General Mathematics - Arithmetic Basics
ordered pair
Two numbers listed in a specific order; it describes a point on the coordinate graph
like fractions
Two or more fractions that have the same denominator
Unit Profit = Sale Price - Unit Cost
Unit Profit = ? or Sale Price = ?
lowercase Variables
Unknown quantities by the first letters of the alphabet (a, b, c, d, etc..); Known quantities by the last letters (u, x, y, etc.)
When would you use the Heavy Division Shortcut, and how do you do it? If the answer is not precise enough, what should you do?
Use the Heavy Division Shortcut when you need an approximate answer to a division problem using decimals that looks complex. ~ Get a SINGLE DIGIT to the left of the decimal in the denominator. Do this by moving the decimals in the numerator & denominator the SAME DIRECTION and round to whole numbers. ~ Focus on the whole number parts of the numerator and denominator and solve. IF the answer's not precise enough, keep one (or 2) more decimal places and do long division: e.g. for 1,530,794 / 314,900, instead of doing 15/3, do 153/31 for more accuracy.
ORIGINAL + CHANGE = NEW CHANGE/ORIGINAL = PERCENT CHANGE
What are the 2 "percent change" equations?
1. Arithmetic Mean (Ave.) = Median ... you can find out the ave. by figuring out the Median (i.e. MIDDLE number) 2. Mean & Median = (First + Last terms) / 2... i.e. the average of the First and Last terms 3. Sum(Elements in Set) = Ave. x #Elements
What are the 3 main formulaic properties of evenly-spaced sets?
2 and/or 5 only
What are the only prime factors that a fraction resulting in a terminating decimals have?
The diagonals of a rhombus are ALWAYS perpendicular bisectors (meaning they cut each other in half at a 90deg. angle)
What are the properties of the diagonals of a Rhombus?
1. NEVER pick 1 or 0, or 100 for % VICS 2. All numbers you pick must be DIFFERENT 3. Pick SMALL numbers 4. Try to pick PRIME numbers 5. Avoid picking numbers that are COEFFICIENTS in several answer choices
What are the rules for picking numbers in VICS?
Sequences of numbers that go up/down by the same amount (the INCREMENT) from one item in the sequence to the next
What is an evenly-spaced set?
ONE
What is the MINIMUM number of multiples of 3 in a set of 3 consecutive integers?
ONE, and therefore it's product is divisible by 8!
What is the MINIMUM number of multiples of 8 in a set of 8 consecutive integers?
(Last - First + 1)
What is the formula for COUNTING consecutive integers?
( (Last - First) / Increment ) + 1
What is the formula for COUNTING consecutive multiples?
Sum of Interior Angles of a Polygon: (n - 2) x 180
What is the formula for the Sum of Interior Angles of a Polygon? ...where n = the number of sides
A = (Base x Height) / 2 , A = (BH)/2
What is the formula for: The Area of a Triangle ?
1
What is the increment of a set of consecutive integers?
Because a polygon can be cut into (n - 2) triangles (where n is the number of sides of the polygon), EACH of which contains 180 deg. Thus, adding all of the interior angles of these triangles gives the sum of the interior angles of the Polygon.
What is the logic behind why the formula for the Sum of Interior Angles of a Polygon as follows: (n - 2) x 180 ?
2
What is the only EVEN prime number?
The distance around the Polgyon... i.e. the sum of the lengths of all the sides.
What is the perimeter of a Polygon?
EVEN
What is the result of ADDING 2 Odds or 2 Evens? e.g. 7 + 11 = 18 e.g. 8 + 6 = 14
ODD
What is the result of ADDING or SUBTRACTING and ODD with an EVEN (or an EVEN with an ODD)? e.g. 7 + 8 = 15 e.g. 13 - 2 = 11
To make sure to solve for BOTH cases.
What rule is essential to follow when solving ABSOLUTE VALUE EQUATIONS?
√6 / 6 because the product of a number and its reciprocal is always 1, multiplying √6 by √6 / 6 = 6/6 = 1 and therefore the fraction above is the reciprocal of √6.
What's the reciprocal of √6 , and why?
EVEN
When MULTIPLYING integers, if ANY integer is even, what is the result - (odd/even)?
ODD
When MULTIPLYING integers, if NO integer is even, what is the result - (odd/even)?
When you are absolutely sure the variable or expression <> 0
When are you allowed to divide by a variable, (or ANY expression) ?
higher terms
When performing routine arithmetic operations with fractions, it is often necessary to convert a fraction to higher terms. This means you multiply both the numerator and denominator by a particular integer value.
prime factors
When the factors of a number are all prime numbers, the factors are said to be the
parallelogram
a quadrilateral with two pairs of congruent, parallel sides
Multitude
a quantity consisting of disconnected parts, as three stones or seven coins. It may be also called a "Discontinued Quantity".
Magnitude
a quantity that is whole and continuous, as a field, a circle, the universe, and so on. It is also called a "Continued Quantity".
axiom
a self-evident statement, that is, one that does not need to be demonstrated.
prism
a solid figure that has two congruent, parallel polygons as its bases. Its sides are parallelograms.
theorem
a statement that needs to be demonstrated and is called in Latin demonstrandum.
