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R.4 Divisibility Rules
Divisibility Rules – A number is divisible by 2 if its ones digit is ___________________________
3 if ____________________________ is divisible by 3
6 if _____________________________
9 if ____________________________ is divisible by 9
10 if ____________________________
5 if ____________________________
4 if ____________________________
8 if ____________________________
Ex a Which numbers are divisible by 2?
17 4,201,122 3801 50,000
Ex b Which numbers are divisible by 3?
29 4,201,122 3801 50,000
Ex c Which numbers are divisible by 6?
29 4,201,122 3801 50,000
Ex d Which numbers are divisible by 9?
387 4,201,122
Ex e Which numbers are divisible by 10? Which are divisible by 5?
295 3,729,231 1620
Ex f Which numbers are divisible by 4? Which are divisible by 8?
9024 387,231 420
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1.1 Place ValueA digit is one of the numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9A number – may have several digits, for example 367
Examples of whole numbers:
Examples of non-whole numbers:
Place Values
1 , 2 3 4 , 5 6 7 , 8 9 0
Use the number 528,174,326,097
Ex a For the number above, what digit is in the hundred thousands place?
Ex b What digit is in the ten billions place?
Ex c What does the digit “0” represent in the number above?
Ex d What does the digit “8” represent in the number above?
Ex e Write “two million, three hundred fifty thousand, sixty seven” using digits.
Ex f Write 56,204 in word form.
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1.2 Introduction to Integers
Integers – positive and negative whole numbers.
Examples:
withdraw $60, deposit $200, 10 degrees below zero, 70 degrees
Note: If a number has no sign:
Ex a Represent the following numbers on the number line: 3, -1, 0, -4, 5
Ex b Use < or > to write a true statement for each pair
3 - 8 0 - 3 - 12 - 5
Absolute Value – the distance from a number to 0 on a number line:
| 5 | | - 7 | |0|
If a number is positive, the absolute value is ______________________
If a number is negative, the absolute value is _________________________
Ex c Simplify:
|-68| |349| |-10,000|
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1.3 Adding IntegersAnalogies: gaining or losing yards in football, depositing or withdrawing money
Ex a 4 + ( - 7) Ex b (-3) + 5
Ex c - 3 + (- 6)
Adding without number lines:1. Add 2 numbers with same sign (both positive or both negative)
a) Add absolute values (amounts)b) Use the common sign
2. Add 2 numbers with opposite signs (1 positive, 1 negative):a) Subtract the amounts, with the larger abs. value on topb) Use the sign of the “dominant” number (greater absolute value)
Ex d Add the integers - 30 + (-14) -11 + 20 13 + (- 41)
Properties of Addition1. Commutative - you can change order without changing value
2. Associative – you can change grouping without changing value
3. Identity Property of Addition – element that makes no change The identity property is sometimes called the _________ property because adding ______to a number leaves it unchanged
_______ + 21 = 21 - 42 + _________ = - 42
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Adding More than 2 IntegersMethod 1: Work left to right
Ex e - 2 + 7 + (-3) + 6 + (-5)
Method 2: Group all positives together, and all negatives together. Add both groups.
Strategic grouping: If you have opposites that cancel, or numbers that make easier sums, use them!
Ex f 9 + (-31) + (-18) + 31 + (-6)
Ex g -2 + (-14) + (-98)
Practice Problems
1. - 13 + (-41) + 13 + (-19)
2. -46 + 71 + (-4) + (-16)
3. – 34 + (-18) + 18
4. – 7 + 28 + (-11) + (-13) + 16
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1.4 Subtracting Integers
Opposites (Additive Inverses)Ex a Find the opposites of the numbers: 17 - 49 0
The sum of a number and its opposite is
Subtraction Procedure: (2 changes that cancel)1. Change (–) symbol to (+) symbol.2. Change the sign of the second number
Subtract negative number
Subtract postive number
Ex b 2 – 6 Ex c
Ex d - 2 – 3 Ex e - 4 – (-3)
Ex f 5 – |-7| – (11 – 3)
Practice Problems1. 6 – 11 2. - 8 – 5
3. 0 – 41 4. -18 – (- 30)
5. -3 - |-6| - 5 6. 19 – (11 – 7) + 1
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1.5 Rounding and Estimating
Ex a Round 29 to the nearest 10
Round - 22 to the nearest 10
Round - 25 to the nearest 10
Rounding Whole Numbers Procedure – for a specific place1. Find the digit in the specified place.2. Look at the digit AFTER that place
3. If the “after” digit is 5 - 9___________________________
If the “after” digit is 0 - 4___________________________
4. Replace the rounded digits with __________5. Keep all digits to the left of the specified place
Ex b Round 3,682,357 to the nearest:
million ten thousand hundred ten
Front-End Rounding- Keep one non-zero digit on the front, replace other digits with 0
Ex c Use front-end rounding to estimate the following amountsRestaurant bill: $ 43.58 Truck: $27,875 House: $239,995
For Examples below, estimate using front-end rounding. Then find the exact values
Ex d 58 + 91 + 37
Ex e 764 – 238
Ex f 829 + 2640
Ex g $492 – $61 + $88 - $179
Note: Front-end rounding works best when the highest place values are similar (differ by 1 or less)1.6 Multiplying Integers
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Product of a positive and a negative number – the result is __________________Tip: Determine the sign and set it aside, then multiply the absolute values separately.
Ex a 7(-6)= (-3)(5)= (−11 )⋅(28 ) =
Product of 2 negative numbers – the result is __________________
Ex b (-5)(-3) = -2(-7) = (−25 )⋅(−12 ) =
Properties of Multiplication1. Commutative - you can change order without changing value
2. Associative – you can change grouping without changing value
3. Identity Property of Multiplication – element that results in no changeThe identity property of multiplication is sometimes called the Multiplication Property of __________ because multiplying ______ by a number leaves it unchanged
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4. Multiplication Property of 0 - multiplying by 0 gives a product of ______________
5. Distributive a(b + c) = ab + ac(a + b)c = ac + bcSimilar to doubling a recipe:
Multiplying More than 2 NumbersEvery pair of multipled negative numbers produces a positive number.For more than 2 negatives:
____________________________________________ produces a positive number.
_____________________________________________ produces a negative number.
Ex c (- 4)(- 3)(2)(-1) =
Ex d (- 5)(4)(- 3)(- 2)(- 1) =
Ex e 2(-47)(-5) =
Ex f Write an integer in each blank to make a true statement:
-8 X (_____) = -24 35 = (_______)(-7)
Ex g Each month, $38 is deducted from a bank account.1) Estimate the amount deducted in a year
2) Calculate the exact amount deducted in a year
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1.7 Dividing IntegersThe rules for signs in division are the same as for multiplication.
Quotient of a positive and a negative number – the result is __________________
Ex a
-305 =
27-9
Quotient of 2 negative numbers – the result is __________________
Ex b
-42-6 = −48÷(−12 )=
Recall Division properties
1. ( -9)¿ 1 =
-91 =
2. (-28)¿ (-28) =
-28-28 =
3. 0¿ (-6) =
0-6 =
4. -11¿ 0 =
-110 =
Why?
Related equations:
-243
=− 8
-240
=x
0-24
=x
Multiplication and Division togetherEx d 8(-50) ¿ (-2 X 5)
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ApplicationsEx e A house needs 300 ft. of baseboard. Baseboard is sold in 8-ft pieces. How many pieces must be purchased?
