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# Comparison of Fractions, Decimals, and Percents

## Use <, > or = to compare fractions, decimals and percents.

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Compare and Order Rational Numbers

### [Figure1] License: CC BY-NC 3.0

Terry is studying the stock market. She notices that in one day, the stock that she is tracking has lost value. It decreased 0.5%. On the next day, it lost value again. This time the decrease was reported as minus 0.45. Which day had the worst decrease?

In this concept, you will learn to compare and order rational numbers.

### Comparing Rational Numbers

To compare and order rational numbers, you should first convert each number to the same form so that they are easier to compare. Usually it will be easier to convert each number to a decimal. Then you can use a number line to help you order the numbers. To order the rational numbers from greatest to least, or least to greatest, you need to use inequality signs. The inequality signs are:

Greater than: \begin{align*}>\end{align*}

\begin{align*}<\end{align*}Less than:

Greater than or equal to: \begin{align*}\ge \end{align*}

Less than or equal to: \begin{align*}\le \end{align*}

Let’s look at an example.

Place the following number on a number line in their approximate locations: \begin{align*}\frac{1}{8}, 0.8,8 \%.\end{align*}

First, convert \begin{align*}\frac{1}{8}\end{align*}and 8% into decimals.

\begin{align*}\frac{1}{8}=0.125\end{align*}

\begin{align*}\begin{array}{rcl} 8 \% &=&\frac{8}{100}\\ &=&0.08 \end{array} \end{align*}

Next, place the three decimals on a number line between 0 and 1.

Then, since you have compared the numbers by placing them on a number line, you can order the rational numbers.

The answer is \begin{align*}8 \% < \frac{1}{8}<0.8.\end{align*}

Here is another example.

Which inequality symbol correctly compares 0.29% to 0.029?

First, convert the percent into a decimal.

\begin{align*}\begin{array}{rcl} 0.29 \% &=&\frac{0.29}{100}\\ &=&0.0029 \end{array} \end{align*}

Next, compare it to 0.029.

\begin{align*}0.29 \% < 0.029\end{align*}The answer is \begin{align*}0.29 \% < 0.029\end{align*} .

Remember, the key to comparing and ordering rational numbers is to be sure that they are all in the same form. You want to have all fractions, all decimals or all percentages so that your comparisons are accurate. You may need to convert before you compare!!

### Examples

#### Example 1

Earlier, you were given a problem about Terry and the stock market reporting.  Terry needs to compare 0.5% with 0.45 in order to determine which day had the biggest decrease.

First, convert them all to the same form. Fractions, decimals or percent could all be used but for this situation, use percent.

Since 0.5% is in the percent form, it is done.

\begin{align*}\begin{array}{rcl} 0.45 &=&\frac{45}{100}\\ &=& 45 \% \end{array}\end{align*}

Next, order the numbers. Be sure to write them as they first appeared.

\begin{align*}0.5 \% < 0.45\end{align*}
The answer is \begin{align*}0.5 \% < 0.45.\end{align*}

The second day was a lot worse on the market.

#### Example 2

Order the following rational numbers from least to greatest.

\begin{align*}0.5 \% ,0.68,\frac{3}{15}\end{align*}

First, convert them all to the same form. You could use fractions, decimals or percent, but for this situation, let’s use percent.

Since 0.5% is in the percent form, it is done.

\begin{align*}\begin{array}{rcl} 0.68 \% &=&\frac{68}{100}\\ &=& 68 \% \end{array}\end{align*}

\begin{align*}\begin{array}{rcl} \frac{3}{15}&=& \frac{1}{5} \\ &=& \frac{20}{100}\\ &=& 20 \% \end{array}\end{align*}

Next, order the numbers. Be sure to write them as they first appeared.

\begin{align*}0.5 \%<\frac{3}{15}<0.68\end{align*}

The answer is \begin{align*}0.5 \% < \frac{3}{15} < 0.68\end{align*}.

#### Example 3

Which inequality symbol correctly compares 0.56 to \begin{align*}\frac{4}{5}\end{align*}?

First, convert them all to the same form. You could use fractions, decimals or percent, but for this situation, let’s use fractions.

Since \begin{align*}\frac{4}{5}\end{align*} is in the fraction form, it is done.

\begin{align*}\begin{array}{rcl} 0.56 &=&\frac{56}{100}\\ &=& \frac{14}{25} \end{array}\end{align*}

Next, get a common denominator so you can compare the fractions.

\begin{align*}\frac{4}{5} \times \frac{5}{5}=\frac{20}{25}\end{align*}

Then, order the numbers. Be sure to write them as they first appeared.

\begin{align*}0.56< \frac{4}{5}\end{align*}

The answer is \begin{align*}0.56 < \frac{4}{5}\end{align*}.

#### Example 4

Which inequality symbol correctly compares 0.008 to 0.8%?

First, convert them all to the same form. You could use fractions, decimals or percent, but for this situation, let’s use percent.

Since 0.8% is in the percent form, it is done.

\begin{align*}\begin{array}{rcl} 0.008 &=&\frac{8}{1000}\\ &=& \frac{0.8}{100}\\ &=& 0.8 \% \end{array}\end{align*}

Next, order the numbers. Be sure to write them as they first appeared.

\begin{align*}0.008 = 0.8 \%\end{align*}The answer is \begin{align*}0.008= 0.8 \%\end{align*} .

#### Example 5

Which inequality symbol correctly compares to \begin{align*}\frac{1}{8}\end{align*}to \begin{align*}\frac{1}{10}\end{align*}?

First, get a common denominator so you can compare the fractions. The common denominator between 8 and 10 is 40.

\begin{align*}\begin{array}{rcl} \frac{1}{8} \times\frac{5}{5} &=&\frac{5}{40}\\ \frac{1}{10} \times \frac{4}{4}&=& \frac{4}{40}\\ \end{array}\end{align*}

Next, order the numbers. Be sure to write them as they first appeared.

\begin{align*}\frac{1}{8} > \frac{1}{10}\end{align*}
The answer is \begin{align*}\frac{1}{8} > \frac{1}{10}\end{align*}.

### Review

Compare each pair of rational numbers putting the \begin{align*}<\end{align*} or  \begin{align*}>\end{align*} signs in the space provided.

1.\begin{align*}0.34 \underline{\;\;\;\;\;\;\;\;\;\;} 0.87\end{align*}

2. \begin{align*}-8 \underline{\;\;\;\;\;\;\;\;\;\;} -11\end{align*}

3. \begin{align*}\frac{1}{6}\underline{\;\;\;\;\;\;\;\;\;\;} \frac{7}{8}\end{align*}

4. \begin{align*}0.45\underline{\;\;\;\;\;\;\;\;\;\;}50 \%\end{align*}

5. \begin{align*}66 \% \underline{\;\;\;\;\;\;\;\;\;\;} \frac{3}{4}\end{align*}

6. \begin{align*}0.78 \underline{\;\;\;\;\;\;\;\;\;\;} 77 \%\end{align*}

7. \begin{align*}\frac{4}{9}\underline{\;\;\;\;\;\;\;\;\;\;} 27 \%\end{align*}

8. \begin{align*}0.989898 \underline{\;\;\;\;\;\;\;\;\;\;} 0.35\end{align*}

9. \begin{align*}0.67 \underline{\;\;\;\;\;\;\;\;\;\;} 32 \%\end{align*}

10. \begin{align*}0.123000 \underline{\;\;\;\;\;\;\;\;\;\;} 0.87\end{align*}

Use the order of operations to evaluate the following expressions.

11. \begin{align*}3x\end{align*}when is \begin{align*}x\end{align*} 0.50
12. \begin{align*}4y\end{align*}when\begin{align*}y\end{align*}is \begin{align*}\frac{3}{4}\end{align*}
13. \begin{align*}5x+1\end{align*} when\begin{align*}x\end{align*} is -12
14. \begin{align*}6y-7\end{align*} when \begin{align*}y\end{align*}is \begin{align*}\frac{1}{2}\end{align*}
15. \begin{align*}3x-4x\end{align*} when \begin{align*}x\end{align*}is -5

16. \begin{align*}5x+8y\end{align*} when \begin{align*}x\end{align*} is 2 and \begin{align*}y\end{align*}is -4

To see the Review answers, open this PDF file and look for section 2.16.

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### Vocabulary Language: English

Inequality Symbols

Inequality symbols are symbols used to compare numbers or quantities that are not necessarily equal. The inequality symbols are $<$, $>$, $\le$, $\ge$ and $\ne$.

Integer

The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3...

Irrational Number

An irrational number is a number that can not be expressed exactly as the quotient of two integers.

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.

Repeating Decimal

A repeating decimal is a decimal number that ends with a group of digits that repeat indefinitely. 1.666... and 0.9898... are examples of repeating decimals.

Terminating Decimal

A terminating decimal is a decimal number that ends. The decimal number 0.25 is an example of a terminating decimal.