Download Multiply Decimals

Class Work #1 – M105 Fall 2011
Due: Thursday, September 29
Adding/Subtracting Decimals
Line up decimal places and add/subtract as normal, bringing down the decimal as it is crossed
Ex 1: 542  237.379
Multiply Decimals
Multiply #’s, Ignore Decimals, Count # of Decimals, Place in Product from right counting in to left
Ex 2: (0.08)(0.0009)
Ex 3: (1.5)2
Dividing Decimals
Quotient (answer to ÷ prob), Dividend (the # being subdivided) & Divisor (# of equal parts dividend is to be subdivided into)
Decimal always placed after ones in dividend, zeros can be added indefinitely to the right
Terminating Decimal (an exact answer is achieved)
Repeating Non-Terminating Decimal (a bar is used over the repeat to show, never use fractions in a decimal, don’t round
unless specifically asked or need to for a real-world application)
Non-Repeating Non-Terminating Decimals are irrational numbers and can’t be achieved by dividing one #
by another
Division by decimal: Move decimal out of divisor, moving it the same number of places (to the right) in the
dividend
Ex 4: 5 ÷ 0.25
Ex 5: 5.2 ÷ 0.3
Ex 6: 277 ÷ 80
Multiplication Property of Zero
Anything times zero is zero:
a•0=0
Ex 7: (-7.88)(0)
Set Notation
Elements or members of a set are the things that can belong to the set
Roster Form is a list of elements using braces, a list, possibly ellipsis notation to indicate a pattern
Set Builder Notation is a description of the elements using braces, a variable to describe the elements, a bar
“|” to indicate such that the elements are described as “blah”, a description using mathematical symbols and
or English words to describe
Sets of Numbers
A group or collection of things (elements or members); In math numbers form sets
Real Numbers ( )
Natural Numbers (N): called counting too, {1, 2, 3, …}, subset of real, whole, integers, rationals
Whole Numbers (W): natural #’s plus zero, {0, 1, 2, …}, subset of real, integers, rationals
Integers (Z): counting #’s & opposites and zero, {…, -2, -1, 0, 1, 2, …},
can be subdivided into positive (whole numbers) and negative integers; subset of reals & rationals
Rational Numbers (Q): quotient of integers where divisor ≠ 0, can’t be described in roster,
{x | x ∈ p/q, q ≠ 0},subset of reals; mutually exclusive of irrationals
Irrational Numbers (I): any real that’s not rational, no roster possible, {x | x  p/q, q ≠ 0}, subset of reals;
mutually exclusive of rationals
Subset is a set contained within another set
Ex 8: . List the numbers in the set that follows that are from the given sets:
{-9, -1.25, 0, 3/4, √3, π, 3.3333, √25, 12 1/2, 98}
a) Natural #’s b) Whole # c) Integers d) Rational #’s e) Irrational #’s
Y. Butterworth
Ch. 1 Concepts Review – Int. Alg. FTHL
1
Union & Intersection
Union is a mathematical “or”; Symbol ∪; It is collects everything in all sets
Intersection is mathematical “and”; Symbol ∩; It is what is in common in all sets
When there are no elements in a set we call it the empty set or null set and use { } or 
Ex 9: A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} & B = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20} & C = {1, 3, 5, 7, 9, 11}
a) Give the set A ∪ B
b) Give the set A ∩ B
c) Give the set A ∩ C
Absolute Value
The distance from zero regardless of direction (the number w/out its sign)
Absolute values DON’T “distribute”; they are grouping symbols – do problem inside and then take abs. val
Comparison of Numbers
< is less than, > is greater than, ≤ is less than or equal to & ≥ is greater than or equal to
How to tell apart: Old way is little eats big, new way is point out the small guy
Order Property of Real #’s: On # line left gets smaller & right gets bigger (neg. are all smaller than pos.)
Decimals: Compare number by number, find the smaller, you’ve found the smaller
Fractions: Cross mult. up (denom to num) and the larger product is larger fraction
Neg. #’s: Larger the number looks without it’s sign the smaller it is!
Ex 10: Use <, > or = to compare. Get 2 numbers to compare where needed.
a) | - 15 |
- ( - 15)
b) | 43,472 |
- | - 43,472 |
c)
0.25
0.251
d) 5/12
e)
-2.1
-2.2
f)
-3/11
7
/17
-4/15
Adding Fractions/Mixed #’s
Must have an LCD to add/subtract
Find LCD by prime factorization and unique primes to highest exponent (find product)
Build higher terms by using Fundamental Theorem of Fractions (mult. old denom by constant to get LCD & mult. old
num by same constant to get new num)
All fractional answer are in lowest terms/reduce. Fundamental Thm of Fractions to divide out GCF
Improper Fractions should be changed to mixed #’s
Ex 11: Fractions must be used to complete fraction problems. Work shown.
