Transcript Chapter 5 Notes - Dripping Springs ISD

Chapter 5 Notes
5-1 Compare/Order Rational
Numbers
The multiple of any number is the product of
that number and any nonzero whole number
LCM = least common multiple
Example:
Multiples of 4: 4,8,12,16,20,24,28,32,36,……
Multiples of 6: 6,12,18,24,30,36,42,….
LCM = 12
Using Upside Down Division
Find LCM of 12 and 40
Other examples: 6, 16
9,15
12, 15, 18
Answers
Find LCM of 12 and 40
– LCM = 120
Other examples: 6, 16 = 48
9,15 = 45
12, 15, 18 = 180
Find LCM of Variable
Expressions
6a2 and 18a3
8m2 and 14m4
25y2 and 15x
Answers
6a2 and 18a3 = 18a3
8m2 and 14m4 = 56m4
25y2 and 15x = 75xy2
Using a number line and
ordering fractions
Graph and compare the fractions in
each pair: -(1/2), -(1/10)
Order -(1/2), 3/4, -1, and 2/5 from least
to greatest
Answers
Graph and compare the fractions in
each pair: -(1/2), -(1/10)
-1/2 < -1/10
Order -(1/2), 3/4, -1, and 2/5 from least
to greatest
-1, -1/2, 2/5, 3/4
Comparing Fractions
2/3 � 7/9
8/17 � (-3/8)
(-4/5) � (-7/8)
Comparing Fractions Answers
2/3 � 7/9
<
8/17 � (-3/8)
>
(-4/5) � (-7/8)
>
5-2 Fractions and Decimals
Terminating decimal = when division
ends with a remainder of zero
5/8
1 7/8
3 3/10
Answers
Terminating decimal = when division
ends with a remainder of zero
5/8 = 0.625
1 7/8 = 1.875
3 3/10 = 3.3
Repeating Decimals
Repeating decimal = the same block of
digits repeats infinitely many times
2/3
Repeating or Terminating decimals:
7/9
21/22
11/8
8/11
Repeating Decimals Answers
Repeating decimal = the same block of
digits repeats infinitely many times
2/3 = 0.666666……
Repeating or Terminating decimals:
7/9
21/22
11/8
8/11
0.77… 0.95454 1.375
0.7272..
Ordering fractions and
decimals
Write the numbers in order from least to
greatest
1/4, -0.2, -3/5, 1.1
0.2, 4/5, 7/10, 0.5
Ordering fractions and
decimals - Answers
Write the numbers in order from least to
greatest
1/4, -0.2, -3/5, 1.1
-3/5, -0.2, 1/4, 1.1
0.2, 4/5, 7/10, 0.5
0.2, 0.5, 7/10, 4/5
Writing a decimal as a fraction
Examples:
1 12/100
Examples:
2.32
0.65
1.12
= 1 6/50
= 1 3/25
Writing a decimal as a fraction
- Answers
Examples:
1 12/100
1.12
= 1 6/50
Examples:
2.32 = 2 8/25
0.65 = 13/20
= 1 3/25
Writing a repeating decimal as
a fraction
Examples:
__
_
__
1. .72
.7
.54
___
.213
Writing a repeating decimal as
a fraction - Answers
Examples:
__
_
__
1. .72
.7
.54
8/11
7/9
6/11
___
.213
71/333
5-3 Adding and Subtracting
Fractions
Adding or subtracting fractions with the
same denominator:
– Add or subtract the numerators and keep
the same denominators
– Reduce if necessary
– Examples: 7/10 - 3/10
11/y + (-5/y)
5-3 Adding and Subtracting
Fractions - Answers
Adding or subtracting fractions with the
same denominator:
– Examples: 7/10 - 3/10 = 2/5
11/y + (-5/y) = 6/y
Add/Subtract Fractions with
Different Denominators
Find a common denominator
Add or subtract the numerators
Keep the same denominator
Reduce if necessary
Examples: 1/8 - 5/6
1/8 - 5x/6
3/7 - 2/m
Add/Subtract Fractions with
Different Denominators Answers
Examples:
1/8 - 5/6 = - 17/24
1/8 - 5x/6 =
3/7 - 2/m =
6-40x/48
3m - 14/7m
Add/Subtract Mixed Numbers
Examples:
2 2/3 + 1 3/4 =
5 3/4 + 7/8 =
5 2/3 - 3 1/6 =
2 1/2 - 1 3/4 =
1 1/2 - 2 5/8 =
Add/Subtract Mixed Numbers
- Answers
Examples:
2 2/3 + 1 3/4 = 4 5/12
5 3/4 + 7/8 = 6 5/8
5 2/3 - 3 1/6 = 2 1/2
2 1/2 - 1 3/4 = 3/4
5-4 Multiplying and Dividing
Fractions
Multiplying Fractions = multiply straight
across -- multiply the numerators
together and multiply the denominators
together
Simplify before multiplying if possible
Change any mixed numbers to improper
fractions before multiplying!!!
