Unit 2: Fractions Ratios Rates

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Unit 2 FractionsRatiosRates
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Understanding Fractions
Parts of a fraction
of equal parts that you are looking at
Numerator
Fraction Bar it represents division
Denominator
Total of equal parts in the whole
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Understanding Fractions
Parts of a fraction
All parts of a fraction are equal in size
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Types of Fractions

• Proper Fractions
• Numerator is less than the denominator
• Reduced to lowest terms

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Types of Fractions

• Improper Fractions
• Numerator is greater than or equal to the
  denominator
• Numerator is equal to the denominator
• Not reduced to lowest terms

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Types of Fractions

• Mixed Numbers
• Whole numbers and fractions are written together
  to represent one amount
• Fractional part is reduced to lowest terms

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5
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Types of Fractions

• Equivalent Fractions
• Fractions with different names that represent
  the same amount

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Simplifying FractionsTypes of Numbers
Knowing the number types can help you factor
those numbers in order to reduce your fractions.
Prime Numbers Numbers whose only factors are 1
and itself. Examples 2, 3, 5, 7, 11, 13, 17, 19
Composite Numbers Numbers that have more than 1
and itself as factors. Examples 4, 6, 8, 9,
10, 12, 14, 15, 16, 18, 20, 21
The number 1 is neither prime nor composite.
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Simplifying FractionsPrime Factorization
Prime Factorization is breaking a number down
into its prime number components.
Prime Factorization can be done using factor
trees or by using inverted division.
12 2 6 2 3

• 12
• 2 6
• 3

12 2 x 2 x 3
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Prime Factorization Practice
Give the prime factorization for the following
numbers
20 32 27
2 x 2 x 5 22 x 5 2 x 2 x 2 x 2 x 2 25 3 x 3
x 3 33
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Simplifying FractionsRules of Divisibility
Knowing the Rules of Divisibility can help with
reducing fractions.
A number is divisible by
if 2 ends in 0, 2, 4, 6, 8 3 sum of
the digits is divisible by 3 5 ends in 0
or 5 10 ends in 0
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Simplifying FractionsReducing Fractions
When reducing, the goal is to divide the
numerator and the denominator by the same factor
until there is no factor that will divide the
numerator and denominator evenly.
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Reducing Fractions Practice
Reduce the following fractions to lowest terms.
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Reducing Fractions Practice
Reduce the following fractions to lowest terms.
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Changing Improper Fractions to Mixed Numbers

• To change improper fractions to mixed numbers
• STEP 1 Divide the numerator by the
  denominator
• STEP 2 Write in mixed number format

Whole Number
11 5
1 5
2


New Numerator
Denominator
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Changing Mixed Numbers to Improper Fractions
To change a mixed number to an improper fraction
STEP 1 Multiply the denominator and the whole
number
STEP 2 Add the numerator to your product
STEP 3 Move the denominator into your answer
2 x 3
1
1 2
7 2
3


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Mixed Number/Improper Fraction Practice
Practice Change each improper fraction to a
mixed or whole number. Remember to reduce the
fraction to lowest terms.
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Mixed Number/Improper Fraction Practice
Practice Change each mixed or whole number to an
improper fraction.
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Multiplying Fractions and Mixed Numbers
There are two methods you can use for multiplying
fractions and mixed numbers.
Method A  Step 1 Change mixed s to
improper fractions (if needed) Step 2
Multiply numerators Step 3 Multiply
denominators Step 4 Reduce to lowest terms
Method B  Step 1 Change mixed s to
improper fractions (if needed) Step 2 Reduce
to lowest terms Step 3 Multiply
numerators Step 4 Multiply denominators
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Multiplying FractionsExamples
Method A
Method B
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Multiplying Mixed Numbers Examples
1 3
1 2
4
x

3
9 2
3 2
1 3
x

1
1 2

1
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Multiplying Fractions/Mixed Numbers and Whole
Numbers
1 2
5
x

becomes
5 1
1 2
5 2
1 2
x


2
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Dividing Fractions and Mixed Numbers
Step 1 Change any mixed numbers to improper
fractions Step 2 Change the ? problem to a
x problem a) the 1st value stays the same b)
change the ? to a x c) flip the 2nd value upside
down Step 3 Solve the multiplication
problem Step 4 Reduce to lowest terms or change
to a mixed number (if needed)
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Dividing Fractions and Mixed Numbers Examples
1 2
1 3
?

