# 2.15: Division to Solve Decimal Equations

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

**Practice**Division to Solve Decimal Equations

Have you ever tried to pole vault?

The pole vault is a track and field event where a student uses a pole to launch themselves over a bar. Then the student lands on a large mat underneath the pole. Each part of the pole vault event is very specific. The height of the bar is specific. The length of the pole is specific, and the dimensions of the mat are specific as well.

The track and field team at Harrison Middle school had a special visitor after practice. Jody a pole vaulter from the nearby college visited to share his experiences with the students. He brought some pictures of himself in different events and took a long time answering student questions.

“Even the mat has specific dimensions,” Jody explained. “They measure the length, width and height of the mat to be sure that it has an accurate volume. The mat that we are using has a volume of 9009 cubic feet. The length of the mat is 16.5 feet and the width is 21 feet.”

Justin and Kara were listening intently to Jody’s explanation of the pole vault event. Justin, who loves numbers, began jotting down the dimensions of the mat on a piece of paper.

9009 cubic feet

16.5 feet in length

21 feet in width

Justin knows that he is missing the height of the mat.

“How high is the mat?” Justin asks Kara showing her his notes.

“Who cares?” Kara whispered looking back at Jody.

“I do,” Justin said turning away.

Justin begins to complete the math. But he can’t remember how to work the equation and the division.

**This is where you come in. Pay attention to this Concept. By the end of it, you will need to help Justin with his dilemma.**

### Guidance

Sometimes division is called the ** inverse**, or opposite, of multiplication. This means that division will “undo” multiplication.

\begin{align*}5 \times 6 &= 30\\
30 \div 6 &= 5\end{align*}

See how that works? You can multiply two factors to get a product. Then when you divide the product by one factor, you get the other factor.

*The Inverse Property of Multiplication***states that for every number \begin{align*}x, x\left (\frac{1}{x} \right )= 1\end{align*}.**

In other words, when you multiply \begin{align*}x\end{align*} by its opposite, \begin{align*}\frac{1}{x}\end{align*} (also known as the ** multiplicative inverse**), the result is one.

**Let’s put whole numbers in place of \begin{align*}x\end{align*} to make the property clear: \begin{align*}3 \times \left ( \frac{1}{3} \right ) = \frac{3}{3} = 1\end{align*}.** The Inverse Property of Multiplication may seem obvious, but it has important implications for our ability to solve variable equations which aren’t easily solved using mental math—such as variable equations involving decimals.

**How does this apply to our work with equations?**

**With equations, the two expressions on either side of the equal sign** *must***be equal at all times. The Inverse Property of Multiplication lets us multiply or divide the same number to both sides of the equation without changing the solution to the equation.** This technique is called an ** inverse operation** and it lets us get the variable \begin{align*}x\end{align*} alone on one side of the equation so that we can find its value. Take a look at how it’s done.

\begin{align*}5x & = 15\\ \frac{5x}{5} &= \frac{15}{5} \rightarrow \text{inverse operation } = \text{divide} \ 5 \ \text{from both sides}\\ x &= 3\end{align*}

Remember how we said that division can “undo” multiplication? Well, this is a situation where that has happened.

Notice how, on the left side of the equation, \begin{align*}\frac{5}{5} = 1\end{align*}, leaving the \begin{align*}x\end{align*} alone on the left side. This is very instinctual to us. Once you have been working with equations, it comes naturally that when you divide a number by itself the answer is 1 and that has the variable be by itself on that side of the equals. After a while, we don’t even think about it, but it is still important to point out.

Solve \begin{align*}2.7x = 3.78\end{align*}

We need to find a value of \begin{align*}x\end{align*} that, when multiplied by 2.7, results in 3.78. **Let’s begin by using inverse operations to get \begin{align*}x\end{align*} alone on the left side of the equation.**

\begin{align*}2.7x & = 3.78\\ \frac{2.7}{2.7} x&= \frac{3.78}{2.7} \rightarrow \text{Inverse Operations} = \text{divide both sides by} \ 2.7\end{align*}

**Now, to find the value of \begin{align*}x\end{align*}, we complete the decimal division.** First we multiply by ten and move the decimal places accordingly. \begin{align*}2.7 \rightarrow 27\end{align*} and \begin{align*}3.78 \rightarrow 37.8\end{align*}

\begin{align*}& \overset{ \quad \ \ 1.4}{27 \overline{ ) {37.8 \;}}}\\ & \quad \ \underline{27\;\;}\\ & \qquad 108\\ & \quad \underline{- \; 108}\\ & \qquad \quad 0\end{align*}

**Our answer is 1.4.**

Use the inverse operation to solve each problem.

#### Example A

\begin{align*}2.3x = 5.06\end{align*}

**Solution: \begin{align*}2.2\end{align*}**

#### Example B

\begin{align*}1.6x= 5.76\end{align*}

**Solution: \begin{align*}3.6\end{align*}**

#### Example C

\begin{align*}4.7x = 10.81\end{align*}

**Solution: \begin{align*}2.3\end{align*}**

Here is the original problem once again.

The pole vault is a track and field event where a student uses a pole to launch themselves over a bar. Then the student lands on a large mat underneath the pole. Each part of the pole vault event is very specific. The height of the bar is specific. The length of the pole is specific, and the dimensions of the mat are specific as well.

The track and field team at Harrison Middle school had a special visitor after practice. Jody a pole vaulter from the nearby college visited to share his experiences with the students. He brought some pictures of himself in different events and took a long time answering student questions.

“Even the mat has specific dimensions,” Jody explained. “They measure the length, width and height of the mat to be sure that it has an accurate volume. The mat that we are using has a volume of 9009 cubic feet. The length of the mat is 16.5 feet and the width is 21 feet.”

Justin and Kara were listening intently to Jody’s explanation of the pole vault event. Justin, who loves numbers, began jotting down the dimensions of the mat on a piece of paper.

9009 cubic feet

16.5 feet in length

21 feet in width

Justin knows that he is missing the height of the mat.

“How high is the mat?” Justin asks Kara showing her his notes.

“Who cares?” Kara whispered looking back at Jody.

“I do, some things are worth figuring out,” Justin said turning away.

Justin begins to complete the math. But he can’t remember how to work the equation and the division.

**To solve this problem of height, we need to remember that the formula for volume is length times width times height. Justin has the measurements for the volume and for the length and the width. He is missing the height. Justin can write the following formula.**

\begin{align*}V &= lwh\\ 9009 & = 16.5 \times 21 \times h\end{align*}

**Next, we multiply \begin{align*}16.5 \times 21\end{align*} to begin our equation.**

\begin{align*}16.5 \times 21 = 346.5\end{align*}

**We can write this equation.**

\begin{align*}9009 = 346.5h\end{align*}

**Now we divide 9009 by 346.5.**

\begin{align*}h = 26 \ feet\end{align*}

**The height of the mat is equal to 26 feet.**

### Vocabulary

Here are the vocabulary words in this Concept.

- Divisor
- the number outside the division box. This is the number that is doing the dividing.

- Dividend
- the number being divided. It is the number inside the division box.

- Quotient
- the answer in a division problem.

- Estimation
- using rounding or dividing leading digits to find an approximate answer.

- Inverse
- the opposite

- Inverse Property of Multiplication
- when you multiply a value by its opposite, the answer is one.

- Inverse Operation
- the opposite operation. The opposite operation to division is multiplication.

### Guided Practice

Here is one for you to try on your own.

Divide.

\begin{align*}4.5x = 12.6\end{align*}

**Answer**

To solve this problem, we simply divide 12.6 by 4.5.

\begin{align*}2.8\end{align*}

**This is our answer.**

### Video Review

Here is a video for review.

- This is a video on the inverse property of multiplication.

### Practice

Directions: Solve the following problems using what you have learned about dividing decimals and equations. Write an equation when necessary.

1. Solve \begin{align*}3.7x = 7.77\end{align*}

2. Solve \begin{align*}3.1x = 10.23\end{align*}

3. Solve \begin{align*}7.2x = 29.52\end{align*}

4. Solve \begin{align*}2.7x = 11.34\end{align*}

5. Solve \begin{align*}1.2x = 6.72\end{align*}

6. Solve \begin{align*}11x = 27.5\end{align*}

7. Solve \begin{align*}6.7x = 42.21\end{align*}

8. Solve \begin{align*}8.2x = 51.66\end{align*}

9. Solve \begin{align*}1.9x = 12.92\end{align*}

10. Solve \begin{align*}5.7x = 54.72\end{align*}

11. Solve \begin{align*}.55x = .31955\end{align*}

12. Solve \begin{align*}9.8x = 114.66\end{align*}

13. In a week of track practice, Rose ran 3.12 times more than Jamie. If Rose ran 17.16 kilometers, how many kilometers did Jamie run? Write an equation and solve.

14. Ling’s flower bed has an area of \begin{align*}23.12 \ m^2\end{align*} and a width of 3.4 meters. What is the length of Ling’s flower bed? Write an equation and solve.

15. A jet airplane travels 6.5 times faster than a car. If the jet travels at 627.51 kilometers per hour, how fast is the car? Write an equation and solve.

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### Image Attributions

Here you'll learn to identify and apply the inverse property of multiplication in decimal operations with numerical and variable expressions.