Ex f If a car loan (with interest) is $17,400 is made over 5 years, how many monthly payments must be made? Estimate the amount of each payment. Then calculate the exact amount of each payment to the nearest dollar.
Practice Problems
1. Simplify:
0-19
23-1
-520
-87-87
2. Simplify: (-24¿ 6)(18 ¿ (-2))
3. How many $3 sodas can be bought with a $50 bill?
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1.8 Exponentials and Order of Operations
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33 = (-3)3 = - 33
52 = (-5)2 = - 52
Ex a Write in exponent form: 7⋅7⋅7 10⋅10⋅10⋅10⋅10⋅10
Ex b Evaluate: 7⋅7⋅7 10⋅10⋅10⋅10⋅10⋅10
Ex c Evaluate (-2)3(-7)2
Simplifying Expressions (Order of Operations for several operations)1. Parentheses (and grouping symbols like { } or [ ])2. Evaluate all exponential expressions3. Multiplication and Division, in order from left to right4. Addition and Subtraction, in order from left to right
Ex c 100 – (68 – 21) (100 – 68) – 21
Ex d 5⋅22 (5⋅2 )2
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2.1 Introduction to Variables
We represent real quantities as variables, because these quantities keep changing.My husband’s age:
variable
constant
expression
Ex a Suppose your job pays $20/hour, but the number of hours you work changes from week to week. Write an expression using the variable “h”, which calculates the salary earned from working “h” hours.
Evaluating an expression – “Evaluate” means “find the number value”Ex b Evaluate the expression above for h = 17
Ex c If 4 more photocopies are made than the number of students:1) Choose a variable for the number of students, and write an expression for the number of copies needed.
2) If there are 38 students in the class, how many copies are made?
3) If there are 44 students, how many copies are made?
Ex d Evaluate the expression 2s + 5b for s = 10, b = 7
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Properties Using VariablesCommutative Prop of Addition: a + b = b + a
Commutative Prop. of Multiplication: ab = ba
Exponents with variables
34 = x4 =
- 7x2y5 =
Ex d Evaluate 2a2b for a = -3, b = 2
Ex e Evaluate |xy – z| for x = -5, y = 3, z = 7
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2.2 Simplifying Expressionsterm – an added or subtracted “piece” in an expression. It may contain numbers, variables, and/or multiplication (glue)
Examples of single terms:
Examples of things that are NOT single terms:
A variable term has 2 parts:
1. ______________________- the number part, including the sign
2. ______________________ - the letters, including exponents
Ex Find the coefficient and variable part of each term below:
A term without a variable part is called ________________________________
Like terms - terms that have exactly the same variable parts (including exponents)Ex Are the following pairs of terms like or unlike?
Combining Like Terms – treat as “types”, with amounts of each type1. Identify each type of variable part2. For “like” types, add the coefficients. For unlike types, keep separate.
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A simplified expression should have:1. No parentheses (use distributive law if needed)2. Multiplied pieces combined3. Like terms combined
Practice Problems
Simplify: 1. 3x2 + 2x + 7x2
2. (-5x3)(2x)
3. 3(4x – y + 5)
4. -2(b + 4) + 7b – 11
5. Evaluate ½ at2 for a = 32, t = 3
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2.3 Solving Equations Using Addition (or Subtraction)
Solve - get a number that makes the equation trueSolution - a number that makes the equation true
Checking to see if a number is a solution - Procedure1. Replace the variable(s) with the number2. Simplify and determine if the equation is true
Addition/Subtraction Property of Equality - you can add or subtract the same amount from both sides without changing the solution
If A = B , then A + C = B + C (also) If A = B, then A – C = B – C
Goal in Solving: Isolate the variable
Solving Equations with added “clutter”1. Combine like terms on each side of the equal sign.2. Look at the variable. Decide what is “clutter”3. Decide how the “clutter” is connected to the variable. Do the reverse to “undo” the
clutter.4. Isolate the variable5. Check
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2.4 Solving Equations Using Division (or Multiplication)Multiplication/Division Property of Equality - you can multiply or divide by the same amount on both sides without changing the solution
If A = B, then A/C = B/C(also) If A = B, then AC = BC
“Undo” clutter by doing the reverse operation
Practice Problems - Solve the following1. 32 + x = 20
2. -t + 19 + 2t = - 6
3. 35 = - k
4. 12 =
y3
5. 160 – 139 = 10x – 3x
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2.5 Solving Equations with Several Steps
What if you have both added & multiplied “clutter”?
Solving Procedure (summary) 1. Simplify if needed (clear parentheses, combine like terms)2. Get all variable terms on one side, by adding or subtracting from both sides; combine
into one variable term3. Get rid of added/subtracted “clutter”4. Get rid of multiplied/divided “clutter”
Variable terms on both sides
Clearing Parentheses
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Practice Problems
1. x + 7 + 3x = 5x – 12
2. 4 – 5(x – 2) = -4(x + 1)
3. 5(p – 3) + 6p – 2 = 60
4. -2(x + 7) – 9 = 5x + 12
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3.1 Perimeter Problems
Perimeter – the distance around the outside of a shape. The operation is:
Rectangle: has 4 angles which are 90o (right angles) Question:
Formula for Perimeter of a rectangle: Formula for Perimeter of square: P = P =
Ex a Find the perimeter of the rectangle:
Irregular objects:
Ex b Find the perimeter of each object:
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Ex c Find the perimeter: 18 ft
22 ft
10 ft
30 ft
Ex d A 9’ X 10’ room is decorated with border paper. If each roll is 12 ft, how many rolls are needed?
Ex e A living room is 18 ft X 12 ft. The doorway into the living room is 6 ft wide. a) If baseboard costs $2.25/foot, what is the cost of installing baseboard?
b) If baseboard is only sold in 8-ft segments for $12 each, what is the cost?
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3.2 Area Problems
Area of a rectangle:
A =
Area of a Square Area of a Parallelogram:
Ex a The area of a rectangle is 240 ft2. The length is 20 ft. Find the width.
Ex b A parallelogram has height a base of 17 inches and height of 11 inches. Find the area.
Exc Find the area of the room below. If the cost of carpet is $4/sq. ft, how much does it cost to carpet the room?
18 ft
22 ft
10 ft
30 ft
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Practice Problems 1. a) Find the perimeter of a square picture frame with 9 inches on each side.
b) Find the area of the picture in the frame.
2. A yard is enclosed with chicken wire fencing. If each roll of 50 ft. costs $26, how much does it cost to enclose a 60 ft X 30 ft back yard?
3. A room is 12 ft X 15 ft.a) How much baseboard is needed to surround the room?
b) How much carpet is needed to cover the floor?
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3.3 Solve Applications - One Unknown
Translating Word Phrases to Algebra 4 important words:
sumthe difference of x and y
productquotient
Additionthe sum of a number and 2
5 more than a number
3 added to t
a number increased by 10
Subtractionthe difference of 3 and a number
3 less than y
7 subtracted from a number
x decreased by 8
Multiplicationthe product of 4 and a number
half of x
twice a number
Divisionthe quotient of a number and 11
4 divided by a number
$15 per 5 gallons
Equal
Ex a Translate and solve: 4 less than a number is 18
Ex b Seven more than the product of 8 and a number yields 47. Find the number.
Ex c Translate and solve: The sum of n and 3 subtracted from 12 times n equals -11 plus the product of 2 and the difference of n and 5.
