Download prealgebra-review concepts

PREALGEBRA REVIEW DEFINITIONS
1. Subtraction is the inverse of addition:
A. If a - b = c, then c + b = a
B. So if 10 - 3 = 7, then 7 + 3 = 10
2. Division is the inverse of multiplication:
A. If a ÷ b = c, then c × b = a
B. So if
45 ÷ 9 = 5, then 5 × 9 = 45
3. Addition and multiplication follow the commutative properties:
A. a + b = b + a, so 8 + 3 = 3 + 8
B. a × b = b × a, so 8 × 3 = 3 × 8
4. Affixing zeroes to the left of a number or to the right of a decimal point will not change the number:
A. So 84 = 084 = 0084 = 84.0 = 84.00, and so on.
B. Also, 26.9 = 026.9 = 26.90 = 26.900, and so on.
C. However, note: 84 does not equal 804, 4.7 does not equal 4.07, and 5.3 does not equal 50.3
5. To multiply a number by 10, 100, and so forth, simply attach the appropriate number of zeroes to the right of the number, or simply
move the decimal point the appropriate number of places to the right.
A. So 26 × 10 = 260, and 95 × 100 = 9,500
B. Also,
8.17 × 10 = 817
. , and 314
. × 100 = 314
6. To divide a number by 10, 100, and so forth, simply move the decimal point of the number the appropriate number of decimal places
to the left:
A. So 34 ÷10 = 3.4, and 691.5 ÷ 100 = 6.915
7. When dividing, you may cancel an equal number of “right-hand” zeros:
A. So 600 ÷ 10 = 60, and 8,000 ÷ 200 = 80 ÷ 2 = 40
8, A PERCENT (%) means “parts of 100.” 63% means 63 parts of 100, or 63/100. To convert to a percent, just multiply by 100. If the
number is a decimal, use the rule from #6 to multiply by 100 and attach the % sign .02 = .02 X 100 = 2%
To convert from a percent to a decimal, just drop the % sign and divide by 100. 2% = 2/100 = .02
9. ORDER OF OPERATIONS (PEMDAS)
Please: do all operations within parentheses and other grouping symbols (such as [ ], or operations in numerators and denominators of
fractions) from innermost outward.
Excuse: calculate exponents
My Dear: do all multiplications and divisions as they occur from left to right
Aunt Sally: do all additions and subtractions as they occur from left to right.
Example:
20 – 2 + 3(8 - 6)2 Expression in parentheses gets calculated first
= 20 – 2 + 3(2)2 Next comes all items with exponents
=
20 – 2 + 3(4) Next in order comes multiplication. Multiplication and Division always come before addition or division, even if to the
right.
= 20 – 2 + 12 Now when choosing between when to do addition and when to do subtraction, always go from left to right, so do 20-2
first, because the subtraction is to the left of the addition.
= 18 + 12 Now finally we can do the addition.
= 30
10. The Test of Reasonableness
In general applying a test of reasonableness to an answer means looking at it in relation to the numbers operated upon to determine if
it’s “in the ballpark.” Put in simple terms, you look at the answer to see if it makes sense. For example, if you determine 10% of $75 to
$750, you should immediately notice something is very wrong, because 10% of something is much smaller than the original amount.
The test of reasonableness comes very much into play in word problems. For example, questions asking for length, dollars, or time
should never give negative answers because those things would not make sense if they were negative (what is -3 feet?). Also, think
about what the answer should be. If I invest $400 dollars in a bank at 8% interest for 3 years, I would expect the balance to be larger
than the original amount that I invested (that’s why we put money in the bank!), so an answer that is smaller than $400 is obviously
wrong.
PREALGEBRA REVIEW DEFINITIONS
Factor - One of two or more quantities that divides a given quantity without a remainder. For example, 2 and 3 are factors of 6; a and b
are factors of ab.
Factor X Factor = Product
Product - The number or quantity obtained by multiplying two or more numbers together.
Factor X Factor = Product
Dividend - A quantity to be divided.
Divisor - The quantity by which another quantity, the dividend, is to be divided.
Quotient - The number obtained by dividing one quantity by another. In 45 ÷ 3 = 15, 15 is the quotient.
