2.4: Multiplying Decimals
Introduction
The Long Jump
Kevin is new at Franklin Marsh Middle School. In his old school, Kevin had worked very hard to become the best at the long jump event in Track and Field. In fact, Kevin held the state record in his previous state. Now Kevin has moved to a new state, and he is feeling nervous about starting over again.
Before the first big meet, Kevin went out to the field to check out the sand pit where he will land for the long jump. Mr. Rend, the groundskeeper, was outside working on raking the rectangular sand pit when Kevin arrived.
“Hello Kevin,” he said. “How are you?”
“I’m great,” Kevin said “But I am a little nervous. Is this pit the usual size?”
“As usual as it gets,” Mr. Rend said. “The dimensions are \begin{align*}4.5 \ m \times 4 \ m\end{align*}.”
Kevin stopped to think for a minute. The sand pit at his old school had an area of 20 sq. m. He started to think about whether or not these two pits were the same size. While watching Mr. Rend rake, Kevin started to do the math in his head.
How about you? Can you figure this out in your head? To do it, you will need to multiply decimals. This lesson will teach you all that you need to know to help Kevin. Pay attention because you will see this problem again at the end of the lesson.
What You Will Learn
By the end of this lesson, you will learn the following skills.
- Multiply decimals with and without rounding.
- Estimate or confirm decimal products by multiplying leading digits.
- Identify and apply the Commutative and Associative Properties of Multiplication in decimal operations, using numerical and variable expressions.
- Solve real-world problems using formulas involving area of rectangles and decimal dimensions.
Teaching Time
I. Multiply Decimals With and Without Rounding
Using addition, subtraction, and multiplication gives us flexibility in solving real-world decimal problems. We have explored decimals: rounding, adding and subtracting, estimating sums and differences, writing equations and solving real-world problems. In this lesson, we will investigate decimals and the properties that govern multiplication. Let’s look at some statements that use multiplication.
Jeff made five times more per hour than Alex; Peter ran twice as many kilometers as Ellen;
Audrey collected three times as many grams of strawberries as Trey; the length is three times as long as the width.
All these scenarios describe multiplication relationships between decimals.
How do we multiply decimals to find a product?
Multiplying decimals is exactly like multiplying whole numbers—with one important difference.
There is a decimal point in your answer.
Follow these steps for decimal multiplication.
- Line up the numbers on the right. Do not line up the decimal points, just the numbers.
- Multiply each digit just like usual
- Count the number of places after the decimal point in each original number.
- Then, in your answer, count the same number of places from right to left and place the decimal point in the product.
Let’s walk through a simple multiplication problem.
Example
\begin{align*}& 3.25 \times 1.2\\ & \quad \ 3.25 \ \rightarrow 2 \ \text{decimal places}\\ & \underline{\times \ \ \ 1.2 \ \ } \rightarrow 1 \ \text{decimal places}\\ & \quad \ \ 650\\ & \underline{+ \ 3250 \ }\\ & \ \ \; 3.900 \ \rightarrow 3 \ \text{decimal places} \ (2 \ \text{decimal places in} \ 3.25 \ \text{and} \ 1 \ \text{in} \ 1.2 = 3 \ \text{decimal places)}\end{align*}
In our final product or answer in a multiplication problem, we do not need to write zeros that appear at the end of the answer. Our answer is \begin{align*}3.25 \times 1.2 = 3.9\end{align*}.
Sometimes, it will make sense to round decimals to a certain place before multiplying. When you do this, you apply the same rules as you used when you were adding and subtracting. The key to rounding is to be sure that you aren’t looking for an exact answer. When you need an approximate answer, rounding is an excellent method of estimating.
Example
Round the numbers to the nearest hundredth then find the product
\begin{align*}3.748 \times 8.095 \end{align*}
This problem asks us to round each number to the hundredths place before multiplying.
As we have done in the past, to make the rounding steps clear, we underline the number we’re rounding to and bold or circle the number directly to the right of it. We’re rounding to the hundredths place, so we’ll round to the second place to the right of the decimal. The bolded number, the thousandths place, is the one to look at when deciding to round up or down.
