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 Dependent Variable

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# Fractions

## 4.1 Introduction to Fractions and Mixed Numbers

4.1.a Identifying Numerators and Denominators

Numerator / Denominator
Numerator รท Denominator

4.1.b Writing Fractions to Represent Shaded Areas of
Figures

4.1.c Graphing Fractions on a Number Line

4.1.c Examples: Graph the following fractions on
a number line:

4.1.d Simplifying fractions of the form:

## Operations on Mixed Numbers

4.1.e Illustrating Mixed Numbers

Mixed Numbers:
A number containing a whole number and
a proper fraction.

4.1.e Writing Mixed Number s as Improper
Fractions

4.1.f Wr iting Improper Fractions as Mixed Number s
or Whole Number s

Divide the denominator into the numerator to produce
a mixed number.
The Whole Number part of the mixed number is the
quotient, the fraction part of the mixed number is the
remainder over the original denominator.

## 4.2 Factors and Simplest Form

4.2.a Writing a Number as a Product of Prime
Numbers

Finding the Prime Factors using
(a) 'factor tree' or \
(b)successive division by increasing larger
prime numbers

Write the prime factorization of each of the following
numbers
1. 12 2. 48 3. 56 4. 45 5. 240

Determining if a number is divisible by 2, 3, or 5

4.2.b Writing a Fraction in Simplest Form

Write the numerator and denominator in prime
factorization form.
Divide common factors in the numerator and
denominator.

Examples:

4.2.c Determining Whether Two Fractions are
Equivalent

Determine if they are equivalent by Simplifying numerator
and denominator into common factors

Determine if 7/9is equivalent to  21/27
Using Prime Factorization of numerator and denominators of both fractions:

4.2.c (determination of equivalent fractions)

Determine if they are equivalent by Cross Multiplication

If ....

then ....
a*T = b*R

4.2.c Examples: Determine if the following fractions
are equivalent.

4.2.d Solving Problems by Writing Fractions in
Simplest Form

Examples:

14. There are 58 national parks in the United States. Six
of these are in Washington state and 2 are in Wyoming.
Determine, in simplest form, the fraction of the parks
located in each state?

15. The outer wall of the Pentagon is composed of 10
inches of concrete, 8 inches of brick, and 6 inches of
limestone. What is the fractional width of (i) the
concrete (ii) the limestone?

## 4.3 Multiplying and Dividing Fractions

4.3.a Multiplying Fractions
Multiply the numerators
Multiply the denominators
Simplify if possible.

Examples:

4.3.b Evaluating Expressions with Fractional
Bases

Examples:

4.3. c Dividing Fractions
Invert the fraction in the denominator
Multiply the numerator and inverted
denominator fractions

Examples:

4.3.d Multiplying and Dividing with Fractional
Replacement Values

Evaluate
(i) x *y and
(ii) x / y

given the following replacement values for x and y
in (x,y) format.

4.3.e Solving Applications by Multiplying and
Dividing Fractions

Examples:
21. How much money is alloted to rent, if 2/3 of the
\$450 weekly income is to be used.
22. If soup weighs 5/4 pound per can and a case
contains 24 cans, what is the weight of the
case?
23. How many gallons of liquid are in a 48 gallon vat
if it is filled 7/8 to the top?

4. 4 Adding and Subtracting Like Fractions and
Least Common Denominator

4.4.a Adding or Subtracting Like Fractions
Note:

4.4.a Examples:

4.4.b. Adding and Subtracting Given Fractional
Replacement Values

Examples:
Evaluate each expression for the given replacement
values (x,y,z)

4.4 c Solving Problems by Adding and Subtracting
Like Fractions

Examples:
Find the perimeter.
16. A rectangle with dimensions :

17. A triangle with sides of length:

18. A square with a side length of

(4.4 c) Using the following table:
What fraction of employees are:
19. Not covered by HMO?
20. Use P-o-S or Traditional Fee?

 Type of Health Plan Fraction of Employees Using This Pla n HMO 6/20 Point-of-Service 4/20 Prefer red Provided 7/20 Traditional fee-for -service 3/20

4.4.d Finding the Least Common Denominator (LCD)
or Least Common Multiple(LCM)

The LCD is the smallest denominator that is evenly
divisible by ALL denominators in a list of fractions.

: The LCD is '12'
When adding or subtracting unlike fractions, all
must be converted to equivalent fractions of the
same denominator.

( 4.4.d )Finding the LCD for a set of fractions.

Method 1:
Determine if the Largest original denominator is
evenly divisible by ALL denominators in the set of
fractions.

( 4.4.d )Method 2:

Use prime factorization of all denominators. Works
for a list of 3 or more fractions.

For example:
Expressing 1/6 as a fraction with a denominator
consisting of a product of only prime numbers is as
follows.
Note that the new denominator is a product of only
prime numbers.

( 4.4.d )The LCD is a number composed of

ALL unique prime factors appearing in ALL
prime factorization of the denominators
Each prime number factor will appear the
same number of times as the maximum
number found in ANY ONE of the prime
factorizations.

( 4.4.d )Example:

Find the 'prime factorization of ALL denominators of
the following fractions:

Prime Factorization of the denominators:

( 4.4.d )Note:
The LCD is the product of Every Different Prime
Number appearing in the prime factor ization of the
denominator s. (2*3*5)

Each of these unique prime number factor s is raised to
a power equal to the Maximum Number of Occurrences
of that prime number in any one of the factorizations.

This is demonstrated by the exponents of all unique
prime number factor s denoting the maximum number
of coinsurances of each.

4.4.d Examples: Find the LCD of the following sets
of fractions: