Sherpa Math
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