eidk_Quant
Fractions - Division
Flip and then multiply
Factors
For "N" to be a factor of "X", N must contain ALL prime factors that X does. Must beware of overlapping factors. All primes must be unique.
Percent Increase
% Increase of ... New % Mult Org Value by.. 10% 110% 1.1 20% 120% 1.2 or 6/5 25% 125% 1.25 or 5/4 50% 150% 1.5 or 3/2 100% 200% 2
Percent Decrease
% decrease of ... New % Mult Org Value by.. 10% 90% 0.9 20% 80% .8 or 4/5 25% 75% .75 or 3/4 50% 50% .5 or 1/2 75% 25% .25 or 1/4
Trapezoid
1. Area = Height * (b1 + b2) / 2
Powers of 10
10^1=10 10^2=100 10^3=1,000
Word Problems contd..
- If you multiply two quanties with units multi the units as well - Likewise, units cancel in the same way as numbers and variables do
Number Properties
0 and 1 are not prime #'s
Number Properties
0 is an even number
Percents
1. "what percent" = x / 100 2. "% of" means multiply (ie. 30% of 200 = .30 * 200)
Root Rules
1. (√10)^2=10 2. √10^2=10 3. √-10^2=10 4. √2= ~1.4 5. √3= ~1.7 6. the √ of a bigger number is alwasy bigger than the √ of a smaller number 7. The √ of a number >1 is smaller than the org number (√2<2) 8. the √ of a number betwee 0 and 1 is bigger than the orginal (√2/3>2/3) 9. √5^12=5^6 (take the √ of a postive number rasied to a power and rewrite the √ as an exponet of 1/2, the mult exponets 10. 125^2/3= 3√125^2
FDP's
1. Convert decimal to a fraction: put the digits to the right of the decimal point over the appropriate power of 10 then simplify (ie. .036 = 36/1000) 2. If deonominator of fraction only contains 2's and 5's as factors convert to a power of 10 (ie. 7/8 = (7*125)/(8*125)=875/1000 = .875 3. Mult. by power of 10 move decimal to the right 4. Divide by a power of 10 move decimal the number of the exponet places to the left (.6 / 10^3 = .0006) 5. Multiplying by a negative power of 10 flip the power and change the sign (ie. .004 * 10^(-3) = .004 / (10^3) = .000004
Consecutive Integers
1. Evenly Spaced Integers - the mean and the median are equal the avg of the first and last terms 2. The sum of the elements in the set are equal to the mean * the number of items in the set 3. Counting Intergers: Last term - First term + 1 4. Counting Intergers for consecutive multiples: (Last - First) / Increment + 1 ( ie. All EVEN intergers between 12 and 24 = (24-12) / 2 + 1 5. To find the sum of consecutive intergers simply find the number of terms and multiply by the median or avg in the set
Fractions - Add/Sub
1. Have addition or sub in the numberation can split into two fractions 2. Add/Sub in denominator can pull out a factor that will cancel in the numeration but NEVER split into two 3. Don't cancel on add and sub fractions
Ratios
1. If you see that the ratio of sharts to dolphins is 3 to 13..write as a proportion ( ie. 3/13) and write each quanity in terms of an unknown multiplier (ie. Sharks = 3x, Dolphins = 13x) 2. If you have two parts that make a whole and have a ratio of 3 to 4...write "Part : Part : Whole" ratio as 3 : 4 : 7 and use unknow multipler as needed (ie. Lefties = 3x, Righities = 4x so people = 7x)
Percent Change
1. Orginal % + Change% = New% (all % of orginal value) 2. %Change = New Value / Org. Value 3. To find a new value from the %change and the org value need to find the new "% of" using Org + Change = New, then multi by org value (ie. $60 + 25% = 100% + 25% = 125% as new %; 125% = 5/4 = (5/4) * 60 = $75 4. To find %more than or less than treat like %increase or decrease prob (ie. $230 is what % more than $200? = 30/200 = 15%)
Triangles - Right Triangles
1. Paythagorean Theorem: a^2 + b^2 = c^2 2. Triplets: (3 - 4 - 5 ), (5 - 12 - 13), (8 - 15 - 17), (or any triple or double of the 3)
Strategies For Problem Solving Q's