Number
a term that expresses quantity definitely and particularly, such as one, five, seven, and so on.
prime number
a whole number that has only one set of factors, itself and 1.
a) Consecutive integer Set A has an integer mean... Does this set have an EVEN or ODD number of elements? Why? b) Consecutive integer Set B has an integer mean, + 1/2. Does this set have an EVEN or ODD number of elements? Why?
a) ODD. If the number of elements is odd, the set of consecutive integers will have an integer mean. This is because there is only one 'middle term' in a set with an odd number of elements. b) EVEN. If the number of elements is EVEN, the set of consecutive integers will have an integer mean + 1/2. This is because there is are 2 'middle terms' in a set with an even number of elements, and thus the median is the average of these 2 middle terms, which is a number/2 therefore resulting in an integer, + 1/2 any set of consecutive integers must have either an integer mean (if the number of integers is odd) or a mean that is an integer + 1/2 (if the number of integers is even).
0.625
5/8 --> Decimal ?
Line Segments
A Polygon is a closed shape formed by
expression
A combination of numbers and variables connected by one or more operations signs
ratio
A comparison of the two values of two numbers
TERMINATING DECIMAL
A decimal which ends without repeating e.g. 0.2 , 0.47, 0.375
scale drawing
A drawing of an object that is different in size (usually smaller than the original) but keeps the same proportions
exactly the same portion
A fraction such as 12/16 might look a lot different from 3/4, but it represents
improper fraction
A fraction with a numerator that is larger than or equal to its denominator.
lowest terms
A fraction with all common factors (other than 1) factored out of the numerator and denominator
radius
A length that is half the diameter of a circle; the distance from the center of the circle to the circle itself
radius
A length that is half the diameter of a circle; the distance from the center of the circle to the circle itself.
diameter
A line segment that passes through the center of a circle and has its endpoints on the circle
diameter
A line segment that passes through the center of a circle and has its endpoints on the circle. It describes how wide the circle is.
equation
A mathematical sentence that uses an equal sign
reflection
A mirror image of a figure shown over a line of reflection
common factor
A number that is a factor of two or more numbers.
common denominator
A number that is a multiple of all denominators in a problem.
composite number
A number that is not a prime number is called a
exponent
A number that tells how many times the base is multiplied by itself
square root
A number that when multiplied by itself results in the original number
squared
A number with an exponent of 2 is often said to be
cubed
A number with an exponent of 3 is often said to be
prime number
A number with only two factors: the number itself and one.
rhombus
A parallelogram with all sides equal and congruent
rectangle
A parallelogram with four right angles
quadrilateral
A polygon that has four sides
quadrilateral
A polygon that has four sides.
pentagon
A polygon with five sides
hexagon
A polygon with six sides.
composite number
A positive whole number with more than two factors. In other words, a number that is not prime. Zero and one are neither composite nor prime.
trapezoid
A quadrilateral with one pair of parallel sides
parallelogram
A quadrilateral with two pairs of congruent, parallel sides.
rate
A ratio that compares two different types of quantities
unit ratio
A ratio that shows the cost per unit of measure
contains only a single number
A sign of grouping can be omitted when it
pyramid
A solid figure that has triangles for its sides and a polygon as its base
prism
A solid figure that has two congruent, parallel polygons as its bases. Its sides are parallelograms
cylinder
A solid figure with two congruent and parallel circular bases
1. Smallest (First) or Largest (Last) number in the set 2. The increment 3. The number of items in the set
An evenly-spaced set is fully-defined if what is known...?
one is positive and the other is negative
Adding integers that have opposite signs means
both integers are positive or both are negative
Adding integers that have the same sign means
integer values
All the positive whole numbers Zero All the negative whole numbers
integers
All whole numbers (both positive and negative) and zero.
combination of addition and subtraction
Always perform combinations of multiplication and division before
from left to right
Always perform the operations
before solving
Always try to FACTOR a quadratic equation
FALSE An ODD number divided by ANY OTHER INTEGER can NEVER produce an EVEN integer !
An ODD number divided by ANY OTHER INTEGER can produce an EVEN integer? TRUE or FALSE
TRUE An ODD number divided by an EVEN integer CANNOT produce an integer. This is because the odd number will NEVER be divisible by the factor of 2 concealed within the EVEN number.
An ODD number divided by an EVEN integer CANNOT produce an integer. TRUE or FALSE?
acute angle
An angle measuing more than 0 degrees and less than 90 degrees
obtuse angle
An angle measuring more than 90 degrees and less than 180 degrees
acute angle
An angle measuring more than zero degrees and less than 90 degrees
straight angle
An angle that measures 180 degrees
right angle
An angle that measures 90 degrees
octagon
An eight-sided polygon
proportion
An equation stating that two ratios are equal
Dividing the Polygon into triangles by cutting them into lines connecting the corners, and using the sum of the interior angles of the triangles.
