Remember the diving that Cameron was doing in the Divide Integers Concept? Take a look at this dilemma.

On the dive boat one morning, Cameron began talking with another boy named Chet. Chet and his family were from Colorado and Chet was just two years older than Cameron. The boys struck up a great conversation about diving and fish and the things that they had seen on their dives.

After a little while, they spotted some dolphins swimming with the boat. This is something that can often happen as dolphins love the rushing that the motor makes on the back of the boat.

“Did you know that they can swim .83 miles in one minute?” Chet asked Cameron.

“Really, no I didn’t know that. I do know that a swordfish can swim almost one-half mile in a minute. I think the exact number is \begin{align*}\frac{9}{20}\end{align*} of a mile.”

“Wow, which one can swim the farthest in one minute?” Chet asked thinking carefully through the math.

By the time they reached the dive site, Cameron had figured out which one can swim the farthest in one minute.

**Have you? The numbers that the boys are using are called rational numbers. When you understand rational numbers, you will also understand how to figure out which one can swim the farthest in one minute. Pay attention, and this Concept on rational numbers will teach you all that you need to know.**

### Guidance

Previously we worked with rational numbers. Now that you know how to identify a rational number, you may need to compare or order them from time to time. For example, what if you have a loss of \begin{align*}\frac{1}{2}\end{align*} compared to a loss of .34. You would need to figure this out.

Placing the numbers on a number line can help you do this.

Let's review the inequality symbols which can help us compare and order rational numbers:

> means *is greater than*.

< means *is less than*.

= means *is equal to*.

Choose the inequality symbol that goes in the blank to make this statement true.

\begin{align*}-2.5 \ \underline{\;\;\;\;\;\;\;} \ -5\end{align*}

First, Draw a number line from -5 to 5.

Place the numbers -2.5 and -5 on that number line. Since \begin{align*}0.5=\frac{1}{2}, \ -2.5\end{align*} will be halfway between -2 and -3 on the number line.

Since -2.5 is further to the right on the number line than -5 is, -2.5 is greater than -5

**The symbol > goes in the blank because \begin{align*}-2.5 > -5\end{align*}.**

Order these rational numbers from least to greatest.

\begin{align*}&\frac{4}{5} \qquad 0.6 \qquad 1 \qquad 0.\bar{6}\end{align*}

It is often fairly easy to place decimals on a number line that is divided into tenths.

So, we can draw a number line from 0 to 1 and divide it into tenths. Then we can place all four numbers on the number line.

First, we should change \begin{align*}\frac{4}{5}\end{align*} to a fraction with a denominator of 10:

\begin{align*}\frac{4}{5} = \frac{4 \times 2}{5 \times 2} = \frac{8}{10}\end{align*}

Since eight tenths is equivalent to \begin{align*}\frac{4}{5}\end{align*}, we can find eight tenths on the number line and place \begin{align*}\frac{4}{5}\end{align*} there.

0.6 means six tenths. So, we can find six tenths on the number line and place 0.6 there.

1 is shown on the number line, so we can add a dashed line to show that number also.

\begin{align*}0.\bar{6}\end{align*} means 0.666... So, \begin{align*}0.\bar{6}\end{align*} is a little greater than six tenths, but less than seven tenths. We can place \begin{align*}0.\bar{6}\end{align*} roughly where it belongs on the number line.

The number line will look like this when we are finished.

From the number line, we can see that \begin{align*}0.6< 0.\bar{6} < \frac{4}{5} < 1\end{align*}.

**So, ordered from least to greatest, the numbers are \begin{align*}0.6, \ 0.\bar{6}, \ \frac{4}{5}, \ 1\end{align*}.**

**Yes. That is how you can be sure that the numbers are in the correct order. Remember, they are all rational numbers!!**

Compare the following rational numbers.

#### Example A

\begin{align*}-.7 \ \underline{\;\;\;\;\;\;\;} \ -\frac{7}{10}\end{align*}

**Solution: =**

#### Example B

\begin{align*}.34 \ \underline{\;\;\;\;\;\;\;} \ \frac{1}{2}\end{align*}

**Solution: <**

#### Example C

\begin{align*}67 \ \underline{\;\;\;\;\;\;\;} \ -10\end{align*}

**Solution: >**

Here is the original problem once again.

On the dive boat one morning, Cameron began talking with another boy named Chet. Chet and his family were from Colorado and Chet was just two years older than Cameron. The boys struck up a great conversation about diving and fish and the things that they had seen on their dives.

