Basic College Mathematics (ALEKS)
Section 2 - 1
Chapter TWO – FRACTIONS AND MIXED NUMBERS:
MULTIPLICATION AND DIVISION
Introduction to Fractions and Mixed Numbers
Fractions – equal parts of the whole.
Numerator – TOP = number of parts
Denominator – BOTTOM = number of parts the whole is divided into.
Proper Fraction = Numerator is LESS than the Denominator.
Improper Fraction = Numerator is GREAT than or EQUAL to the
Denominator.
Mixed Numbers = SUM of a Whole Number and a Fraction.
Procedure – Changing a Mixed number to an Improper Fraction
1) Multiply the Whole Number by the Denominator.
2) Add the result to the Numerator.
3) Keep the Denominator
=
∗
5
10 ∗ 8 + 5
80 + 5
85
10 =
=
=
8
8
8
8
15 ∗ 2 + 1
31
1
=
15 =
2
2
2
Procedure – Changing Improper Fraction to Mixed Number
1) Divide Numerator by Denominator.
2) Whole Number = Quotient
Numerator = Remainder
Denominator or Divisor = Denominator
1
ALEKS –Chapter TWO
Basic College Mathematics (ALEKS)
Chapter TWO – FRACTIONS AND MIXED NUMBERS:
MULTIPLICATION AND DIVISION
14
4
=24=2
5
5
7
95
=47=4
22
22
Section 2 – 2
Prime Numbers and Factorization
AxB =C
A and B are FACTORS of C
Factorization:
of 12
1x12
of 17
1x17
2x6
3x4
2x2x3
Find all the FACTORS of 72
1 * 72 a) left side gets bigger
2 * 36 b) right side gets smaller
3 * 24 c) let small numbers find larger numbers
4 * 18 d) where numbers pass stop NO reason to find numbers twice
5 * ___ e) Factors = 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
6 * 12 f) Always starts with “1” and ends with “Number”
7 * ___
8*9
Divisibility RULES
By 2 - number ends in a 0, 2, 4, 6, 8 (even)
1. 84
YES
2. 59
NO
3. 998
YES
4. 2225
NO
By 3 – sum of the digits is divisible by 3
5. 111 = 1+1+1 = 3 ÷ 3 = 1
6. 1111 = 1+1+1+1 = 4 ÷ 3 = 1 R 1
7. 309 3+0+9=12 ÷ 3 = 4
2
ALEKS –Chapter TWO
YES
NO
YES
Basic College Mathematics (ALEKS)
8.
17216
By 10 – ends with a 0
17. 305
18. 300
19. 847
20. 8760
By 5 – ends in 0 or 5
21. 5780
22. 3427
23. 34678
24. 7775
By 7 – no rule just divide
Chapter TWO – FRACTIONS AND MIXED NUMBERS:
MULTIPLICATION AND DIVISION
1+7+2+1+6=17÷3=5 R 2
NO
NO
YES
NO
YES
YES
NO
NO
YES
PRIME NUMBER – is a whole number greater than “1” that has ONLY
TWO factors, itself and “1”.
PRIMES from “2” to “157”
2, 3, 5, 7, 11, 13, [13x13=169] 17, 19,
23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 71, 73, 79, 83, 89, 97, 101, 107,
109, 113, 127, 131, 137, 139, 149, 151, 157
COMPOSITE NUMBER – is a whole number greater than “1” that is
not PRIME.
Note:
“0” and “1” are NEITHER prime or composite.
PRIME FACTORIZATION – the factorization in which every factor is a
prime number. Every number has JUST ONE unique prime
factorization.
Factor out “2s”, then “3s”, then “5s”, etc.
3
ALEKS –Chapter TWO
Basic College Mathematics (ALEKS)
Chapter TWO – FRACTIONS AND MIXED NUMBERS:
MULTIPLICATION AND DIVISION
Factor Tree
220
220
11
10
22
2
5
2
20
2
11
10
2
5
220
220
2
4
55
2
2
5
110
2
11
55
5
11
Successive Division – divide by primes until a prime remains.
220 2 = 110
110 2 = 55
55 5 = 11
220 = 2 * 2 * 5 * 11 = 22 * 5* 11
168
2
84
2
42
2
21
3
4
7
2 * 2 * 2 * 3 * 7 = 23 * 3 * 7
ALEKS –Chapter TWO
Basic College Mathematics (ALEKS)
Chapter TWO – FRACTIONS AND MIXED NUMBERS:
MULTIPLICATION AND DIVISION
990
10
2
99
5
3
33
3
11
2 * 3 * 3 * 5 * 11 = 2 * 32 * 5 * 11
Determine if Prime, Composite, or Neither
39 3*13 Composite
0
Neither
41 1*41 Prime
Section 2 – 3
SIMPLIFYING TO LOWEST TERMS
Equivalent Fractions = equal each other.
3
2
1
= =
6
4
2
Determining if Equivalent
"
=
if a * d = b * c then equivalent if NOT then $.