Another way to find the sum of the interior angles in a Polygon, apart from using the formula, is ? E.G. a Hexagon can be divided into 4 triangles by 3 lines connecting the corners. Therefore the sum of its angles is 4(180) = 720deg.
factor
Any number multiplied to form a product. A product can be divided by one factor to find the other factor.
equal to 1 divided by that number with a positive exponent
Any number with a negative exponent
Evaluating Powers With Negative Exponents
Any number with a negative exponent is equal to 1 divided by that number with a positive exponent
equal to 1
Any number with an exponent of 0
equal to itself.
Any number with an exponent of 1
When the product in the 1's column is greater than 9
carry the 10's digit of the product to the top of the 10's column of factors.
The RATIO of ANY TWO of the following: Original, Change and New
For Data Sufficiency problems involving percent change, all you need to compute a percent change is ____ ?
When the addends have the same sign both + or both -
Step 1: Add the absolute values of the addends Step 2. Give the result the sign that is common to the addends
congruent
Having the same size and shape
equivalent
Having the same value
ONE... the number 2
How many EVEN primes are there?
When you are trying to figure out the algebraic manipulation method of solving a VIC, but get stuck, what should you do?
IMMEDIATELY switch to a number-picking strategy! N.B. NEVER give up on a VIC problem before picking numbers... Sometimes very difficult VIC problems are easily solved with test numbers.
they have the same absolute value
If 2 numbers are OPPOSITES of each other
EVEN
The SUM of n consecutive integers is NOT divisible by n if n is
ODD
The SUM of n consecutive integers is divisible by n if n is
enclosing the numbers in a pair of vertical lines | |
The absolute value of numbers is indicated by
difference
The amount that remains after one number has been subtracted from another
To convert any fraction to higher terms
multiply both the numerator and denominator by the same integer value.
Relative multitude
multitude viewed in relation to something else, as greater, smaller, half, double, and so on.
reduced fraction
no integer (except 1) that divides evenly into both the numerator and denominator.
fraction or broken number
that which is referred to Unity as a Part to a Whole as, 1 half, 2 thirds, 1 third, 3 fourths, etc..
integer or whole number
that which is referred to Unity as a Whole to a Part as, 1, 2, 3, 4, etc..
improper fraction
the absolute value of the numerator is greater than, or equal to, the absolute value of the denominator.
proper fraction
the absolute value of the numerator is smaller than the absolute value of the denominator.
Computation
the action of the mind whereby a quantity is measured by Unity or a Unit.
Always completed first
Operations enclosed in a sign of grouping
x/100
" x percent" = ?
inverse operations
Operations that do the exact opposite of each other; they undo each other (addition and subtraction, for example)
Explain why this is true: "For any x, √x2 = |x|"
"For any x, √x2 = |x|" is true because: For example, x2 = 25 and |x| = 5 share the same solution for x... Namely, x = (+/-) 5 When you square a variable x, the result is positive, no matter what the sign of the base.Remember, even exponents hide the sign of the base. Therefore, the square root of the square of the variable x (again regardless of the sign of the base) will always be positive, and therefore is equal to the absolute value of x, which again is always positive no matter whether x is positive or negative.
1 - (y/100)
"y percent less than" = ?
What is the formula for COUNTING consecutive multiples?
( (Last - First) / Increment ) + 1
What is the formula for COUNTING consecutive integers?
(Last - First + 1)
62.5%
.625 --> Percent?
0
0 to any power is equal to
5/8
0.625 --> Fraction ?
1
1 to any power is equal to
mixed fractions
11/2, 2 3/4, 6 5/8, -4 1/4
What's the sum of all the integers from 20 to 100, inclusive?
1.) Average the first and last term to find the median of the set (which equals the average) = (100 + 20)/2 = 60 2) Count the number of terms ( 100 - 20 + 1 = 81) 3. Sum = Ave. x Number of terms = 60 x 81 = 4860 Answer = 4860
proper fractions
1/2, 1/3, 2/3, -5/8
The rule for adding negative integers is the same as the rule for adding positive integers:
1: Add the absolute values of the addends 2. Give the result the sign that is common to the addends
A number is increased by 25%... what must you reduce the new number by to get the old number again?
20%... because 5/4 = 125% and 4/5 = 80% (reciprocal of 5/4)... therefore 80% of the new number is the old number, therefore you must reduce the new number by 20% to get this amount...
1. By shifting the midpoint - and re-compensating... i.e. the midpoint (x) here is -1, so you must add 1 to it to compensate. 2. find the centre of the range (the average of the endpoints) then use that to test the endpoints... 3. test the end-points in the answer choices. when they both produce an equal value, that is the correct answer.
3 ways to solve an absolute value inequality
improper fractions
3/2, 8/3, -16/5, 7/7
2
A sum of 2 primes is ODD ... One of those primes must be the number __ ?
equilateral triangle
A triangle that has three equal sides and three equal angles
right triangle
A triangle with one right angle
scalene triangle
A triangle with sides of different lengths and no two angles are the same
scalene triangle
A triangle with sides of different lengths and no two angles are the same.
isosceles triangle
A triangle with two equal sides and two equal angles
square unit
A unit for measuring area
median
A value found by ordering a group of data from least to greatest and choosing the middle value of the group.
