# 2.8: Number Properties

**At Grade**Created by: CK-12

## Introduction

*The Week of Sales*

Once the school store had been up and running for a few weeks, the students saw what a huge success it was. Students began to rely on getting their school supplies and school and the student council was overjoyed at the amount of money that they would have for events.

“We are doing terrific!” Kelly said at the meeting one week.

“Yes. Here are the total sales that we have done for the month.” Trevor said taking out his paper. “We made 130.25, 75.18, and 85.00 for a grand total of 380 dollars and 43 cents.”

“Wait a minute, there are four weeks in the month. You only said three amounts,” Mallory pointed out.

Trevor looked back down at his paper.

“You’re right. I missed week 3. But I know the total is correct.”

**This is where you come in. Based on the information that Trevor gave the group, you should be able to figure out how much money was made in the missing week. This lesson will show you how to use properties to work with variable and numerical expressions. At the end of this lesson, you will write an equation and use properties to figure out the sales for the third week.**

*What You Will Learn*

In this lesson, you will learn to use the following skills.

- Identify and apply the associative property of addition and multiplication in rational number operations, using numerical and variable expressions.
- Identify and apply the commutative property of addition and multiplication in rational number operations, using numerical and variable expressions.
- Identify and apply the distributive property in rational number operations, using numerical and variable expressions.
- Model and solve real-world problems using equations involving rational numbers.

*Teaching Time*

In this chapter, you have learned about all different kinds of rational numbers and the different number properties. It’s important to know how and when to use these properties when given a real-world situation. Let’s take a look at how properties can help you solve problems.

First, let’s review all of the properties you have learned.

**Associative Property of Addition**

The grouping of addends does not affect the sum: \begin{align*}4.5+(2.1+9.6)=(4.5+2.1)+9.6\end{align*}

**Associative Property of Multiplication**

The grouping of numbers does not affect the product: \begin{align*}4.5 \times (2.1 \times 9.6)=(4.5 \times 2.1) \times 9.6\end{align*}

**Commutative Property of Addition**

The order of addends does not change the sum: \begin{align*}6.3+8.7=8.7+6.3\end{align*}

**Commutative Property of Multiplication**

The order of numbers does not change the product: \begin{align*}6.3 \times 8.7 = 8.7 \times 6.3\end{align*}

**Distributive Property**

The product of a number and a sum is equal to the sum of the individual products of addends and the number: \begin{align*}3.2 (1.5 + 8.9)=(3.2 \times 1.5)+(3.2 \times 8.9)\end{align*}

**Additive Identity**

The sum of any number and zero is that number: \begin{align*}\frac{3}{11}+0=\frac{3}{11}\end{align*}

**Additive Inverse**

The sum of any number and its inverse is zero: \begin{align*}\frac{3}{4}+ \left(-\frac{3}{4}\right)=0\end{align*}

**Multiplicative Identity**

The product of any number and one is that number: \begin{align*}\frac{3}{11} \times 1=\frac{3}{11}\end{align*}

**Multiplicative Inverse**

The product of any number and its reciprocal is one: \begin{align*}\frac{3}{4} \times\frac{4}{3}=1\end{align*}

**Zero Property**

The product of any number and zero is zero: \begin{align*}\frac{4}{7} \times 0=0\end{align*}

**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.

I. **Identify and Apply the Associative Property of Addition and Multiplication in Rational Number Operations, using Numerical and Variable Expressions**

**We can start by remembering that when we apply the associative property that we can move the parentheses in an expression to help us with our work.**

Let’s look at an example of a numerical expression.

Example

\begin{align*}\left(\frac{1}{3}+\frac{1}{4}\right)+\frac{2}{4}\end{align*}

**Here we have an expression that has three rational numbers in it. We can see that there are two different denominators. However, look at the groupings. If we move the parentheses to group the fourths together, it will help us with the addition.**

**This is an example of the Associative Property of Addition.**

\begin{align*}\frac{1}{3}+\left(\frac{1}{4}+\frac{2}{4}\right)\end{align*}

**Now we can find a solution.**

\begin{align*}\frac{1}{3}+\frac{3}{4}=\frac{4}{12}+\frac{9}{12}=\frac{13}{12}=1 \frac{1}{12}\end{align*}

**This is our answer.**

Now let’s look at an example of a variable expression where the associative property would be useful.