After a little while, they spotted some dolphins swimming with the boat. This is something that can often happen as dolphins love the rushing that the motor makes on the back of the boat.

“Did you know that they can swim .83 miles in one minute?” Chet asked Cameron.

“Really, no I didn’t know that. I do know that a swordfish can swim almost one-half mile in a minute. I think the exact number is \begin{align*}\frac{9}{20}\end{align*} of a mile.”

“Wow, which one can swim the farthest in one minute?” Chet asked thinking carefully through the math.

By the time they reached the dive site, Cameron had figured out which one can swim the farthest in one minute.

**To figure out which one can swim the farthest in one minute, we will need to compare these two rational numbers.**

**A dolphin = .83 of a mile in one minute**

**A swordfish \begin{align*}= \frac{9}{20}\end{align*} of a mile in one minute**

**To figure this out, we first need to change the fraction into a decimal so that both numbers are in the same form.**

\begin{align*}\frac{9}{20}=\frac{45}{100}=.45\end{align*}

**Next, we compare .83 to .45.**

**.83 > .45**

**A dolphin can swim farther than a swordfish in one minute.**

### Vocabulary

- Rational Number
- any number positive or negative that can be written as a ratio.

- Ratio
- a comparison between two quantities. Can be written using the word “to”, using a colon, or using a fraction bar.

- Terminating Decimal
- a decimal that has a definite ending.

- Repeating Decimal
- a decimal where some of the digits repeat themselves.

- Irrational Number
- a decimal that does not terminate or repeat but continues indefinitely.

### Guided Practice

Here is one for you to try on your own.

Compare these two rational numbers.

\begin{align*}-\frac{4}{5}\end{align*} and \begin{align*}-.25\end{align*}

**Answer**

To accurate compare these two values, we need to write them into the same form.

We can convert -.25 into a fraction or \begin{align*}-\frac{4}{5}\end{align*} into a decimal.

Let's convert -.25 into a fraction.

\begin{align*}-\frac{25}{100} = -\frac{1}{4}\end{align*}

Now we can compare.

This is tricky! Negative four-fifths is almost a whole. It is less because negative one-fourth is closer to one.

\begin{align*}-\frac{4}{5} < -.25\end{align*}

**This is our answer.**

### Video Review

- This is a Khan Academy video on comparing rational numbers.

### Practice

Directions: Choose the inequality symbol (>, <, or =) that goes in the blank to make each statement true.

1. \begin{align*}1.1 \ \underline{\;\;\;\;\;\;} \ 1 \frac{1}{10}\end{align*}

2. \begin{align*}-2 \ \underline{\;\;\;\;\;\;} \ 1 \frac{1}{3}\end{align*}

3. \begin{align*}\frac{2}{5} \ \underline{\;\;\;\;\;\;} \ 0.3\end{align*}

4. \begin{align*}-.34 \ \underline{\;\;\;\;\;\;} \ -\frac{3}{10}\end{align*}

5. \begin{align*}5.6 \ \underline{\;\;\;\;\;\;} \ 5.7\end{align*}

6. \begin{align*}-8 \ \underline{\;\;\;\;\;\;} \ -12\end{align*}

7. \begin{align*}-\frac{3}{4} \ \underline{\;\;\;\;\;\;} \ -.75\end{align*}

8. \begin{align*}18.4 \ \underline{\;\;\;\;\;\;} \ 18.2\end{align*}

9. \begin{align*}-.356 \ \underline{\;\;\;\;\;\;} \ -1 \frac{1}{10}\end{align*}

10. \begin{align*}5.678888 \ \underline{\;\;\;\;\;\;} \ 6 \frac{5}{10}\end{align*}

11. \begin{align*}-.509 \ \underline{\;\;\;\;\;\;} \ -\frac{509}{1000}\end{align*}

12. \begin{align*}.87 \ \underline{\;\;\;\;\;\;} \ \frac{8}{10}\end{align*}

13. \begin{align*}-4 \ \underline{\;\;\;\;\;\;} \ -14\end{align*}

14. \begin{align*}1.212 \ \underline{\;\;\;\;\;\;} \ 1.232\end{align*}

Directions: Place each rational number on the number line below. Then list these rational numbers in order from greatest to least.

15. \begin{align*}\frac{1}{2} \qquad 0.9 \qquad 0 \qquad 0.\bar{9}\end{align*}