4
Make the changes
9 2
3 1
27 2

x
1 2

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Multiplication/Division Practice
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Multiplication/Division Practice
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Multiplication/Division Practice
2
6
8
2
6
24
4
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Multiplication/Division Practice
3
10
2
1
17
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Equivalent Fractions
Remember that Equivalent Fractions are fractions
that represent the same amount but have different
names.
Equivalent fractions are found when adding or
subtracting fractions (and/or mixed numbers)
where the fractions have different denominators.
Equivalent fractions are also found
when reducing/simplifying fractions to
their simplest form.
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To Make a Larger Equivalent Fraction
The reason for finding equivalent fractions with
larger denominators is so that the fractions
(that you are adding or subtracting) will have
common denominators.
To find equivalent fractions with larger
denominators Multiply the numerator and
denominator by the same value
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Examples of how to find equivalent fractions with
larger denominators
1 3
1 2


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Finding equivalent fractions with smaller
denominators
To find fractions with smaller denominators is
done at the end of any fraction problem. This
is also known as reducing or simplifying.
To find equivalent fractions with smaller
denominators Divide the numerator and
denominator by the same value
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Example of how to find equivalent fractions with
smaller denominators
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Comparing Fractions
When you compare fractions, look at the two
fractions to see which is larger and which is
smaller.
Symbols for comparing are lt - Less than gt -
Greater than
When comparing fractions, the fractions must
have common denominators
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Comparing Fractions
Steps for comparing fractions
Step 1 Check to make sure you have common
denominators (you may have to find equivalent
fractions)
Step 2 Compare the numerators
Step 3 Put in the correct sign
8 12
3 12
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Ordering Fractions
Steps for ordering fractions
Step 1 Check to make sure you have common
denominators (you may have to find equivalent
fractions)
Step 2 Compare the numerators
Step 3 Put in the correct order from least to
greatest (or greatest to least)
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Adding and Subtracting Fractions
The steps for adding and subtracting fractions
are basically the same. The only difference is
for addition problems you add and for
subtraction problems you subtract.
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Adding Fractions
(Write in vertical format)
Step 1 Make sure the fractions have common
denominators (you may need to find equivalent
fractions)
Step 2 Add the numerators
Step 3 Move your denominator into your answer
(NEVER add the denominators)
Step 4 Reduce your answer to lowest terms
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Addition of Fractions Examples
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Subtracting Fractions
(Write in vertical format)
Step 1 Make sure the fractions have common
denominators (you may need to find equivalent
fractions)
Step 2 Make sure you can subtract the
numerators
Step 3 Subtract your numerators
Step 4 Move your denominator into your answer
(NEVER subtract the denominators)
Step 5 Reduce your answer to lowest terms
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Subtraction of Fractions Examples
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Adding Mixed Numbers
(Write in vertical format)
Step 1 Make sure the fractions have common
denominators (you may need to find equivalent
fractions)
Step 2 Add the numerators
Step 3 Move your denominator into your answer
(NEVER add the denominators)
Step 4 Add the whole numbers
Step 5 Reduce your answer to lowest terms
(sometimes an improper fraction must be
changed into a mixed number and added to the
whole number)
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Addition of Mixed Numbers Examples
4
4
7
7
5 12
4