Dividend/ Divisor = Quotient
or Dividend ÷ Divisor = Quotient
quotient
divisor dividend
Remainder - The number left over when one integer is divided by another: The remainder plus the product of the quotient times the
divisor equals the dividend.
If there is a remainder, then
Dividend = Quotient X Divisor + Remainder
Least Common Multiple (LCD) - The smallest quantity that is divisible by two or more given quantities without a
remainder: 12 is the least common multiple of 2, 3, 4, and 6. Also called lowest common multiple.
Greatest Common Factor (GCF) - The largest number that divides evenly into each of a given set of numbers. The
greatest common divisor is useful for reducing a fraction into lowest terms. Consider for instance
where we cancelled 14, the greatest common factor of 42 and 56.
Prime number is a whole number that has exactly two factors: itself and 1.
Composite number is a whole number that has more than two factors.
PREALGEBRA REVIEW DEFINITIONS
A set is a collection of objects. The objects in the set are called the elements of the set. The roster method of
writing sets encloses a list of the elements in braces.
Example: The set of even natural numbers less than 10 can be written like this: {2,4,6,8}.
The set of natural numbers is {1,2,3,4,5,6,7,….}. These are basically the “counting numbers.”
The set of whole numbers is the set of natural numbers and the number, 0. {0, 1, 2, 3, 4, 5, ….}
The set of integers is the set of whole numbers and their opposites. {…. -4,-3,-2,-1,0,1,2,3,4,….}
The number 0 is an integer, but it is neither negative nor positive. For any two different places on the number line, the
integer on the right is greater (>) than the integer on the left.
The absolute value (using the | | symbol) of a number is its distance from zero on the number line. The absolute value
of a number is ALWAYS POSITIVE (or 0).
| 5| = 5,
|-5| = 5,
|3-8| = |-5| = 5, |3-3| = |0| = 0
|-4|=4
-4 is 4 units away
from 0
|4|=4
+4 is 4units away
from 0
PREALGEBRA REVIEW DEFINITIONS
Adding and Subtracting Integers
When adding two integers with the same sign, just ignore the signs, then attach them on the answer.
-3 + -5 = - (3 + 5) = -8
When adding two integers with different signs, take the absolute values (make both numbers positive), and subtract the smaller one
from the larger one. The sign of the integer with the larger absolute value will be the sign of your answer.
7 + (-8)
The absolute value of 7 is |7| = 7
The absolute value of -8 is |-8| = 8 larger
8– 7=1
Remember in the original problem, the integer whose absolute value was 8 was -8, so our answer is negative.
7 + (-8) = -1
Subtracting a negative integer is the same as ________ a _______ integer.
3 – (-5) = 3 + 5 = 8
-2 – (-3) = -2 + 3 = 1
Subtracting a positive integer is the same as _____ a _____ integer.
5 – 3 = 5 + (-3) = 2
-2 – 3 = -2 + (-3) = 1
Multiplying and Dividing Integers
(positive integer) X (positive integer) = (_____ integer)
(positive integer) ÷ (positive integer) = (______ integer)
(negative integer) X (negative integer ) = (______ integer)
(negative integer) ÷ (negative integer ) = (_______ integer)
(positive integer) X (negative integer) = (______ integer)
(positive integer) ÷ (negative integer) = (_______ integer)
(negative integer) X (positive integer ) = (negative integer)
(negative integer) ÷ (positive integer ) = (negative integer)
PREALGEBRA REVIEW DEFINITIONS
Memory Tip:
When something bad happens to a good person, that’s bad.
When something good happens to a bad person, that’s bad.
When something good happens to a good person, that’s good.
When something bad happens to a bad person, that’s good.
When a negative integer is raised to an even power, the result is _____.
(-2)4= (-2)(-2)(-2)(-2) = 16
When a negative integer is raised to an even power, the result is ______.
(-2)5= (-2)(-2)(-2)(-2)(-2) = -32
Exponents are only applied to the number directly diagonally left of it. If a negative integer is raised to a power, it must be in
parentheses () in order for the exponent to apply to the negative number.