"five squared"
A value such as 5^2 can be called
"six cubed"
A value such as 6^3 can described as
mixed number
A value that combines a whole number and a fractional amount
putting that number over 1
A whole number can be expressed as an improper fraction by
Whenever you square an equation to solve it, what should you do?
ALWAYS check the solutions you get in the original euqation! Squaring both sides can actually introduce and extraneous solution.
product
Any whole number can be expressed in terms of the
A = (D1 x D2) / 2
Area of a Rhombus is? The diagonals of a rhombus are ALWAYS perpendicular bisectors (meaning they cut each other in half at a 90deg. angle).
Step 2 of Converting Improper Fractions to Mixed Numbers
Assemble the mixed number. The whole-number part of the mixed number is the whole-number part of the quotient. The numerator of the fraction part of the mixed number is the remainder from the quotient. The denominator of the fraction part of the mixed number is the denominator of the original improper fraction.
Change / Original Formula?
CHANGE + - ORIGINAL = NEW
PERCENT CHANGE
CHANGE/ORIGINAL =
Be careful not to assume that a quadratic equation always has TWO SOLUTIONS. Always FACTOR quadratic equations to determine their solutions. This will enable you to see whether a quadratic equation has ONE or MORE solutions.
Be careful not to assume that a quadratic equation always has _____ _____. Always _____ quadratic equations to determine their solutions. This will enable you to see whether a quadratic equation has ____ or ____ solutions
Whole Numbers
Begins with zero and counts upward through tens, hundreds, thousands, millions, and so on. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ... The scale on the number line begins with zero and runs to the right ("from zero to infinity").
prime factorization
Breaking down a composite number until all of the factors are prime
CHANGE + - ORIGINAL = NEW
Change / Original Formula
EVEN and ODD
Consecutive Integers alternate between ___ and ___ ? e.g. 2, 3, 4, 5, 6, 7 E,O,E,O,E
How do you find out, easily, if one fraction is bigger than another?
Cross-multiply
INCREASES the value.
Decreasing the DENOMINATOR of a fraction INCREASES/DECREASES the value?
Involves: 1. Picking numbers for all or most of the unknowns in the problem 2. Using those numbers to calculate the ANSWER (i.e. the TARGET) to the problem 3. Plugging in each number you've picked into each answer choice to see which answer choice yields the same value as your target.
Describe the VIC solving method of Picking Numbers & Calculating a Target... When is this method useful?
1. Pick numbers for each variable. Can be helpful to use a chart. 2. Answer the question, walking through the logic with the numbers that we've picked. This answer is the TARGET. 3. Test EACH answer choice, EVEN if you've already found one that equals your target value...
Describe the steps for solving a VICS problem using "Pick Numbers & Calculate a Target"
Step 1 of Converting Improper Fractions to Mixed Numbers
Divide the denominator into the numerator. Use ordinary whole-number division that produces a quotient and a remainder.
Inserting a zero at the left end of a whole number
Does not affect its value at all. Zeros that are used at the left end of a number are called leading zeros, and are used only for special reasons.
EVEN
EVEN +/- EVEN = ? e.g. 10 + 20 = 30 e.g. 2 + 6 = 8
EVEN, ODD or NON-INT
EVEN / EVEN = ? e.g. 12/2 = 6 e.g. 12/4 = 3 e.g. 12/8 = 1.5
EVEN or NON-INT
EVEN / ODD = ? e.g. 12/3 = 4 e.g. 12/5 = 2.4
EVEN div. by 4
EVEN X EVEN = ? ... and is div. by ?
AXIOM VI.
Every lesser homogeneous number is contained in a greater either as an aliquot or an aliquant part.
AXIOM VIII.
Every lesser number is contained in a greater more than once.
AXIOM VII.
Every number is contained in itself once.
AXIOM II.
Every quantity is equal to itself.
AXIOM I.
Everything may be assumed as unity.
commutative law of multiplication
Factors may be multiplied in any order.
similar figures
Figures that have the same shape but different sizes; their sides are proportional, while their corresponding angles are equal
When numbers do not divide evenly:
Find the largest number of times the divisor will divide into the dividend. This is the quotient. To determine the remainder, multiply the quotient by the divisor, then subtract the result from the dividend.
Because sum = ave. x # items... the average for an ODD # items is an integer, so the SUM is a MULTIPLE of the number of items. e.g. The average of {13,14,15,16,17} is 15, so 15 x 5 = 13 + 14 + 15 +16 + 17 i.e. 13 + 14 + 15 + 16 + 17 = 15 x 5
For a set of consecutive integers with an ODD number of items, the sum of ALL the integers is ALWAYS a multiple of the number of items... Why is this so?
EVERY integer between 1 and X, inclusive, must be a factor of X
For there to be X unique factors of X, what must be true?
When should you use fractions and when should you use decimals ?
Fractions * Use to cancel factors. * Also fractions are the best way of exactly expressing proportions that don't have clean decimal equivalents such as 1/7. * In some cases it might be easier to compare a bunch of fractions by giving them all a common denominator rather than converting them all to decimals or percents. Decimals/Percents * Use to estimate or compare quantities - the implied denominator is 100 so you can easily compare percents (of the same whole) to each other.