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Applications- construct an equation using a variable for the unknown quantity
Ex d Ben’s beginning bank balance was $258. He deposited $360, made a withdrawal, and his final balance was $146. How much did he withdraw?
Ex e The Greens had a 20-lb bag of bird seed. Mice ate some of the seed. The Greens then bought an 8-lb bag of seed and put all the seed in a metal container. They now have 24 lb. How much did the mice eat?
A shipment of 30 crates of cereal boxes was received. If 284 boxes were sold and 76 boxes were left, how many boxes were in each crate?
Practice Problems1. Four times the sum of a number and 3 results in that number. Find the number.
2. The difference of 31 and the product of 4 and a number equals 3. Find the #.
3. Don weighs 184 lb. His weight is 2 lb less than 6 times his daughter’s weight. What is her weight?
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3.4 Solve Applications – 2 Unknowns
Applications with 2 unknowns -- Procedure for solving1. Let one quantity be an expression “built around” x, and the other be x.2. Assign quantities to the expressions3. Make an equation using the expressions4. Solve and ANSWER THE QUESTION5. Check (optional)
Ex a Amy & Sam pick 63 pounds. of peaches. If Amy picks 7 more pounds than Sam how many pounds each person pick?
Common constructions (algebraic expressions) for 2 types1. “one side is 4 ft more than the other side”
2. “there are twice as many children as adults”
Ex b The length of a field is 6 more ft. than twice the width. If the perimeter is 132 ft, find the dimensions.
Ex c A board is 48 inches long. It is cut into 2 pieces, where one piece is 14 inches longer than the other piece. How long is each piece?
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4.1 Introduction to Signed FractionsFraction – part out of a whole
partwhole
=numeratordenominator
Ex a What fraction is represented by the shaded portions?
Proper fraction – numerator amount (absolute value) is less than the denominator amount (abs. val.)
Improper fraction - numerator abs. value is greater than or equal to the denominator abs. value.
Ex b Circle all proper fractions in the list: 1717, −3
5, 5
2, −7
3, 1
−5
Ex b What improper fraction is represented by the shaded portions?
Graphing a fraction on a number line
Ex c Graph the number
35
Ex d Graph the number −1
4
3 Ways to Write a Signed Fraction: −1
4 =
Absolute Value of a Fraction
Ex e Find the absolute values: |35|
|−14|
|07|
29
30
Equivalent Fractions – Reducing and Building Upreducing - dividing/cancelling common factorsEx Write an equivalent fraction with the new denominator:
68 4
building up – multiply by the missing factor
Using Division to Simplify Fractions
Ex
-77
246
−181
Practice:1. Write 2 fractions equivalent to ½
2. Write an equivalent fraction with the new denominator:
a)
206
=2
710
=40
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4.2 Writing Fractions in Lowest TermsEquivalent fractions – have the same value:
Lowest terms – numerator and denominator have no common factors (except 1)
Writing in Lowest Terms (Reducing) – dividing by common factors
Ex a Simplify:
1824
Ex b Simplify: –
6072
Prime number - a natural number greater than 1, which can’t be “broken down” to smaller factors.
Ex c Circle the prime numbers:
1 2 3 4 5 6 7 8 9 10 11 12 1314
All numbers above 1 that are not prime are __________________________
The _____________________ numbers from this list are:
Prime Factorization - Breaking down numbers to the smallest possible factors
Ex d Find the prime factorization of 72 using a factor tree
Listed factors:
Exponent form:
32
Try: Find the prime factorization of 60
Reducing by cancelling prime factors
Ex e Reduce by listing and cancelling prime factors: –
6072
Variables
Ex f Reduce by listing and cancelling factors: 24x3
16x4
Practice ProblemsWrite in lowest terms using any method.
1.
1230
2. –
1854
3.
11701200
4.
3xy3 z2
2xyz
33
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4.3 Multiplying and Dividing Signed Fractions
Find
13
of 14
The word “of” usually means _____________________
Multiplying a Fraction by a Fraction
ab⋅cd=acbd
1. Multiply the 2 numerators – keep in numerator2. Multiply the 2 denominators – keep in denominator
Ex a Multiply (−4
7 )(−35 )
Ex b Multiply ( 2
3 )(−43 )
Simplifying (before multiplying)
Ex c
35⋅10
9
Procedure1. Put numerator and denominator factors together in the num. & denom., but don’t
actually multiply out the numbers2. Factor the numerator and denominator3. Cancel common factors.4. Multiply out the products to get a single number in numerator & denominator.
Ex d Find
38
of 1627
35
Variables
Ex e 8x2⋅ 7
2x5
Multiply a Fraction by an Integer
Ex f Multiply
Reciprocals - Pairs of fractions whose product = 1. We find a reciprocal by________
Ex g Find the reciprocal of
Dividing Fractions
ab÷ cd= ab⋅dc=adbc
1. Keep
2. Change
3. Flip (use reciprocal)
Ex h Divide:
23÷8
9
Ex i Divide: 24÷1 1
2
36
Ex j Divide: (− x2 y
z )÷(−xyz2 )
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Applications
Ex h Financial aid covers
35 of a student’s expenses. If expenses are $4500, how
much is covered by financial aid?
Ex i How many 3/4 ounce servings of chips can be made from a 12 ounce bag?
Practice Problems
1. 9⋅( 5
12 )2.
57⋅ 110 3.
( 425 )⋅(15
16 )
4.
314
÷ 67 5.
6÷( 43 )
6. (−6a
11 )÷(− a22 )
7. The pitch of a screw is
116 inch (this is how far it moves with every full turn). How
far into a piece of wood will it go when makes 12 full turns?
8. If 2/3 of all students in the Central Valley graduate from high school, and 1/6 of HS graduates earn a BA/BS degree, what fraction of all CV students earn BA/BS degrees?
38
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4.4 Adding and Subtracting Signed FractionsLike Denominators1. Add numerators2. Keep same denominator3. Reduce if possible
Ex a Add:
18+ 3
8 = It doesn’t say reduce – should we?
Different Denominators – find and build to (Common) Denominator
Easy Case Least Common Denominators (LCDs) :
1. If all denominators have no common factors, the LCD is the product of denominators:
Ex b Find the LCD of
18
and 13 , and add
18
+ 13
2. If one number is a perfect multiple of all the numbers, the “big” number is the LCM.
Ex c Find the LCD of
14
and 316 , and subtract
14
− 316
Hard case LCDs – denominators have common factors, but are not perfect multiplesUse Prime Factorization method
1. Find the prime factorization of every number2. Choose each unique base3. Write largest exponent for each base4. Multiply the numbers to get the LCM
Ex d Find the LCD of
59
and 215 , and add
59
+ 215
40
Ex e Find the LCD of
1112
and 518 , and subtract
1112
− 518
Another way to find Hard LCDs – Listing Multiples Method1. List multiples of the larger number2. See if these are also multiples of the smaller number
Ex e (again) Find the LCD of
1112
and 518 by listing multiples.
Last Resort: If you can’t find the Least Common Denominator easily, you can find some common denominator (not always least), by taking the product of denominators. You’ll need to reduce at the end.
Variables
Ex f
25y+ 1
2y
Whole Numbers and Fractions
Ex g
57−3
Applications
Ex i I have ½ lb. butter, and use 1/3 lb. How much is left?
41
Ex h A nurse records the following fluids gained or lost (in pints). What is the net gain or loss of fluid?
Practice Problems
1.
34−2
5 =
2.