(-2)2 ? - 22
(-2)2 = (-2)(-2) = 4
- 22 = - (2)(2) = -4
PREALGEBRA REVIEW DEFINITIONS
Exponent - A number or symbol, as 3 in (x + y)3, placed to the right of and above another number, variable, or expression (called the
base), denoting the power to which the base is to be raised. Also called power.
The exponent (or power) tells how many times the base is to be multiplied by itself.
Example 1:
(x + y)3 = (x + y)(x+y)(x+y)
Example 2:
(-3)4 = (-3)(-3)(-3)(-3) = 81
Properties of Exponents
If m and n are integers and
If m and n are integers, then
x x =x
m
n
m+ n
x ≠ 0 , then
xm
= x m− n
n
x
If m and n are integers, then
( x m ) n = x m⋅n
If x is a real number and
0
x ≠ 0 , then
If n is a positive integer (-n is negative) , and
then
x −n =
x =1
1
xn
and
If n is an integer and
If m, n, and p are integers, then
( xy ) = x y , and
( x m y n ) p = x m⋅ p y n⋅ p
n
n
n
n
x ≠ 0,
1
= xn
−n
x
b ≠ 0 , then
a
 a
  = n
 b
b
n
If n is a positive integer (-n is negative) , and
then
−n
n
 a
 
 b
 b
= 
 a
b ≠ 0,
PREALGEBRA REVIEW DEFINITIONS
FRACTIONS, DECIMALS , and PERCENTS
A fraction is just a division problem.
Numerator
= Numerator ÷ Denominator
Denominato r
A fraction is in LOWEST TERMS when the numerator and denominator have no common factors.
6
is not in lowest ter ms,
10
3
is in lowest ter ms
5
3
Mixed Number - A number, such as 6 , consisting of an integer and a fraction. A mixed number is just the sum of a whole
5
number and a fraction.
Improper Fraction - A fraction in which the numerator is larger than the denominator. Converting from mixed number to improper
fraction:
3 6 ⋅ 5 + 3 33
6 =
=
5
5
5
Mixed numbers must be converted to improper fractions or decimals before doing ANY MULTIPLICATION OR DIVISION
OPERATIONS on them.
A fraction can be converted into a decimal by dividing: Numerator ÷ Denominator
denominato r numerator
When ADDING or SUBTRACTING FRACTIONS, they must have the same denominator, then you just add the numerators and leave
the denominator the same.
1 3
+ =
6 8
The denominators, 6 and 8, are not the same, so we must find the LEAST COMMON DENOMINATOR to convert these fractions into
equivalent ones with the same denominator. The LCM of 6 and 8 is the SMALLEST NUMBER THAT BOTH 6 and 8 can go
into. Choose the larger denominator (which is 8 in this case) and start taking multiples until 6 can go into it.
Does 6 go into 8? NO
Does 6 go into 8x2? NO
Does 6 go into 8x3? Yes, 6 goes into 24. Therefore 24 is the LCM.
Multiply the numerator and denominator of each fraction by whatever it takes to get the LCM as the new denominator.
Then once you have the same denominators in each fraction, just add the numerators and leave the denominator the same.
1  4 3 3 4
9 13
+
=
 +  =
6  4  8  3  24 24 24
MULTIPLYING FRACTIONS
Multiplying Fractions uses a different rule. When multiplying fractions, you multiplying the numerators AND the denominators.
3
 1  3  1 × 3
=
   =
 6  8  6 × 8 48
This fraction can simplified by canceling out common factors in the numerators and denominators (tops and bottoms)before
multiplying across. 6 can be rewritten as 2x3, and the 3’s can be cancelled out.
1× 3
1
 1  3  1 × 3
=
=
   =
 6  8  6 × 8 2 × 3 × 8 16
DIVIDING FRACTIONS
The rule for dividing fractions is simple. Just take the RECIPROCAL(flip the top and bottom) of the DIVISOR (the fraction to the right of
the ÷ symbol) multiply
 1   3   1   8  1× 8 1× 2 × 4 4
=
=
  ÷  =  ×   =
 6   8   6   3 6× 3 2× 3×3 9
PREALGEBRA REVIEW DEFINITIONS
PERCENT = part of a hundred
5
= .05
5% =
100
When converting from Percent to a Decimal, drop the % sign and move the decimal place to the
LEFT TWO PLACES.