8
7
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Subtracting Mixed Numbers
(Write in vertical format)
Step 1 Make sure the fractions have common
denominators (you may need to find equivalent
fractions)
Step 2 Make sure you can subtract numerators
(You may have to regroup from the whole number)
Step 3 Subtract the numerators
Step 4 Move the denominator into the answer
Step 5 Subtract the whole numbers
Step 6 Reduce your answer to lowest terms
(sometimes an improper fraction must be changed
into a mixed number and added to the whole
number)
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Subtraction of Mixed Numbers Examples
7
7
7
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Subtraction of Mixed Numbers Examples
5
6
6
5
6
5
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Adding Mixed Numbers Practice
3
5
1
6
3
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Subtracting Mixed Numbers Practice
3
5
1
4
3
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Subtracting Mixed Numbers Practice
3
5
2
2
4
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Raising a Fraction to a Power
Remember that exponents are a short cut for
repeated multiplication. Exponential notation
applies to both numerators and denominators.
3
(
)
3 4
3 x 3 x 3 4 x 4 x 4
(
)
27 64


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Complex Fractions
2 3 4 5
2 3
4 5
Means
Write the problem horizontally then solve the
division problem
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Practice Problems
15 16
12
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Practice Problems
8 125
49 64
1 64
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Changing Fractions and Mixed Numbers to Decimals
Step 1 Divide the numerator by the denominator
Step 2 Keep dividing until the problem ends,
you see a pattern, or you are asked to round
If working with mixed numbers, the whole number
in the mixed number STAYS as a whole number in
the decimal valuethe fraction part of the mixed
number is changed to a decimal
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Examples of Changing Fractions Mixed Numbers to
Decimals
Examples
1 3
2 5
3 8

0.33
4

0.4

4.375
5
7
3
0.
3 3
0.
4
0 .
0
0
0
3.
8
0 0
1.
3
0
2 .
5
4
-2
0
6

9

-
0
2
-
6
5
-
1 0 - 9 1
0
0
4
0
4
-
0
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Changing Fractions Mixed Numbers to Decimals
Practice
Practice
7 9
1 8
2 5

0.77

3.125

0.4
3
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Changing Fractions Mixed Numbers to Decimals
Practice
Practice
3 8
3 4

0.375

5.75
5
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Order of Operations
Please Excuse My Dear Aunt Sally
P Parenthesis E Exponents M Multiply D
Divide A Addition S Subtraction
Do whichever one comes first working from left
to right
Remember to use the rules that apply to whole
numbers, signed numbers, and fractions
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Order of Operation Practice
Solve each of the following
¼ - 3(½ ¾)
-3½
19 64
2? ? (1¼ - ?)
2
5 ½ ¾ ¾ ½
1 6

4
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Order of Operation Practice
Solve each of the following
? (? ¾)
35 64
7 8
7? (¾ - ¼)2
7
2(5½ 4¾) 3(¾ - ½)
1 3

27
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Ratios and Rates
Both ratios and rates compare two quantities
Ratios comparison of two quantities with the
same unit name
Rates comparison of two quantities with
different unit names
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Ratio and Rate Formats
Ratios and rates can be written in three
different formats
X Y Fraction Format
X to Y Word Format
X Y Colon Format
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Ratio and Rate Formats
When you write a ratio in simplest form, you are
simply reducing.
12 to 18 2 to 3
2 4 1 2
20 45 4 9
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Simplifying a Rate
When you write a rate in simplest form, it is
written in terms of one item.
385 miles on 22 gallons (simplified) 17.5 miles
per gallon
40 for 8 books (simplified) 5 for 1 book
If a rate involves a size (quantity) and money,
it is always money divided by size (quantity)
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Simplifying Ratios Practice
Write each ratio in simplest form.
69 50 16 to 24 72 85 84 520 21
to 15
23 2 to 3 14 7 to 5
10 6 17 7
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Simplifying Rates Practice Problems
Write each rate in simplest form.
520 in 40 hours 1935 words on 9 pages 780
miles in 12 hours 2.88 for 16 oz can 72
pounds for 4 people 414 miles on 23 gallons
13 per hour 215 words per page 65
miles per hour 0.18 per ounce 18 lbs per
person 18 miles per gallon
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Review the things that you need to review.
Study the things that you need to spend more time
on.
Ask questions about things you dont understand.
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