215
+ 49+ 1
6 =
3. − 7
12 + 2 =
4. A salad dressing recipe calls for
23 cup oil,
14 cup vinegar, and
116 cup soy sauce.
How much dressing is made?
5.
st9
−2st3
42
43
4.5 Mixed Numbers and EstimatingWhat fraction is represented below?
Ex a Write as a mixed numeral: 11 +
34 = 9 +
35 =
Convert Mixed Numbers to Improper Fractions 1. Build the whole number to the denominator2. Add the new numerator to the fraction numerator3. Put the sum over the denominator
Ex b Convert to an improper fraction
4
35 5
23
Convert Improper Fractions to Mixed Numbers – use long division
Ex c Convert to mixed numbers:
957
1583
Mixed Number Multiplication & Division –must convert to improper form, then back to mixed
Ex d (1 3
5 )(2 1
4 ) Ex e 8 2
3÷2 3
5
44
Mixed Number Addition & Subtraction – convert to improper OR keep whole number & fractions separate
Ex f 1 3
5+4 2
3
Ex g 5 ½ - 1 ¾
Rounding/Estimating Mixed Numbers to Nearest Whole Number
In the fraction, compare the numerator to half of denominatorIf more than half (or same as half), round fraction to 1. If less than half, round fraction to 0
Ex h Estimate, rounding to the nearest whole number
3 58
−14 13
6 311
−12 78
Estimates and OperationsTo add & subtract, you can estimate any fraction or mixed number to a whole number.To multiply or divide, if a fraction rounds to zero, keep the fraction instead of rounding to
zero. Rounding to any other whole number is OK (multiplication or division by zero should be avoided in estimation.
Ex i Estimate each term to the nearest whole number, then perform the operations:
7 319
+5 45−231
37−1
3
45
Ex j Estimate the product. (suppose you are one out of 6 children inheriting a business
that operated for 8 3
4 years and earned 16 1
6 million dollars per year)
( 16 )⋅(8 3
4 )(16 16 )
Practice Problems
1. Estimate the difference: 8 5
6 – 2 1
12
, then find the exact value.
2. A plumber joins a pipe of length 51 5
16 inches, and a pipe of length 34 3
4 inches. Estimate, then find the exact amount of pipe used.
Applications
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4.6 Exponents, Order of Operations, Complex Fractions
Exponents with Fractions:
Ex a Simplify (−2
3 )2
(−23 )
3
Order of operations: Recall PEMDAS
For multiplying – improper fractions are bestFor adding – either mixed numbers or improper fractions can be used
Ex
Complex fractions (more than 2 layers) - can be written as a division problem
47
Practice Problems:
1. Evaluate 2x –
13 y for x =
37 and y =
−23
2. Evaluate b2 – 4ac for a = ½ , b = -5, and c = 3
3. Simplify:
58
3 4
4. Simplify:
23
6
48
4.7 Solving Equations with Fractions
Some problems may have “one step clutter” – easier.Ex a Ex b
Two-step clutter
Alternate Solving Procedure for 2 kinds clutter – counterintuitive1. Find LCD2. Multiply both sides by LCD to GET RID of fractions (do NOT build up to keep
fractions)3. Solve as before
Practice Problems – Solve the following:
1. 2 = -7 +
54x
2.
34x+ 1
5=1
2
49
3. −5+ y3=9 4.
45p+ 1
3= 7
15
50
4.8 Geometry Applications : More Area and VolumeArea of a Triangle: A = ½ bh
Ex a Find the areas:
Rectangular Solids (fill with cubes)
Formula for Volume of a box: V =
Ex b Find the volume of a box whose dimensions are 8” X 10” X 3”
(We will skip volume of a pyramid).
51
Length, Area, and Volume (Perimeter is a )
What type of quantity is represented by each of the following?
1. carpet
2. baseboard
3. gas in a car
4. distance to LA
5. amount of fencing
6. house size
7. refrigerator size
8. grass/sod
9. compost
Recall of Perimeter, Area, and Volume formulas
Perimeter of a square:
Perimeter of a rectangle:
Perimeter of a triangle:
Area of a square:
Area of a rectangle:
Area of a triangle:
Area of a parallelogram:
Volume of a box:
Practice Problem: Find the volume of a cube where each side has length = ½ inch.
52
5.1 Reading and Writing Decimal Numbers
Decimal Values – correspond to fractions
0.8 represents
0.26 represents
Place Values 1 2 3 . 4 5 6 7
Decimal Notation and Word Names – decimal words are similar to the fraction they represent
1. Number left of decimal point:
2. Point:
3. Number right of point:
Ex b Give the word name of
1) 1.73
2) 0.064
3) 12.408
Ex c In the number 347.29186, what digit is in each place?
tens tenths hundreds hundredths ten-thousandths
Graphing on a number line
Ex d Graph -3.7 and 2.3 (approximately) on a number line
__________________________________________Converting Decimals to Fractions or Mixed Numbers:
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1. Write the digits in the numerator2. Count the number of decimal places3. Write that power of 10 in the denominator (same # of zeroes as power of ten)4. Keep the sign and reduce if needed
Ex e Convert to a fraction:
0.357 – 0.0182 23.41
Note: Whole number parts on the left of the decimal point cause the fraction to be
________________________
Ex e Write as a fraction in lowest terms:
– 0.75 11.8 6.05
Practice1. Convert to a fraction without reducing
0.3 0.03 0.003
- 0.74 0.0321 -0.485
2. Write as a fraction or mixed number and reduce if needed.
0.8 7.65 22.13
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5.2 Rounding Decimal NumbersRounding1. Look at the specified place; draw a “cut off” after that place2. Look at the next digit after the cut off3. If the next digit is
0 – 4, keep the desired digit (before the cutoff)5 – 9, round up
4. Keep digits to the left of the specified place
Ex a Round to the nearesta) thousandth
b) hundredth
c) whole number (unit)
d) ten
e) hundred
Ex b Round to the nearesta) hundred
b) ten
c) whole number (unit)
d) tenth
e) hundredth
Rounding Dollars and Cents
Ex c Round $16.6667 to the nearest:
dollar: cent:
Ex d: Round $3.999 to the nearest:
dollar: cent:
Ex e Ty’s GPA is 3.7193. Round to the nearest:thousandth hundredth tenth whole number (unit)
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5.3 Add & Subtract Signed Decimals
Procedure:1. Line up decimal points! (most important)2. Fill in zeroes at the end of decimals if needed
Ex a Add: 2.68 + 11.3 + 0.009 + 5
Ex b 6 + (– 4.27)
Ex c Subtract 16 from 8.32
Ex d Last month’s electric bill was $116.43, and there was a $22.39 increase this month. What is the new electric bill?
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Ex e A $50 bill was used to pay for $23.61 in groceries. How much change was given?
Practice Problems - Perform the operations
1) 7 – 2.381 2) 14.843 + 0.34 + 1.9 + 10 3) – 13.7 – 2.843
4) On a shopping trip, Mia buys items costing $38.95, $129.99 and $9.77. Estimate the cost by rounding to the nearest ten, then find the exact value.
5) A $491.79 tablet is discounted by $109.21. Estimate the final price, then find the exact value.
