2.1: Properties of Rational Numbers
Suppose that you and two of your friends went out to lunch and that you ate \begin{align*}\frac{3}{5}\end{align*} of a personal pan pizza, one of your friends ate \begin{align*}\frac{5}{8}\end{align*} of a personal pan pizza, and your other friend ate \begin{align*}\frac{4}{3}\end{align*} of a personal pan pizza. Could you tell which of your fractions of pizza were proper and which were improper? Also, could you put the fractions of pizza in order from smallest to largest? In this Concept, you will answer questions such as these for any group of fractions, also known as rational numbers.
Guidance
Rational numbers include fractions, integers and whole numbers. The definition below shows that all rational numbers can be written in the form of a fraction:
Definition: A rational number is a number that can be written in the form \begin{align*}\frac{a}{b}\end{align*}, where \begin{align*}a\end{align*} and \begin{align*}b\end{align*} are integers and \begin{align*}b \ne 0\end{align*}.
An integer, such as the number \begin{align*}3\end{align*}, is also a rational number because it can written as \begin{align*}\frac{3}{1}\end{align*}.
A Review of Fractions
You can think of a rational number as a fraction of a cake. If you cut the cake into \begin{align*}b\end{align*} slices, your share is \begin{align*}a\end{align*} of those slices. For example, when we see the rational number \begin{align*}\frac{1}{2}\end{align*}, we imagine cutting the cake into two parts. Our share is one of those parts. Visually, the rational number \begin{align*}\frac{1}{2}\end{align*} looks like this.
There are three main types of fractions:
- Proper fractions are rational numbers where the numerator is less than the denominator. A proper fraction represents a number less than one. With a proper fraction you always end up with less than a whole cake!
- Improper fractions are rational numbers where the numerator is greater than the denominator. Improper fractions can be rewritten as a mixed number – an integer plus a proper fraction. An improper fraction represents a number greater than one.
- Equivalent fractions are two fractions that give the same numerical value when evaluated. For example, look at a visual representation of the rational number \begin{align*}\frac{2}{4}\end{align*}.
The visual of \begin{align*}\frac{1}{2}\end{align*} is equivalent to the visual of \begin{align*}\frac{2}{4}\end{align*}. We can write out the prime factors of both the numerator and the denominator and cancel matching factors that appear in both the numerator and denominator.
\begin{align*}\left (\frac{2}{4}\right ) = \left (\frac{\cancel{2}\cdot 1}{\cancel{2}\cdot 2 \cdot 1}\right )\end{align*} We then re-multiply the remaining factors. \begin{align*}\left (\frac{2}{4}\right ) = \left (\frac{1}{2}\right )\end{align*}
Therefore, \begin{align*}\frac{1}{2} = \frac{2}{4}\end{align*}. This process is called reducing the fraction, or writing the fraction in lowest terms. Reducing a fraction does not change the value of the fraction; rather, it simplifies the way we write it. When we have canceled all common factors, we have a fraction in its simplest form.
Example A
Classify and simplify the following rational numbers.
a) \begin{align*}\left (\frac{3}{7}\right )\end{align*}
b) \begin{align*}\left (\frac{9}{3}\right )\end{align*}
c) \begin{align*}\left (\frac{50}{60}\right )\end{align*}
Solution:
a) Because both 3 and 7 are prime numbers, \begin{align*}\frac{3}{7}\end{align*} is a proper fraction written in its simplest form.
b) The numerator is larger than the denominator; therefore, this is an improper fraction.
\begin{align*}\frac{9}{3}= \frac{3 \times 3}{3}= \frac{3}{1}=3\end{align*}
c) This is a proper fraction; \begin{align*}\frac{50}{60}= \frac{5 \times 2 \times 5}{6 \times 2 \times 5}= \frac{5}{6}\end{align*}
Ordering Rational Numbers
To order rational numbers is to arrange them according to a set of directions, such as ascending (lowest to highest) or descending (highest to lowest). Ordering rational numbers is useful when determining which unit cost is the cheapest.
Example B
Cans of tomato sauce come in three sizes: 8 ounces, 16 ounces, and 32 ounces. The costs for each size are $0.59, $0.99, and $1.29, respectively. Find the unit cost and order the rational numbers in ascending order.
Solution: Use proportions to find the cost per ounce: \begin{align*}\frac{\$0.59}{8} = \frac{\$0.07375}{ounce}; \ \frac{\$0.99}{16} = \frac{\$0.061875}{ounce}; \ \frac{\$1.29}{32} = \frac{\$0.0403125}{ounce}\end{align*}. Arranging the rational numbers in ascending order, we have: 0.0403125, 0.061875, 0.07375.
Example C
Which is greater, \begin{align*}\frac{3}{7}\end{align*} or \begin{align*}\frac{4}{9}\end{align*}?