1
a)
/6 + 3/10
b)
542 13/36  237 23/24
Multiplying Integers
+ • + = +, – • – = +, – • + = – or + • – = –
Addition of Real Numbers (Integer Addition)
Subtraction is not allowed, change to addition by adding the opposite of the number following the
subtraction symbol
Like Signs when adding the numbers add and you keep the sign
Unlike Signs when adding the big minus the small and bigger #’s sign is sign of answer
Ex 12:
a) -13
Ex 13:
Ex 14:
Change each problem to addition
– -29
b) -19 – 13
c) 13 – -19
Solve each addition problem created in Ex 12
5
/6  1 1/3
Y. Butterworth
Ch. 1 Concepts Review – Int. Alg. FTHL
2
Order of Operations
PEMDAS – Parentheses, Exponents, Mult/Division (left to right order), Add/Subt. (left to right order)
Parentheses is generic for grouping symbols which include parentheses, brackets, braces, absolute values,
radicals, fractions bars
Most common errors: add/subt. before mult. divide & mult. before dividing
Ex 15: Simplify using strict order of operations. Show work in each step.
a) 24 ÷ 6  2 + 2
b) 2  8  23  4 + 2(13  10 + 5)2
c)
-2 | 6  27 |  3 + 4
16  32  23
Multiplying Fractions
Cancel if possible (dividing out a common factor from num & denom)
Mult. numerators & mult. denominators
If improper change to mixed number
Check for common factors, especially if you didn’t try to cancel
Ex 16: Use fractions to multiply
a)
( 3/16 )( 4/7 )
b) (1 1/3)2
Dividing Fractions/Mixed #’s
Change all mixed
Multiply the dividend by the reciprocal of the divisor (dividend ÷ divisor = quotient!!)
Follow multiplication process
Ex 17: Use fractions to divide
a) 4/30  5/14
b)
5 ÷ 2 2/3
Exponents
Represents repeated multiplication (base used as a factor number of times indicated by exponent)
Exponents only apply to the number that they are written above & to the right of
Fractions: numerator & denominator to exponent (if lowest terms to start will be in lowest terms in end)
Decimals: see mult. decimals
Grouping Symbols: Simplify inside 1st then take single number to power
One to any power is one: 1n = 1
Negative number to even power (parentheses around the negative #) is always positive
Neg. # to odd power (parentheses around neg. #) is always negative
-an ≠ (-a)n when n is even
-an is read as: The opposite of a to the nth power
(-a)n is read as: A negative number used as a factor n times
Anything to the zero power is one: a0 = 1
Ex 18: Expand and simplify.
a)
c)
2 –1
d)
(-2xy)0
g)
(2xy) -1
(-3)2
e)
b)
2xy0
- 32
f)
2xy -1
Evaluation
Put in the values given for the variables, using parentheses to replace the variables with the values
Simplify using order of operations
Distributive property should never be used in lieu of order of operations
This is taught as a first step in a check for an equation, and it’s use in many solution methods
Ex 19: Evaluate a) x + y – z
b) |x + z| – y if x = 2, y= 1 & z = -5
Ex 20: Evaluate
Y. Butterworth
/y + z – y/x
x
if x = 2, y = 5 & z = 7/30
Ch. 1 Concepts Review – Int. Alg. FTHL
3
Properties of the Real Numbers
Commutative Property of Addition and of Multiplication (move addend/factors around)
Associate Property of Addition and of Multiplication (group addends/factors in different orders)
Identity Property of Addition and of Multiplication (gives back the identity using identity element)
Identity Element of Addition: ZERO
Identity Element of Multiplication: ONE
Inverse Property of Addition and of Multiplication (inverse is used to give back identity element)
Distributive Property (Multiplication distributes over add/subt)
Division by Zero: UNDEFINED
Zero Divided by Anything: ZERO (division is multiplication by a recip so becomes zero times anything)
Multiplication by Zero: ZERO
Ex 21: Simplify
a)
0  -251
b)
502  0
Ex 22: Name the property used in each of the following
a) y  (5  2) = (5  2)y
b) (2 + 3) + 8 = 8 + (2 + 3)
c) -6  1 = -6
d) 5 + -5 = 0
e) ¼ (2 + 4z) = ½ + z
Translation
There is a whole separate sheet with all the translation nuances
Ex 23: Translate each of the following into expressions or equations. If there is an
unknown number let that number be x. Do not simplify.
a)
The difference of some number and the product of fifteen and the sum of the
number and negative 9.
b)
The quotient of nine subtracted from some number and 27.
c)
Twenty-three less some number is equivalent to five more than twice that number.
d)
Three less than some number is divided by the product of the number and nine.