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5.4 Multiplying Signed DecimalsDecimals have fractional equivalents:
Ex a 0.03 X 0.9
Procedure:1. Multiply digits as if they were whole numbers2. Find the total number of decimal places in all factors and move the point left that
# of places.3. Fill in zeroes if needed4. Find the correct sign
Ex b Multiply: (6.7)(-0.038)
Estimating Products – operation is- Round each factor to one non-zero digit in highest place value (front-end rounding)
Ex c Use front-end rounding of each factor to estimate the product:(13.692847)(0.3812984)
Ex d Use the fact that 32 X 7 = 224 to calculate the following:
(-3200)(70) = (0.32)(0.7)= (32)(0.007)
Ex d Coffee costs $3.61 (including tax). Estimate how much is spent in a 31-day month, then find the actual amount. Of the guesses: $10/month, $100/month, $1000/month, which are reasonable?
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Multiply by 0.1, 0.01, 0.001, etc. (small numbers)Ex e Multiply 18.47 X 0.001
Multiply by 10, 100, 1000, etc. (large numbers)Ex f Multiply 18.47 X 1000
Ex g Multiply 0.389 X 400
Practice Problems
1) 4.6 X (- 0.9)
2) 0.01 X 821.37
3) Shrimp costs $8.95/lb, and 6.245 lb are bought. Estimate the cost of the purchase.
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5.5 Dividing Signed Decimals
Divide decimals by whole numbers – similar to whole number long division1. Put the decimal point in the quotient2. Divide as if whole numbers3. Tack on extra zero to at the end of decimal if needed
Ex a 15|25 .5 Ex b
0 .38
Ex c Divide
311 and write as a repeating decimal
Ex d Divide 92 1
7 and round to the nearest hundredth
Decimal divisors (denominators) - make fraction and move point to get a whole number in denominator.Ex e 2.734¿ 0.04
Divide by 10, 100, 1000, etc. (large numbers)
Ex f Divide 128.54 ¿ 1000
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Divide by 0.1, 0.01, 0.001, etc. (small numbers)
Ex g Divide
0 .0630 .001
Ex h Dan is paid $892.12 for 11 days. Estimate his daily pay, then calculate the exact amount to the nearest cent.
Order of OperationsEx i (0.9)2 + 3(5 – 3.7)
Practice Problems
1)
14 . 310 .01
2) 11.2 ¿ 4
3) Simplify: 4.37 + (0.6)2 – 2(0.8)
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5.6 Fractions and Decimals
Ex a Write 78 as a decimal Ex b Write 52 1
6 as a decimal and round to the
nearest hundredth.
Common Equivalent Fractions and Decimals
Comparing Decimals/Arranging in order
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5.7 Statistics: Mean, Median, and Mode
Mean (Average) – Procedure for calculating1. Find the sum of values2. Divide by how many values
Ex a Trey’s test scores are 88, 92, 79, and 84. What is the average?
Weighted Mean - sometimes, certain categories count more “heavily” than othersGPA (grade point average) A = 4.0, B = 3.0, C = 2.0, D = 1.0How heavily a class is weighted depends on the number of units(credits).GPA = total grade points/# of units
Ex b A student takes earns an “A” in a 3-unit English course and a “C” in a 4-unit math course. What is the GPA?
Class Units Grade
English 3 A (4.0)
Math 4 C (2.0)
Ex In a class, exams are weighted 50%, homework is weighted 25%, and the final is weighted 25%. If Pat’s exam avg. is 80, the final exam is 75, and homework is 40, what is her final average and grade?
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Median – simple case for an odd number of values 1. Rewrite the list of values in order (smallest to largest or largest to smallest)2. Choose the middle value (the median)
Ex d The assessed value of 7 neighborhood houses is below. Find the median.
250,000 176,000 220,000 6,000,000 206,000 192,000 220,000
Median – a more sophisticated approach for an even number of values1. Rewrite the list in order2. Look at the middle 2 values.3. Find the “halfway point” of the middle 2 values (average of 2 values, not average
of all values)
Ex e Find the median of the 8 neighborhood houses below.
250,000 176,000 220,000 6,000,000 206,000 192,000 220,000 210,000
ModeThe most common value is the mode. If there are 2 (or more) most common values, there are 2 (or more) modes. If no value is more common than any other, there is no mode.
Ex f Find the mode of the ages of students:
22, 20, 19, 20, 18, 35, 19, 58, 21, 19, 28
Find the mode of these ages:
9, 9, 9, 12, 15, 15, 15
Find the mode:
9, 9, 12, 12, 15, 15, 17, 17
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5.8 Pythagorean Theorem and Square Roots
Square Root - the number which, squared, gives the final number (also, the length of a square with a given area)
( x )2 = 25
Every positive number has 2 square roots
Principal Square root –
When a square root is written as a symbol, we use the __________________________
Ex
Some perfect squares (helpful to memorize)
Root number
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Square 1 4 9 16 25 36 49 64 81 100 121 144
169 196 225
Square Roots of Other Numbers (not perfect squares
1. precisely – use
2. approximately
Pythagorean Theorem In a right triangle, the formula: applies
legs – 2 sides touching the right angle
hypotenuse – side opposite the right angle
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Ex For the right triangle shown, solve for x. Give your answer as:
1) a simplified radical
2) a range of 2 whole numbers that x falls between
3) a decimal answer, obtained by calculator
Ex To get from home to school, Ira walks 4 blocks south and 7 blocks east. Approximately how many blocks (to the nearest 2 whole numbers) is the diagonal shortcut between home and school?
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5.9 Solving Equations
2 Methods for isolating the variableMethod 1: Get rid of all decimals by moving the point the same number of places in EVERY term.
Method 2: Keep all decimals until the end
ApplicationsEx A phone plan charges $50 for the base rate, with unlimited talk and text, but no data. Data costs $1.50 per unit of data, where a unit is 1/10 GB. a) Write a formula where x = number of units used, and C = total cost.
b) If Cy’s phone bill is $78.50, how many units of data did he use? How many GB is this?
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5.10 Circles, Cylinders, and Surface Area
Diameter vs Radius Formulas:
Ex Find the diameter and radius of each circle below
Circumference – the distance around the _____________________________
Formula for Circumference: C =
Also: C =
Estimates for
Ex A spa is advertised as 25 ft (circumference). If its radius is 3 ft., what is the actual circumference?
Ex A robot has wheels that make 2 rotations.1) If 4-inch wheels are used, how far does it travel?
2) If 3-inch wheels are used, how far does it travel?
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Formula for Area of a CircleA =
Ex Find the area of a circle with radius = 3 ft. Use 3.14 as an estimate for
Ex Find the area of the half-pizza shown:
Volume of a cylinder/straight-sided object
Ex Find the volume of a can whose diameter is 16 cm and whose height is 10 cm.
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Surface Area – rectangular box
SA = 2LW + 2WH + 2HL
Ex Find the surface area of a box with dimensions 3 in. X 6 in. X 10 in.
Practice Problems:1. How many cubic feet is a refrigerator that is 2 ft wide, 1½ ft deep, and 5 ft tall?
2. What is the surface area of the refrigerator above?
3. A water tower has a height of 40 m and diamter of 20 m. Find the volume.
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6.1 Ratios - a ratio is a comparison of 2 quantities
Ex A club has 16 old members and 7 new members. Write the ratio of new to old members3 ways to write:
1) 2) 3)
Ratios may use like units or unlike unitsLike units - units cancel; some “units” are implied e.g., male:female ratio
Ex A computer screen is 16” wide X 10” tall. Find the width to height ratio
Ex A $50 pair of shoes is on sale for $35. Find the ratio of the discount to the original price.