The new qty. is (100 + x)% of the original... i.e. a 15% increase produces a quantity that's 115% of the original... I.E. ORIGINAL*(1 + PCT INCREASE/100 ) = NEW
If a quantity is increased by x percent, then what, in algebraic terms, is the new quantity as a percent of the original?
the Complement of that part to the whole
If any one Part of a Whole is assumed, then the rest of the parts are called the Complement of that part to the whole.
2^3 = 8
If there are 3 EVEN integers in a set of integers being multiplied together, what is the result divisible by (in terms of power/base)? e.g. 2 x 5 x 6 x 10 = 600 E x O x E x E = div. by 23
When will a decimal terminate and why?
If, after being fully reduced, the denominator ONLY has factors of 2 and/or 5, the decimal WILL TERMINATE
When will a decimal NOT terminate and why?
If, after being fully reduced, the denominator has any prime factors OTHER than 2 or 5, the decimal WILL NOT TERMINATE. This is because a terminating decimal ONLY arises as a result of an integer being divided by a power of 10. i.e. the denominator should only have prime factors of 2 and/or 5
dividend
In a division problem, it's the number being divided
divisor
In a division problem, the number that an amount is divided by
mode
In a group of values, the value that occurs most often
mode
In a group of values, the value that occurs most often.
cross product
In an equation made up of two fractions, the numerator of one fraction times the denominator of the other fraction.
roots of numbers
In order to understand and use exponents that are fractions or decimals, you must first know about
BOTH
Is ZERO positive or negative, both, or none?
Equality
Is the agreement of things in Quanity.
The SUM of n consecutive integers is divisible by n. What does this tell us about n, and why?
It means that n is ODD. This is because the sum of n consecutive integers divided by n is the average/mean of that set of integers. Because the average is itself an integer, n can only be odd. This is because the average of an odd number of consecutive integers will always be an integer, because the median/ave. (i.e. "middle number") will be a single integer.
parallel lines
Lines in the same plane that do not intersect. The symbol //
angle
The figure formed when two rays meet at a common endpoint called a vertex.
When a denominator of a fraction is 9, 99, 999 or another power of 10 minus 1, what's an easy way of determining the REPEATING DIGITS of the decimal equivalent of the fraction?
Look at the numerator... This will give you the repeating digits (perhaps with leading zeroes) if the denominator of the fraction is 1 less than a power of 10. e.g. 1/11 = 9/99 = 0.09 (rep.) 3/11 = 27/99 = 0.27 (rep.) 4/9 = 0.4 (rep.) 23/99 = 0.23 (rep.)
Long Division - Dividing by two digit numbers
Make use of estimation to assist in finding the quotient. Do this by rounding both the target digits of the dividend and the factoring divisor. For example turn 18 into 20 and 147 into 150 and divide 150/20 in your head to get 7.5 ish. This being the case guess how close 18 will go into 147 and try it out.
Shift DP 2 places right
Method: convert Decimal to Percent?
Shift DP 2 places left
Method: convert Percent to Decimal?
A = (Diagonal1 x Diagonal2) / 2
The formula for the: Area of a Rhombus is?
Arithmetic, Music, Geometry and Astronomy
The four Mathematical arts are:
Multiply the numerator of a positive, proper fraction by 1/2 INCREASE.
Multiply the numerator of a positive, proper fraction by 1/2
Step 1 of Converting mixed numbers to improper fractions
Multiply the whole number by the denominator and add the numerator. This becomes the numerator of the improper fraction.
2
Multiplying several EVEN integers together results in higher and higher powers of ...? Because each even number will contribute at LEAST one 2 to the factors of the product
ODD
ODD +/- ? = EVEN e.g. 3 + 5 = 8 e.g. 13 + 19 = 32
EVEN
ODD +/- ____ = ODD
NON-INT
ODD / EVEN = ? e.g. 9/6 = 1.5
ODD or NON-INT
ODD / ODD = ? e.g. 15/5 = 3 e.g. 15/25 = 0.6
EVEN
ODD x ? = EVEN
ODD
ODD x ODD = ? e.g. 3 x 3 = 9 e.g. 5 x 11 = 55 e.g. 9 x 3 = 27
NEW
ORIGINAL + CHANGE =
Percent Decrease Formula
ORIGINAL x (1 - x/100) = NEW
Greater
Of two Unequal Magnitudes, one that has a part Equal in Magnitude with the Whole of the other Magnitude.
More
Of two Unequal Multitudes, one that has a part equal in Multitude with the Whole of the other Multitude.
x = (+-) a
Once we have an equation of the form |x| = a, and x>0, what do we know about x ?
farther to the right on the number line.
One number is said to be greater than (>) another when it is
farther to the left on the number line
One number is said to be less than (<) another when it is
quadrant
One of the four regions formed by the intersection of the axes of a coordinate graph
PERCENT/100
PART/WHOLE =
sample
Part of the population that is studied to find the characteristics of the whole population.
ORIGINAL x (1 - x/100) = NEW
Percent Decrease Formula ?
What must you do in a VIC problem, using the Pick Numbers and Calculate a target strategy, when you CANNOT pick a value for each variable?