1. Picking Numbers - when you come across a variable or other unknown, consider Picking Numbers 2. When you see a percent question with uspecified values, pick 100 for the unknown 3. Pick numbers when you have variables in the answer choices 4. On "which of the following" questions start with E and work up 5. When numbers are given in the answer choices try to backslove (start with B or D first) - if you first answer is incorrect, its helpful to know whether the correct answer is larger or smaller
Geometry Word Problems
1. Redraw, fill in, label target 2. Spot relationships & write equations 3. Slove for what you can 4. Make inferences
Triangles
1. Sum of any two sides will always be greater than thrid 2. 3rd side must be less than the sum of the other two sides but greater than the difference of the other two sides 3. Sum of all angles = 180 4. Longest side is opposite the largest angle, smallest side opposite smallest angle 5. same sides = same angles 6. Area = (1/2)B * height
Imporant Quadratic Rules
1. To factor -2x^2 + 16x - 24 factor out -2 from all terms. The facotr normally. = -2(x-6)(x-2). Same applies is the term is negative. -x^2+9x-18 = -(x^2-9x+18) 2. If you cannot pull out a common factor from x^2 then keep coefficient on one or even both x's. (i.e 2x^2 - x - 15 = (2x+..)(x+...) experiment with factor pairs of the constant 3. To solve (x - 7)^2 = 625 take the positive and negative sq root of both sides. (ie x - 7 = 25 or x-7 = -25) (x=32 or x=-18) 4. To solve x^3=x must factor. (ie x^2 - x = 0...x(x^2 - 1)=0...x=0 or -1 or 1. **x^3 should alert that there could be 3 solutions** 5. Expressions that only differ by a sign change are only different by a factor of -1 (ie y-x / x-y = -(x-y) / (x-y)...cancel and get -1 6. (y+7)^2 + (y+7) / (y+8) = (y+7)[(y+7)+1)] / (y+8) ...can cancel and solve
Exponent Rules
1. a^2*a^3=a^5 (Mult with same base add exponents) 2. a^5/a^3=a^2 (Div with same base subtract exponents) 3. a^0=1 (anything to zero=1) 4. a^-2= 1/a^2 (negative exponents are the recipricoal) 5. 2a^-2/3=2/3a^2 (good to manipulate equations - when move a term from top to bottom of fraction switch the sign of the exponent) 6. (a^2)^4=a^8
Finding the number of factors of an interger
1. make a prime factorazation n=a^p * b^q * c^r ; where a,b,c are prime factors # of factors= (p+1)(q+1)(r+1)
Cordinate Plane
1. y= Mx + B (M=slope, B = y-intercept) 2. Slope: (y2 - Y1) / (X2 - X1)
Roots contd...
11. √a * √b = √ab 12.√a/√b=√a/b 13. to simplfy √ factor our sqs (ex: √50=√25*2=√25*√2=5√2 14. add or sub under √ factor out a sq factor from the sum or difference (EX: √4^14+4^16=√4^14(1+4^20=√4^14*√1+16=4^7√17 15. 3√x=x^1/3
Squares
1^2=1 2^2=4 3^2=9 4^2=16 5^2=25 6^2=36 7^2=49 8^2=64 9^2=81 10^2=100 11^2=121 12^2=144 13^2=169 14^2=196 15^2=225 16^2=256 17^2=289 18^2=324 19^2=361 20^2=400 30^2=900
Cubes
1^3=1 2^3=8 3^3=27 4^3=64 5^3=125 6^3=216 7^3=343
Number Prop
2 is the only EVEN prime
Powers of 2
2^1=2 2^2=4 2^3=8 2^4=16 2^5=32 2^6=64 2^7=128 2^8=256 2^9=512 2^10=1,024
Powers of 3
3^1=3 3^2=9 3^3=27 3^4=81
Powers of 4
4^1=4 4^2=16 4^3=64
Powers of 5
5^1=5 5^2=25 5^3=125
FDP's contd...
6. Multiply decimal and a big number then trade decimal places from the big number to the decimal (ie. 50,000 * .007 = 50*7) 7. Divide two decimals move the decimal points in the same direction to eliminate decimals as far as you can (.002 / .0004 = 20 / 4) ( always go with larger number of moves, can always add zeros to the other number)
Consecutive Inters Contd
6. The average of an odd number of consecutive intergers will always be and integer 7. The average of an even number of consecutive intergers will NEVER be an integer 8. The product of K consecutive integers ias always divisible by K factorial (K!) 9. For any set of consecutive integers with an ODD number of items, the sum of all integers is ALWAYS a multiple of the number of items 10. For any set of consecutive integers with an EVEN number of items, the sum of all the items is NEVER and multiple of the number of items 11. For any odd number of consecutive integers, the sum of those integers is divisible by the number of integers
Exponet Rules Contd..
