Basic College Mathematics (ALEKS)
Section 3 – 1
Chapter THREE – FRACTIONS AND MIXED NUMBERS:
ADDITION AND SUBTRACTION
Addition and Subtraction of Like Fractions
Like Fractions – have same denominators – common denominators
Procedure
Step 1:
Add or Subtract the Numerators = new Numerator
Step 2:
Keep common Denominator
Step 3:
Simplify, if possible
2 4
6
2‫ݔ‬3
2
+ = =
=
9 9
9
3‫ݔ‬3
3
7
5
11
23
+
+
=
12 12 12
12
2 5
7
+ =
3 3
3
14
8
6
−
=
11 11
11
2
5 3
− = =1
2 2
2
Order of Operations
3
2 ଶ
5 ଶ
1 ଶ
1 1
1
( +
) =( ) = ൬ ൰ = ‫= ݔ‬
10 10
10
2
2 2
4
3
2
4
3
2 5
3
5
8
2‫ݔ‬4
4
+ ÷ =
+ ‫= ݔ‬
+
=
=
=
14 7
5
14 7 4
14 14
14
2‫ݔ‬7
7
1
ALEKS - Chapter THREE
Basic College Mathematics (ALEKS)
Chapter THREE – FRACTIONS AND MIXED NUMBERS:
ADDITION AND SUBTRACTION
Page 166
all distances are in meters = m
3
2
3
1
3
12
2‫ݔ‬6
6
݉+ ݉+ ݉+ ݉+ ݉=
݉=
݉= ݉
10
10
10
10
10
10
2‫ݔ‬5
5
1
=1 ݉
5
5 7 3 12 3
+ − =
−
଼
଼
8 8 8
8 8
ଷ
Then uses (-) ݈݃ܽ, ‫ݓ‬ℎܽ‫?ݐ݂݈݁ ݏ݅ ݐ‬
9
1
଼
= ‫ ݎ݋‬1 ݈݃ܽ
8
8
Section 3 – 2
Least Common Multiple
ହ
଻
Jamie mixed (+) ݈݃ܽ ‫ݐ݅ݓ‬ℎ ݈݃ܽ
Multiples of a Number = take number and multiple by 1, 2, 3 …
2
1x2=2
2x2=4
3x2=6
4x2=8
5x2=10
6x2=12*
7x2=14
8x2=16
12x2=24**
2
3
1x3=3
2x3=6
3x3=9
4x3=12*
5x3=15
6x3=18
7x3=21
8x3=24**
12x3=36
4
1x4=4
2x4=8
3x4=12*
4x4=16
5x4=20
6x4=24**
7x4=28
8x4=32
12x4=48
ALEKS - Chapter THREE
Basic College Mathematics (ALEKS)
Chapter THREE – FRACTIONS AND MIXED NUMBERS:
ADDITION AND SUBTRACTION
Note that both 12 and 24 are multiples of 2, 3, and 4 BUT 12 is the
LEAST COMMON MULTIPLE (LCM) of all three.
Finding the LCM by Listing Multiples:
For 15 and 25
15 25
30 50
45 75
60 100
75 125
90 150
105 175
75 is the LCM
4 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60
6 6, 12, 18, 24, 30, 36, 42, 48, 54, 60
10 10, 20, 30, 40, 50, 60
60 is the LCM
Find the LCM by Using Prime Factors:
1) Write each number as a product of prime factors
2) LCM is the product of unique prime factors from all numbers.
Use repeated factors the maximum number (MOST) of times they
appear in a factorization.
LCM of 9 and 24
9 = 3x3
24 = 2x2x2x3
2’s 3’s
9 0 2**
24 3* 1
Max 2’s = 3 and max 3’s = 2
LCM = 2x2x2x3x3 = 72
3
ALEKS - Chapter THREE
Basic College Mathematics (ALEKS)
LCM of 16 and 9
Chapter THREE – FRACTIONS AND MIXED NUMBERS:
ADDITION AND SUBTRACTION
16 = 2x2x2x2
9 = 3x3
2’s 3’s
16 4 0
9 0 2
LCM = 2x2x2x2x3x3 = 144
LCM of 36, 42, 30
36 = 2x2x3x3
42 = 2x3x7
2’s 3’s 5’s 7’s
36 2 2 0 0
42 1 1 0 1
30 1 1 1 0
LCM = 2x2x3x3x5x7 = 1260
Finding the LCM by Division by Primes:
Plan: use 2 then 3 then 5 etc. etc.