Pick a value for ALL BUT ONE of the variables and then solve for the value of the remaining variable. THEN, plug the numbers we've selected into the original expression to get the TARGET value, and TEST EACH ANSWER CHOICE.
Profit = Revenue ($) - Cost ($)
Profit = ?
AXIOM III.
Quantities which are both equal to one and the same third are equal to one another.
inverse
Reversed position or direction
order of operations
Rules that tell which steps to follow when solving an expression
order of operations
Rules that tell which steps to follow when solving an expression.
DECREASES
SQUARING a positive proper fraction/percent INCREASES/DECREASES the value? e.g. 1/4 x 1/4 = 1/16
How do you take a power or a root of a decimal? Give some examples of each... Also, what shortcuts can be deduced from this?
Split the decimal into 2 parts: an integer, and a power of ten... e.g. (0.5)4 = (5x10-1)4 = 54x10-4 = 625 x 10-4 = 0.0625 e.g. 3√0.000027 = (27x10-6)1/3 = 271/3x10-2 = 3x10-2 = 0.03 You can take a shortcut by counting decimal places. For example, the number of decimal places in the result of a cubed decimal is 3 times the number of decimal places in the original decimal. ALSO, the number of decimal places in a cube root is 1/3 the number of decimal places in the original decimal...
Step 2 of Converting mixed numbers to improper fractions
Set the denominator of the improper fraction equal to the denominator of the fraction in the mixed number
Inserting a zero at the right end of a whole number
Shifts all the others upward one place value. The result is exactly ten times larger than before the zero is added.
The inner group is enclosed in parentheses ( ) The outer group is enclosed in brackets [ ]
Signs of grouping may be nested — signs of grouping placed within other signs of grouping.
the ratio of integers that results in a terminating decimal
Some integer/Some Power of 10 (i.e. some integer that can be expressed in the form 2^x5^y where x and y are integers) e.g. 0.2 = 2/10 = 1/5 e.g. 0.375 = 375/1000 = 3/8
Homogenous or Heterogenous
Species and Number may be
Known (Given) and Unknown (Sought)
Species are distinguished into
lowercase letters
Species of Quantities are signified by
To add integers that have the same sign (both positive or both negative):
Step 1: Add the absolute values of the addends Step 2. Give the result the sign that is common to the addends
To add integers that have the same sign both positive or both negative:
Step 1: Add the absolute values of the addends Step 2. Give the result the sign that is common to the addends
supplementary angles
Two angles whose sum is 180 degrees
There are two parts in the procedure for subtracting signed integers:
Step 1: Change the subtraction sign to the addition sign, and then switch the sign of the subtrahend the number that immediately follows the operation sign you just changed. Step 2: Add the result according to the procedures for adding signed integers.
Subtracting Signed Integers
Step 1: Change the subtraction sign to the addition sign, and then switch the sign of the subtrahend the number that immediately follows the operation sign you just changed. Step 2: Add the result according to the procedures for adding signed integers.
Converting Improper Fractions to Mixed Numbers
Step 1: Divide the denominator into the numerator. Use ordinary whole-number division that produces a quotient and a remainder. Step 2: Assemble the mixed number. The whole-number part of the mixed number is the whole-number part of the quotient from Step 1. The numerator of the fraction part of the mixed number is the remainder from the quotient in Step 1. The denominator of the fraction part of the mixed number is the denominator of the original improper fraction.
When solving combinations of addition, subtraction, multiplication, and division in the same expression:
Step 1: Do the multiplication and division first, from left to right. Step 2: Do the addition and subtraction last, from left to right.
Reducing: The Brute-Force Method
Step 1: Find any integer greater than 1 that can be divided evenly into both the numerator and denominator. Step 2: Divide the numerator and denominator by the integer from Step 1. Repeat Steps until the fraction is completely reduced. Can the result be reduced?
Converting mixed numbers to improper fractions.
Step 1: Multiply the whole number by the denominator and add the numerator. This becomes the numerator of the improper fraction. Step 2: Set the denominator of the improper fraction equal to the denominator of the fraction in the mixed number.
When the addends have opposite signs one is + and the other is -
Step 1: Subtract the absolute values of the addends Step 2. Give the result the sign of the addend that has larger absolute value
To add integers that have opposite signs:
Step 1: Subtract the absolute values. Step 2. Write the sum with the sign of the larger number.
area
Surface space that is measured in square units
area
Surface space that is measured in square units.
factors of the multiplication operation
Taken together, the multiplicand and multiplier are known as
1 and itself
The ONLY possible factors for a prime number are
n! According to the Factor Foundation Rule, every number is divisible by all the factors of its factors. The product of any set of n consecutive integers is divisible by n. e.g. the product of 3 consecutive integers will always be a multiple of 3, and a multiple of 2, and a multiple of 1, therefore 3 x 2 x 1 = 6 e.g. the product of 5 consecutive integers will always be a multiple of 5 (as one of those integers will always be a multiple of 5), and a multiple of 4 (i.e. two 2's), and a multiple of 3, and a multiple of 2, therefore 5 x 4 x 3 x 2 x 1 = 6
The PRODUCT of n consecutive integers is divisible by ? Why?
