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# 2.12: Commutative Property of Multiplication with Decimals

Difficulty Level: At Grade Created by: CK-12
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Practice Commutative Property of Multiplication with Decimals

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Have you ever been on a long car trip?

One day during their visit, Marc and Kara went on a ride out to the Berkshires with their grandparents. The car that they drove in gets 25 miles per gallon and they traveled 160.5 miles. They traveled the same distance on the way back.

How many gallons of gas were needed for the trip?

Do you know how to write this as an expression?

Can you solve it?

This Concept is about using the commutative property of multiplication in decimal operations. You will be able to answer these questions at the end of the Concept.

### Guidance

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 also works for four, five, six factors. It works for decimal factors, too. Let’s test the property using simple whole numbers.

2×3=63×2=62×3×4=242×4×3=243×2×4=243×4×2=244×3×2=244×2×3=24\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 change no matter 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. It is one that we can answer by thinking about expressions and equations. We have learned how to simplify expressions and how to solve equations. These two properties are extremely useful for simplifying expressions and solving equations.

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 this.

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 these expressions 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 and add the variable to the end.

Our answer is \begin{align*}363.32x\end{align*}.

Notice that we are simplifying here not solving!!

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 is that we are solving this equation, not simplifying. Next, we apply the properties. The Associate Property of Multiplication tells us that we can ignore the parentheses. 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 the decimal places clearer.

Now that we know the product of two 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*}

Use the two properties as you simplify and solve for the missing variable.

#### Example A

Solve \begin{align*}4.5x (3) = 27\end{align*}

Solution:\begin{align*}x = 2\end{align*}

#### Example B

Simplify \begin{align*}3.45y\end{align*} when \begin{align*}y = 2.3\end{align*}

Solution:\begin{align*}7.935\end{align*}

#### Example C

Solve \begin{align*}3.4x = 23.8\end{align*}

Solution:\begin{align*}x = 7\end{align*}

Here is the original problem once again.

One day during their visit, Marc and Kara went on a ride out to the Berkshires with their grandparents. The car that they drove in gets 25 miles per gallon and they traveled 160.5 miles. They traveled the same distance on the way back.

How many gallons of gas were needed for the trip?

Do you know how to write this as an expression?

Can you solve it?

First, let's write an expression to describe the situation.

We know the number of miles per gallon and the number of gallons is our unknown.

\begin{align*}25g\end{align*}

Then we know how many miles were traveled. This helps change our expression to an equation.

\begin{align*}25g = 160.5\end{align*}

We can solve it by using the inverse property of multiplication. We divide 160.5 by 25. This will tell us how many gallons were needed for the trip.

\begin{align*}6.42\end{align*}

It was a little less than 6 and a half gallons of gasoline for the trip.

### Vocabulary

Here are the vocabulary words in this Concept.

Product
the answer in a multiplication problem.
Estimation
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.

### Guided Practice

Here is one for you to try on your own.

Simplify the following expression.

\begin{align*}4.5(9.2)y\end{align*}

To simplify this expression, we can multiply the two decimals first.

\begin{align*}4.5(9.2) = 41.4\end{align*}

Our final answer is \begin{align*}41.4y\end{align*}.

### Video Review

Here are videos for review.

### Practice

Directions: Simplify the following expressions.

1. \begin{align*}(4.21 \times 8.8) \times p\end{align*}

2. \begin{align*}16.14 \times q \times 6.2\end{align*}

3. \begin{align*}3.6(91.7x)\end{align*}

4. \begin{align*}5.3r(2.8)\end{align*}

5. \begin{align*}5.6x(3.8)\end{align*}

6. \begin{align*}2.4y(2.8)\end{align*}

7. \begin{align*}6.7x(3.1)\end{align*}

8. \begin{align*}8.91r(2.3)\end{align*}

9. \begin{align*}5.67y(2.8)\end{align*}

10. \begin{align*}4.53x(2.2)\end{align*}

11. \begin{align*}5.6(2.8x)\end{align*}

12. \begin{align*}9.2y(3.2)\end{align*}

13. \begin{align*}4.5x(2.3)\end{align*}

14. \begin{align*}15.4x(12.8)\end{align*}

15. \begin{align*}18.3y(14.2)\end{align*}

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