Multiply Decimals
Multiply #’s, Ignore Decimals, Count # of Decimals, Place in Product from right counting in to left
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
Multiplication Property of Zero
Anything times zero is zero:a • 0 = 0
Sets of Numbers
A group or collection of things (elements or members); In math numbers form sets
Real Numbers (ℜ)
Natural Numbers (N): subset of real, whole, integers, rationals
Whole Numbers (W): subset of real, integers, rationals
Integers (Z): can be subdivided into positive (whole numbers) and negative integers; subset of reals & rationals
Rational Numbers (Q): subset of reals; mutually exclusive of irrationals
Irrational Numbers (I): subset of reals; mutually exclusive of rationals
Subset is a set contained within another set
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!
Adding Fractions
Must have an LCD to add
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
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
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
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
Adding/Subtracting Decimals
Line up decimal places and add/subtract as normal, bringing down the decimal as it is crossed
Multiplying Integers
+ • + = +, – •– = +, – • + = – or + • – = –
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
Exponents
Represents repeated multiplication (base used as a factor number of times indicated by exponent)
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
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
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
Translation
There is a whole separate sheet with all the translation nuances
Solving Equations
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
Solving an Equation for 1 Variable
Follow the process for solving an equation, only focusing on the variable of interest
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
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!
Addition
Word / Phrasing / SymbolsSum / The sum of 7 and 2 / 7 + 2
more than / 5 more than 10 / 10 + 5
added to / 6 added to 10 / 10 + 6
greater than / 7 greater than 9 / 9 + 7
increased by / 4 increased by 20 / 4 + 20
years older than / 15 years older than John. John is 20. / 20 + 15
total of / The total of 6 and 28 / 6 + 28
plus / 8 plus 281 / 8 + 281
Subtraction
Word / Phrasing / Symbolsdifference of / The difference of 5 and 2
The difference of 2 and 1 / 5 2
2 1
*years younger than / Sam's age if he is 3 years younger than John. John is 7. / 7 3
diminished by / 15 diminished by 9
21 diminished by 15 / 15 9
21 15
*less than / 17 less than 49
7 less than 17 / 49 17
17 7
decreased by / 29 decreasedby 15
15 decreased by 7 / 29 15
15 7
*subtract(ed)
from / Subtract 13 from 51
Subtract 51 from 103 / 51 13
103 51
take away / 79 take away 61 / 79 61
subtract / 54 subtract 2 / 54 2
less / 16 less 4 / 16 4
* - Means that the numbers come in opposite order than they appear in the sentence.
Multiplication
Word / Phrasing / Symbolsproduct / The product of 6 and 5 / 65
times / 24 times 7 / 24(7)
twice / Twice 24 / 2(24)
multiplied by / 8 multiplied by 15 / 8*15
at / 9 items at $5 a piece / ($5)9
"fractional part" of / A quarter of 8 / (¼)(8) or 8/4 .
"Amount" of "$" or "¢" / Amountof money in 25 dimes
(nickels, quarters, pennies, etc.) / ($0.1)(25) or (10)(25) ¢
percent of / 3 percent of 15 / 0.03(15)
Division
Word / Phrasing / Symbolsdivide / Divide 81 by 9 / 81 9
quotient / The quotientof 6 and 3
The quotient of 24 and 6 / 6 3
24 6
divided by / 100 divided by 20
20 divided by 5 / 100 20
20 5
ratio of / The ratio of 16 to 8
The ratio of 8 to 2 / 16 8
8 2
shared equally among / 65 apples shared equally among 5 people / 65 5
Note: Division can also be written in the following equivalent ways, i.e. x 6 = x/6 = 6x = x
6
Y. ButterworthTranslation Problems Review1
Exponents
Words / Phrasing / Algebraic Expressionsquared / Some number squared / x2
square of / The square of some number / x2
cubed / Some number cubed / x3
cube of / The cube of some number / x3
(raised) to the power of / Some number (raised)to the power of 6 / x6
Equality
Words / Phrasing / Algebraic Equationyields / A number and 7 yields 17. Let x = #. / x + 7 = 17
equals / 7 and 9 equals 16 / 7 + 9 = 16
is / The sum of 5 and 4 is 9. / 5 + 4 = 9
will be / 12 decreased by 4 will be 8. / 12 4 = 8
was / The quotient of 12 and 6 was 2. / 12 6 = 2
Note: Any form of the word “is” can be used to mean equal.
Parentheses
Parentheses are indicated in four ways.
The first is the use of a comma, such as:
The product of 5, and 16 less than a number.
The second is the use of two operators' phrases next to one another, such as:
17 decreased by the sum of 9 and 2.
*Notice how decreased by is followed by the sum of and not a number, this indicates that we will be doing the sum first; hence a set of parentheses will be needed.
Next, you may notice that the expected 'and' between the two numbers being operated on is after a
prepositional phrase[A phrase that consists of a preposition (usually “of” in our case) and the noun it
governs (usually number in our case) and acts like an adjective or adverb]. Such as:
The sum of 9 times a number and the number.
*Usually we would see the 'and' just after the number 9, but it does not appear until after the prepositional phrase 'of 9 times a number'. If you think of this in a logical manner, what you should see is that you have to have two numbers to operate on before you can complete the operation, which would require the use of parentheses to tell you to find a number first!
Finally, you may notice a phrase containing another operator after the 'and' where you would
expect a number. An example here might be:
The difference of 51 and the product of 9 and a number.
*The note about thinking in a logical manner applies here too! You must have two numbers to operate on!
Chapter 1 Pretest
Circle the best answer. Support with work whenever possible.
1.(-6.4)(3)(0) = (5)(0)(5)
a)Trueb)False
2.-12 < -13
a)Trueb)False
3.Simplify:(-4/9) – (-3/4)
a)11/36b)-1 7/36c)-1/9d)19/36
4.Simplify:| 5 – 12 | + | 24 – 11 |2
a)18b)12c)32d)16
5.Simplify:5(-0.2) – (0.1)(2)
a)-1.2b)0.8c)-0.8d)-0.3
6.Simplify:11 + -54 ÷ 6 – 2(3)
a)0b)-4c)14d)-7
7.Simplify:4(-4 /5) – 32(2) + (2/3)2
a)-22 8/9b)1/81c)-7/36d)-20 34/45
8.Simplify: (-3 + 1)2 + | 7 – 9 |
24 ÷ 6 + 2(4)
a)-1/2b)-1/6c)3/5d)1/2
9.Evaluate when x = 6, y = -1 and z = 0: x + 6y – z
6x – y + z
a)1/37b)0c)5 12/37d) undefined
10.Name the property illustrated:3(x + y) = 3(y + x)
a)associative prop. of mult.b)communtative prop. of add.
c)distributive propertyd)associative prop. of add.
11.Name the property illustrated:(a + b) • 0 = 0
a)multiplicative identity prop.b)assoiciative prop. of add.
c)multiplicative prop. of zerod)distributive prop.
12.Translate the statement using mathematical symbols.
The quotient of seven and the sum of x and two is equal to four.
a) 7 = 4b)7(x + 2) = 4
x + 2
c) 7 = 4d) 7 = 4
x – 2 x + 2
13.Translate the statement using mathematical symbols.
The difference of 3 and twice x, multiplied by 4, is 12.
a)4(3 – 2x) = 12b)3 – 2x = 4(12)
c)3 – 4(2x) = 12d)4(3 + 2x) = 12
14.Translate the statement using mathematical symbols.
Eight less than x is twice x.
a)8 – x = 2xb)x – 8 = x2
c)x – 8 = 2xd)8 – x = x2
15.Translate the statement using mathematical symbols.
Three times x squared subtracted from 5 is the product of 8 and x.
a)3x2 – 5 = 8xb)3(x2 – 5) = 8x
c)5 – 3x2 = 8xc)3(5 – x2) = 8/x
16.Solve:-4(2 – 3x) = -(4 + x) – 3(x + 2)
a)x = -1/8b)x = -1c)x = 5/7d)x = 1
17.Solve:3x – 5 + 8(x – 4) = 5(2x – 7)
a)x = 4b)x = -26c)x = -4d)x = 0
18.Solve: 2x + 5 = 4x – 3
3 8
a)x = -3 3/8b)x = -7 3/4c)x = -12 1/4d)x = -1 3/4
19.Solve:6 – 5m = 5 – 3(m + 1) 2m + 4
a)No Sol.or ∅b)m = 5/6c)m = 0d)All Real #’s
20.Solve for y:5x + 11y = 6
a)y = 5x/11 + 6/11b)y = 11/6 – 5x/6
c)y = 6/11 – 5x/11d)y = 11/5 – 6x/5
21.Solve for y:4/x + 3/y = 1/z
a)y = 4xz b)y = 3xz
z – 3x 4z – x
c)y = 3xz b)y = 3xz
4x – z x – 4z
22.Jose is deciding whether to accept a sales position at an electronics store. He is
offered a salary of $1200 monthly plus a 8% commission on his sales. If his sales
are $12,000, what is his pay?
a)$2160b)$960c)$1056d)$2560
Y. ButterworthCh. 1 Concepts Review – Int. Alg. FTHL1