3.748 \begin{align*}\rightarrow\end{align*} rounded to the hundredths place \begin{align*}\rightarrow\end{align*} 3.75
8.095 \begin{align*}\rightarrow\end{align*} rounded to the hundredths place \begin{align*}\rightarrow\end{align*} 8.1
Now that the numbers are rounded, we ignore the decimal points, align the numbers to the right, and multiply.
\begin{align*}& \quad \ \ \ 3.75\\ & \ \ \underline{\times \ \ \ 8.1\ }\\ & \qquad 375\\ & \underline{+ \ 30000 \ }\\ & \quad 30.375\end{align*}
To place the decimal point in the answer after multiplying, count the decimal places in the original numbers and transfer that sum into the answer. In this problem, 3.75 has two decimal places and 8.1 has one decimal place. So, once we have our answer, we count over three numbers from the right, and place our decimal point between the 0 and the 3.
2I. Lesson Exercises
Find each product.
- \begin{align*}1.23 \times 6.7\end{align*}
- \begin{align*}4.56 \times 1.34\end{align*}
- Round to the nearest tenth then multiply, \begin{align*}5.67 \times 4.35\end{align*}
Now check your work by comparing your answers with a neighbor. Did you put the decimal point in the correct place? Correct any mistakes before moving on.
Write down the steps for multiplying decimals before moving on to the next section.
II. Estimate or Confirm Decimal Products by Multiplying Leading Digits
With addition and subtraction of decimals, you have seen how estimation works to approximate a solution. It is a good idea to get in the habit of estimating either before or after solving a problem. Estimation helps to confirm that your solution is in the right ballpark.
With multiplication, rounding decimals before multiplying is one way to find an estimate. You can also simply multiply the leading digits.
Remember how we used front-end estimation to approximate decimal sums and differences? Multiplying leading digits works the same way. The leading digits are the first two values in a decimal. To estimate a product, multiply the leading digits exactly as you have been - insert the decimal point into the solution based on the sum of places in the front end numbers used.
Example
Estimate the product, \begin{align*}6.42 \times 0.383\end{align*}.
First we need to identify the leading digits, being careful to preserve the placement of the decimal point in each number. In 6.42, the leading digits are 6.4. In 0.383, the leading digits are .38. Note that 0 is not one of the leading digits in the second decimal. Because zero is the only number on the left side of the decimal point, we can disregard it. Now that we have the leading digits, we multiply.
\begin{align*}& \quad \ \ 6.4\\ & \ \ \underline{\times \; .38\ }\\ & \quad \ \ 512\\ & \underline{+ \; 1920 \ }\\ & \quad 2.432\end{align*}
Example
Find the product. Then estimate to confirm your solution. \begin{align*}22.17 \times 4.45\end{align*}.
This problem asks us to perform two operations—straight decimal multiplication followed by estimation. Let’s start multiplying just as we’ve learned: aligning the numbers to the right and multiplying as if they were whole numbers.
\begin{align*}& \quad \ \ \ 22.17\\ & \quad \ \underline{\times \; 4.45 \ }\\ & \quad \ \ 11085\\ & \quad \ \ 88680\\ & \underline{+ \; 886800 \ }\\ & \ \ 98.6565\end{align*}
Note the placement of the decimal point in the answer. The original factors both have two decimal places, so once we have our answer, we count over four places from the right, and place the decimal point between the 8 and the 6.
Now that we have multiplied to find the answer, we can use estimation to check our product to be sure that it is accurate. We can use the method multiplying leading digits. First, we reduce the numbers to their leading digits, remaining vigilant as to the placement of the decimal point. The leading digits of 22.17 are 22. The leading digits of 4.45 are 4.4. Now we can multiply.
\begin{align*}& \quad \ 22\\ & \underline{\times \; 4.4 \ }\\ & \quad \ 88\\ & \underline{+ \; 880 \ }\\ & \ \ 96.8\end{align*}
If you look at the two solutions, 98.6 and 96.8, you can see that they are actually very close. Our estimate is very close to the actual answer. We can trust that our answer is accurate.