LCM 20, 36, 15
Divide by
2
2
3
3
5
20 36 15
10 18 15*
5 9 15
5 3 5
5 1 5
1 1 1
LCM = 2x2x3x3x5 = 180
4
ALEKS - Chapter THREE
30= 2x3x5
Basic College Mathematics (ALEKS)
Chapter THREE – FRACTIONS AND MIXED NUMBERS:
ADDITION AND SUBTRACTION
Application of the LCM
On an oval track, One runner takes
60 sec, second takes 75 sec, and
third takes 90 sec. When will they
all be back to the start again? Each
time they complete a lab it is a
multiple of their time. Therefore,
LCM is the answer.
60 75 90
2
30 75 45
2
15 75 45
3
5
25 15
3
5
25 5
5
1
5
1
5
1
1
1
LCM = 2x2x3x3x5x5 = 900 sec
60 = 2x2x3x5
75 = 3x5x5
90 = 2x3x3x5
LCM = 2x2x3x3x5x5 = 900 sec
Equivalent Fractions and Ordering Fractions
Equivalent Fractions have the same value:
ଶ
ସ
=
ଶ௫ଵ
ଶ௫ଶ
=
ଵ
ଶ
‫݋ݏ‬
ଶ
ସ
݅‫݋ݐ ݐ݈݊݁ܽݒ݅ݑݍ݁ ݏ‬
ଵ
ଶ
Write the fractions with the indicated denominator.
ଶ
ଷ
=
?
ଵଶ
Find a least common denominator (LCD) = 12
2 4
8
‫= ݔ‬
?= 8
3 4 12
ହ
=
ହ
‫= ݔ‬
଺
଺
5
ଽ
ଽ
?
ହସ
2
8
݁‫݋ݐ ݐ݈݊݁ܽݒ݅ݑݍ‬
3
12
LSD = 54
ସହ
ହସ
? = 45
ହ
଺
݁‫݋ݐ ݐ݈݊݁ܽݒ݅ݑݍ‬
ALEKS - Chapter THREE
ସହ
ହସ
Basic College Mathematics (ALEKS)
Chapter THREE – FRACTIONS AND MIXED NUMBERS:
ADDITION AND SUBTRACTION
Order of Fractions
1) If the denominator is the same just compare numerators
2) If different convert to fractions with the LCD
ସ
଻
ሾሿ
ହ
LCD of 7 and 9 = 63
ଽ
4 9
5 7
‫ ݔ‬ሾ ሿ ‫ݔ‬
7 9
9 7
ଷ଺
଺ଷ
ሾ ሿ
ଷହ
଺ଷ
ଷ଺
଺ଷ
>
ଷହ
଺ଷ
Rank the fractions from least to greatest
LCD of 9, 15, 5
9 = 3x3
ହ
ଽ
,
଼
,
ଷ
ଵହ ହ
15 = 3x5 5 = 5 LCD = 3x3x5 = 45
8
3
15
5
5 5
25 8 3 24 3 9 27
‫= ݔ‬
‫= ݔ‬
‫= ݔ‬
9 5
45 15 3 45 5 9 45
2
1
3
5
9
Section 3 – 3
Addition and Subtraction of Unlike Fractions
1) Convert all fractions to LCD equivalent
2) Add or Subtraction Numerators
ଵ
ଵ଴
+
ଵ
ଵହ
LCD of 10, 15 LCD = 30
1 3 1 2
3
2
5
5‫ݔ‬1
1
‫ ݔ‬+ ‫= ݔ‬
+
=
=
=
10 3 15 2
30 30
30
5‫ݔ‬6
6
6
ALEKS - Chapter THREE
Basic College Mathematics (ALEKS)
Chapter THREE – FRACTIONS AND MIXED NUMBERS:
ADDITION AND SUBTRACTION
1 2
1 5 2 3
5
6
11
+ = ‫ ݔ‬+ ‫= ݔ‬
+
=
3 5
3 5 5 3
15 15
15
9 7
9 2 7
18 7
