2.7: Rational Numbers
Introduction
Boxes of Erasers
Kelly and Mallory are comparing the number of boxes of erasers that each one of them has sold. Kelly works in the store one day and Mallory works in the store the next day. They decide to write a riddle about their sales and give it to Trevor to figure out. When Trevor comes in the store on the day after Mallory has worked he finds this riddle.
Mallory sold 4 more boxes of erasers than the three-fourths that Kelly sold. If Mallory sold 13 boxes, how many boxes did Kelly sell?
Trevor is puzzled. He knows that there has to be a variable because the number of boxes that Kelly sold is unknown. He also knows that he needs to write an equation and that this equation will have a rational number, three-fourths in it.
He starts to work.
Using rational numbers is a skill that you will need as you approach higher levels of mathematics. To solve this problem, you will need to understand rational numbers. Let’s begin with this lesson and at the end, you can help Trevor with this problem.
What You Will Learn
In this lesson you will learn how to complete the following skills.
- Find equivalent forms of rational numbers given in assorted forms.
- Compare and order rational numbers on a number line and with inequality symbols.
- Apply properties and use order of operations to evaluate numerical and variable expressions involving rational numbers.
- Model and solve real-world problems using simple equations involving rational numbers.
Teaching Time
I. Find Equivalent Forms of Rational Numbers Given in Assorted Forms
You have learned about different kinds of numbers. You have learned about decimals, fractions, integers and whole numbers, now we are going to investigate rational numbers.
What is a rational number?
A rational number is a number that can be written in fraction form.
That is a good question. It means that the number can be written as a fraction, not that it necessarily is a fraction right away.
We can think of fractions first. A fraction is a rational number because it is written in fraction form. Rational numbers can also be positive or negative. Think about if you had lost one-half. Then the number would be negative.
\begin{align*}\frac{-1}{2} \ and \ \frac{1}{2}\end{align*}
Both of these are rational numbers.
What other numbers can be written in fraction form and are therefore rational numbers?
Well, you can think of integers. Remember that an integer is the set of whole numbers and their opposites. We can write any integer as a fraction over 1. This makes all integers rational numbers.
\begin{align*}-4 &= \frac{-4}{1}\\ 13 &= \frac{13}{1}\end{align*}
These integers are rational numbers too.
What about decimals?
A decimal is related to a fraction. Most decimals can be written as fractions. When you understand how to convert a decimal to a fraction and a fraction to a decimal, you will be able to determine whether or not the decimal is a rational number. Let’s look at an example.
Example
Is .34 a rational number?
If you look at this example, what is the value of the decimal? It is 34 hundredths. We can write this decimal as a fraction with a denominator of 100.
\begin{align*}.34 = \frac{34}{100}\end{align*}
This is a rational number. Regular decimals are also rational numbers.
Example
Is .3434343434 a rational number?
This is a tricky one because we have a repeating decimal. However, because it has an end to it, we can find its fraction equivalent. All repeating decimals are also rational numbers.
Example
Is .35678921 a rational number?
This is not a repeating decimal, but it is a terminating decimal. A terminating decimal has an end to it. As long as it has an end, it is a rational number.
Which decimals are not rational numbers?
Decimals that do not have an end are not rational numbers. We can think of an irrational number in this way. An example of an irrational number is pi. We say pi is equal to 3.14, but really it goes on and on and on.
Pi or 3.14... . These values are not rational numbers.
Take a few minutes and write these examples in your notebook.
We can also convert rational numbers into different forms.
Example
What decimal is equivalent to the fraction \begin{align*}\frac{7}{8}\end{align*}?
Divide to find the equivalent decimal.
\begin{align*}7 \div 8=0.875\end{align*}
The decimal 0.875 is equivalent to \begin{align*}\frac{7}{8}\end{align*}.
To convert from a decimal to a fraction, place the numbers in the decimal over the appropriate place value. For example, to convert 0.875 back to a fraction, count the decimal places. There are three decimal places, so there will be three zeros in the denominator.
\begin{align*}0.875=\frac{875}{1000}\end{align*}
Now simplify the fraction to lowest terms.
\begin{align*}\frac{875}{1000}=\frac{175}{200}=\frac{35}{40}=\frac{7}{8}\end{align*}
Example
What fraction is equivalent to the decimal 0.3125?