7. If you have diff bases that are numbers try breaking the bases down to prime factors. Can express everything in terms of one base (Ex: 2^2*4^3*16=2^3*(2^2)^3*2^4=2^12 8.(ab)^3=a^3b^3 9. (a/b)^4=a^4/b^4 10. a^3b^3=(ab)3 11. 2^3+2^5=2^3(1+2^2) (add or subtract terms with same base pull out common factor) 12. 2^3+6^3=2^3*(3*2)^3=2^3*3^3*2^3=2^3*(1+3^3) (add or subtract terms with diff bases break down bases and pull out common factor)
Fractions within fractions
Always work out from the deepest level
Find LCM
Consider the multiple of the larger integer until you find on thats of the smaller EX: 12&40 40*1=40 40*2=80 40*3=120 (120 is a multiple of 12 and is the LCM) or find the prime factorization 40= 2,2,5,2 12=2,2,3 LCM = 2*2*2*3*5=120
Distance Formula
Distance = Rate x Time *Multiply a rate by the denominators unit and youll get the numerators unit* - always put time in the demoninator
Divisibility Rules
Divisible by...: 2 - Last digit even 3 - Digits add up to a multiple of 3 4 - All numbers that end w/ "00" or Last two digits multiples of 4 or divisible by 2 twice 6 - multiple of 2 & 3 (divisible by 2 & 3) 7 - take the LAST digit and double. Subtract answer from remaining digits- If answer is divisible by 7 or 0 (EX: 161= 1*2 = 16-2 = 14) 8 - last 3 digits=multiple of 8 9 - if digits add to multiple of 9 12 - multiple of 3&4
Odd and Even Rules
Even +/- Even = even even +/- odd = odd odd +/- odd = even Even * even = even even * odd = even odd * odd = odd ** Only way to get odd is "even +/- odd" or "odd*odd"
PEMDAS - Exponents
Exponets come before everything else EXCEPT (). EX: -3^2=-(3^2)=-9 but (-3)^2=9
Circles - Sectors
Figure out the fraction of the circle that the sector represents and you can figrure out everything: 1. (Central angle / 360) 2. (Sector area / circle area) 3. (arc length / circumfrence)
Fraction Conversions
Fraction Decimal Percent 1/100 0.01 1.0% 1/20 0.05 5.0% 1/10 0.1 10.0% 1/8 0.125 12.5% 1/5 0.2 20.0% 3/8 0.375 37.5% 2/5 0.4 40.0% 3/5 0.6 60.0% 5/8 0.625 62.5% 4/5 0.8 80.0% 7/8 0.875 87.5% 6/5 1.2 120.0% 5/4 1.25 125.0% 3/2 1.5 150.0%
Divisibility
If a number is NOT divisible by 3 it is NOT divisible by 6 or 9
Absolute Value
If you have a value inside of absolute value signs (isolate value on one side)..drop the absolute value and set up two equations, on positive and one negative (ie. IzI=4, z=4 and -z=4 .. solve)
Consecutive Interger DS Problem
Is k^2 off? 1) K - 1 is divisible by 2 2) The sum of consecutive integers is divisible by K Statemement (1) tells us that K must be odd so K^2 will be odd Statement (2) tesll us that the sum of K consecutive integers is divisible by K. Therefore, this sum divided by K is an integer. Moreover, the sume of K consecutive integers divided by K is the avg of that set of K integers. As a result, statement (2) tells us that the avg of the K consecutive intergers is and integer. Therefore, must be ODD.
Ratio Problems
Manipulate ratio problems to get unit answer needed for question (set up to cancel out others)
Fractions - Multiplication
Mulitply tops and bottoms, CANCELLING COMMON FACTORS FIRST. Can even break fractions down in to their units (ie. 20/10 = (5*4)/(5*2)...can cancel the 5's)
Successive % Change
Multiply org value by the "new percents" for BOTH percent changes (ie. $50 + 10% then +20% = $50(11/10)(6/5) = $66
Age Problems
On age problems must add number of years to BOTH sides of equations
FDP's equation
Part = Fraction * Whole
Fractions - Square's
Proper fractions produce smaller numbers (ie (1/3)^2 = (1/9)
45 - 45 - 90 Triangles
Side Ratio: x : x : x√2 (hypotnuse) Half of a sq, rect, Quad is 45-45-90 triangle
30 - 60 - 90 Triangles
Side ratio: x (short) : x√3 (middle) : 2x (hypotnuse) Half of equilaterall tri = 30-60-90
Special Quadratic Expressions
Square of a sum : (x+y)^2 = x^2 + 2xy + y^2 Square of a difference: (x-y)^2 = x^2 - 2xy = y^2 Difference of Squares: (x+y)(x-y) = x^2 - y^2
GCF
Take the lowest common power of the prime factorization of both numbers (the GCF of two numbers cannot be larger than the difference between the two numbers.
Inequalities
The only numbers that make two inequalities true are those that are true for BOTH inequalitites
LCM of Large Numbers
To find LCM of large #'s use prime factorization and multiply greatest power of the common multiples EX: 150&225 150= 2*3*5^2 225= 3^2*5^2 LCM=2*3^2*5^2 = 2*9*25 = 450 = LCM
Parallelogram
When divided gives you two right triangles. 1. Area = Base * Height 2. Perimeter = 2 (s1 + s2)
Word Problems
Use these steps on every problem: 1. Identify what they want 2. Identifiy what they give you 3. Represent relationships as equations 4. Slove the algebra
Circles
When you know one thing you can find everything: 1. Circumfrence = ∏*D or (2∏r) 2. Diameter = 2r 3. Area = ∏r^2
Work Formula
Work = Rate x Time For combined work problems simply add each individuals rate togeter to plug into W=R*T
consecutive even integers
n,n+2,n+4,n+6...