2J. Lesson Exercises
Find each product. Use multiplying leading digits to check your answer.
- \begin{align*}67.9 \times 1.2\end{align*}
- \begin{align*}5.3 \times 2.3\end{align*}
- \begin{align*}8.71 \times 9.12\end{align*}
Take a few minutes to check your work with a friend. Review how you each estimated to check your answers.
III. Identify and Apply the Commutative and Associative Properties of Multiplication in Decimal Operations, Using Numerical and Variable Expressions
Now that you are adept at multiplying decimals, we can apply two mathematical properties which govern multiplication. Remember the Commutative and Associative properties of Addition? Well, both properties also relate to multiplication.
The Commutative Property of Multiplication states that the order of the factors does not change the product. The Commutative Property of Multiplication works regardless of the number of "numbers" we have, whether it be four, five, or six factors. It works for decimal factors, too. Let’s test the property using simple whole numbers.
\begin{align*}&2 \times 3 = 6 && 2 \times 3 \times 4 = 24 && 3 \times 2 \times 4 = 24 && 4 \times 3 \times 2 = 24\\ &3 \times 2 = 6 && 2 \times 4 \times 3 = 24 && 3 \times 4 \times 2 = 24 && 4 \times 2 \times 3 = 24\end{align*}
As you see, when there are two factors (2 and 3) and when there are three factors (2, 3, and 4), the order of the factors does not change the solution.
Many people commute to work each day. If you think about it, the Commutative Property applies here too. It doesn’t matter whether people take a train, subway, bus or car the end destination is the same. The result doesn’t regardless of how you get there.
What about the Associative Property of Multiplication?
From the order of operations, we know that operations in parentheses must be completed before any other operation, but the Associative Property of Multiplication states that the way in which factors are grouped does not change the product. The Associate Property of Multiplication works for multiple addends as well as decimal addends. Once again, let’s test the property using simple whole numbers.
\begin{align*}(2 \times 3) \times 4 = 24 && (2 \times 4) \times 3 = 24 && (3 \times 4) \times 2 = 24\end{align*}
That is a great question. We have learned how to simplify expressions and how to solve equations. These two properties are extremely useful for simplifying expressions and solving equations that also contain decimals.
Recall that an algebraic expression is a mathematical phrase involving letters, numbers, and operation symbols. An equation is a mathematical statement that two expressions are equal.
Let’s apply these properties to a few examples to see how this works.
Example
Simplify \begin{align*}29.3(12.4x)\end{align*}
When you simplify an expression, you perform all the operations you can—without knowing the value of the variable. Remember, the Commutative Property of Multiplication states that it doesn’t matter what order we multiply the factors and the Associative Property of Multiplication states that the way in which the factors are grouped doesn’t change the product. To simplify this expression then, we want to multiply all the decimals we can and put the variable beside the product.
If we simplify this expression, we multiply the number parts (29.3 x 12.4) and add the variable to the end.
Our answer is \begin{align*}363.32x\end{align*}.
Notice that we are simplifying here not solving!!
Example
Solve for the value of \begin{align*}x\end{align*} in the following equation, \begin{align*}(0.3x) \times 0.4 = 0.144\end{align*}
The first thing to notice in this problem, because of the equals sign, we are going to be solving this equation, not simplifying. Next, we apply the properties. The Associative Property of Multiplication tells us that the parentheses may be ignored, because it is a multiplication problem. The Commutative Property of Multiplication tells us that it doesn’t matter which order we multiply. So let’s start by multiplying \begin{align*}0.3 \times 0.4\end{align*}. As usual, we align the numbers to the right and ignore the decimal points until we have our answer.
\begin{align*}& \ \ \ 0.3\\ & \underline{\times \; 0.4\ }\\ & \ \ 0.12\end{align*}
Because each of the factors has one decimal place, our answer has two decimal places. We add the zero in the ones place to make it more obvious that we are working with a decimal.
Now that we know the product of the decimals on the left side of the equation, we can write a simpler equation.
\begin{align*}0.12x = 0.144\end{align*}.
