Basic College Mathematics (ALEKS)
Section 1 - 1
Chapter ONE – WHOLE NUMBERS
Introduction to Whole Numbers
Ones
Tens
Hundreds
Ones
Tens
Hundreds
Ones
Tens
Hundreds
Ones
Tens
Hundreds
Ones
Tens
Hundreds
Whole Numbers – Positive Whole (NO FRACTIONS) with ZERO
Digit – 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Periods – groups of three
Place Value Chart – (Page 2)
Trillions
Billions
Millions
Thousands
Ones
1
3
0
6
3
1
3
8
1
2
1 Billion, 306 Million, 313 Thousands, 812 Ones
Standard Notations – 4,123
Expanded Notations – Four Thousands, One Hundred, Two Tens, Three Ones
Word Names – Four Thousand, One Hundred, Twenty-Three
What does the digit “2” mean in each number 1 – 6?
526,555
265,789
42,789,654
24,789,654
8924
5,643,201
2 ten thousands
2 hundred thousands
2 millions
2 ten millions
2 tens
2 hundreds
Write expanded notation
280,219
2 hundred thousands; 8 ten thousands; zero thousands;
2 hundreds; 1 ten; 9 ones
1 thousand + 8 hundreds + 9 tens + 5 ones
2 ten thousands + 3 thousands + 4 hundreds + 1 ten +
6 ones
1895
23,416
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ALEKS Chapter 1
Basic College Mathematics (ALEKS)
3031
Chapter ONE – WHOLE NUMBERS
3 thousands + 3 tens + 1 one (DON’T NEED THE ZERO
HUNDREDS)
4 thousands + 1 hundred + 8 tens
1 hundred thousand + 5 ten thousands + 4 thousands +
6 hundreds + 1 ten + 6 ones
4180
154,616
Write a word name:
49
forty-nine
16
sixteen
38
thirty-eight
204
two hundred four
45,155
forty-five thousand, one hundred fifty-five
1,879,204
one million, eight hundred seventy-nine thousand, two hundred
four
6,449,000,000
six billion, four hundred forty-nine million
Write Standard notation: two hundred thirteen million, one hundred five thousand,
three hundred twenty-nine:
213,105,329
The Number Line and Order (number to right are larger)
----+----+----+----+----+----+----+----+----+
0
1 2 3
4 5
6 7 8
< = less than
> = greater than
points to smaller number!!!
9?5
9>5
8 ? 19
8 < 18
Section 1 - 2
Addition of Whole Numbers and Perimeter
Sum – The total value of items to be added.
Addend – The items to be added.
Additive Identity – is ZERO. A+0=0+A=A.
Associative Law of Addition – If you are just adding, you can do it in any
order.
A+(B+C)=(A+B)+C
Commutative Law of Addition – If you add A+B you get the same value
when you add B+A.
A+B = B+A
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ALEKS Chapter 1
Basic College Mathematics (ALEKS)
Chapter ONE – WHOLE NUMBERS
Perimeter – the distance around an object is its perimeter.
Words / Phases that IMPLY Addition
Hints
Add Sum
Sum
1+9 10
Added To
2+8 10
Increased By
3+7 10
More Than
4+6 10
Plus
5+5 10
Total Of
7968 + 5497 = 13,465
6203 + 3542 = 9745
9804 + 6378 = 16,182
1932 + 6723 + 9878 + 8941 = 27,474
Find perimeter:
5 in + 6 in + 9 in + 5 in + 4 in = 29 in
16 ft
15 ft
15 ft
16 ft
16 ft + 15 ft + 16 ft + 15 ft = 62 ft
“3 by 5 inch” index card 3 + 5 + 3 + 5 = 16 inches
“5 by 8 inch”
5 + 8 + 5 + 8 = 26 inches
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ALEKS Chapter 1
Basic College Mathematics (ALEKS)
Section 1 – 3
Chapter ONE – WHOLE NUMBERS
Subtraction of Whole Numbers
Minuend – The number from which another number is subtracted.
Subtrahend – The number being subtracted.
Difference – Value after subtraction.
