2.2: Multiplying and Dividing Decimals
Introduction
Ordering Pencils
The students decided to divide up the tasks of ordering. Each pair needed to work on figuring out a reasonable purchase price and quantity for the item that they were assigned. Then keeping the budget in mind the students would present their item, purchase price and quantity purchased to the team.
Mallory and Trevor are working on ordering pencils.
“Wow, there are a lot of different ones to choose from,” Mallory said looking through a catalog of pencils.
“There sure is, but I think that we should go for the number 2 ones with our school name on them. Those are sure to sell,” Trevor said pointing to a blue and red pencil in the catalog.
“I agree. Now let’s figure out the cost. It says here that we can buy a case of 25 boxes of pencils and each box has 144 pencils in it for $196.08. That seems like a good deal.”
“Yes, but we are going to need to order two cases so that we are sure that we have enough,” Trevor said as he began multiplying. “Why don’t you work on figuring out the cost per box if we order 25 for $196.08?”
“Okay, you do the other part,” Mallory said as she began her division problem.
There are two parts to this problem. One involves multiplying decimals that is the part that Trevor is working on and the other requires a division of decimals. Mallory is working on the second part of the problem. To complete these two problems, you will need to know about multiplying and dividing decimals. Estimation may also be very useful to you. Take some notes because you will be required to finish this problem at the end of the lesson.
What You Will Learn
In this lesson you will learn how to do the following skills.
 Multiply and divide decimals with and without rounding.
 Estimate decimal products and quotients using leading digits.
 Identify and apply the properties of multiplication and division in decimal operations, using numerical and variable expressions.
 Model and solve realworld problems using simple equations involving decimal multiplication and division.
Teaching Time
I. Multiply and Divide Decimals with and without Rounding
In the last lesson you learned how to add and subtract decimals, now we are going to look at multiplying and dividing decimals. Let’s start with multiplying decimals.
Do you remember how to multiply whole numbers with several digits?
Well, we are going to multiply decimals in the same way. Don’t worry about the decimal point in the beginning. We will work with it in the product.
Example
Multiply: \begin{align*}34.67 \times 8.2\end{align*}
We can multiply decimals like we multiply whole numbers. First, ignore the decimal points and line up the numbers from the right.
\begin{align*}& \quad 34.67\\
& \underline{\;\; \times 8.2 \;}\end{align*}
Now multiply each digit in the top number by each digit in the bottom number, just like whole numbers.
Now place the decimal point in the product by counting the number of decimal places in each of the numbers that were multiplied. The first number has two decimal places, and the second number has one decimal place. So move the decimal point three places.
The product is 284.294.
What about division?
We can divide decimals too. When we divide decimals, we have to pay attention to the decimal point. Let’s look at an example.
Example
Divide: \begin{align*}253.26 \div 4.5\end{align*}
We can divide decimals like we divide whole numbers. First, move the decimal point so that the divisor, 4.5, is a whole number. Then move the decimal point in the dividend, 253.36, the same number of places and write out the problem in long division. Notice that the first number is the dividend. The dividend is the number being divided so it goes into our division box. The other number is the divisor. The divisor is the number doing the dividing.
\begin{align*}\overset{\ \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}}{45 \Big ) 2532.6}\end{align*}
Now ignore the decimal point and divide as you would divide whole numbers. Then place the decimal point in the quotient directly above the decimal point in the dividend.
Notice that we had to add a zero into the dividend so that our quotient would be even.
The answer is 56.28.
You have probably worked on multiplying and dividing decimals in past math classes. However, it is always a skill worth practicing because of how many times you will work with decimals in reallife. We can also estimate products and quotients of decimals.
II. Estimate Decimal Products and Quotients using Leading Digits
To estimate products and quotients with decimals, you need to first round the numbers so that they are easier to work with. To round to the nearest whole number, look at the digit in the tenths place. If it is less than 5, round down. If it is 5 or greater, round up.
Remember that an estimate is an answer that is not exact, but is approximate and reasonable.
Let’s look at an example.
Example
Estimate the product: \begin{align*}11.256 \times 6.81\end{align*}
First, we round the first number. Since there is a 2 in the tenths place, 11.256 rounds down to 11.
Now round the second number. Since there is an 8 in the tenths place, 6.81 rounds up to 7.
Now multiply the rounded numbers.
\begin{align*}11 \times 7 = 77\end{align*}
A good estimate for the product is 77.
Example
Estimate the quotient: \begin{align*}91.93 \div 4.39\end{align*}
First we round the first number. Since there is a 9 in the tenths place, 91.93 rounds up to 92.
Now round the second number. Since there is a 3 in the tenths place, 4.39 rounds down to 4.