The BOTTOM side of the triangle
The base of a triangle refers to?
the denominator of the original improper fraction
The denominator of the fraction part of the mixed number is
range
The difference between the least and greatest values in a set of numbers
range
The difference between the least and greatest values in a set of numbers.
circumference
The distance around a circle (the perimeter of a circle)
circumference
The distance around a circle (the perimeter of a circle).
perimeter
The distance around a figure
perimeter
The distance around a figure.
whether the addends have the same sign or opposite signs
The exact procedure for adding signed integers depends upon
Axiom IX.
The greater any number is in comparison to another, the more equal parts will it contain of that other.
x-axis
The horizontal number line of a coordinate graph
reciprocal
The inverse of a fraction; when multiplied by the original fraction, it results in a product that equals one
greatest common factor (GCF)
The largest single factor for two or more numbers.
probability
The likelihood that an event will occur. The probability that an event will occur is 0, 1, or somewhere between 0 and 1.
denominator
The lower number in a fraction is the
Axiom X.
The nearer any lesser number approaches a greater number, the less often will it be contained in that greater number.
If a quantity is decreased by x percent, then what, in algebraic terms, is the new quantity as a percent of the original?
The new qty. is (100 - x)% of the original... i.e. a 15% decrease produces a quantity that's 85% of the original... I.E. ORIGINAL*(1 - PCT INCREASE/100 ) = NEW
dividend
The number being divided is called the
multiplicand
The number being multiplied is called the
divisor
The number doing the dividing is called the
multiplier
The number to be multiplied by is called the
quotient
The numerator of the fraction part of the mixed number is the remainder from the
numerator
The part of a fraction that stands for how many parts of a whole or group are included in the fraction.
denominator
The part of a fraction that stands for the number of equal parts a whole or group is divided into.
vertex (vertices - plural)
The point of intersection for two sides of a plane figure, three sides of a solid figure, or the endpoints of two rays that form an angle.
factoring
The process of breaking a number down into its factors is called
k! e.g. 4x3x2x1
The product of k consecutive integers is ALWAYS divisible by what?
change the operation from subtraction to addition
The purpose of the first step in Changing Integer Subtraction to Integer Addition is to
percent
The ratio of a number to 100 (per one hundred). The symbol %
percent
The ratio of a number to 100 (per one hundred); the symbol %
quotient
The result of dividing one number by another; the solution to a division problem
product
The result of muliplying two or more numbers
product
The result of multiplying two or more numbers.
quotient
The result of the division called the
product
The result of the multiplication is called the
least common multiple (LCM)
The smallest multiple that two or more numbers have in common
mean
The sum of a group of numbers divided by the number of numbers. Also known as the average.
mean
The sum of a group of numbers divided by the number of numbers; also known as the average
sum
The total of two or more numbers being added
Magnitude at Rest and Magnitude in Motion
The two kinds of Magnitude are
Absolute and Relative
The two kinds of Multitude
Multitudes and Magnitudes
The two kinds of Quantity are
Aliquot and Aliquant Parts
The two kinds of parts
numerator
The upper number in a fraction is the
x-coordinate
The value on the x-axis used to locate a point on the coordinate graph. It is the first value in an ordered pair.
y-coordinate
The value on the y-axis used to locate a point on the coordinate graph. It is the second value in an ordered pair.
pi
The value that shows the relationship of a circle's circumference to its diameter; it has an approximate value of 3.14
y-axis
The vertical number line of a coordinate graph
AXIOM V.
The whole is equal to all of its parts taken together.
AXIOM IV.
The whole is more or greater than its part.
quotient
The whole-number part of the mixed number is the whole-number part of the
The Decimal Numbering System
The whole-number system uses only ten characters -0 through 9. They are the characters of our familiar decimal numbering system. Note that the deci- in decimal means ten -- the total number of digits (fingers and thumbs) on our two hands. 0 1 2 3 4 5 6 7 8 9 Every number we might ever want to express can be written as a combination of these ten, simple digits.
referred to the same Unit
Things are Equal in Magnitude when they are
referred to Unity in the same way
Things are Equal in Multitude when they are
Adding negative integers will always produce a negative sum
This is an addition problem. Although the addends both have negative values, you still add their absolute values.
the same point on the number line
Two numbers are said to be equal (=) when they are at
polygon
Three or more line segments in a plane that forms a closed figure. The line segments never cross but meet at their endpoints.
Explain how the Last Digit Shortcut works, using this example: (72)(33)(92)
To find the units digit of a product, or a sum of integers, ONLY pay attention to the units digit of the numbers you're working with. Drop any other digits. This shortcut works because only units digits contribute to the units digit of the product. e.g. (72)(33)(92): Step 1: 7x7 = 49 Step 2: 9x9 = 81 Step 3: 3x3x3 = 27 Step 4 (final step): 9x1x7 = 63
simplify
To make a fraction easier to work with by taking out common factors
simplify
To make a fraction easier to work with by taking out common factors. In an expression, combining variables that have like unknowns.
Total Earnings ($) = Wage Rate ($ per hr) x Hrs worked
Total Earnings ($) = ?