Converting Unlike Units
U.S. Measurement UnitsLength Weight
12 inches (in) = 1 foot (ft) 3 feet (ft) = 1 yard (yd)5280 feet (ft) = 1 miles
16 ounces (oz) = 1 pound (lb)2000 pounds (lb) = 1 ton
Capacity (Volume) Time8 fl. oz = 1 cup (c)2 cups (c) = 1 pint (pt)2 pints (pt) = 1 quart (qt)4 quarts (qt) = 1 gallon (gal)
60 seconds (sec) = 1 minute (min)60 minutes (min) = 1 hour (hr)24 hours (hr) = 1 day7 days = 1 week
Ex A TV movie is shown in a 2-hour time slot, and there are 30 minutes of commercials. Find the ratio of commercials to the actual movie length, represented in minutes and reduced.
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Unit Ratio – denominator = 1
Ex At a college, there are 3500 students and 200 faculty. Find the student to faculty ratio, reduced to lowest terms.
Ex Express the ratio above as a unit ratio
Ex Express as a ratio of whole numbers in lowest terms
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6.2 Rates - a rate is a comparison of unlike units
Ex a A car drives 500 miles on 16 gallons of gas. Write as a rate, reduced to lowest terms.
unit rate = denominator number is 1
Ex b Write the rate above as a unit rate.
Ex Ty earns $150 in 8 hours. Write as a rate, reduced to lowest terms, then as a unit rate.
Unit price – cost per unit (cost/unit)
Ex A bag of chips costs $4.29 for 11 oz. Find the unit cost
Ex Which of the following jars of peanut butter is the better buy?Brand A is 40 oz. and costs $5.00 Brand B is 28 oz. and costs $3.00
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6.3 Proportions
Proportion – 2 ratios that equal each other:
ab= cd
Reducing:
Cross Multiplying (cross products)
If
ab= cd , then ad = bc
Testing whether ratios are proportions (cross multiply)
Solving Proportions1. Cross multiply to eliminate fractions2. Solve as before
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Mixed Numbers and Decimals
Practice Problems:
1. Which is a better buy: See’s candy at $10.50 for a ½ box, or Turtles candy at $5 for 4 ounces?
2. Solve for x:
2. 515
= x0 .3
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6.4 Applications with Proportions
Setting Up a ProportionCase 1 Case 2
Quantity 1 ORQuantity 2
Quantity 1 Quantity 2Case 1Case 2
Cancelling Shortcuts1. You can cancel straight up/down (this is the same as reducing a fraction)2. You can cancel straight across (same as multiplying both sides of an equation)3. You can’t cancel diagonally – can’t “cross cancel”
Ex It costs $3000 to carpet a 1200 sq. ft. house. How much does it cost to carpet an 1800 sq. ft. house?
Ex A recipe calls for 1 lb. of chicken for 6 servings. How many lb. of chicken are needed for 20 servings?
Ex An 8” X 12” sheet of paper is reduced so its width is 3”. What is its length?
Practice Problem: A 20-ft tree casts a 30-ft shadow. How long is the shadow cast by a 6-ft man?
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7.1 The Basics of Percent
percent – out of 100 Percents can also be represented as fractions or decimals
57% =
A fire is 27% contained. Percent notation:
Fraction notation:
Decimal notation:
Converting percent to decimal – Replace % with ___________ or ______________This causes you to remove _______, make number ____________
Ex a Convert to decimal:
58% 7.2% 150% 0.03%
Ex b Convert 5 2
3%
to decimal:
Convert decimal to percent – multiply by ___________ Does this change the value?
Ex c Write percent notation for
0.27 0.735 0.4 2.7 0.0009
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Converting a fraction to percent1. First, convert fraction to decimal 2. Next, convert decimal to percent
Ex d Convert 7/8 to a percent
Ex e Convert 7 2
3 to a percent
Shortcut - Only works when denominator is a factor of 1001. Multiply top and bottom to build the denominator to 100.2. Change /100 to %
Ex f Convert to a percent:1320
310
4750
100%, 50%, and 10%
Ex g In a class of 40 students
100% have had a cold at some time
50% have a cold now
10% are taking cold medication now
How many students do these percentages represent?
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Practice Problems
1. Find percent notation for:
0.7 0.3891
512
725
2. Find decimal notation for:
57% 1.5% 22 ½ % 240%
3. Find fraction notation for:
57% 1.5% 22 ½ % 240%
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7.2 The Percent Proportion (Dividing Equation)
percent =
partwhole =
amountbase
percent =
percent number100 =
p100
Ex a In a math class, 26 out of 40 students pass. What percent of students pass?
Ex b A cooler has 11 colas and 14 other sodas. What percent are colas?
Ex What percent of 40 classes is 7 classes?
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7.3 The Percent Equation (Multiplying Equation)
Translations: of out of is
percent of whole is part OR part is percent of whole
Word phrase Equation Unknowna) 15% of 50 is what number?
b) 200% of what number is 14?
c) What percent of 40 is 12?
Note: When solving or performing calculations, represent percent as a decimal
Solve:
Ex What is 30% of $19.95?
Ex 20% is 30 out of how many?
Ex 7 of 9 is what percent?
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Ex 16 2
3 % of what number is 5?
How do we know when to choose 1) part = percent ¿whole vs. 2) percent =
partwhole ?
Quick Calculations with 100%, 10% and 1%
Estimates with 50% and 25%
Practice Problems1. What is 6.5% of 80?
2. What percent of 40 is 23?
3. 30% of what number is 18?
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7.4 Applications with PercentWe will make use of the relationships:
1) part = percent of whole and 2) percent =
partwhole
Ex A college has 18,000 students and 60% receive financial aid. How many students receive financial aid?
Ex A college accepts 40% of all applicants. If 500 students were accepted, how many applied?
Ex 2000 DVD’s are produced, and 37 are damaged. What percent can be sold (undamaged)?
Ex Sue missed 6 questions on a test and got 80%. How many questions were on the test?
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Practice Problems:
1) A bag of 40 potatoes is 5% rotten. How many potatoes are rotten?
2) Ana has taken 8 math units, 15 English units, and 17 other units. What percent are math units?
Percent Increase and DecreaseNote: Both increase and decrease are a “part” of the original base
Ex Ana’s electric bill was $135 last month, and is $160 this month. Find the percent of increase.
Ex A tent was originally priced at $200, and it now sells for $145. What is the percent decrease?
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7.5 Consumer Applications: Sales Tax, Tips, Discounts, Simple Interest
Rate of tax, interest, or discount = percent of tax, interest, or discount
Ex If I want to leave a 15% restaurant tip and the food cost is $60, how much should the tip be?
Ex Sales tax adds $12.74 to the price of a fire pit. If the sales tax rate is 8%, find the original price.
Ex A backpack is discounted and sells for $25. If the amount of discount is $15, find the original price and rate of discount.
Ex A $60 pair of shoes is marked 25% off. If an additional 10% is taken off, what is the final price? Is this the same as 35% off?
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Simple Interest Formula: I = PrtP = Principal – amount invested I = Interest– amount added to principal r = interest rate, – percent used to calculate interest – usually annualt = time – usually in years, if the interest rate is annual
Ex How much interest accumulates on a loan of $15,000 borrowed for 1 year at 20% simple interest?
Ex How much interest accumulates on the same loan above for 5 years?
Practice Problems1. A $40 pair of shoes is charged 7.35% sales tax. How much tax is charged, and
what is the final price?