Simplifying Algebraic Expressions
Expressions have no equal sign; they can’t be solved; they can only be evaluated &/or simplified
Terms are parts of an algebraic expression separated by addition or subtraction; parentheses are like
presents – you can’t see inside them to see any terms (use distributive property to see terms)
Like terms are determined by the variable portion being EXACTLY the same
Numeric coefficients are the number multiplied by a variable(s)
1st use distributive property to get individual terms
Physically or “mentally” use commutative & associative properties to group like terms
Add numeric coefficients of like terms & rewrite as a “sum” of unlike terms
1
Ex 24: Simplify
/3 + 2/5(x  1/3)
Solving Equations in 1 Variable
Equations are the equality of 2 expressions
Equations can be solved to find a value that creates a true statement
Solutions to an equation can be checked through evaluation
Simplify: 1) Distribute 1st 2) Clear fractions/decimals 3) Combine like terms
Addition Property of Equality used to move things that are add/subt. from one another across the equal sign
(can be used twice)
Multiplication Property of Equality used to remove numeric coefficient of variable (last step used only once)
Give answers as x = #, or as a solution set in roster form
3 Types of Equations: 1) Conditional 2) Identity 3) Contradictions
3 Types of Solutions from 3 Types of Eq.: 1) Single Solution 2) All Real Numbers 3) No Solution
Ex 25: Solve a)
2(x  5) + 3x = 2  3(x + 4)
1
2
c)
/3 + /5(x  1/3) = 1/10 x + 3/5
e) 5(x  4) + 4x = 9(x  7) + 43
Y. Butterworth
b)
2x + 5 = 2(x + 5)
d) 9x  6(x  2) = 2(x + 6)
f)
5½
= x
9
¾
Ch. 1 Concepts Review – Int. Alg. FTHL
4
Solving an Equation for 1 Variable
Follow the process for solving an equation, only focusing on the variable of interest
h = g  7e
4
-12 + 2y = 3x
Ex26: Solve the following for “e”
Ex 27: Solve the following for “y”
Percentage Problems
Percentage to decimal conversion: Move the decimal 2 places left (remember that decimal always comes after ones
position)
Decimal to Fraction Conversion: Read the decimal and write what you read or count the number of
decimal places and put the number in the decimal over a factor of 10 with the number of decimal places
that you just counted
Set up as algebra problem: _____% of ______ (whole) is ________(part) where percent as a decimal is
multiplied by the whole and is equal to the part
Set up as a proportion: is over of equals some part of one hundred
Simple Interest: PRT = I
% Increase/Decrease Problems: Original Price (op) is unknown and % is a known, and final result is known
(price after increase or decrease) _____% of ________ (op) is _______ (increase/decrease) and then an equation
results: op ± increase/decrease = price after which can be solved for op
Ex 28: Convert each to the other 2 equivalent forms. Every answer must be exact and in
the lowest terms where needed.
a) 1/9
b) 0.375
c) 225%
d) 66 2/5%
e) 1.222
Ex 29: Solve the following problems. Give ___% of ___ is ___ and then use algebra to
solve the problem. Conversions of percentages to decimals or fractions and
algebra need to be shown.
a)
What is 40% of 55?
b)
80 is what percent of 120?
c)
60% of what number is 99?
Ex 30: Malory goes to the store and finds a 40% off sale on her favorite dishes. If
Malory pays $75 for the set, what would Malory pay for the dishes if there was no
sale? Algebra must be used, and work using decimals or fractions needs to be shown.
Radical Basics
A radical finds the base when presented with the answer and the exponent
In Other Words: A square root “undoes” a square
Radicand is under the radical sign(original answer), Index is number outside to left or radical(original exponent;
missing when original exponent was 2), Answer is root(original base); EG Original 22 = 4  Radical √4 = 2
If an index is even and set of numbers drawn from is real #’s, then radicand can’t be negative or root is No
 Solution
If index is odd and radicand is negative then the root is negative
Ex 31: Give the real root
3
4
a) -64
b) -81
Y. Butterworth
c)
16
121
Ch. 1 Concepts Review – Int. Alg. FTHL
5
Word Problems
Read, Understand, Write Basic Info in Phrase (no #’s at 1st), Form Mathematical Plan w/ Operators &
Words/Phrases, Fill in #’s & Variables & Form Expressions, Substitute #’s, Variables & Expressions into
Mathematical Equation/Expression, Solve, Answer w/ appropriate units
Ex 32: The height of Mt. Shasta is 14,162 ft above sea level and Death Valley is 282 ft
below sea level (a negative number). What is the difference in the altitude between Mt.
Shasta and Death Valley?
Ex 33: The Ringers play 5 more games than the Setters during a regular season. If
together they play a total of 51 games how many does each team play?
This is not all the concepts in Chapter 1, but this is an adequate review. I do not have time to cover every
concept in the detail that I would like. I will leave it to you to review on your own and to look over my
supplementary notes (Ch. 1 on my web page). Please do not put the review on the back burner, for it may
become very important sometime in the very near future!
Y. Butterworth
Ch. 1 Concepts Review – Int. Alg. FTHL
6