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 Dependent Variable

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# 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 two-thirds of a class (meaning two of every three students) or three-fourths 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 three-fourths, 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?

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?

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?