We know that the equal sign indicates that both sides of the equation have the same value. So \begin{align*}x\end{align*} must be a factor that, when multiplied by 0.12 equals 0.144.
Let’s solve using mental math. What do you recognize about 0.12 and 0.144? \begin{align*}12 \times 12 = 144\end{align*}, so \begin{align*}x\end{align*} must be some version of 12. The product (0.144) has three decimal places, so \begin{align*}x\end{align*} must have one decimal place, because the other factor (0.12) has two decimal places.
\begin{align*}x\end{align*} must be 1.2. Let’s double-check.
\begin{align*}& \ \ \ 0.12\\ & \ \underline{\times \; 1.2 \ }\\ & \quad \ \ 24\\ & \underline{+ \; 120\ }\\ & \ 0.144\end{align*}
\begin{align*}x = 1.2\end{align*}
Our answer is 1.2.
2K. Lesson Exercises
Use the two properties as you simplify and solve.
- Solve \begin{align*}4.5x (3) = 27\end{align*}
- Simplify \begin{align*}3.45y (2.3)\end{align*}
Take a few minutes to check your answers with a neighbor.
IV. Solve Real-World Problems Using Formulas Involving Area of Rectangles and Decimal Dimensions
We would like to think that all measurement problems have nice whole number measurements. However, this is rarely the case. Because different dimensions are often not whole numbers, we must be able to calculate with decimals to determine measurements. Area is one such measurement that often has decimal dimensions. Also, we use multiplication to find the area of a figure, so figuring out the area of a figure will involve multiplying decimals.
What is area?
Area refers to the space inside a figure. Rectangles are one of the most common figures that we work with when it comes to area. Think about all of the things that are the shape of a rectangle, from a pool to a park to a garden plot. We use the following formula for determining the area of a rectangle.
\begin{align*}A=lw\end{align*}
This is where decimals come in. If the length or the width or both of a rectangle are decimal dimensions, then we will need to apply what we have learned about multiplying decimals to our work with area.
Example
Janet’s backyard patio has a length of 6.35 meters and a width of 4.4 meters. What is the area of Janet’s patio?
The problem asks us to find the area of Janet’s patio given the length and width. We can substitute the patio’s dimensions into the formula for area and solve.
\begin{align*}A & = lw\\ A &= 6.35(4.4)\end{align*}
Now we multiply the length and width.
\begin{align*}& \qquad 6.35\\ & \quad \ \underline{\times \; 4.4 \ }\\ & \quad \ \ \; 2540\\ & \underline{+ \ \; 25400 \ }\\ & \quad 27.940\end{align*}
\begin{align*}A = 27.94 \ m^2\end{align*}
Notice that the unit of measurement is listed in square units. Area is always measured in square units, so you will need to add this in whether you are working in feet, inches, miles, etc.
Take a few notes on area of rectangles before moving on to the introductory problem.
Real Life Example Completed
The Long Jump
Here is the original problem once again. Reread the problem and underline any important information.
Kevin is new at Franklin Marsh Middle School. In his old school, Kevin had worked very hard to become the best at the long jump event in Track and Field. In fact, Kevin held the state record in his previous state. Now Kevin has moved to a new state, and he is feeling nervous about starting over again.
Before the first big meet, Kevin went out to the field to check out the sand pit where he will land for the long jump. Mr. Rend, the groundskeeper, was outside working on raking the rectangular sand pit when Kevin arrived.
“Hello Kevin,” he said. “How are you?”
“I’m great,” Kevin said “But I am a little nervous. Is this pit the usual size?”
“As usual as it gets,” Mr. Rend said. “The dimensions are \begin{align*}4.5 \ m \times 4 \ m\end{align*}.”
Kevin stopped to think for a minute. The sand pit at his old school had an area of 20 sq. m. He started to think about whether or not these two pits were the same size. While watching Mr. Rend rake, Kevin started to do the math in his head.
To solve this problem, we need to figure out if the two sand pits are the same size. First, we can round the dimensions to make the multiplication easier. This will give us an estimate and not an exact answer.