!
#
%&
'(
,
(
5
)*
)*
+
%%
-(
'(
13*11 [ ] 24*6 143 [ ] 144
$
9*24 [ ] 4*54
=
216 [ ] 216
ALEKS –Chapter TWO
Basic College Mathematics (ALEKS)
Chapter TWO – FRACTIONS AND MIXED NUMBERS:
MULTIPLICATION AND DIVISION
Simplifying to LOWEST TREMS
/∗0
/ 0
/
/
=
= ∗ = ∗1=
1∗0
1 0
1
1
.
1) Prime factor the numerator and denominator.
2) Cancel any common factors in both.
3) Multiply remaining factors in the numerator and denominator.
15
3∗5
3 5 3
3
=
= ∗ = ∗1=
35
5∗7
7 5 7
7
2 ∗ 13
2
2
26
=
=
=
3 ∗ 5 ∗ 13
3∗5
15
195
150
2 ∗ 3 ∗ 5 ∗ 5 2 ∗ 5 10
=
=
=
105
3∗5∗7
7
7
Greatest Common Factor – the largest factor of both the Numerator and
Denominator. (If you see it; prime factorization always works.)
'2
=
(∗3
&,
=
&∗%&
=
%-
=
%-
=
&+
&
,5
(∗,
=
&∗%
+∗%-
3
'4'43
,
'4'4&4&
%&
%
= 13
+
=
=
3
&4&
=
3
,
(Just divide)
%
+
(When you run out of factors, you can always multiply by 1).
Simplifying factors of 10, 100, 1000, etc
630
63 10
63
=
∗
=
190
19 10
19
1300
13 100
1 ∗ 13
1
=
∗
=
=
52000
520 100
13 ∗ 40
40
6
ALEKS –Chapter TWO
Basic College Mathematics (ALEKS)
Chapter TWO – FRACTIONS AND MIXED NUMBERS:
MULTIPLICATION AND DIVISION
21 1000
21
3∗7
3
21000
=
∗
=
=
=
35 1000
35
5∗7
5
35000
(You can cancel the same number of trailing ZEROs in both the
Numerator and Denominator.
Joanne planted 77 seeds and 55
sprouted. Geoff (Jeff) planted 140
seeds and 80 sprouted.
a) What fraction sprouted for
Joanne? What fraction sprouted
for Geoff?
b) Who had a greater portion
sprout?
Section 2 – 4
Procedure:
a)
b)
Joanne =
33
25
=
-∗%%
3∗%%
2
=
-
3
'∗(
(
Geoff =
=
=
=
%(5
%(
'∗3
3
As the Denominators are the same,
Just compare Numerators.
Joanne did better.
Multiplication of Fractions and Applications
Multiplying Fractions (Note NOT Mixed numbers)
Multiply Numerators = new Numerator
Multiply Denominators = new Denominators
0
∗0
∗ =
/ /∗
1
1
1 1
1∗1
1
6 = ∗ =
=
2
4
2 4
2∗4
8
2 5
10
∗ =
3 9
27
7
7 11 77
∗ 11 =
∗
=
12
12 1
12
7
--
ALEKS –Chapter TWO
Basic College Mathematics (ALEKS)
Chapter TWO – FRACTIONS AND MIXED NUMBERS:
MULTIPLICATION AND DIVISION
Multiplying and Simplifying Fractions
multiplying)
(Often easier to do before
4∗7
7
7 4 28
∗ =
=
=
20 3 60 4 ∗ 15 15
7 4
7∗4
7
7
∗ =
=
=
20 3 4 ∗ 5 ∗ 3 3 ∗ 5 15
2∗3∗3∗5
1
6 15
∗
=
= =
25 18 5 ∗ 5 ∗ 2 ∗ 3 ∗ 3 5 5
(when you run out of factors “* by 1”)
2
9∗2
3∗3∗2
6
97 8 =
=
=
15
15
3∗5
5
3 4 6
3∗2∗2∗3∗2
1
1
7 87 87 8 =
=
=
16 9 5
2 ∗ 2 ∗ 2 ∗ 2 ∗ 3 ∗ 3 ∗ 5 2 ∗ 5 10
Order of Operations
4
4 4
16
( )' = ∗ =
3
3 3
9
6 1
1
1 1 1
1
( ∗ )& = ( )& =
∗
∗
=
5 12
10
10 10 10
1000
%
AREA of a Triangle A = ∗ ;< ∗ =
>ℎ =
'
%
%
'
b = 8m, h = 5m A = ∗ 8 ∗ 5 = 20 = 20 '
'
(
b = 2 ft, h = 6
&
%
(
A = ∗ 26 ∗ 6 =
'
&
KITE (page128)
8
ALEKS –Chapter TWO
(
&
6 '
%
BH = /ℎ
'
Basic College Mathematics (ALEKS)
base =
%%
'
&
+ =
'
%(
'
Chapter TWO – FRACTIONS AND MIXED NUMBERS:
MULTIPLICATION AND DIVISION
= 76 ℎ
>ℎ = 26
Kite = 2*triangle
%
A = 2* ∗ 76 ∗ 26 = 14ft2
'
Section 2 – 5
Division of Fractions and Applications
Procedure:
Reciprocal
/
1
1
→
→ →
→ →
/
1
1
!