25
5‫ݔ‬5 5
+
= ‫ ݔ‬+
=
+
=
=
=
10
5‫ݔ‬2 2
5 10 5 2 10 10 10
7 1
7 2 1 3 14 3
11
− =
‫ ݔ‬− ‫= ݔ‬
−
=
12 8
12 2 8 3 24 24
24
଻
ଵ଼
+
ସ
ଵହ
−
ଵ଻
ଷ଴
LCD of 18, 15, 30 18= 2x3x3 15=3x5 30=2x3x5
LCD = 2x3x3x5 = 90
Missing factors 18 = 5 15 = 2x3 30 = 3
To Convert use the missing factors
4 6
17 3
35 + 24 − 51 59 − 51
8
7 5
=
=
൬ ‫ ݔ‬൰+൬ ‫ ݔ‬൰−൬ ‫ ݔ‬൰=
18 5
15 6
30 3
90
90
90
2‫ݔ‬4
4
=
=
2‫ݔ‬45 45
ଵ
ଶ଴
ହ
+ −
଼
଻
ଶସ
LCD of 20, 8, 24 20=2x2x5 8=2x2x2 24=2x2x2x3
LCD = 2x2x2x3x5=120 Missing factors 20=6 8=15 24=5
1 6 5 15 7 5 6 + 75 − 35 81 − 35
46
2‫ݔ‬23
‫ ݔ‬+ ‫ݔ‬
− ‫= ݔ‬
=
=
=
20 6 8 15 24 5
120
120
120 2‫ݔ‬60
23
=
60
7
ALEKS - Chapter THREE
Basic College Mathematics (ALEKS)
Chapter THREE – FRACTIONS AND MIXED NUMBERS:
ADDITION AND SUBTRACTION
Order
2 1 ଶ
14 3 ଶ
11 ଶ 11 11 121
( − ) =( − ) =( ) =
‫ݔ‬
=
21 21
21
21 21 441
3 7
4 2 1
4 5 1 2 1 4 1 3 3‫ݔ‬1 1
÷ − =
‫ ݔ‬− = − = − = =
=
15 5 6
15 2 6 3 6 6 6 6 3‫ݔ‬2 2
ଶ
ଵ
Monday, inch rain; Tuesday,
ଷ
ହ
inch, How much more (-) Monday
than Tuesday.
ଷ
Maggie mixed gal of paint with
ସ
ଷ
gal (+). Then uses of (*) the
ସ
mixture.
a) How much paint did she use?
b) How much paint is left over?
ଵ
ଷ
2
1
10 3
7
݅݊ − ݅݊ =
−
=
݅݊
3
5
15 15 15
3 1
9
4
13
+ =
+
=
4 3
12 12 12
13 3
13‫ݔ‬3
13
∗ =
=
12 4 3‫ݔ‬4‫ݔ‬4 16
13 13 52 39 13
−
=
−
=
12 16 48 48 48
3
9
3
5
7
12 9 6 10 7 44
݉݅ + ݉݅ + ݉݅ + ݉݅ + ݉݅ =
+ + +
+ =
8
4
4
8
8 8 8 8 8
8
2
4‫ݔ‬11 11
1
=
=
‫ ݎ݋‬5 ݈݉݅݁‫ݏ‬
4‫ݔ‬2
2
2
8
ALEKS - Chapter THREE
Basic College Mathematics (ALEKS)
Section 3 – 4
Chapter THREE – FRACTIONS AND MIXED NUMBERS:
ADDITION AND SUBTRACTION
Addition and Subtraction of Mixed Numbers
Add the fractions and then the whole numbers:
7
2
1
2
1
2
1
3
1
+2
=7+
+2+ = 7+2+
+
=9+
=9
15
15
15
2
15 15
15
5
ଶ
7
+2
9
ଵହ
ଵ
ଵହ
ଷ
ଵହ
ଵ
6
+3
ଵଷ
=9
ଷ௫ଵ
ଷ௫ହ
ଶ
ଵଵ
ଶଶ
ଶ
ଶଶ
ଶ
ଶଶ
ଵଵ
ଵ
=9
+
ଵ
ହ
ଵଵ
ଶଶ
=
ଵଷ
ଶଶ
9
ଶଶ
Carry:
8
18 40
58
13
13
2
+
= 12 +
= 12 + 1 +
= 13
5 + 7 = 12 +
5
9
45 45
45
45
45
Adding by using improper Fractions
ଵ
ଷ
12 + 4 =
ଷ
ସ
ଷ଻
ଷ
+
ଵଽ
ସ
=
ଵସ଼
ଵଶ
+
ହ଻
ଵଶ
=
ଶ଴ହ
ଵଶ
= 17
ଵ
ଵଶ