Reduce if necessary
Examples
3/7 x 4/5
9/15 x -(5/9)
y/4 x 8/11
-(5/15) x 21/25
Examples - Answers
3/7 x 4/5 = 12/35
9/15 x -(5/9) = - 1/3
y/4 x 8/11 = 2y/11
-(5/15) x 21/25 = - 7/25
Dividing Fractions
To divide fractions = keep the 1st fraction the
same, change the division sign to a
multiplication sign, flip the 2nd fraction, and
multiply across
Reciprocal
To divide mixed numbers = change all mixed
numbers to improper fractions, then follow the
same rules as when dividing fractions
Examples
2/9 ÷ 2/5
x/3 ÷ x/4
1 3/4 ÷ (-2 5/8)
Examples
2/9 ÷ 2/5 = 2/9 x 5/2 = 10/18 = 5/9
x/3 ÷ x/4 = 1 1/3
1 3/4 ÷ (-2 5/8) = -2/3
5-7 Solving Equations by Adding
or Subtracting Fractions
You solve equations with fractions the same
way you solve equations with integers and
decimals by doing the opposite operation you
already have.
Example:
1/4 + n = 1/3
-1/4
-1/4
n = 1/3 - 1/4
4/12 - 3/12
n = 1/12
Examples
y + 8/9 = 5/9
2/3 = x + 3/5
C + 3/10 = 11/15
6/7 = x - 2/7
3 7/18 = a + 1 1/3
Y + 4 7/8 = 2
A - 2 1/12 = 3 1/12
Examples - Answers
y + 8/9 = 5/9
2/3 = x + 3/5
C + 3/10 = 11/15
6/7 = x - 2/7
3 7/18 = a + 1 1/3
Y + 4 7/8 = 2
A - 2 1/12 = 3 1/12
y = -1/3
x = 1/15
c = 13/30
x = 1 1/7
a = 2 1/18
y = -2 7/8
a = 5 1/6
5-8 Solving Equations by
Multiplying Fractions
You can undo multiplication by dividing
each side of an equation by the same
number.
You can also multiply each side of an
equation by the reciprocal to undo
multiplication.
Multiplying by a reciprocal
You can get a variable by itself by
multiplying by a reciprocal
Example:
5a = 1/7
Multiplying by a reciprocal Answers
You can get a variable by itself by
multiplying by a reciprocal
Example:
5a = 1/7
(1/5) 5a = (1/5) (1/7)
a = 1/35
Multiplying by the negative
reciprocal
Example:
-(14/25)k = 8/15
Try these:
-(6/7)r = (3/4)
-(10/13)b = -(2/3)
Multiplying by the negative
reciprocal - Answers
Example:
-(14/25)k = 8/15
(-25/14)(-14/25)k = (8/15)(-25/14)
k = -20/21
Try these:
-(6/7)r = (3/4)
r = -7/8
-(10/13)b = -(2/3) b = 13/15
Solving Equations with Mixed
Numbers
Change mixed numbers to improper
fractions before multiplying
Example:
1 5/8n = 25
Solving Equations with Mixed
Numbers - Answers
Change mixed numbers to improper
fractions before multiplying
Example:
1 5/8n = 25
13/8n = 25
(8/13)(13/8)n = 25(8/13)
n = 15 5/13
Solving Equations with Mixed
Numbers
Examples:
3 1x = 28
2
-2 3h = (-12 1/2)
4
(-7/20) = 1 1y
6
Solving Equations with Mixed
Numbers - Answers
Examples:
3 1x = 28
2
(-7/20) = 1 1y
6
-2 3h = (-12 1/2)
4
x=8
y = -3/10
h = 4 6/11