635% = 6.35
When converting from a Decimal to a Percent, move the decimal place to the RIGHT TWO PLACES and then attach a % sign to the
end.
.076 = 7.6%
When converting a fraction to a percent, first convert the fraction to a decimal (see previous page).
Percent Problems:
Amount = Percent * Base
4
is 40% of 10
Example 1:
What is 30% of 60?
Amt = Percent X Base
Amt = 30% X 60
Convert 30% to decimal.
30% = .30
Amt = .30 X 60 = 20
Amt
Percent
Base
Example 2:
200% of what is 400?
200% is the Percent
Converting Percent to a Decimal gives 200% = 2
400 is the Amount
“What” is the Base.
Base = Amt / Percent = 400 / 2 = 200
Example 3:
3 is what percent of 45?
Amt = 3
Base = 45
Percent = Amt/ Base
Percent = 3/45 = 0.06666…..
Converting to Percent using % symbol gives 6.66666…% which rounds to 6.67%
PREALGEBRA REVIEW DEFINITIONS
GEOMETRY
x- intercept - point where graph crosses x-axis. y = 0 at this point.
y-intercept – point where graph crosses y-axis. x = 0 at this point.
slope = rise/run of a line. If linear equation is in the form
y = mx + b, m is the slope, and b is the y-intercept.
y= - ½ x +1
straight angle = 180 degrees.
Adjacent angles that form a straight angle add up to 180 degrees.
A
B
A + B = 180°
right angle = 90 degrees. Complementary angles area adjacent angles that form
a right angle and add up to 90 degrees.
acute angle = less than 90 degrees
obtuse angle = greater than 90 degees.
Vertical angles (opposite angles formed from intersecting lines) are congruent (the same).
c
a
b
a 2+b 2=c 2
Pythagorean Theorem:
The sum of the squares of the legs of a
right triangle is equal to the square of the
hypotenuse.
POLYGONS
QUADRILATERALS (4-sided polygons):
Parallelogram
Rhombus
Rectangle
Square Trapezoid (only 2 sides are parallel)
(opposite sides parallel) (all sides same size) (each angle 90°)
(each angle 90°
and all sides same size)
Triangles
Scalene Triangle
(no sides the same)
Isosceles Triangle
Equilateral Triangle
(2 sides the same) (3 sides the same)
Angles of a triangle add up to 180 degrees.
Perimeter – distance around the edges of an object
Perimeter of a square = 4s, where s = length of one side
Perimeter of a rectangle = 2W + 2L, where W = width and L = length
Perimeter of a triangle = side 1 + side 2 + side 3
Area – amount of surface covered by an object.
Area of a square = s 2
Area of a rectangle = WL
Area of a triangle = ½ bh, where b = length of base and h = height
Area of a parallogram = bh b = length of base and h = height
Right Triangle
(one angle is 90°)
CIRCLES:
A circle is the set of all points that are a fixed distance from the center.
The fixed distance is called the radius, often abbreviated by the variable, r.
The diameter starts at one side of the circle, goes through the center and ends on the other side.
So the Diameter is twice the Radius:
Circle Sector and Segment
Diameter = 2 × Radius = 2r
Slices
There are two main "slices" of a
circle.
Center
The "pizza" slice is called a
Sector.
And the slice made by a chord
is called a Segment.
Given this property, you can make a formula for
the circumference by multiplying both sides of this equation
by the diameter.
Common Sectors
The Quadrant and Semicircle are two special types of Sector:
Circumfere nce
= π ( Diameter )
Diameter
Circumfere nce = π (Diameter )
Circumfere nce = π (2 r ) = 2πr
(Diameter)
Quarter of a
circle is
called a
Quadrant.
Half a circle
is called a
Semicircle.
Area of a Circle:
If you make a parallelogram out of a circle by piecing together the
slices of pie, you will use half the circumference for one base and
half the circumference for the other base (so base = ½(2pr)=pr).
The height of the parallelogram will be the radius, r.
Now we now the area of a parallelogram is base X height.
So the area of a circle is (pr)(r).
Area of a Circle = pr2