Subtraction of Mixed Numbers
Subtract the fractions and then the whole numbers:
3
1
9
4
5
6 −2 =6 −2
=4
4
3
12
12
12
When Borrowing from the Whole Number subtract 1. 1 =
9
ALEKS - Chapter THREE
௅஼஽
௅஼஽
Basic College Mathematics (ALEKS)
ଶ
24 − 8
଻
ହ
଻
23
଻
-8
ହ
଻
Chapter THREE – FRACTIONS AND MIXED NUMBERS:
ADDITION AND SUBTRACTION
ଶ
ଽ
+ =
଻
଻
ହ
− =
଻
ଶ
଻
଻
ଶ
15
଻
Borrowing:
ଷ
ଶ
9 -8 =9
ସ
ଷ
ଵ
଼
ଵଶ
-8
଺
ଽ
ଵଶ
ଶ଴
=8
ଵଶ
ଵ
10 - 3 = 9 − 3 = 6
଺
଺
଺
− 8
ହ
ଽ
ଵଶ
=
ଵଵ
ଵଶ
଺
Subtracting by using Improper Fractions
2
5
74 23
148 69 79
7
8 − 3 =
−
=
−
=
=4
9
6
9
6
18 18 18
18
ଵ
ଵ
ଷ
ଵ
ଶ
6 feet. How much did it increase
ଶ
(-).
Section 3 – 5
ହ
଼
ଵ
ଶ
ଶ
3 ÷4*( )=
ସ
ହ
ଷ
ଶ
ଵ
ଷ
଺
଺
଺
Order of Operations and Applications of
Fractions and Mixed Numbers
ହ
ଷ
ଷ
‫ݔ‬
ଷ
ଶ
ଶ௫ଵଷ
=
(4 - )ଶ = ( − )ଶ = ( )ଶ =
ଶ
ଵ
6 − 4 = 6 − 4 = 2 feet
Snow base was 4 feet later it was
ଶ
ଵଷ
ସ
ଵ
ଶ
∗ ∗ =
ସ
ହ
ଶ
ଶ௫ସ଴
ଶ
=
ଵଷ
ଽ
ସ
‫ ݎ݋‬2
ସ଴
ଵ
ସ
2
1
2 1
2 6
7 2
21 10
3 − ( )ଶ ∗ 6 = 3 − ∗ 6 = 3 − = 2 − = 2 −
5
3
5 9
5 9
5 3
15 15
11
=2
15
10
ALEKS - Chapter THREE
Basic College Mathematics (ALEKS)
Chapter THREE – FRACTIONS AND MIXED NUMBERS:
ADDITION AND SUBTRACTION
Discus Throw in Olympic Games
Women
Men
ଷ
1960
180 ft
2000
224
1960
194 ft
2000
ସ
ହ
ଵଶ
ଵ
ft
଺
ଵ
227 ft
ଷ
How much farther (-) was the
men’s throw than the women’s
throw in 2000?
What was the average throw for
women?
ଵ
ହ
ଵଵ
227 - 224 = 2 ft
ଷ
ଵଶ
ଵଶ
5
3
14
224 + 180
404
12
4=
12
2
2
7
= 202
12
Sylvia made $20,000. Std
$20,000 - $5,150 = $14,850
ଵ
deduction = $5,150 (-). of (*) the 1 ∗ $14,850 = $2970
ହ
adjusted income = taxes. Taxes ?? 5
11
ALEKS - Chapter THREE
Basic College Mathematics (ALEKS)
Chapter THREE – FRACTIONS AND MIXED NUMBERS:
ADDITION AND SUBTRACTION
The above yard is to be fenced
(perimeter) and covered with sod.
(area). Area is made up of two
triangles 4 by 12 and a rectangle
ଵ
10 by 12.
ଷ
12
ଵ
ଵ
Area = 2( ∗ 5 ∗ 12) + 10 ∗ 12
ଶ
ଷଵ
ଷ
Area = 60 ‫ ݀ݕ‬ଶ + ∗ 12 ‫ ݀ݕ‬ଶ
ଷ
2
Area = 184 yd
1
1
ܲ = 5 + 10 + 5 + 13 + 10
3
3
2
+ 13 = 56 ‫ݏ݀ݕ‬
3
ALEKS - Chapter THREE