Sherpa Math

Ace your homework & exams now with Quizwiz!

dividing decimals

"decimal fraction" - put the division in fraction form and "slide" the decimal point an equal number of times until neither numerator or denominator contains a decimal

How to Find All Factors

(1) Find the perfect square immediately below ex. 72-> 64 (2) Take square ex. 8 and list out the numbers up to that in one column; cut the number up in the other

Math Tips Strategy 101

(1) Skip around (2) Find your favorite 15 (3) Guess on questions you don't understand (4) Don't take more than 2.5 minutes on a single question (5) Never rush

Math Tip

(1) Take info pieces one at a time (2) Label (3) Don't assume - remember not drawn to scale!

From two points to line equation

(1) establish slow and plug it into y=mx+b (2) plug coordinate values of either of the given points into y=mx+b and solve for b

minimization/maximization problems

(1) identify greatest and smallest value possible for each input (2) test out every combination of those inputs ex. if -5 ≤p≤ 14 and -9≤q≤11 to find greatest possible value of pq p = -5 or 4; q = =9 or 11, so multiply (-5)(-9) and so on....

alternative multiplication and comparing numbers

(1) to multiply, break it down ex. 9x12 -> 9(10+7)-> 90+63 (2) when comparing numbers, you can remove common elements

(2/3)x

(2/3)(x/1) -> 2x/3....NOT 2/3x

Rates - Quantity Mixture

(Rate x Quantity) + (Rate x Quantity) = Total Cost

% Change

(difference/original) (100)

finding percent change

(difference/original) x 100

What percent of a is B

(is/was B / of A ) x 100

Interior ∠s

(n-2)180°; INTERIOR ∠ is any ∠ located inside a polygon 3->180°; 4->360°; 5->540°; 6->720°

(x²-y²)

(x+y)(x-y)

Three Simple Equations

(x+y)(x-y) <--> (x²-y²) (x+y)² <--> x²+2xy+y² (x-y)² <--> x²-2xy+y²

x²+2xy+y²

(x+y)²

x²-2xy+y²

(x-y)²

x(a/b)

(x/1)(a/b) = ax/b ......NOT ax/bx

(x+y)(x-y)

(x²-y²)

Lines and Quadrants

- Points to the right of origin + X; Points to left of origin - X; Points above origin +Y; Points below origin -Y

Units Digits 1234.5678

1 thousands; 2 hundreds; 3 tens; 4 units; (.) 5 tenths; 6 hundredth; 7 thousandth

fractions and mixed numerals

1) determine how many times the denominator goes into the numerator 2) then place remainder over denominator

mixed numeral conversion

1) multiply the denominator and whole number 2) add numerator 3) place result over denominator

Area of a ∆

1/2(base)(height) where BASE is horizontal length and HEIGHT is vertical distance from top to base (base and height are ⊥)

2x³ labels

2 - COEFFICIENT x - BASE and ³- EXPONENT

Bell Curve

2%/ m-2d/14%/ m-d/34% /m/34%/m+d/14%/m+2d/2% where m is mean and d is standard deviation

Circumference

2πr (or πd)

Cylinder Surface Area

2πrh + 2πr² --> 2πrh (rectangle) + 2πr² (circles)

Factorials

4! = 4 x 3 x 2 x 1; Start with most restrictive condition; important to note whether ordering or scrambling; Permutations Totals/Winners and Losers

Cube Surface Area

6s² (units²)

Similar ∆s

= Any two ∆s whose ∠s are equal or whose sides are PROPORTIONAL; if angles are equal, corresponding sides are proportional, and vice versa (note it is specifically the sides that correspond the are proportional)

absolute value

= distance from zero

Special Right ∆s - Right Isosceles

= right ∆ that has 2 equal ∠s and sides (one 90°∠ and 2 45°∠s) LEGS are x:x:√2; SHORTCUT if the x√2 does not contain a √2 --> can be obtained in two ways **FAST: Cut hypotenuse in half and multiply the quotient by √2** or SLOW: set hypotenuse equal to x√2 and solve for x...