Example

\begin{align*}\left(x+\frac{4}{5}\right)-\frac{2}{5}\end{align*}, when \begin{align*}x\end{align*} is \begin{align*}\frac{3}{7}\end{align*}

**First, we can substitute three-sevenths into the expression for the variable.**

\begin{align*}\left(\frac{3}{7}+\frac{4}{5}\right)-\frac{2}{5}\end{align*}

**Next, you can see that it makes much more sense to work with the fifths and then the sevenths. Let’s regroup these fractions and find a solution.**

\begin{align*}& \frac{3}{7}+\left(\frac{4}{5}-\frac{2}{5}\right)\\ & \frac{3}{7}+\frac{2}{5}=\frac{15}{35}+\frac{14}{35}=\frac{19}{35}\end{align*}

**This is our answer.**

II. **Identify and Apply the Commutative Property of Addition and Multiplication in Rational Number Operations, using Numerical and Variable Expressions**

**We can also apply the Commutative Property when we evaluate expressions with rational numbers in them. Rearranging the numbers we are working with can help us to simplify our efforts.**

Let’s look at an example.

Example

\begin{align*}.56+\frac{1}{2}+.24\end{align*}

**In this example, we have three rational numbers that we are adding. If you look, it doesn’t make sense to add the decimal and the fraction and then the decimal. It makes more sense to use the commutative property to rearrange the values. We add the decimals first and then deal with the fraction.**

\begin{align*}.56+.24.+\frac{1}{2}\end{align*}

**Now we can add the decimals together.**

\begin{align*}& .56 + .24 = .80\\ & .80+\frac{1}{2}\end{align*}

**Here we are trying to add a decimal and a fraction. We need to convert the fraction to a decimal so that the rational numbers are in the same form. Then we can easily add them.**

\begin{align*}\frac{1}{2} &= .50\\ .80+.50 &= 1.3\end{align*}

**This is our answer.**

Now let’s look at an example with a variable expression and multiplication.

Example

\begin{align*}\frac{1}{2} \cdot (.56) \cdot \frac{1}{3}\end{align*}

Here is a numerical expression with two fractions and a decimal. Notice that the decimal is in the middle. It makes the most sense to rearrange the fractions so that you can multiply them together, and then multiply the decimal.

\begin{align*}\frac{1}{2} \cdot \frac{1}{3}=\frac{1}{6}\end{align*}

Now let’s convert .5 to a fraction and multiply it by one-sixth.

\begin{align*}\frac{1}{6} \cdot \frac{1}{2}=\frac{1}{12}\end{align*}

**This is the answer.**

III. **Identify and Apply the Distributive Property in Rational Number Operations, using Numerical and Variable Expressions**

**We can use the Distributive Property to help us simplify expressions with parentheses and operations within the parentheses.** Let’s look at an example.

Example

\begin{align*}\frac{1}{2} \left(\frac{3}{6}+8\right)\end{align*}

**In this example, we need to multiply the term outside of the parentheses with both of the terms inside the parentheses. Then we can simplify the expression further.**

\begin{align*}\frac{1}{2} \cdot \frac{1}{2} &= \frac{1}{4}\\ \frac{1}{2} \cdot 8 &= 4\\ 4+\frac{1}{4} &= 4 \frac{1}{4}\end{align*}

**This is the answer.**

We can also use the Distributive Property to simplify variable expressions.

Example

\begin{align*}x(3+7)\end{align*}

**Now we can distribute the variable to both of the terms in the parentheses and then we can simplify the expression.**

\begin{align*}& 3x+7x\\ & 10x\end{align*}

**This is the answer.**

IV. **Model and Solve Real – World Problems Using Equations Involving Rational Numbers**

**Let’s look at using what we have learned to solve some real-world problems with rational numbers.**

Example

*Nina owns a printing shop. The expression \begin{align*}\frac{1}{2}x+\frac{22}{3}\end{align*} represents her profits each hour, where \begin{align*}x\end{align*} is the number of pages she prints. If she made $15 in the last hour, how many pages did she print?*

First set up the equation. You know the profit she made and the expression that represents the profit. Set them equal to each other.

\begin{align*}\frac{1}{3}x+\frac{22}{3}=15\end{align*}

Now solve for \begin{align*}x\end{align*}. The fractions make this equation more difficult to solve. So, use properties to take away the fractions. The reciprocal property states that any number multiplied by its reciprocal is equal to 1. Since both fractions are thirds, multiply both sides of the equation by the reciprocal, \begin{align*}\frac{3}{1}\end{align*}, to take away the fraction.

\begin{align*}\frac{3}{1} \left(\frac{1}{3}x+\frac{22}{3}\right)=\frac{3}{1}(15)\end{align*}

Now use the distributive property to simplify the left side of the equation.

\begin{align*}\left(\frac{3}{1}\right)\left(\frac{1}{3}x\right)+\left(\frac{3}{1}\right)\left(\frac{22}{3}\right)=\frac{3}{1}(15)\end{align*}

\begin{align*}x+22=45\end{align*}

Now you can easily solve the equation. Subtract 22 from both sides to isolate the variable.

\begin{align*}x+22-22 &= 45-22\\ x+0 &= 23\\ x &= 23\end{align*}

Note: By subtracting 22 from both sides, you are applying the additive inverse property to the left side of the equation. You can prove that this isolates \begin{align*}x\end{align*} because of the additive identity property.