Total Sales or Revenue = Unit Price x Qty. Sold
Total Sales or Revenue = ?
UnitPrice ($/unit) x Qty.Purchas'd (units)
TotalCost($) =
Fill in the missing parts of the following equations: Total Cost ($) = ? Total Sales or Revenue = ? Profit = ? Unit Profit = ? or Sale Price = ? Total Earnings ($) = ?
TotalCost($) = UnitPrice ($/unit) x Qty.Purchas'd (units) Total Sales or Revenue = Unit Price x Qty. Sold Profit = Revenue ($) - Cost ($) Unit Profit = Sale Price - Unit Cost Sale Price = Unit Cost + Markup Total Earnings ($) = Wage Rate ($ per hr) x Hrs worked
Explain the concept of trading decimal places and how it works: e.g. 0.0003 x 40,000
Trading decimal places refers to moving the decimals in the opposite direction the same number of places, when multiplying a very large number and a very small number. The reason this technique works is that you're multiplying, and then dividing, by the same power of ten. i.e., you're trading decimal places in one number for decimal places in another. e.g. 0.0003 x 40,000 = (3 x 10-4) x (4 x 104) = 3/10,000 x (4 x 10,000) = 3 x 4 = 12
complementary angle
Two angles whose sum equals 90 degrees
complementary angles
Two angles whose sum equals 90 degrees.
always clear the innermost groups first
When working with nested signs of grouping
Equal
Whole is equal in Multitude to a Part of the other.
Because they hide the sign of the base, and can have a POSITIVE and a NEGATIVE solution!
Why are EVEN EXPONENTS dangerous?
when there are explicit or implicit equations in the problem:
You can NEVER pick a value for EVERY variable e.g. when the variables are related to each other through an equation
Whole
a collection of things taken as a Unity. A bushel of wheat is a whole.
Unity, or a Unit
a known quantity we refer to as One.
absolute value
an integer is its value without regard to the sign, Or is its distance from the origin (zero) on the number line.
Quantity
anything that may be increased or diminished
Negative-value integers
are located to the left of the zero on the integer number line. Negative-value integers use the same symbols as the whole number system, but are distinguished by the use of a negative sign ( - ). Numbers 5 and - 5, for example, might resemble one another in most respects, but they are two entirely different values. Refer to the integer number line, and you will see that 5 and - 5 are located in two entirely different places.
Positive-value integers
are located to the right of the zero on the integer number line. Positive integers are sometimes indicated with a positive sign ( + ). More often, however, we omit the positive sign. So when you see an integer value that does not have a sign, you can rightly assume it is a positive value.
Parts
are those things collected in a whole.
Unknown quantities
by the first letters of the alphabet (a, b, c, d, etc..)
Known quantities
by the last letters (u, x, y, etc.)
Composite number
can have many different combinations of factors
mixed number
expresses fractional parts that are greater than 1.
Zero (0)
has no sign value
mixed fraction
includes an integer as well as a fractional part
mixed number
includes an integer as well as a fractional part
exponent
indicates the number of times the base is to be multiplied
base
indicates the number to be multiplied
The height of a triangle
is ALWAYS perpendicular (at 90 deg. to) the base!
Aliquant Part
is a part which, being repeated a number of times, always exceeds or falls short of the whole, as 5 is of the numbers 8 and 12.
Aliquot Part
is a part which, being repeated a number of times, becomes equal to the whole; as 4 is of the numbers 8 and 12.
Species
is a term used to express quantity indefinitely and universally, such as "a certain number", "some" etc...
Any value divided by one
is equal to the original value
Any value multiplied by one
is equal to the original value. a x 1 = 1
Zero multiplied by any value
is equal to zero. 0 x a = 0
Inequality
is the disagreement of things in Quantity.
Reducing fractions
is the opposite of raising fractions to higher terms.
Zero divided by any whole number (except 0)
is zero
Proof
or demonstration, is a connection of arguments used to demonstrate the truth or falsehood of a statement.
dividing the numerator and denominator by the same number
reducing fractions is
Reducing by the Largest Common Factor (LCF)
the largest integer that can be divided evenly into both the numerator and denominator.
Power notation
the method for indicating the power of a number—has two parts, the Base and the Exponent
improper fraction
the numerator is greater than, or equal to, the denominator
proper fraction
the numerator is smaller than the denominator
Some integer/Some Power of 10
the ratio of integers that results in a terminating decimal
Mathematics
the study of quantity
Quantity is expressed
the terms Species and Number
A plus sign (+) is used for two entirely different purposes:
to indicate the addition operation to indicate a positive integer value
A minus sign ( - ) is used for two entirely different purposes:
to indicate the subtraction operation to indicate a negative integer value
Different Units
units that are not understood under the same notion, such as a pound of stones and a ton of stones, or an inch of string and a foot of string.
Same units
units that are understood under the same notion, such as a pound of stones and a pound of feathers, or an inch of string and an inch of wood.
Fractions allow you to plot values between whole numbers and integers.
where fractions occur on a number line?
ODD
x and y are primes... What values (ODD/EVEN) must x and y be for x + y = ODD?