Place the numbers in the decimal over the appropriate place value. Count the decimal places. There are four decimal places, so there will be four zeros in the denominator.
\begin{align*}0.3125=\frac{3125}{10000}\end{align*}
Now simplify the fraction to lowest terms.
\begin{align*}\frac{3125}{10000}=\frac{125}{400}=\frac{5}{16}\end{align*}
The fraction \begin{align*}\frac{5}{16}\end{align*} is equivalent to 0.3125.
That is a great question. Because a percent also represents a part of a whole, percents can also be rational numbers. We can convert a percent to a decimal and to a fraction too. Remember that a percent means out of 100.
Example
What is 30% as a fraction? As a decimal?
We can start by knowing that 30% means 30 out of 100. Now we can write it as a fraction and as a decimal.
\begin{align*}30\% = \frac{30}{100}=.30\end{align*}
Percents that have been converted to decimals and fractions can also be considered rational numbers.
II. Compare and Order Rational Numbers on a Number Line and with Inequality Symbols
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.
Example
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.
Example
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!!
III. Apply Properties and Use Order of Operations to Evaluate Numerical and Variable Expressions Involving Rational Numbers
Let’s start by thinking back to the order of operations. The order of operations can help you simplify numerical and variable expressions. Here is a reminder of the order of operations.
- First evaluate expressions in parentheses.
- Then evaluate exponents.
- Then multiply and divide in order from left to right.
- Finally, add and subtract in order from left to right.
Now let’s look at an example.
Example
If \begin{align*}x = 4.5\end{align*}, what is the value of \begin{align*}4x(x-10)\end{align*}?
Substitute the value for \begin{align*}x\end{align*} into the expression.
\begin{align*}4(4.5)(4.5-10)\end{align*}
Evaluate the expression in parentheses: \begin{align*}4(4.5)(4.5-10)=4(4.5)(-5.5)\end{align*}
There are no exponents, so multiply in order from left to right: \begin{align*}4(4.5)(-5.5)=18(-5.5)=-99\end{align*}
The answer is -99.
Notice that in this example, we used the Distributive Property to work with the values outside of the parentheses. When we multiplied those values, we were able to get rid of the parentheses.
Also, you can see that we could use the Commutative Property of Multiplication at the end of the problem. It does not matter which order we multiply those values, the product is still the same.
Now let’s look at an example with fractions.
Example
If \begin{align*}x=\frac{1}{3}\end{align*} and \begin{align*}y=\frac{1}{2}\end{align*}, what is \begin{align*}2xy-6x\end{align*}?
Look at the expression. Multiplying fractions several times can be difficult and time consuming. The Distributive Property can help you simplify the expression before substituting the values for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.
\begin{align*}2xy-6x=2x(y-3)\end{align*}
Now substitute the values for \begin{align*}x\end{align*} and \begin{align*}y\end{align*} into the expression.
\begin{align*}2 \left(\frac{1}{3}\right) \left(\frac{1}{2}-3\right)\end{align*}
Now use the order of operations to simplify the expression.
Evaluate the expression in parentheses: \begin{align*}2 \left(\frac{1}{3}\right) \left(\frac{1}{2}-3\right)=2 \left(\frac{1}{3}\right) \left(-\frac{5}{2}\right)\end{align*}
Multiply in order from left to right: \begin{align*}2 \left(\frac{1}{3}\right) \left(-\frac{5}{2}\right)=\frac{2}{3} \left(-\frac{5}{2}\right)=-\frac{5}{3}=-1\frac{2}{3}\end{align*}
The answer is \begin{align*}-1 \frac{2}{3}\end{align*}.
IV. Model and Solve Real – World Problems Using Simple Equations Involving Rational Numbers
You have seen how fractions, decimals, and integers can be used to solve real-world problems. You can use rational numbers to write and solve equations to represent real-world problems.
Example
Candy is 5 years older than one-third Liam’s age. If Candy is 16, how old is Liam?
Choose a variable to represent Liam’s age. Let \begin{align*}l\end{align*} represent Liam’s age.
Write an equation using the information in the problem.
The phrase “5 years older” translates to “+5.” The phrase “one-third Liam’s age” translates to “\begin{align*}\frac{1}{3} \cdot l\end{align*}.” Since the problem tells you that Candy is 16, set the expression that shows Candy’s age equal to 16.
Put the parts together to form an equation.
\begin{align*}\frac{1}{3}l+5=16\end{align*}
Now solve the equation for \begin{align*}l\end{align*}. Remember that any operation you perform on one side of the equation must also be performed on the other side.
First, subtract 5 from both sides to isolate the variable term on one side of the equation.
\begin{align*}\frac{1}{3}l+5-5 &= 16-5\\ \frac{1}{3}l &= 11\end{align*}
Now you need to remove the \begin{align*}\frac{1}{3}\end{align*} from the variable term. In order to eliminate a fractional coefficient, multiply by the reciprocal.
\begin{align*}\frac{3}{1} \cdot \frac{1}{3} l &= 11 \cdot \frac{3}{1}\\ l &= 33\end{align*}
Liam is 33 years old.
Now let’s go back to the problem in the introduction and use what we have learned to solve it.
Real-Life Example Completed
Boxes of Erasers
Here is the original problem once again. Use what you have learned to solve it. You will need to write an equation, solve it for the number of boxes that Kelly sold, and write an inequality comparing the two quantities. There are three parts to your answer.
Kelly and Mallory are comparing the number of boxes of erasers that each one of them has sold. Kelly works in the store one day and Mallory works in the store the next day. They decide to write a riddle about their sales and give it to Trevor to figure out. When Trevor comes in the store on the day after Mallory has worked he finds this riddle.
Mallory sold 4 more boxes of erasers than the three-fourths that Kelly sold. If Mallory sold 13 boxes, how many boxes did Kelly sell?
Trevor is puzzled. He knows that there has to be a variable because the number of boxes that Kelly sold is unknown. He also knows that he needs to write an equation and that this equation will have a rational number, three-fourths in it.
He starts to work.
Now work the problem through in your notebook. Remember that there are three parts to the answer.
Solution to the Real – Life Example
First, you need to write an equation. Here is how each part of the equation breaks down.
\begin{align*}x =\end{align*} Kelly’s boxes
\begin{align*}+ 4 =\end{align*} the four more boxes that Mallory sold
\begin{align*}\frac{3}{4} =\end{align*} three-fourths of Kelly’s boxes
\begin{align*}13 =\end{align*} the number that Mallory sold
Here is the equation.
\begin{align*}\frac{3}{4} x+4=13\end{align*}
Next, we can solve it.
Start by subtracting four from both sides. This makes perfect sense in simplifying.
\begin{align*}\frac{3}{4} x=9\end{align*}
Next, we want to get the variable alone, so we can multiply by the reciprocal. This is the Inverse Property of Multiplication. Any number multiplied by its inverse or reciprocal is equal to 1.
\begin{align*}\frac{4}{3} \cdot \frac{3}{4} x & =9 \cdot \frac{4}{3}\\ x & = \frac{36}{3}=12\end{align*}
Now we know that Kelly sold 12 boxes and Mallory sold 13 boxes.
We can write an inequality the compares Kelly’s boxes to Mallory’s boxes.
\begin{align*}12 < 13\end{align*}
Vocabulary
Here are the vocabulary words that are found in this lesson.
- 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 >.
Time to Practice
Directions: Identify whether each is a rational number or not. Write yes or no for your answer. Then identify the form of the number: integer, decimal, repeating decimal, fraction, terminating decimal, irrational number
- \begin{align*}.456\end{align*}
- \begin{align*}\frac{2}{3}\end{align*}
- \begin{align*}- 45\end{align*}
- \begin{align*}567\end{align*}
- \begin{align*}- 8,970\end{align*}
- \begin{align*}.3434343434\end{align*}
- \begin{align*}.234\ldots\end{align*}
- \begin{align*}.234567\end{align*}
- \begin{align*}-.876\end{align*}
- \begin{align*}-\frac{2}{7}\end{align*}
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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