Missing Addend:
2 + [] = 7 7 – 2 = difference
Words/Phrases that IMPLY Subtraction
Minus
Difference
Decreased By
Less Than
Subtract From
(Note – NO ASSOCIATE OR COMMUTATIVE LAWS – not any way you
want)
7893 – 4092 =
8686 – 2358 =
7145 – 2398 =
70 – 14 =
503 – 298 =
7007 – 6349 =
6000 – 3149 =
9035 – 7480 =
3801
6328 check 6328 + 2358 = 8686
4747 check 4747 + 2398 = 7145
56
205
658
2851
1546
Section 1 – 4
Rounding and Estimating
Rounding –
a)
Locate the digit to be rounded
b)
Consider the next digit to the right
c)
If it is (0,1,2,3,4) down (5,6,7,8,9) up
d)
Change all the digits to the right to ZEROs
= means equal to
≈ means is approximately equal to.
687 ≈ 690
≠ means not equal to
687 ≠ 690
Estimating –
Inequality –
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ALEKS Chapter 1
Basic College Mathematics (ALEKS)
Chapter ONE – WHOLE NUMBERS
Round to nearest ten:
37 40
52 50
73 70
98 100
35 40
75 80
85 90
137 140
473 470
235 240
285 290
Round to nearest hundred:
641 600
759 800
750 800
9325 9300
Round to nearest thousands:
7896 8000
8459 8000
19,343 19,000
68,500 69,000
Round to ten, hundred, and thousand:
48,968
ten: 48,970;
hundred: 49,000;
269,582
ten: 269,580;
hundred: 269,600;
thousand: 49,000
thousands: 270,000
Estimate by Round to ten first then add
74 + 23 + 35 + 66 =
70 + 20 + 40 + 70 = 200
Estimate by Round to hundred
650 + 685 + 238 + 168 =
700 + 700 + 200 + 200 = 1800
9285 – 6739 =
9300 – 6700 = 2600
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ALEKS Chapter 1
Basic College Mathematics (ALEKS)
Chapter ONE – WHOLE NUMBERS
Estimate by Round thousands
23,278 – 11,698 =
23,000 – 12,000 = 11,000
Chapter 1 - 5
Multiplication
Multiplication = Repeated Addition
3 X 5 = 5 + 5 + 5 = 5 X (1 + 1 + 1) = 5 X (3) = 15
Factors are the numbers to be multiplied and the result is the Product.
Notation: 3 X 5 = 3 • 5 = (3)(5) = 3(5) = 15
Multiplying by ZERO: A • 0 = 0
Multiplication Identity: A • 1 = A
Distributive Law of Multiplication: A • (B + C) = (A • B) + (A • C)
Commutative Law of Multiplication: A • B = B • A
Associative Law of Multiplication: A • (B • C) = (A • B) • C
AREA: number of square units to fill object.
Rectangular objects:
Area = Length X Width = L • W
Words / Phases Implying Multiplication
Product
Times
Multiply … by
“OF”
Repeated addition
58 x 2 =
37 x 4 =
823 x 6 =
1348 x 5 =
45 x 23 =
48 x 63 =
746 x 62 =
245 x 837 =
472 x 306 =
408 x 704 =
2344 x 6005 =
472 x 830 =
116
148
4938
6740
1035
3024
46,252
205,065
144,432
287,232
14,075,720
391,760
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ALEKS Chapter 1
Basic College Mathematics (ALEKS)
Chapter ONE – WHOLE NUMBERS
2344 x 7400 =
17,345,600
100 x 562 =
56,200
1000 x 562 =
562,000
ESTIMATE Product – ROUND FIRST then do Math
Round ten
837 x 245
840 x 250 = 210,000
Round hundred
837 x 245
800 x 200 = 160,000
Area of table tennis table 9 ft by 5 ft
A = L x W = 9 ft x 5 ft = 5 ft x 9 ft = 45 sq ft
Chapter 1 – 6
Division
Division = Repeated Subtraction
20 ÷ 5 = 4 Twenty divided by 5 equals 4
Dividend – number being divided
(20)
Divisor – number doing the dividing (5)
Quotient – answer
(4)
Remainder: What is left over, when the Quotient is not a Whole number.
