Try the Free Math Solver or Scroll down to Tutorials!












Please use this form if you would like
to have this math solver on your website,
free of charge.


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

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

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

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
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


4.2.c Determining Whether Two Fractions are

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


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.


4.3.b Evaluating Expressions with Fractional


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


4.3.d Multiplying and Dividing with Fractional
Replacement Values

(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

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
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

4.4.a Examples:

4.4.b. Adding and Subtracting Given Fractional
Replacement Values

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

4.4 c Solving Problems by Adding and Subtracting
Like Fractions

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

( 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
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

( 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: