2.16: Compare and Order Rational Numbers
Have you ever tried to compare rational numbers? Take a look at this dilemma.
Terry is studying the stock market. She notices that in one day, the stock that she was tracking has lost value. It decreased \begin{align*}.5\%\end{align*}. On the next day, it lost value again. This time decreasing \begin{align*}.45\end{align*}.
Which day had the worst decrease? Comparing rational numbers will help you with this task.
Guidance
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.
Take a look at this situation.
Place the following number on a number line in their approximate locations: \begin{align*}8\%, \frac{1}{8}, 0.8\end{align*}
Convert each number to a decimal.
\begin{align*}8\% &= 0.08\\ \frac{1}{8} &= 1 \div 8=0.125\\ &0.8\end{align*}
All of the numbers are between 0 and 1. You can use place value to find the correct order of the numbers. Since 0.08 has a 0 in the tenths place, 8% is the least number. Since 0.125 has a 1 in the tenths place, \begin{align*}\frac{1}{8}\end{align*} is the next greatest number. Since 0.8 has an 8 in the tenths place, it is the greatest number.
We wrote these three values on a number line. This is one way to show the different values. We can also use inequality symbols.
Inequality symbols are < less than, > greater than, \begin{align*}\le\end{align*} less than or equal to, and \begin{align*}\ge\end{align*} greater than or equal to.
Here is another one.
Which inequality symbol correctly compares 0.29% to 0.029?
Change the percent to a decimal. Then use place value to compare the numbers.
Move the decimal point two places to the left.
\begin{align*}0.29\%=0.0029\end{align*}
Now compare the place value of each number. Both numbers have a 0 in the tenths place. 0.029 has a 2 in the hundredths place, while 0.0029 has a 0 in the hundredths place. So 0.0029 is less than 0.029.
\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!!
Example A
\begin{align*}.56\end{align*}____\begin{align*}\frac{4}{5}\end{align*}
Solution: \begin{align*} < \end{align*}
Example B
\begin{align*}.008\end{align*}____\begin{align*}.8\%\end{align*}
Solution: \begin{align*}=\end{align*}
Example C
\begin{align*}\frac{1}{8}\end{align*}____\begin{align*}\frac{1}{10}\end{align*}
Solution: >
Now let's go back to the dilemma from the beginning of the Concept.
To figure out this dilemma, you have to compare \begin{align*}.5\%\end{align*} and \begin{align*}.45\end{align*}.
First, let's convert them both to percents.
\begin{align*}.5\%\end{align*} is already a percent.
\begin{align*}.45\end{align*} becomes \begin{align*}45\%\end{align*}
Now let's compare.
\begin{align*}.5\%\end{align*} < \begin{align*}45\%\end{align*}
The second day was definitely worse.
Vocabulary
- Rational Number
- number that can be written in fraction form.
- Integer
- the set of whole numbers and their opposites.
- Percent
- number representing a part out of 100.
- Terminating Decimal
- a decimal that has an ending even though many digits may be present.
- Repeating Decimal
- a decimal that has an ending even though many digits may repeat.
- Irrational Number
- a decimal that has no ending, pi or 3.14... is an example.
- Inequality Symbols
- symbols used to compare numbers using < or >.
Guided Practice
Here is one for you to try on your own.
Order the following rational numbers from least to greatest.
\begin{align*}.5\%, .68, \frac{3}{15}\end{align*}
Solution
First, let's convert them all to the same form. We could use fractions,decimals or percents, but for this situation, let's use percents.
\begin{align*}.5\%\end{align*} stays the same.
\begin{align*}.68 = 68\%\end{align*}
\begin{align*}\frac{3}{15} = \frac{1}{3} = 33.3\%\end{align*}
Now we can easily order them. Be sure to write them as they first appeared.
\begin{align*}.5\%, \frac{3}{15}, .68\end{align*}
This is our answer.
Video Review
Khan Academy Compare and Order Rational Numbers
Practice
Directions: Compare each pair of rational numbers using < or >.
- \begin{align*}.34 \ \underline{\;\;\;\;\;\;\;\;\;\;} \ .87\end{align*}
- \begin{align*}-8 \ \underline{\;\;\;\;\;\;\;\;\;\;} \ -11\end{align*}
- \begin{align*}\frac{1}{6} \ \underline{\;\;\;\;\;\;\;\;\;\;} \ \frac{7}{8}\end{align*}
- \begin{align*}.45 \ \underline{\;\;\;\;\;\;\;\;\;\;} \ 50\%\end{align*}
- \begin{align*}66\% \ \underline{\;\;\;\;\;\;\;\;\;\;} \ \frac{3}{4}\end{align*}
- \begin{align*}.78 \ \underline{\;\;\;\;\;\;\;\;\;\;} \ 77\%\end{align*}
- \begin{align*}\frac{4}{9} \ \underline{\;\;\;\;\;\;\;\;\;\;} \ 25\%\end{align*}
- \begin{align*}.989898 \ \underline{\;\;\;\;\;\;\;\;\;\;} \ .35\end{align*}
- \begin{align*}.67 \ \underline{\;\;\;\;\;\;\;\;\;\;} \ 32\%\end{align*}
- \begin{align*}.123000 \ \underline{\;\;\;\;\;\;\;\;\;\;} \ .87\end{align*}
Directions: Use the order of operations to evaluate the following expressions.
- \begin{align*}3x\end{align*}, when \begin{align*}x\end{align*} is \begin{align*}.50\end{align*}
- \begin{align*}4y\end{align*}, when \begin{align*}y\end{align*} is \begin{align*}\frac{3}{4}\end{align*}
- \begin{align*}5x+1\end{align*}, when \begin{align*}x\end{align*} is \begin{align*}-12\end{align*}
- \begin{align*}6y-7\end{align*}, when \begin{align*}y\end{align*} is \begin{align*}\frac{1}{2}\end{align*}
- \begin{align*}3x-4x\end{align*}, when \begin{align*}x\end{align*} is \begin{align*}-5\end{align*}
- \begin{align*}6x+8y\end{align*}, when \begin{align*}x\end{align*} is 2 and \begin{align*}y\end{align*} is \begin{align*}-4\end{align*}
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Term | Definition |
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Inequality Symbols | Inequality symbols are symbols used to compare numbers or quantities that are not necessarily equal. The inequality symbols are , , , and . |
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. |
Image Attributions
Here you'll learn to compare and order rational numbers.