%
∗ = 1 ∗ = 1 $ 0 / $ 0
!
7
10
→
10
7
1
4
→ →4
4
1
7→
1→
9
7
1
→
1
7
1
→1
1
ALEKS –Chapter TWO
Basic College Mathematics (ALEKS)
Chapter TWO – FRACTIONS AND MIXED NUMBERS:
MULTIPLICATION AND DIVISION
Division of Fractions – to divide by a Fraction, you multiply by the
reciprocal of the fraction (divisor).
4
3
4 5
20
= ∗ =
5
7 3
21
7
2
1
2 5
10
= ∗ =
3
5
3 1
3
1
2
1 5
5
= ∗ =
( 1 <
CD
61 D ℎ 0
C0D)
4
5
4 2
8
3
9
3 10
3∗2∗5
5
= ∗
=
=
8 10
8 9
2∗4∗3∗3
12
15 1
3∗5
3
15
10 =
∗
=
=
4 10
4∗2∗5
8
4
12
20 5
4∗5∗5
25
20 =
∗
=
=
5
1 12
4∗3
3
Order of Operations
3
6 1
3
3 14
6
2 = ∗ = ∗
=2
7
14
7 2
14
7 3
4
8
4 3
3
9
( )' = ( ∗ )' = ( )' =
7
3
7 8
14
196
'
2 2 2
Recipe requires package
20
−
− − − …
5
5 5
Of chips for each batch of
Repeated sub = Cookies. How many batches
2
5
can be made with 20
20 = 20 ∗
5
2
packages?
= 50 /0ℎ<
10
ALEKS –Chapter TWO
Basic College Mathematics (ALEKS)
'-
Chapter TWO – FRACTIONS AND MIXED NUMBERS:
MULTIPLICATION AND DIVISION
-
yd ditch will be filled with yd
'
(
lengths of pipe. How many
segments are needed?
School will cost $20,000,000 to
&
build. State will pay of cost.
School district sells bonds to cover
(
of non-state costs?
a) state costs?
b) non-state costs?
c) district bond sales?
%
of police department is female.
&
of these were promoted. What
fraction were promoted
Section 2 – 6
numbers
25 5
− −
2 4
25 5
=
2 4
5
− …
4
25 4
∗ = 10
2 5
&
) of $20,000,000
of = times
3
∗ $20,000,000 = $12,000,000
5
b) $20,000,000-$12,000,000 =
$8,000,000
(
c) ∗ $8,000,000 = $6,400,000
-
'
3
1
2
1 2
2
6 = ∗ =
3
7
3 7 21
Multiplication and Division of MIXED
Procedure:
a) Change each Mixed number to a Fraction.
b) Multiply or Divide
c) Convert back to Mixed if needed.
5
23 35
23 ∗ 7
161
5
3
74 8 75 8 = 7 8 7 8 =
=
= 26
5
6
5
6
6
6
6
1
7
33 40
3 ∗ 11 ∗ 2 ∗ 20
716 8 73 8 = 7 8 7 8 =
= 60
2
11
2 11
2 ∗ 11
11
ALEKS –Chapter TWO
Basic College Mathematics (ALEKS)
Chapter TWO – FRACTIONS AND MIXED NUMBERS:
MULTIPLICATION AND DIVISION
1
43 10
43 ∗ 2 ∗ 5
215
2
77 8 (10) = 7 8 7 8 =
=
= 71
6
6
1
2∗3
3
3
1
5
31
17
31 6
31 ∗ 2 ∗ 3 62
11
710 8 72 8 = 7 8 7 8 =
∗
=
=
=3
3
6
3
6
3 17
3 ∗ 17
17
17
4
24 8 5
8∗5 5
2
84 =8
= ∗
=
= =1
5
5
1 24 8 ∗ 3 3
3
4
112 8 112 1 14 ∗ 8 14
5
12 8 =
=
∗ =
=
=1
9
9
1
9 8
9∗8
9
9
&
Recipe requires 2 cups of flour.
(
%
How much flour for 2 recipes?
&
%
2 ∗2 =
(
'
%%
(
-
∗ =
'
-2
3
= 6 cups
2
'
Costs $2 to wrap a present. Ribbon $2 – nothing to the problem.
2 ft long is used for each present. Repeated sub
2
How many presents can be
wrapped with 168 ft roll?
12
5 168 21
168 2 =
8
1
8
168 8
=
∗
1 21
21 ∗ 8 ∗ 8
=
= 64
21
ALEKS –Chapter TWO