Solution: Begin by creating a common denominator for these two fractions. Which number is evenly divisible by 7 and 9? \begin{align*}7 \times 9 = 63\end{align*}, so the common denominator is 63.
\begin{align*}\frac{3 \times 9}{7 \times 9} = \frac{27}{63} && \frac{4 \times 7}{9 \times 7} = \frac{28}{63}\end{align*}
Because \begin{align*}28 > 27, \ \frac{4}{9} > \frac{3}{7}\end{align*}.
Video Review
Guided Example
For the fractions \begin{align*}\frac{5}{9}\end{align*} and \begin{align*}\frac{10}{20}\end{align*}:
a.) Determine whether these are proper or improper fractions.
b.) Simplify if necessary.
b.) Order the two fractions.
Solution:
a.) For both fractions, the numerator is smaller than the denominator, so they are both proper fractions.
b.) For \begin{align*}\frac{5}{9}\end{align*}, since 5 is prime and 9 is not a multiple of 5, this fraction cannot be simplified.
For \begin{align*}\frac{10}{20}\end{align*}, both 10 and 20 are multiples of 10, so we rewrite them as
\begin{align*}\frac{1 \cdot 10}{2 \cdot 10}=\frac{1}{2}.\end{align*}
c.) To order the fractions, we'll use the simplified versions:
\begin{align*}\frac{5\cdot 2}{9\cdot 2}=\frac{10}{18}\end{align*} and \begin{align*}\frac{1\cdot 9}{2\cdot 9}=\frac{9}{18}.\end{align*}
Since
\begin{align*}\frac{9}{18}< \frac{10}{18}\end{align*}
then
\begin{align*}\frac{1}{2}< \frac{5}{9}.\end{align*}
Practice
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Integers and Rational Numbers (13:00)
- What are the three types of fractions?
In 2 – 4, determine what fraction of the whole each shaded region represents.
In 5 – 8, place the following sets of rational numbers in order from least to greatest.
- \begin{align*}\frac{1}{2}, \frac{1}{3}, \frac{1}{4}\end{align*}
- \begin{align*}\frac{11}{12}, \frac{12}{11}, \frac{13}{10}\end{align*}
- \begin{align*}\frac{39}{60}, \frac{49}{80}, \frac{59}{100}\end{align*}
- \begin{align*}\frac{7}{11}, \frac{8}{13}, \frac{12}{19}\end{align*}
In 9 – 14, find the simplest form of the following rational numbers.
- \begin{align*}\frac{22}{44}\end{align*}
- \begin{align*}\frac{9}{27}\end{align*}
- \begin{align*}\frac{12}{18}\end{align*}
- \begin{align*}\frac{315}{420}\end{align*}
- \begin{align*}\frac{19}{101}\end{align*}
- \begin{align*}\frac{99}{11}\end{align*}
Mixed Review
- Evaluate the following expression: \begin{align*}\frac{5}{6} d + 7a^2\end{align*}; use \begin{align*}a=(-1)\ \text{and } \ d=24\end{align*}.
- The length of a rectangle is one more inch than its width. If the perimeter is 22 inches, what are the dimensions of the rectangle?
- Determine if \begin{align*}x=-2\end{align*} is a solution to \begin{align*}4x+7 \le 15\end{align*}.
- Simplify: \begin{align*}\frac{(7+3) \div 2 \times 3^2-5}{(58-8)}\end{align*}.
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Term | Definition |
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Equivalent Fractions | Equivalent fractions are fractions that can each be simplified to the same fraction. An equivalent fraction is created by multiplying both the numerator and denominator of the original fraction by the same number. |
improper fractions | These are rational numbers where the numerator is greater than the denominator. Improper fractions can be rewritten as a mixed number – an integer plus a proper fraction. An improper fraction represents a number greater than one. |
proper fractions | Rational numbers where the numerator is less than the denominator. A proper fraction represents a number less than one. |
rational number | A rational number is a number that can be expressed as the quotient of two integers, with the denominator not equal to zero. |
common denominator | The common denominator is the least common multiple of the denominators of multiple fractions. Each fraction can be rewritten as an equivalent fraction using the common denominator. |
Denominator | The denominator of a fraction (rational number) is the number on the bottom and indicates the total number of equal parts in the whole or the group. has denominator . |
Irrational Number | An irrational number is a number that can not be expressed exactly as the quotient of two integers. |
Least Common Denominator | The least common denominator or lowest common denominator of two fractions is the smallest number that is a multiple of both of the original denominators. |
Least Common Multiple | The least common multiple of two numbers is the smallest number that is a multiple of both of the original numbers. |
Lowest Common Denominator | The lowest common denominator of multiple fractions is the least common multiple of all of the related denominators. |
Mixed Number | A mixed number is a number made up of a whole number and a fraction, such as . |
Numerator | The numerator is the number above the fraction bar in a fraction. |
proper fraction | A proper fraction has a numerator that is a lesser absolute value than the denominator. Proper fractions always represent values between -1 and 1. |
reduce | To reduce a fraction means to rewrite the fraction so that it has no common factors between numerator and denominator. |
Image Attributions
Here you'll learn how to distinguish between rational numbers written as proper and improper fractions. You'll also learn how to simplify these fractions and decide if one is less than or greater than another.