Special Right ∆s - 30°-60°-90°

= right ∆ that has angles measuring 30°-60°-90°; LEGS are x:x√3;2x; ∠s across from corresponding legs x(30°): x√3(60°);2x(90°); TIP: two of these ∆ form an equilateral ∆

Parallelogram

A slanted quadrilateral whose opposite sides are parallel and equal in length AREA = (base)(height); diagonals are equal (within set); diagonals bisect each other, but two sets of of diagonals NOT equal

Rhombus

A slanted quadrilateral with 4 sides of equal length and ‖ sides (1) opposite ∠s are equal (2) adjacent ∠s add up to 180°; AREA = [(Diagonal 1)(Diagonal 2)]/2; slanted square; diagonals intersect at 90°∠; split rhombi into 4 right ∆s

Sphere

A three dimensional circle; the distance from a point on the surface to the center is the RADIUS; every cross section is a circle; forms a larger circle toward the middle than toward the surface; HEMISPHERE is half a sphere

Cones

A three dimensional shape that has a circle as a base and sides taper to form a point; for a RIGHT circular cone, apex (highest point) runs to its base, or bottom circle; cross section (horizontal slice) is always a circle

Pyramid

A three dimensional shape that has a polygon for a base and triangular sides with a common vertex; often quadrilateral base but can be any sort of polygon

Conversions

Always do unit conversions first

Circle Tip - Label Your Radii

Always label all your radii - it is an easy way to unmask hidden relationships

Trapezoids

Any quadrilateral with one set of opposing sides that are parallel but unequal in length; AREA = avg of bases x avg of heights (can also obtain by carving into a rectangle or square and one or more right ∆s)

Square

Area = s²; Diagonals are equal and cut each other into full equal halves; diagonals intersect each other at a 90°∠ and bisect the corner ∠s of the square; diagonal split a square into 4 45°-45°-90° ∆s

Asswholes

Assign whole numbers

bisectors and midpoints

BISECTOR = any line that splits an ∠ or a line segment in half; MIDPOINT = any point that lines at the middle of a line segment

Cylinder Band Area

Circle portion unnecessary - use band height 2πrh

Rectangular Solids

Composed of 6 rectangles aka FACES, each side of a face is an EDGE; each point at which two edges meet is a VERTEX

If a line intersects two parallel lines...

Creates 4 big ∠s (>90°) and 4 small ∠s (<90°); (1) all 4 big ∠s are equal; all 4 small ∠s are equal; (2) the sum of any big ∠ and any small ∠ is 180°, since together they form a straight line (parallel lines are often hidden in parallelograms...extend the lines!)

Quantitative Comparison

Cross-multiply up; fraction under larger product will always be larger

Distance

DISTANCE is length of line between any two points; to find distance consider the two points of the hypotenuse of a right ∆ - calculate legs by measuring rise and run; look for special right ∆s

Shaded Regions

Determine area of shaded region: (1) determine area of larger region (2) determine area of smaller region within it (3) find the difference

Rectangle

Diagonals equal and cut each other into 4 equal halves; diagonals do NOT intersect each other at 90°∠ and do NOT bisect the corner angles of the rectangle

Polygon Perimeter

Distance around Polygon is perimeter = sum of length of its sides

Exterior ∠s of a ∆

Ext ∠ of a ∆ is any ∠ formed by extending the side of a ∆; exterior ∠ is ALWAYS equal to the sum of its two opposite interior ∠s

Even and Odd Rules

E±O=O; (O)(O)=O

factor vs. multiple

FACTOR - something a number is divisible by; MULTIPLE - take n and multiply by every integer

factoring vs distribution

FACTORING - removal of the greatest common factor to two or more items (1) determine max factor then (2) remove it; DISTRIBUTION - distributes a common element to two or more terms

dividing or multiplying an inequality by a negative number

FLIP the sign - beware of ×/÷ using a variable unless you know whether it represents a positive or negative number

FOIL

First Outer Inner Last; Order of multiplication for distribution of factors to form a quadratic ex. (x+5)(x-2); then combine the inner and outer terms (two co-coefficients for x)

∆ Inequality Theorem

For a triangle to exist, the length of a given side must be less than the sum of the other two sides but greater than the difference between them ex. legs are 3 and 5, x can be 3, 4, 5, 6 or 7 because x< 5+3 =8 but x >5-3 = 2

Quadratics and Factors

If a problem is in quadratic form, consider factoring and vice versa; if factors already set equal to zero don't FOIL, just find solutions

Inscribed Shapes

Inscribed shapes have the same center

Ratios & Proportions

Inverse relationships - multiply; Direct relationships - division

Finding the intersection (line and point)

LINE AND POINT: (1) plug the coordinate value of the point into the equation (2) true statement means they intersect

Rectangular Solid Volume

Length x Width x Height

Statistics Terms

MEDIAN - 50th percentile (excludes outliers); RANGE - highest-lowest; INTER-QUARTILE RANGE - Q₃-Q₁; STANDARD DEVIATION - reflects distribution; MEAN/AVERAGE = Sum/#