2. A sweater originally costing $75 is marked down to $45. What is the rate of discount?
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8.1 U.S. Measurement Units (English Units)
U.S. Measurement UnitsLength Weight
12 inches (in) = 1 foot (ft) 3 feet (ft) = 1 yard (yd)5280 feet (ft) = 1 miles
16 ounces (oz) = 1 pound (lb)2000 pounds (lb) = 1 ton
Capacity (Volume) Time8 fl. oz = 1 cup (c)2 cups (c) = 1 pint (pt)2 pints (pt) = 1 quart (qt)4 quarts (qt) = 1 gallon (gal)
60 seconds (sec) = 1 minute (min)60 minutes (min) = 1 hour (hr)24 hours (hr) = 1 day7 days = 1 week
Converting Large units to Small units (short cut)
Ex a 5 1
3 yards = ___________ inches
Note relationship between units and numbers
Converting between units: Multiply by fractions that equal 1 (unit fractions)
Ex b 12 yds. = ________________ ft
Conversion Procedure (also called dimensional analysis)1. Write equations containing the original and desired quantities2. Write the original quantity on top3. Multiply by unit fractions, arranging units to cancel4. Cancel units, put numbers together
Ex 10 miles = ________ inches
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Ex Convert 4 ½ gallons to cups.
Ex How many tons is 64,000 oz?
Ex 150 min = ______________ hr.
Bonus Problems
Ex Convert 60 mph to ft/sec
Ex Convert 2 ½ sq. ft to sq. inches
Ex How many sq. yards of carpet are needed for a 12 ft. X 20 ft. room?
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8.2 The Metric System – Length
Metric Length UnitsPrefix kilo-
meterhecto-meter
deka-meter meter
deci-meter
centi-meter
milli-meter
Meaning 1000 meters
100meters
10meters
1meter
1/10meter
1/100meter
1/1000meter
Symbol km hm dam m dm cm mm
Reasonable units
1. A newborn baby’s head is 12 ____________ across.
2. My husband is 1.7 ____________ tall.
3. Ira runs 5 ________________ in 30 minutes.
4. A wedding ring has a thickness of about 1 ______________.
5. The swimming pool is 50 __________ long.
Converting Units
Ex Convert 8.6 m to cm
Ex 46,900 mm = ______________ km
Ex 0.352 cm = _____________ mm
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8.3 The Metric System – Capacity and Mass
Metric Capacity (Volume) UnitsPrefix kilo-
litehecto-liter
deka-liter liter
deci-liter
centi-liter
milli-liter
Meaning 1000 liters
100liters
10liters
1liter
1/10liter
1/100liter
1/1000liter
Symbol kL hL daL L dL cL mL
Note: 1 mL = 1 cubic centimeter = 1 cc (medical)
Metric Weight UnitsPrefix kilo-
gramhecto-gram
deka-gram gram
deci-gram
centi-gram
milli-gram
Meaning 1000 grams
100grams
10grams
1gram
1/10gram
1/100gram
1/1000gram
Symbol kg hg dag g dg cg mg
Reasonable Units
1. A football player weights 100____________.
2. Sal drank a 250 ____________ soda for lunch.
3. An allergy medicine has 25 ____________ of antihistamine.
4. The car’s gas tank holds 50 ______________.
5. The burgCer contains 30 ___________ of fat.
Conversions can be done with unit fractions or counting.
Ex Convert 320 mg to g
Ex Convert 42.3 mL to L
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Ex 0.0024 kg = _______________ mg
Ex During a marathon, an athlete can sweat up to 1.5 L. How many mL is this?
Related Prefixes
kilobytes (KB)
megabytes (MB)
gigabytes (GB)
terabytes (TB)
milliseconds (msed)
microseconds (sed)
nanoseconds (nsec)
picoseconds (psec)
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8.4. Metric Applications
Ex A dose of cough medicine is 25 mL. How many doses are in a 0.5 L bottle?
Ex A certain (legal) medical product costs $15 per gram. What is the value of a 3 kg shipment of this product?
Ex A runner has a stride length of 1.2 meters. How many strides are taken in a 2 km race?
Ex A picture frame is made from two 1.5 m pieces and two 80 cm pieces. The price of the wood is $6/meter, plus 7% sales tax. How much does the wood for the frame cost?
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8.5 Metric – U.S. Conversions
Metric to U.S. Units U.S. to Metric Units1 kilometer ¿ 0.6214 mile 1 meter ¿ 1.094 yards1 meter ¿ 3.281 feet1 centimeter ¿ 0.3937 inch
1 mile ¿ 1.609 kilometers 1 yard ¿ 0.9144 meter1 foot ¿ 0.3048 meter1 inch ¿ 2.54 centimeters
1 liter ¿ 0.2642 gallon 1 liter ¿ 1.057 quarts
1 gallon ¿ 3.785 liters 1 quart ¿ 0.946 liter
1 kilogram ¿ 2.2 pounds1 gram ¿ 0.0353 ounce
1 pound ¿ 0.454 kilograms1 ounce ¿ 28.35 grams
Temperature formulas: , where F = Fahrenheit temp., C = Celsius temp.
C=59(F−32) F=9
5C+32
For easier calculations without a calculator, you may round to 2 significant figures. When using a calculator, it’s best to keep more digits for precision.
Ex Convert 3 yards to cm
Ex How many cups are in a 2-liter bottle?
Ex A speed limit sign in Canada reads “100 kph.” How many mph is this?
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Ex Convert 2 kg to ounce
Ex Convert 68o F to Celsius
Bonus: “Irregular” conversions
Ex A vet administers Butorphanol (a sedative) to a 1000 lb. horse. The recommended dosage is 0.02 mg/kg. A vial of the drug has a concentration of 10 mg/mL. How much of the drug should be injected?
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Conversion Charts for Exams
U.S. Measurement UnitsLength Weight
12 inches (in) = 1 foot (ft) 3 feet (ft) = 1 yard (yd)5280 feet (ft) = 1 miles
16 ounces (oz) = 1 pound (lb)2000 pounds (lb) = 1 ton
Capacity (Volume) Time8 fl. oz = 1 cup (c)2 cups (c) = 1 pint (pt)2 pints (pt) = 1 quart (qt)4 quarts (qt) = 1 gallon (gal)
60 seconds (sec) = 1 minute (min)60 minutes (min) = 1 hour (hr)24 hours (hr) = 1 day7 days = 1 week
Metric Length UnitsPrefix kilo-
meterhecto-meter
deka-meter meter
deci-meter
centi-meter
milli-meter
Meaning 1000 meters
100meters
10meters
1meter
1/10meter
1/100meter
1/1000meter
Symbol km hm dam m dm cm mm
Capacity and weight units can be found by replacing “meter” with “liter” or “gram”Abbreviations replace “m” with “L” or “g”
Note: 1 mL = 1 cubic centimeter = 1 cc (medical)
Metric to U.S. Units U.S. to Metric Units1 kilometer ¿ 0.6214 mile 1 meter ¿ 1.094 yards1 meter ¿ 3.281 feet1 centimeter ¿ 0.3937 inch
1 mile ¿ 1.609 kilometers 1 yard ¿ 0.9144 meter1 foot ¿ 0.3048 meter1 inch ¿ 2.54 centimeters
1 liter ¿ 0.2642 gallon 1 liter ¿ 1.057 quarts
1 gallon ¿ 3.785 liters 1 quart ¿ 0.946 liter
1 kilogram ¿ 2.2 pounds1 gram ¿ 0.0353 ounce
1 pound ¿ 0.454 kilograms1 ounce ¿ 28.35 grams
Temperature formulas where F = Fahrenheit temp., C = Celsius temp.