Now divide the rounded numbers.
\begin{align*}92 \div 4=23\end{align*}
A good estimate for the quotient is 23.
Did you notice which numbers we multiplied? We multiplied the whole digits or the digits that were leading the entire number. We did the same thing when we divided.
Yes. We work with the digits that are in the “lead”. With those digits, we can find a reasonable estimate.
Now let’s see how the properties of multiplication and division can help us when working with decimals.
III. Identify and Apply the Properties of Multiplication and Division in Decimal Operations, using Numerical and Variable Expressions
To work with multiplication and division of decimals, we are going to use a few properties that you are probably already familiar with. Here are the properties.
Associative Property of Multiplication
The grouping of numbers does not affect the product: \begin{align*}4.5\times(2.1\times9.6)=(4.5\times2.1)\times9.6\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 \cdot 1.5)+(3.2 \cdot 8.9)\end{align*}
The first thing that we can look at is identifying these properties in some examples. Notice how the property is applied.
Example
Which of the following shows the Commutative Property?
a. \begin{align*}y+7.2=7.2y\end{align*}
b. \begin{align*}y \div 7.2=7.2\div y\end{align*}
c. \begin{align*}y \times 7.2=7.2 \times y\end{align*}
Consider choice a.
This equation states that a number added to 7.2 is equal to that number multiplied by 7.2. This is not correct.
Consider choice b.
This equation states that the quotient of a number and 7.2 is equal to the quotient of 7.2 and a number. This is not correct.
Consider choice c.
This equation states that the product of a number and 7.2 is equal to the product of 7.2 and a number. The Commutative Property states that the order of numbers does not change the product, so this is the correct equation.
The answer is choice c.
You can use the properties of multiplication and division to simplify numerical expressions.
Example
\begin{align*}5(2\cdot 3)\cdot9\end{align*}
First, notice that we can use the order of operations here. We find the product of the terms inside the parentheses.
\begin{align*}& 2 \times 3 = 6\\
& 5(6) \cdot 9\\
& 30 \cdot 9\end{align*}
The answer is 270.
Now you could have also worked with this example by changing the grouping through the associative property. Take a look.
\begin{align*}(5 \cdot 2)\cdot 9 \cdot3\end{align*}
The product would have been \begin{align*}10 \times 27\end{align*}
The product is 270.
We can also use properties to simplify variable expressions.
Example
Simplify: \begin{align*}2.5(2.1x+4.3y)\end{align*}
The addends inside the parentheses cannot be combined because two different variables are being used, so you can use the distributive property to help you simplify the expression.
Apply the distributive property: \begin{align*}2.5(2.1x+4.3y)=(2.5 \times 2.1x)+(2.5 \times 4.3y)\end{align*}
Then simplify: \begin{align*}(2.5 \times 2.1x)+(2.5 \times 4.3y)=5.25x+10.75y\end{align*}
This is our answer.
IV. Model and Solve Real – World Problems using Simple Equations Involving Decimal Multiplication and Division
Think of all of the places where decimals appear in the realworld. You will encounter many different situations where you will need to multiply and divide decimals. Let’s look at an example.
Example
Avi bought five new telephones for the school office. They cost $61.35 each. If the price includes tax, about how much did Avi spend? What is the exact amount that Avi spent?
Round each decimal to a number that is easy to multiply. Then find the sum.
5 does not need to be rounded.
61.35 rounds down to 60.
\begin{align*}5 \times 60=300\end{align*}
Avi spent about $300. This is our estimate.
To find the exact amount Avi spent, write a simple equation to represent the problem. Let \begin{align*}x\end{align*}
\begin{align*}x &= 5 \times 61.35\\
&=306.75\end{align*}
Avi spent exactly $306.75. This is our exact answer.
Example
The City Orchestra received a total of $1,891.50 in donations. This needs to be divided evenly among six different departments. How much will each department receive?
Notice that the key word “each” tells us that we are going to need to divide. The money is being split up and that means division.
Divide to find the amount each department will receive: \begin{align*}1891.5 \div 6 = 315.25\end{align*}
Each department will receive $315.25.
Now let’s apply the information from this lesson to the problem in the introduction.
RealLife Example Completed
Ordering Pencils
Here is the original problem once again. Reread it and then work on solving the problem in your notebook. You should have four answers when completed. An estimate and exact product for Trevor’s part of the problem, and an estimate and exact quotient for Mallory’s part of the problem.
The students decided to divide up the tasks of ordering. Each pair needed to work on figuring out a reasonable purchase price and quantity for the item that they were assigned. Then keeping the budget in mind the students would present their item, purchase price and quantity purchased to the team.