22 ÷ 6 = 3 + R
Dividend ÷ Divisor = Quotient + Remainder
20 ÷ 5 = [] related equation 5 • [] = 20 (missing factor)
A ÷ B (BUT B CANNOT EQUAL ZERO) = C (Unique) and A = B • C
A ÷ B = C and A ÷ C = B
Word / Phases that Implying Division
Divide
Quotient
Per
Divides into
Shared equally
Repeated subtraction
Properties of Division
Divide by “1”
A÷1=A
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ALEKS Chapter 1
Basic College Mathematics (ALEKS)
Chapter ONE – WHOLE NUMBERS
Any non-zero number divide by self
A ÷ A = 1 (A ≠ 0)
Zero divided by any non-zero number
0 ÷ A = 0 (A ≠ 0)
Divide by zero
A ÷ 0 = NOT DEFINED (we just don’t divide by zero)
54 ÷ 6
or
54/6
Repeated Subtraction
54 ÷ 9
54, 45, 36, 27, 18, 9, 0 (6)
61 ÷ 9
53 ÷ 12
157 ÷ 24
239 ÷ 4
8855 ÷ 6
5075 ÷ 5
6030 ÷ 45
3288 ÷ 52
4846 ÷ 6
7616 ÷ 7
9724 ÷ 27
44847 ÷ 56
6R7
4R5
6 R 13
59 R 3
1475 R 5
1015
134
63 R 12
807 R 4
1088
360 R 4
800 R 47
Section 1 – 7
6 x 9 = 54
6 x 9 = 54 + 7 = 61
2 x 12 = 48 + 5 = 53
6 x 24 = 144 + 13 = 157
Exponents, Square Roots, and Order of
Operations
Exponents:
Base: is the number, which is going to be raised to the Power.
Exponent: is the Power to which the base is going to be raised.
3 • 3 • 3 • 3 = 34
Any number to the one power is itself. 31 = 3
3A•3B = 3A+B
3A÷3B = 3A-B
2nd power is that same as Squared
3rd power is Cubed
Area of a Square = Length (side) squared = L2 = S2
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ALEKS Chapter 1
Basic College Mathematics (ALEKS)
Chapter ONE – WHOLE NUMBERS
105 = 10 x 10 x 10 x 10 x 10 = 100,000
105 ≠ 10 x 5
Write exponential notation
5x5x5x5
54
5x5x5x5x5
55
10 x 10
102
10 x 10 x 10 x 10
104
Evaluating Exponential Expressions
104
10 x 10 x 10 x 10 = 10,000
2
10
10 x 10 = 100
3
8
8 x 8 x 8 = 512
5
2
2 x 2 x 2 x 2 x 2 = 32
√
√
Square Roots:
Reverse of the process of Squaring
Radical sign = positive answers
√ = A * A
A
0
1
2
3
4
5
6
7
8
9
10
11
12
13
0
1
4
9
16
25
36
49
64
81
100
121
144
169
0
1
2
3
4
5
6
7
8
9
10
11
12
13
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ALEKS Chapter 1
Basic College Mathematics (ALEKS)
Chapter ONE – WHOLE NUMBERS
Simplifying Expressions: What if there is adding, subtracting, multiplying
and dividing all in the same problem.
Rules For Order of Operations:
1.
Do all calculations within parentheses (), brackets [], or braces {} before
operations outside. If parentheses are within parentheses, do innermost ones first.
There is no differences in (), [], or {}, they just help clarify the problem.
( [ { = opening
) ] } = closing
find first closing then back up!
2.
Evaluate all exponential expressions.
3.
Do all multiplication and divisions in order from left to right.
4.
Do all additions and subtractions in order from left to right.
P Please
E Excuse
M My*
D Dear*
A Aunt**
S Sally**
*Same level from left to right
**Same level from left to right
93 – 14 x 3
93 – 42 = 51
104 ÷ 4 + 4
26 + 4 = 30
25 x 26 – (56 + 10)
25 x 26 – 66
650 – 66 = 584
75 ÷ 5 + (83 – 14)
75 ÷ 5 + 69
15 + 69 = 84
64 ÷ (32 ÷ 2)
(64 ÷ 32) ÷ 2
64 ÷ 16 = 4
2÷1=2
(28 + 13) + 11
28 + (13 + 11)
41 + 11 = 52
28 + 24 = 52
9 x 4 – (20 + 4) ÷ 8 – (6 – 2)
36 – 24 ÷ 8 – 4
36 – 3 – 4
10
9 x 4 –24 ÷ 8 – (6 – 2)
33 – 4 = 29
9 x 4 –24 ÷ 8 – 4
ALEKS Chapter 1
Basic College Mathematics (ALEKS)
Chapter ONE – WHOLE NUMBERS
5 x 5 x 5 + 26 x 71 – (16 + 75)
5 x 5 x 5 + 26 x 71 – (16 + 25 x 3)
5 x 5 x 5 + 26 x 71 – 91
