FRACTIONS
A fraction is the quotient of two quantities and is
of the form, 
, where a and b are any numbers, 
. a is the numerator
and b is the denominator
Fractions are either proper or improper:
Proper Fraction: the numerator is less than the denominator
Examples: 
Improper Fraction: the numerator is greater than or equal to the denominator
Examples: 
Equivalent Fractions: 
is equivalent to 
or 
if 
Example:
Equivalent fractions are formed by:
(1)
Multiplying both the numerator and denominator by the same
nonzero number, i.e. 
Examples:
(a) 
(b)
(2) Dividing both the numerator and denominator by the same nonzero number, i.e.
Examples:
(a)
(b) 
Problems
Find the missing value of x.
(1)
(7)
(2) 
(8)
(3) 
(9)
(4) 
(10)
(5) 
(11)
(6) 
(12)
Answers
(1) 12 (2) 16 (3) 3 (4) 16 (5) 27 (6) 35 (7) 4 (8) 2 (9) 1 (10) 8
(11) 5 (12) 4
Mixed Numbers
A mixed number is a whole number added to a proper fraction
Examples:
(1) 
(2) 
(3) 
We can convert mixed numbers to improper fractions and improper fractions to mixed numbers.
To convert a mixed number to an improper fraction:
(1) Multiply the whole number by the denominator of the fraction
(2) Add the numerator of the fraction to the product
(3) Place your answer over the denominator of the fraction
Examples:
(a) Convert 
to an improper
fraction
(1) 
(2) 
(3) 
(b) Convert 
to an improper
fraction
(1) 
(2) 
(3) 
To convert an improper fraction to a mixed number
(1) Divide the numerator by the denominator. This gives the whole number part of the answer
(2) The remainder, if any, is placed over the denominator. This forms the fractional part of the answer.
Examples:
(a)
Convert 
to a mixed number
(1)
(2) 
(b)
Convert 
to a mixed number
(1)
(2) 
Prime Number: an integer greater than 1 that is divisible only by itself and 1.
Examples: 2 (the only even prime), 3, 5, 7, 9, 11,…
Composite Number: an integer greater than 1 that is divisible by a number other than itself and 1
Examples: 4, 6, 8, 9, 10, 12, 14,…
4 is not only divisible by itself and one it is also divisible by 2.
Any composite number can be written as a product of prime numbers or prime factors (numbers that are multiplied together):
Examples:
(a)
(b)
(c)
To Find the Least Common Denominator (LCD)
(1) Write each denominator as a product of its prime numbers
(2) Multiply the highest power of each prime number
Example:
(a) Find the LCD of 6 and 9
(1) 
(2) LCD 
(b) Find the LCD of 8 and 12
(1) 
(2) LCD 
(c) Find the LCD of 10, 15, and 20
(1) 
(2) LCD 
Problems
Find the LCD of
(1) 12 and 18
(2) 9 and 15
(3) 6, 9, and 18
Answers
(1) 36 (2) 45 (3) 36
Addition and Subtraction of Fractions
(1) 
If the denominators of the two fractions that are to be combined by addition or subtraction are the same, combine the numerators and place the answer over the common denominator. Reduce the answer to simplest terms if possible.
Examples:
(a) 
(b) 
(c) 
(d) 
(2) 
If the denominators of the fractions that are to be combined by addition or
subtraction differ:
(A) Find a LCD
(B) Rewrite each fraction as an equivalent fraction with the LCD
(C) Combine the numerators and place the answer over the LCD
(D) Reduce the answer to simplest terms
Examples:
(a)
(b)
(c)
(d)
(e)
(add the fractional parts then the whole number parts)
(f)
(g)
Since 
is greater than 
borrow
1 from the 4 and add to 
Problems
Perform the indicated operations and leave your answers in simplest form.
(1) 
(8)
(2) 
(9)
(3) 
(10)
(4) 
(11)
(5) 
(12)
(6) 
(13)
(7) 
(14) A student completed 
of a paper one day
and 
the next day. How much of
the paper did he complete and how much is left for him to complete?
(15) A marathoner strives to run 30 miles per week. She runs 
miles, 
miles,
miles and 
miles. How many miles has she run and how many
miles
must she run to reach her goal?
Answers
(1) 
(8)
(2) 
(9)
(3) 
(10)
(4) 
(11)
(5) 
(12)
(6) 
(13)
(7) 
(14) 
(15) 
Multiplying Fractions
Let 
and 
represent two
fractions: 
To Multiply Fractions:
(A) Multiply the numerators
(B) Multiply the denominators
(C) Reduce the product to simplest terms
Examples:
(a) 
(b) 
(c) 
In Example (c) we can simplify the fraction first and then multiply.
(d) 
(e) 
Convert mixed numbers to improper fractions, then multiply.
(f) 
(g) 
Dividing Fractions
Let 
and 
represent any two
fractions: 
(A) Invert the divisor (the fraction immediately following the division sign)
(B) Multiply the fractions
(C) Reduce the answer to simplest terms
Examples:
(a) 
(b) 
(c) 
(d) 
(e) 
Change mixed numbers to improper fractions, then divide.
(f)
(g)
Problems
Perform the indicated operations and leave your answers in simplest form.
(1) 
(7)
(2) 
(8)
(3) 
(9)
(4) 
(10)
(5) 
(11)
(6) 
(12)
(13) A graduating class consists of 90 seniors of which 
are business
majors. How
many business majors are graduating?
(14) A person earns $1000 per week. He pays 