Midpoint

MIDPOINT is point halfway between any two points; (x₁+x₂/2 , y₁+y₂/2)

Special Right ∆s

MOST right ∆s are special - pyth. th. usually a trap 3-4-5 -> multiples are 6-8-10, 9-12-15, 12-16-20, 15-20-25 5-12-13 -> rare

Percentage Tip

Make sure to apply sequentially

Negative Exponents

Negative Exponents are unhappy where they are ex. 10⁻³ -> 1/10³ -> 1/1000 ex. 123456.7x10ⁿ -> 1234.567 => n=⁻²

Probabilities

OR + − AND ÷ ×; important to note whether there is replacement i.e. could the events have occurred together? Also watch out for option of zero i.e. no desert (pg. 40)

Percentage Tip- 90%

Often easier to subtract 10%

% Change - Final

Original (1±change) = FINAL

order of operations

PEDMAS (Please Excuse My Dear Aunt Sally) Parens; Exponents; x/÷; +/−

Parallel & Perpendicular Lines

Parallel (‖) lines have the same slope; Perpendicular (⊥) lines - the product of the slopes = -1

PUQs -

Percentages with Unspecified Quantities - use 100

Pythagorean theorem and long legs

Presence of long legs in a right ∆ usually signifies that a missing side can be determined by special equation x²-y² = (x+y)(x-y)

Three Dimensional Shapes

Prisms, Cubes, Cylinders, Pyramids and Cones - SURFACE AREA - collective space on surface of a 3D object (units²); VOLUME - quantity of material that a 3D object can hold (units³)

Circle Lines

RADIUS - line segment that runs from center of circle to point on the circle; DIAMETER - line segment that runs directly from one point of a circle to another and passes through the center of that circle (longest possible line); CHORD - Straight line drawn from one point on a circle to another - diameter is longest possible chord; a radius intersecting a chord cuts it in half; TANGENT - straight line drawn outside a circle that intersects the circle at a single point...radius drawn to point of tangency intersects it at 90°∠

triangle types

RIGHT ∆ - has a right ∠; side opposite the right angle is the hypotenuse EQUILATERAL ∆ - has three sides of equal length; each ∠ = 60° ISOSCELES ∆ - least 2 sides of equal length; ∠ opposite the equal legs are also equal

Rate Formula

Rate x Time = Distance or Work

Slope

Rise/Run = Difference between y coordinates / difference between x coordinates

multiple equations: substitution vs. elimination

SUBSTITUTION: (1) isolate the variable (2) substitute the results into the other equation. If asked to solve for a specific variable, always isolate the other vs. ELIMINATION (1) stack (2) arrange variables (3) add or subtract to eliminate one of the variables

Polygon Area

Space enclosed by sides of polygon is area = measured in square units

Polygon Exterior ∠s

Sum is always 360°; Exterior ∠ is any angle formed by extending a side of a polygon; sum of an interior ∠ and an exterior ∠ is always 180° (because together they form a straight line)

Rectangular Solid Surface Area

Sum of Area of Faces OR 2(LWxWHxHL)

Finding the intersection (two lines)

TWO LINES: (1) set two equations equal to each other and solve for x (2) plug x coordinate into either of initial equations and solve for y

Even exponents

Taking the root of an EVEN exponent always results in absolute value ex. x² = 16 -> x=±4; plug it back in to check

Visualizing Shapes

Try to recognize every possible shape and look for a shape that has all ∠s defined- may be key to solving

Timing - Be Careful of Quick Wins

Unless you've used one of the math tricks, be wary of a problem you can do in 5 seconds: may be a trap

Overlapping Sets

Use an it/not matrix with totals

Fives and Tens

We don't like counting nickles but we do like counting dimes ex. 44/5 -> 88/10

Coordinate Geometry Strategy

When in doubt (1) Establish the equation of any line that you can (2) Plug in any given points into that equation (or another equation)

Missing ∠ Trick

When working with polygons formed by crossing lines, always label their interior ∠s; rewrite an ∠ on a line outside the polygon (x) as "180-x" within the polygon

Integers

Whole number can be +,-, or zero; Inclusive (within) or Exclusive (between)

Exponent Rules - same base; x

add exponents

rule of opposites

addition gets rid of subtraction/subtraction gets rid of addition; multiplication gets rid of division; division gets rid of multiplication

decimals between 0 and 1

always behave in opposite manner ex. multiplication -> smaller; division -> bigger; squaring -> smaller; square root - > bigger

cross multiplication

an equation in which one fraction equals another can be simplified by multiplying the numerator of each fraction with the denominator of the other ex. (5x-1)/4 = 3x+2/2 -> 2(5x-1)=4(3x+2)

reciprocals

any two numbers whose product equals 1

Inscribed Angles

any ∠ whose vertex lies on perimeter of a circle is known as an inscribed ∠; three special properties (1) inscribed ∠ of an arc is always half the measure of the central ∠; (2) inscribed ∠s drawn to the same arc have equal measures; (3) inscribed ∠s of equal measures have arcs of equal length and vice versa

Pythagorean theorem

a²+b²=c² can be used for right ∆; only if proof of right ∆, and usually a trap - look for special ∆s; can be used to verify whether a ∆ is indeed a right ∆

Polygon

closed figure formed by 3 or more line segments (known as sides) - each side intersects with exactly two other sides at their endpoints, known as vertex/vertices

comparing √ to non √ expressions

consider putting the non √ expression into square root form (1) square the non exp (2) simplify or whatever 3) put under √ at the end (easier than trying to simplify √ expression)

quadratic equations

contain a variable that is raised to both the 1st and 2nd power - typically ax²±bx±c=0 but can look different solve by (1) set equal to zero (2) make sure x² term is positive (3) set up brackets (x )(x ) (4) choose two #s that multiply to the last number and add to the middle number (5) determine solutions aka roots by setting each factor equal to zero

inequalities

crocodile is hungry opens his mouth to eat bigger number!

multiples of 8

cut in half 3 times

Third side of triangle

difference of 1 and 2 < 3rd side < sum of 1 and 2

∆ ∠s and sides

direct correlation between size of angle and size of side it faces (smallest ∠ opposite smallest leg and so on)

multiples of 9

divisible by 3 twice; divide sum of digits twice

multiples of 7

easier to just divide by 7/memorize 7s 7,14,21,28,35,42,49,56,63,70,77,84

multiples of 5

ends in zero or five

anything⁰

equals 1!

multiples of 6

even number; and sum of digits is a multiple of 3

multiples of 2

even numbers; can cut in half once

equations with exponents & roots

exponents and roots are opposite operations (1) isolate the radical (2) multiple both sides of the equation by the corresponding exponent OR (1) Isolate the exponent and (2) take the corresponding root for both sides

function

f(x)=x² means hits graph once ex. f( :D ) = :D²

Exponent Rules - same base; +/-

factor

dividing fractions

flip the second fraction and multiple

behavior of 0<x<1

fractions between 1 and 0 do the opposite of whole numbers

complex numerator shortcut

if numerator contains +/−, can be split out ex. 27²+27/27 => 27²/27 + 27/27 = 27+1 = 28

fractions and properties of zero

if numerator is 0 fraction equals zero ex. 0/5 = 0; if denominator is zero, fraction is undefined ex. 5/0 = undefined

vertical ∠s

if two lines intersect, their opposite angles = vertical ∠s --> (1) opposite ∠s are equal; (2) ∠s that form a straight line have a joint measure of 180° (straight line can be thought of as half a circle 360°

(x)(x)

is x² ......NOT 2x

three part inequalities

isolate the given variable by doing the same thing to ALL THREE sides of the inequality

solving for x

isolate the variable on one side of the equal sign by performing identical operations to both sides of the equation (1) combine terms with the same variable (2) add or subtract numbers not directly attached to variable (3) multiple or divide out numbers directly attached to variable

number lines

label the segments algebraically, which will allow you to set up equations - you can then plug into one another to solve

multiples of 4

last 2 digits can be cut in half twice

absolute value and inequalities

make sure to flip the sign of the negative equation!

decimals multiplication - 10³

moves three places to the right

decimals multiplication = ÷10

moves to the left

decimals multiplication - x10

moves to the right (by # of zeros)

Exponent Rules - Same Exponent x/÷

multiple base and keep exponent OR divide base and keep exponent

getting rid of denominators when equations lack +/-

multiply both sides of the equation by the reciprocal of the co-efficient ex. 2/3x=4 -> 3/2(2/3x) = 3/2(4)

multiplying decimals

multiply first then worry about decimals; number of decimal places in the product will equal the sum of the decimal spots in the two factors

to convert a fraction into a percent

multiply it by 100

powers of 10 and decimals

multiplying by a power of 10 shifts decimal to the right; dividing to the left; # of shifts corresponds to the change in the number of zeros

fractions of 9 -> decimals

note all the numbers repeat ex. 3/9-> .333; 6/9->.666; 8/9-> .888

Fraction Labels ? / ?

numerator/denominator

when to round (decimal problems)

only if a problem contains #s that are hard to reduce AND a phrase such as "approx" or "closest to"

units digit of a product

only need to multiply the units digit of each number to find

applying percent changes

original x (1±Change) = Final

converting decimals to fractions

place decimal over a 1 with a number of zeros equal to decimal places ex. 0.123 -> 123/1,000

Sector

portion of area of a circle Sector = (πr²)(x/360) where x equals central ∠

Arc

portion of circumference of circle Arc = (2πr)(x/360) where x equals central ∠

Interpreting Equations

positive m -> ascending line; negative m -> descending line; positive b -> y-int is above origin; negative b -> y-int above origin; equation with no b-term -> y-int @ origin (ex. y=3x); equation with no m-term - slope of 1 (ex. y=x+2)

prime numbers

prime = only two distinct factors; 2 is the lowest prime number because 1 x 1 =1 is only one number and you need two distinct numbers

Quadrilaterals

quadrilateral = shape with opposite sides of equal length and 4 90°∠s; AREA = Length x Width

Plug Ins

quantitative comparison question that compares algebraic expressions - whenever possible use special #s (1,0,-1); remove any element shared by two expressions, then plug in!

absolute value equations

refers to the non-negative value of a number w/in absolute value bracket; equations have two solutions; |expression|=answer -> expression =± answer; always plug the two solutions back in to validate them

multiplying fractions

rip them up and cancel them out

get rid of denominators

simplify by getting rid of the denominator (1) multiply your equation by the denominator - use lowest common denominator (2) distribute and cancel out ex. 12 (3/4x-5 = 1/4x + 6)

Math Timing

skip around and find favorite 14; no more than 2.5mins per question; work for a minute and if you're not getting anywhere move on-- if you look at it for 30 seconds and you are not sure, then guess, flag and skip

In Terms of

solve for the variable that the question (1) ask about or (2) tells you to solve for; never solve for the "in terms of" variable

Exponent Rules - same base; ÷

subtract exponents

multiples of 3

sum of digits is a multiple of 3 ex 81

Cube Volume

s³ (units³)

Cube Diagonal

s√3 (units¹)

Cube

three dimensional square; 4 sides traditionally labeled s - only need one s (because they are all the same)

simplify fractions within fractions

to simplify, multiple the numerator by the "flip" of the denominator ex. (2/5)/3/8 -> 2/5x8/3

Cylinder

two circles and a rectangle joined together to make a right circular cylinder

0⁰

undefined

plug in problems

use 1,0,-1 if you can; don't assume variables are difference numbers!

xa+xb/x+xd

x(a+b)/x(1+d)....NOT (a+b/d)

Finding x-intercepts

x-intercept = point where line intersects with x axis; let y = zero and solve for x

Vertical Line

x=a; undefined slow b/c rise but never run; i.e. slope = infinity/zero -> undefined

4x²+πx²

x²(4+π)....NOT 4π²

(x+y)²

x²+2xy+y²

(x-y)²

x²-2xy+y²

Line Equation

y = mx + b, where m is the slope and b is the y-intercept

Horizontal Line

y= b; slope is zero because it runs but never rises; slope = 0/infinity => 0

when you multiply a negative across an inequality...

you have to flip the sign!

Roots are fractional exponents

y√= x∧(1/y); 2∧½ = √2 = 1.4; 2¹=2; 2²=4

Circle Area

πr²

Cylinder Volume

πr²h, where h is height

Rectangular Solid Diagonal

√(L²+W²+H²); partial diagonal use same formula but halves

π

≈3.14 - if problem says approx. use 3.1 or 22/7; It is customary to leave π as part of the answer

complex fractions and cancelling

≠ can only cancel within simplifed fractions - must add or subtract first!

perpendicular and parallel lines

⊥ lines that intersect at a 90° ∠ ‖ lines lie in the same plane but never intersect


Related study sets

Chapter 11 review 1, Computer User Support - Chapter 11, Chapter 11, A Guide to Customer User Support Quiz 11, Chapter 11 Quiz Tech Cust, Tech Support Admin. Ch. 11, Tech Sup Chapter 11

View Set

RIM_10_Vital Records and Business Continuity (Class)

View Set

ACCA F4 - Question Bank - 5. Capital & financing

View Set

Abdomen problem solving & positioning errors

View Set

Sexuality/Reproduction Practice Questions

View Set

Glass Fracture - Entrance vs Exit

View Set