > 4/9 (answer)

Practice word problem with fractions Cathy's salary is 3/7 of Nora's salary and is 5/4 of Teresa's salary. Nora's salary is what fraction of Teresa's salary?

a. C= (3/7)N b. C= (5/4)T c. (3/7)N = (5/4)T d. N= (7/3)x(5/4)T e. N= (35/12)T

n/n

n/n always equals 1. Doesn't matter how ugly a. 8/8 = 1 b. 0.045/0.045= 1 c. (3^3)/(3^3)= 1

{[5+(5/8)]/[4+(1/2)]}=

{[5+(5/8)]/[4+(1/2)]}= (45/8)/(9/2)= (45/8)x(2/9)= (5/4)x1= 5/4= 1+(1/4) (answer)