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?

### Rational Numbers

Rational numbers include fractions, integers and whole numbers. 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**.

#### Let's classify and simplify the following rational numbers:

- \begin{align*}\frac{3}{7}\end{align*}

Because both 3 and 7 are prime numbers, \begin{align*}\frac{3}{7}\end{align*} is a proper fraction written in its simplest form.

- \begin{align*}\frac{9}{3}\end{align*}

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*}

- \begin{align*}\frac{50}{60}\end{align*}

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.

#### Let's practice ordering rational numbers in the following problems:

- 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.

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.

- Which is greater, \begin{align*}\frac{3}{7}\end{align*} or \begin{align*}\frac{4}{9}\end{align*}?

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*}.

### Examples

#### Example 1

Earlier, you were told 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 pizza, one of your friends ate \begin{align*}\frac{5}{8}\end{align*} of a personal pizza, and your other friend ate \begin{align*}\frac{4}{3}\end{align*} of a personal pizza. Which of these fractions is proper? Which of these fractions is improper? Order the fractions from least to greatest.

Since their numerators are smaller than their denominators, \begin{align*}\frac{3}{5}\end{align*} and \begin{align*}\frac{5}{8}\end{align*} are proper fractions. \begin{align*}\frac{4}{3}\end{align*} is improper because the numerator is bigger than the denominator.

To order these fractions, we need to find the common denominator. The common denominator for these three fractions is 120.

\begin{align*}\frac{3\times8\times3 }{5\times8\times3}=\frac{72}{120}&& \frac{5\times5\times3 }{8\times5\times3}=\frac{75}{120}&& \frac{4\times5\times8 }{3\times5\times8}=\frac{160}{120}\end{align*}

Because \begin{align*}72<75<160\end{align*}, \begin{align*}\frac{3}{5}<\frac{5}{8}<\frac{4}{3}\end{align*}.

#### Example 2

For the fractions \begin{align*}\frac{5}{9}\end{align*} and \begin{align*}\frac{10}{20}\end{align*} classify, simplify, and order the two fractions.

For both fractions, the numerator is smaller than the denominator, so they are both proper fractions.

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*}

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*}

### Review

- 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*}.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 2.1.