Example

*Ben bought one share of stock for $40. He watched the price of his stock every day for a week. On Monday, the stock moved 2.5 points up. On Tuesday, it moved 1.75 points down. On Wednesday, it moved p points. On Thursday and Friday, the stock moved 0.75 points up each day. If the price of the stock was $45 at the end of the week, what is the value of \begin{align*}p\end{align*}?*

Use the information in the problem to write an equation to represent the price of the stock.

Use the order of operations to simplify the equation. First, multiply. Then add and subtract in order from left to right.

\begin{align*}40+2.5-1.75+p+2(0.75)=45\end{align*}

Now apply the commutative property in order to combine the decimals.

\begin{align*}40+2.5-1.75+p+1.5 &= 45\\ 42.5-1.75+p+1.5 &= 45\\ 40.75+p+1.5 &= 45\end{align*}

Finally, isolate the variable by subtracting 42.25 from both sides. You can then apply the commutative and additive identity properties to the left side of the equation to solve for \begin{align*}p\end{align*}.

\begin{align*}42.25+p-42.25 &= 45-42.25\\ p+42.25-42.25 &= 45-42.25\\ p+0 &= 2.75\\ p &= 2.75\end{align*}

The stock moved 2.75 points up on Wednesday.

**Now let’s go back to the original problem from the introduction and work on solving it.**

## Real-Life Example Completed

*The Week of Sales*

**Here is the original problem once again. Reread it and then write an equation to show the information that Trevor reported on group sales. After writing the equation, solve it for the missing sales for week 3.**

Once the school store had been up and running for a few weeks, the students saw what a huge success it was. Students began to rely on getting their school supplies and school and the student council was overjoyed at the amount of money that they would have for events.

“We are doing terrific!” Kelly said at the meeting one week.

“Yes. Here are the total sales that we have done for the month.” Trevor said taking out his paper. “We made 130.25, 75.18, and 85.00 for a grand total of 380 dollars and 43 cents.”

“Wait a minute, there are four weeks in the month. You only said three amounts,” Mallory pointed out.

Trevor looked back down at his paper.

“You’re right. I missed week 3. But I know the total is correct.”

*Now work on this problem. Remember there are two parts to your answer.*

*Solution to Real – Life Example*

**To begin, we can write an equation that shows what we know and what we don’t know. We know the total sales for weeks one, two, and four. We are missing week 3, and we know the sum.**

\begin{align*}130.25 +75.18 + x + 85.00 = 380.43\end{align*}

**Notice that we used a variable to represent the missing quantity.**

**Now we can solve the equation. You can use the commutative property of addition in this work. By rearranging the quantities that you have, you can find a sum that does not include the variable. Here it is.**

\begin{align*}290.43 + x = 380.43\end{align*}

**Next, we can subtract the 290.43 from the 380.43.**

**The amount of sales in week 3 was $90.00.**

## Time to Practice

Directions: Use what you have learned to work with each problem concerning rational numbers.

- Write an equation that shows the commutative property of addition.
- Write an equation that shows the zero property of multiplication.
- What is the difference between the additive identity and the multiplicative identity properties?
- The following shows a student’s work: \begin{align*}\left(\frac{3}{3}\right) \left(\frac{1}{5}\right)+ \left(\frac{5}{5}\right) \left(\frac{2}{3}\right)=\frac{3}{15}+\frac{10}{15}\end{align*}. What property did the student use to find a common denominator?
- The expression \begin{align*}2p^3-4m\end{align*} can be used to find the sales profit of a company where \begin{align*}p\end{align*} is the number of products they sell and \begin{align*}m\end{align*} is the number of miles they travel. If they sold 5 products and traveled 10 miles, what was their profit?
- The expression \begin{align*}\frac{11s}{2}+7t\end{align*} can be used to find the group admission price, where \begin{align*}s\end{align*} is the number of students and \begin{align*}t\end{align*} is the number of teachers. If there are 20 students and 4 teachers, what is the group admission price?
- Brooke needs to save $146 for a trip. She has $35 in her savings account. She saves $15.75 each week. She also has to spend $15 to buy a present for a friend. How many weeks will Brooke need to save to have enough for her trip?
- Vinnie is \begin{align*}\frac{1}{2}\end{align*} as old as Julie. In 6 years, Julie will be 24. How old is Vinnie?
- Manuel starts with $30. He earns $8.50 per hour plus an additional bonus of $12 each day. He spends $7.50 for lunch. If he has $94 at the end of the day, for how many hours did he work?
- A formula for the perimeter of a rectangle is \begin{align*}P=2(l+w)\end{align*}, where \begin{align*}P\end{align*} is the perimeter, \begin{align*}l\end{align*} is the length, and \begin{align*}w\end{align*} is the width. If the perimeter of a rectangle is 312 centimeters and the width is 67.3 centimeters, what is the length?

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