4.5 rounds up to 5
4 stays the same
\begin{align*}5 \times 4 = 20 \ sq. meters\end{align*}
If we rounded, the pits would be the same size. Since the dimension is 4.5 and not 5, we know that the pit is smaller than the one at Kevin’s old school. We can multiply \begin{align*}4.5 \times 5\end{align*} and get the exact dimension of the pit.
\begin{align*}& \ \ 4.5\\ & \underline{\times \ \ 4 \ }\\ & \; 18.0\end{align*}
The area of the sand pit is 18 sq. meters.
Kevin thought about this for a minute. The difference wouldn’t be that large. Kevin would be fine during the competition!
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Product
- the answer in a multiplication problem.
- Estimation
- finding an approximate answer through rounding or multiplying leading digits
- Commutative Property of Multiplication
- states that the order in which you multiply the factors does not affect the product of those factors.
- Associative Property of Multiplication
- states that the groupings in which you multiply factors does not affect the product of those factors.
- Algebraic Expression
- a mathematical phrase made up of numbers, operations and variables
- Equation
- a mathematical statement where the value on one side of the equal sign is the same as the value on the other side of the equal sign.
- Simplify
- to make smaller
- Area
- the measurement inside a figure. Area is always measured in square units.
Technology Integration
James Sousa, Multiplying Decimals
James Sousa, Example of Multiplication of Decimals
James Sousa, Another Example of Multiplication of Decimals
Other Videos:
http://www.mathplayground.com/howto_multiplydecimals.html – This is a video on multiplying decimals.
Time to Practice
Directions: Find the products.
1. \begin{align*}12.7 \times 0.8\end{align*}
2. \begin{align*}0.552 \times 0.3\end{align*}
3. \begin{align*}6.09 \times 3.34\end{align*}
4. \begin{align*}25.6 \times 0.72\end{align*}
5. \begin{align*}56.71 \times .34\end{align*}
6. \begin{align*}.45 \times 4.3\end{align*}
7. \begin{align*}1.234 \times 7.8\end{align*}
Directions: Find the product after rounding each decimal to the nearest tenth.
8. \begin{align*}33.076 \times 5.228\end{align*}
9. \begin{align*}9.29 \times 0.6521\end{align*}
10. \begin{align*}4.5513 \times 4.874\end{align*}
11. \begin{align*}12.48 \times 7.95\end{align*}
12. \begin{align*}14.56 \times 4.52\end{align*}
13. \begin{align*}8.76 \times 1.24\end{align*}
14. \begin{align*}9.123 \times 6.789\end{align*}
Directions: Estimate the products by multiplying the leading digits.
15. \begin{align*}7.502 \times 0.9281\end{align*}
16. \begin{align*}46.14 \times 2.726\end{align*}
17. \begin{align*}0.39828 \times 0.16701\end{align*}
18. \begin{align*}83.243 \times 6.517\end{align*}
19. \begin{align*}5.67 \times .987\end{align*}
20. \begin{align*}7.342 \times 1.325\end{align*}
Directions: Simplify the following expressions.
21. \begin{align*}(4.21 \times 8.8) \times p\end{align*}
22. \begin{align*}16.14 \times q \times 6.2\end{align*}
23. \begin{align*}3.6(91.7x)\end{align*}
24. \begin{align*}5.3r(2.8y)\end{align*}
Directions: Evaluate or solve each problem as required.
25. Evaluate \begin{align*}(2.2m)7.1\end{align*} if \begin{align*}m = 3.3\end{align*}.
26. Evaluate \begin{align*}9.2 \times p \times 4.5\end{align*} if \begin{align*}p = 2.1\end{align*}.
27. Use mental math to solve for the value of \begin{align*}x\end{align*} in the following equation: \begin{align*}3.2x = 0.64\end{align*}.
28. What is the area of a sandbox with a length of 4.6 yards and a width of 3.21 yards?
29. What is the area of a placemat with a length of 27.2 inches and a width of 13.5 inches?
30. Mr. Duffourc’s flower bed has an area of \begin{align*}9.9 \ m^2\end{align*} and a width of 1.1 meters. What is the length of Mr. Duffourc’s flower bed?
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