25 x 5 + 26 x 71 – 91
125 + 26 x 71 – 91
125 + 1846 – 91
1971 – 91= 1880
6 x 2 + 10 x 20 + 8 x 8 – 23
30 ÷ 5 x 2 + 10 x 20 + 8 x 8 – 23
12 + 10 x 20 + 8 x 8 – 23
12 + 200 + 8 x 8 – 23
12 + 200 + 64 – 23
276 – 23 = 253
212 + 64 – 23
95 – 2 x 2 x 2 x 5 ÷ (24 – 4)
95 – 2 x 2 x 2 x 5 ÷ 20
95 – 8 x 5 ÷ 20
95 – 40 ÷ 20
95 – 2 = 93
95 – 4 x 2 x 5 ÷ 20
53 + 26 x 71 – (16 + 25 x 3)
53 + 26 x 71 – (16 + 75)
53 + 26 x 71 – 91 125 + 26 x 71 – 91
125 + 1846 – 91
1971 – 91= 1880
(1 + 3)3 + 10 x 20 + 82 –23
64 + 10 x 20 + 64 – 23
328 – 23 = 305
43 + 10 x 20 + 82 –23
64 + 200 + 64 –23
81 – 32 x 2 ÷ (12 – 9)
81 - 32 x 2 ÷ 3
81 – 18 ÷ 3
81 – 6 = 75
23 x 28 ÷ 29
2048 ÷ 512 = 4
8 x 28 ÷ 29 8 x 256 ÷ 29
9 x 5 + {6 ÷ [14 – (5 + 3)]}
9 x 5 + {6 ÷ 6}
9x5+1
64 + 10 x 20 + 82 –23
264 + 64 – 23
81 - 9 x 2 ÷ 3
8 x 256 ÷ 512
9 x 5 + {6 ÷ [14 – 8]}
45 + 1 = 46
[18 – (2 + 7) ÷ 3] – (31 – 10 x 2)
[18 – 9 ÷ 3] – (31 – 20)
[18 – 3] – (31 – 20)
15 – (31 – 20)
15 – 11 = 4
Average (mean): is the sum of the items divided by the number of Items.
What is the average weight of these three people: 201, 175, and 155 pounds.
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ALEKS Chapter 1
Basic College Mathematics (ALEKS)
Chapter ONE – WHOLE NUMBERS
Solution = (201 + 175 + 155) ÷ 3 = 531 ÷ 3 = 177
Find Average of 1670, 1483, 1450, and 1381
(1670 + 1483 + 1450 + 1381) ÷ 4 = 5984 ÷ 4 = 1496
Section 1 – 8
Problem Solving Strategies
Five Steps for Problem Solving
1.
Familiarize yourself with the situation.
a.
Read and reread until you understand problem.
b.
Draw a diagram or see if a formula applies.
c.
Assign a variable to the UNKNOWN (understand the Units).
2.
Translate problem into an equation.
3.
Solve the equation.
4.
Check the answer in the original wording of the problem (Units).
5.
State the answer to the problem clearly with appropriate Units.
The odometer of a car read 24,316 miles 37,134 miles – 24,316 miles =
12,818 miles
last year. This year the reading is
37,134. How many miles was the car
driven during the year.
(What is the difference between the two
readings = subtraction.)
One page of print in a book contains 48 48 lines of test. How many lines of text are ∗ 21 =
in one chapter containing 21 pages?
1008 lines
(48 first page + 48 second page ….
Repeated addition = multiplication.)
A vat of flour at a food distributor holds 580 lb ÷ 5 lb = 116 bags
580 lb of flour. How many 5-lb of flour
can be filled from the van.
(580 – 5 = 575 first bag
575 – 5 = 570 second bag ….
Repeated subtraction = division.)
12
ALEKS Chapter 1
Basic College Mathematics (ALEKS)
Chapter ONE – WHOLE NUMBERS
Danielle buys a new entertainment
center with a new plasma television for
$4240. She pays $1000 down, and the
rest is paid off in equal payments over 2
years. What is the monthly payment?
(Two questions: how much is left to pay
and how many payments is that amount
divided into.)
Taylor makes $18 per hour for the first
40 hours worked each week. His
overtime rate is $27 per hour. If his
total salary is $963, determine the
number of hours overtime worked?
(normal pay = $18 * 40 hours
Overtime pay = Total pay – normal pay
Overtime – 27 first hour
-27 second hour = repeated sub =
divide.)
13
Amount to pay: $4240 - $1000 = $3240
2 years *
$
!"#
=
!"#$
%&'(
$*+,
+ !"#
= 24 months
= $135 per month
Normal pay = $18*40 = $720
Overtime pay = $963 - $720 = $243
$243
= 9 ℎ1234
$27
ALEKS Chapter 1