Mallory and Trevor are working on ordering pencils.
“Wow, there are a lot of different ones to choose from,” Mallory said looking through a catalog of pencils.
“There sure is, but I think that we should go for the number 2 ones with our school name on them. Those are sure to sell,”
Trevor said pointing to a blue and red pencil in the catalog.
“I agree. Now let’s figure out the cost. It says here that we can buy a case of 25 boxes of pencils and each box has 144 pencils in it for $196.08. That seems like a good deal.”
“Yes, but we are going to need to order two cases so that we are sure that we have enough,” Trevor said as he began multiplying. “Why don’t you work on figuring out the cost per box if we order 25 for $196.08?”
“Okay, you do the other part,” Mallory said as she began her division problem.
Now it is time for you to solve this problem in your notebook. Remember that you should have four answers when finished.
Solution to Real – Life Example
Now let’s look at the solution to this problem.
We can start with Trevor. Trevor needed to find a product. To estimate, he could round the cost of the case of pencils. Then he wants to order two cases, so he would multiply this rounded dollar amount by 2.
$196.08 rounds to $200.00
\begin{align*}200 \times 2 = 400\end{align*}
The estimate is about $400.00 for two cases.
Now let’s look at the actual product.
\begin{align*}& \$ 196.08\\
& \underline{\times \;\;\;\;\;\;\; 2}\\
& \$ 392.16\end{align*}
You can see that our estimate was reasonable for the actual answer.
Next, we can work with Mallory. Mallory needed to figure out the price per box if there are 25 boxes in a case for $196.08.
It makes sense for her to round the dollar amount first to find an estimate.
$196.08 rounds to $200.00
\begin{align*}200 \div 25 = 8\end{align*}
Each box costs roughly $8.00.
Now let’s find the quotient.
\begin{align*}196.08 \div 25 = \$ 7.84 \ per \ box\end{align*}
You can see that our estimate was reasonable given the quotient.
Vocabulary
Here are the vocabulary words that are found in this lesson.
 Dividend
 the number being divided in a division problem. It is often the first number in a problem written horizontally.
 Divisor
 the number doing the dividing in a division problem.
 Estimate
 an approximate answer that is reasonable and makes sense for the problem.
 Leading Digits
 the first digits in a decimaloften the whole number part of the decimal.
Time to Practice
Directions: Estimate each product using rounding.

\begin{align*}2.67 \times 3.10 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
2.67×3.10=−−−−− 
\begin{align*}4.15 \times 8.09 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
4.15×8.09=−−−−− 
\begin{align*}6.67 \times 7.10 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
6.67×7.10=−−−−− 
\begin{align*}8.21 \times 9.87 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
8.21×9.87=−−−−− 
\begin{align*}5.86 \times 5.13 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
5.86×5.13=−−−−− 
\begin{align*}5.86 \times 5.13 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
5.86×5.13=−−−−− 
\begin{align*}6.35 \times 12.01 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
6.35×12.01=−−−−− 
\begin{align*}4.13 \times 9.87 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
4.13×9.87=−−−−− 
\begin{align*}8.12 \times 9.15 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
8.12×9.15=−−−−− 
\begin{align*}16.21 \times 9.94 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
16.21×9.94=−−−−−
Directions: Estimate each quotient using rounding.

\begin{align*}21.87 \div 2.1 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
21.87÷2.1=−−−−− 
\begin{align*}32.14 \div 8.03 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
32.14÷8.03=−−−−− 
\begin{align*}36.07 \div 8.83 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
36.07÷8.83=−−−−− 
\begin{align*}16.20 \div 7.92 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
16.20÷7.92=−−−−− 
\begin{align*}34.87 \div 5.03 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
34.87÷5.03=−−−−− 
\begin{align*}18.08 \div 3.14 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
18.08÷3.14=−−−−− 
\begin{align*}21.10 \div 3.17 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
21.10÷3.17=−−−−− 
\begin{align*}44.82 \div 8.60 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
44.82÷8.60=−−−−−  \begin{align*}120.02 \div 58.72 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
 \begin{align*}139.87 \div 69.81 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
Directions: Multiply or divide to find each product or quotient.
 \begin{align*}14.50 \times 2.1 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
 \begin{align*}13.64 \div 2.2 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
 \begin{align*}21.35 \div 6.1 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
 \begin{align*}5.2 \times 6.3 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
 \begin{align*}6.7 \times 4.3 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
 \begin{align*}.437 \times 2.1 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
 \begin{align*}5.42 \times 3.3 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
 \begin{align*}16.26 \div 3 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
 \begin{align*}15.18 \div 2.2 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
 \begin{align*}16.39 \div 2.2 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
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