Introduction to Fractions
What Fractions Are and Why They Are Important
A fraction can mean part of a whole. Just as a whole number
answers the question, How many?, a fraction answers the question,
What part of? Every day we use fractions in this sense. For
example, we can speak of twothirds of a class (meaning two of
every three students) or threefourths of a dollar (indicating
that we have split a dollar into four equal parts and have taken
three of these parts).
A fraction can also mean the quotient of two whole numbers. In
this sense, the fraction tells us what we get when we divide the
whole number 3 by the whole number 4.
Definition
A fraction is any number that can be written in the form , where a
and b are whole numbers and b is not zero.
Explain why in this definition b cannot be 0.
From this definition, are all fractions.
When written as a fraction has three components.
 The denominator (on the bottom) stands
for the number of parts into which the whole is divided.
 The numerator (on top) tells us how many
parts of the whole the fraction contains.
 The fraction line separates the
numerator from the denominator, and stands for “out
of ” or “divided by.”
Alternatively, a fraction can be represented as either a
decimal or a percent.
Fraction Diagrams and Proper Fractions
Diagrams help us work with fractions. Each diagram represents
the fraction threefourths, or .
Note that in each diagram the whole has been divided into 4
equal parts, with 3 of the parts shaded.
The number is an example of a proper fraction because its
numerator is smaller than its denominator. Let’s consider
some other examples of proper fractions.
Explain why a fraction whose numerator is smaller than its
denominator must have a value less than 1.
EXAMPLE 1
What fraction does the diagram represent?
Solution
In this diagram, the whole is divided into 9 equal parts, so
the denominator of the illustrated fraction is 9. Four of these
parts are shaded, so the numerator is 4. The diagram illustrates
the fraction .
PRACTICE 1
The diagram illustrates what fraction?
Answer
EXAMPLE 2
A manufacturing company plans to lay off 71 of its 230
workers. What fraction of its employees does the company plan to
lay off?
Solution
There are 230 workers altogether, so the denominator of our
fraction is 230. Because we are concerned with 71 of these
workers, 71 is the numerator. The company plans to lay off of its
employees.
PRACTICE 2
The annual tuition at a college is $2,451. If a student paid
$1,000 toward this tuition, what fraction of the tuition did the
student pay?
Answer
EXAMPLE 3
The U.S. Senate approved a foreign aid spending bill by a vote
of 83 to 17. What fraction of the senators voted against the
bill?
Solution
First, we find the total number of senators. Because 83
senators voted for the bill and 17 voted against it, the total
number of senators is 83 + 17, or 100. So of the
senators voted against the bill.
PRACTICE 3
You have read 125 pages of a novel assigned by your English
instructor. If 39 pages remain, what fraction of the book have
you read?
Answer