C=59(F−32) F=9
5C+32
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9.1 Interpreting Data from Tables and Pictographs
Tables
Ex a See table, Objective 1, page 638
1) What percent of United’s flights were on time?
2) Which airline had the worst on-time performance?
3) Which airline had the best luggage handling record?
4) What is the ratio of United’s to Jetblue’s luggage problems?
5) Which airline would you be most likely to use? Least likely?
Pictographs
Ex b See pictograph, Objective 2 (example 3), page 640
1) What is the population of Atlanta?
2) What is the population of Dallas?
3) What is the difference in population between Atlanta and Dallas?
4) Moving from Atlanta to Dallas, what is the percent increase in population?
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9.2 Circle Graphs (aka Pie Graphs)
Ex a From page 646, “Where Did the Oil Go?”
1) What percent of the leaked oil was chemically dispersed?
2) If 5 million barrels were leaked, how much oil was chemically dispersed?
3) What are the 2 most likely outcomes of the leaked oil?
4) What is the least likely outcome of the leaked oil?
Ex b Draw a circle graph for the following data:
Age of student
< 18 2%
18-24 61%
25 – 34 20%
35 -44 9%
45+ 8%
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9.3 Bar Graphs and Line Graphs
Ex a From the bar graph under Objective 2, page 6541) How many new high-speed connections were made in the 2nd quarter of 2015?
2) In what quarter was the number of new high-speed connections least?
3) In the 3rd quarter, what is the amount of increase in new high-speed connections from 2015 to 2016?
4) What is the percent increase in high-speed connections in the 3rd quarter, from 2015 to 2016?
Ex b Make a bar graph of class data showing favorite technology applications
Number of users
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Line Graphs
Ex c From the line graph, Objective 4, page 656
1) In what year were about 2 million trucks sold?
2) Between which years did sales of cars and trucks remain essentially unchanged?
3) What was the change in car sales from 2014 to 2015?
4) What was the percent increase in car sales from 2014 to 2015?
Ex d Make a line graph of class data showing the number of siblings of the students in the class.
How many siblings? Number of responses
0
1
2
3
4
5
6
7 or more
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9.4 The Rectangular (Cartesian) Coordinate System
Real-life graphs have points (e.g. line graph on page 656) that connect 2 quantities:
Sales of Cars
Year
Cartesian Coordinate Graph
axis – number line used to locate a pointpoint – a location in spaceordered pair (x,y) – the coordinates of a point (x is always first)
Ex a Find the coordinates of the points shown
Ex b Plot the points
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Quadrants - 4 regions defined by the x and y axes
Ex c Which quadrant is each point in?
(3, -2)
(-10, - 20)
(4, 0)
(0, -7)
(-52, 37)
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9.5 Intro to Graphing Linear Equations (2 variables)
A solution is a number or set of numbers that gives a true equation. For x + 8 = 12, the solution looks like:
For x – y = 5, a solution looks like
Ex Are the following ordered pairs solutions of y = - ½ x + 3?
(2, 2)
(4, -5)
(0, 3)
Finding a Solution (there may be more than one solution)1. Choose any number for either x or y.2. Plug that value into the equation to calculate the number for the other variable. Write
as an ordered pair.
Ex Find 3 solutions of the equation 3x – y = 2
x y
Graphing a Linear Equation (procedure)1. Find 2 (or more) solutions to the equation2. Plot the solutions as points3. Draw the line through the points
Ex Graph the 3 solutions from above
Note: The solutions of a linear equation form _______________________Ex Graph the equation y = ½ x – 3
102
This equation has ___________________ slope because the direction is __________
Ex Graph the equation 2x + 5y = 0
This equation has ___________________ slope because the direction is __________
Practice Problems1. Graph the equation x – 2y = 4, and give the direction of its slope
2. Graph the equation y = - 3x + 6, and give the direction of its slope
10.1 Product and Power Rules for Exponents
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Recall some exponent examples:
- 24
(-2)4
Product Rule of Exponents
am ¿ an = am+n
Ex
Cautions: 1) 75⋅73
=
2) 75+73
Product Rule: Several factors1. Gather numbers, then like variables together2. Multiply coefficients, then each type of variable separately.
Power to a Power Rule of Exponents – Power Rule (a) in your text
(am)n = amn
Ex
Product to a Power Rule - Power Rule (b) in your text
104
(ab )m=am⋅bm
1. Raise the coefficient to the power and evaluate2. Raise each variable to the power and simplify
Caution:
Fraction to a Power Rule
( ab )
n=a
n
bn
105
10.2 Integer Exponents & the Quotient Rule
Quotient Rule of Exponents
am
an=am−n
Zero Rule of Exponents – any number raised to the zero power =
a0 =
Negative Exponents
a−n= 1
an
Quotients and Negative Exponents
106
10.3 Scientific Notation
Scientific Notation – a special form of writing numbers.It is most useful for very large (or very small) numbersForm:
Exactly one non-zero digit left of decimal point Zero or more digits right of decimal point Multiplied by power of 10 to express place value
Ex Which of the following are in scientific notation?
Converting Scientific Notation to Place Value Form1. Decide if exponent will make the decimal larger or smaller (pos. or neg. exponent)2. Move decimal point the number of places in exponent3. Fill in zeroes if necessary
Converting Place Value Form to Scientific Notation1. Put decimal point after the first non-zero digit2. Count how many places you moved3. Write that number as the exponent – positive exp. for large numbers (>1),
negative exp. for small numbers (<1)
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10.4 Adding and Subtracting Polynomials
Recall like terms – have exactly the same variable parts (including exponents)
Ex a Combine the like terms:1) 7a + 8b – 2a
2) 6x2 + 15x – 2x2
Polynomial - a sum (or difference) of terms with the form: axn
Is a polynomial:
Not a polynomial:
Descending order – exponents start with highest power and decrease
Ex Write the polynomial 3x2 – x4 – 5 + 2x in descending order
Degree of a term – exponent on that term
Degree of a polynomial – highest exponent of all terms in the polynomial
Ex For the polynomial 3x2 – x4 – 5 + 2x , write each term, its coefficient, variable part, and degree. Then find the degree of the polynomial.
Term Coefficient Variable Part Degree of Term Degree of Polynomial
Adding Polynomials
Vertical Method - Stack like terms in descending order, then combineEx
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Horizontal Method - Gather like terms in descending order, then combineEx
Subtracting Polynomials - Change to an addition problem by1. Changing subtract to add2. Changing the sign of ALL terms in 2nd parentheses (subtracted terms) - distribute
the subtraction sign to all terms
Ex
Perimeter
Ex Find the perimeter of the rectangle:
109
10.5 Multiply Polynomials
Multiplying a Polynomial by a Monomial1. Use the distributive law to multiply each term inside the parentheses2. Simplify as above.
Multiplying 2 Polynomials: Horizontal Method 1. Multiply first term in first polynomial by second polynomial2. Multiply each of the other terms by second polynomial3. Gather like terms
Multiplying 2 Polynomials: Vertical Method1. Stack polynomials, with “like terms” in a column2. Start with the smallest term in the second row; multiply each term in top row3. Go to the next term in second row; multiply each term in top row; shift left4. Repeat if needed5. Add like terms
