# 2.15: Division to Solve Decimal Equations

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

**Practice**Division to Solve Decimal Equations

Peter has decided to open up a special savings account at the bank. He wants to earn interest on the money he has saved. He has $178 saved up that he is planning to deposit into his new account. He learns that his account will earn 3.2% simple interest each year. The amount of interest, \begin{align*}I\end{align*}, he will earn in \begin{align*}t\end{align*} years is given by the following formula:

\begin{align*}I=5.696t\end{align*}

Peter's goal is to get his account to $200. How can Peter determine approximately how many years that will take?

In this concept, you will learn how to apply the inverse property of multiplication to solve decimal equations.

### Using Division to Solve Decimal Equations

Recall that **inverse operations** are operations that reverse one another. Addition and subtraction are inverse operations. Multiplication and division are also inverse operations. For example, if you take any number and multiply it by 3 and then divide the result by 3, you will be back to the original number. The division reversed the multiplication.

The **Inverse Property of Multiplication** states that the product of any non-zero number and its reciprocal is one. In symbols, it says that for any non-zero number \begin{align*}a\end{align*}:

\begin{align*}a \cdot \frac{1}{a} = 1\end{align*}

The **multiplicative inverse** of a number is another word for the reciprocal of a number. The multiplicative inverse of \begin{align*}a\end{align*} is \begin{align*}\frac{1}{a}\end{align*} as long as \begin{align*}a\end{align*} is not equal to zero.

You can use the inverse property of multiplication to help you to solve equations that would be difficult to solve using mental math. Remember that when you are solving an equation your goal is to **isolate** the variable, which means you want to get the variable by itself on one side of the equation.

To use the inverse property of multiplication, if your variable is being multiplied by a number, you can divide both sides by that number to isolate the variable.

Here is an example.

Solve the following equation for \begin{align*}x\end{align*}.

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

First, notice that \begin{align*}x\end{align*} is being multiplied by 2.7 on one side of the equation. Divide both sides by 2.7 to isolate \begin{align*}x\end{align*}. This is the same as multiplying both sides by \begin{align*}\frac{1}{2.7}\end{align*} .

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

Next, simplify the left side of the equation. \begin{align*}\frac{2.7}{2.7}\end{align*} is equal to 1, which leaves you with \begin{align*}1x\end{align*} or \begin{align*}x\end{align*} on the left side of the equation.

\begin{align*}\begin{array}{rcl} 1x &=& \frac{3.78}{2.7} \\ x &=& \frac{3.78}{2.7} \end{array}\end{align*}

Now, simplify the right side of the equation. Use what you have learned about decimal division. Move the decimal points 1 place to the right to change 2.7 into 27 and 3.78 into 37.8. Then, divide using long division.

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

\begin{align*}x=1.4\end{align*}

The answer is \begin{align*}x=1.4\end{align*}.

You can check the solution by substituting that value for \begin{align*}x\end{align*} back into the original equation and verifying that it makes both sides equal. Use what you have learned about decimal multiplication.

\begin{align*}\begin{array}{rcl} 2.7x &=& 3.78 \\ 2.7(1.4) &=& 3.78 \\ 3.78 &=&3.78 \end{array}\end{align*}

The answer is correct.

### Examples

#### Example 1

Earlier, you were given a problem about Peter and his new savings account.

He is depositing $178 into the account. The amount of interest, \begin{align*}I\end{align*}, he will earn in \begin{align*}t\end{align*} years is given by the formula \begin{align*}I=5.696t\end{align*}. Peter wants to know how many years it will take before he has $200 in his account.

First, Peter needs to figure out how much interest he would need to earn in order to have $200 total. If he starts with $178, he would need \begin{align*}$200-$178=$22\end{align*} in interest in order to have $200 total.

Now, Peter should substitute $22 for \begin{align*}I\end{align*} in the formula.

\begin{align*}22=5.696t\end{align*}

Next, Peter should solve the equation for \begin{align*}t\end{align*}. Notice that \begin{align*}t\end{align*} is being multiplied by 5.696 on one side of the equation. He needs to divide both sides by 5.696 to isolate \begin{align*}t\end{align*}.

\begin{align*}\frac{22}{5.696} = \frac{5.696t}{5.696}\end{align*}

Now, Peter should simplify the right side of the equation. \begin{align*}\frac{5.696}{5.696}\end{align*} is equal to 1, which leaves him with \begin{align*}t\end{align*} on the right side of the equation.

\begin{align*}\frac{22}{5.696}=t\end{align*}

Next, Peter should simplify the left side of the equation. He should use what he has learned about decimal division. Move the decimal points 3 places to the right to change 5.696 into 5696 and 22 into 22000. Then, he can divide using long division.

\begin{align*}\begin{array}{rcl}
& \ \ \overset{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3.8...}{5696 \overline{ ) {22000.00 \;}}}\\
& \ \ \quad \underline{-17088\;\;\;\;\;}\\
& \qquad 49120 \\
& \qquad \underline{-45568\;\;}\\
& \qquad 35520
\end{array}\end{align*}

Because Peter only wanted to know approximately how many years it will take, he can stop his division here. \begin{align*}t\end{align*} is approximately equal to 3.8, which means it will take about 4 years.

The answer is it will take approximately 4 years for Peter to have $200 in his account.

#### Example 2

Solve the following equation for \begin{align*}x\end{align*}.

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

First, notice that \begin{align*}x\end{align*} is being multiplied by 4.5 on one side of the equation. Divide both sides by 4.5 to isolate \begin{align*}x\end{align*}.

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

Next, simplify the left side of the equation. \begin{align*}\frac{4.5}{4.5}\end{align*} is equal to 1, which leaves you with \begin{align*}1x\end{align*} or \begin{align*}x\end{align*} on the left side of the equation.

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

Now, simplify the right side of the equation. Use what you have learned about decimal division. Move the decimal points 1 place to the right to change 4.5 into 45 and 12.6 into 126. Then, divide using long division.

\begin{align*}\begin{array}{rcl} & \ \ \overset{ \ \ \ \ \ \ \ \ \ \ \ \ 2.8}{45 \overline{ ) {126.0 \;}}}\\ & \ \ \quad \underline{-90\;\;\;\;\;}\\ & \qquad 360 \\ & \qquad \underline{-360\;\;}\\ & \qquad 0 \end{array}\end{align*}

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

The answer is \begin{align*}x=2.8\end{align*}.

Now, check the answer.

\begin{align*}\begin{array}{rcl} 4.5x &=& 12.6 \\ 4.5(2.8) &=& 12.6 \\ 12.6 &=& 12.6 \end{array}\end{align*}

The answer is correct.

#### Example 3

Solve the following equation for \begin{align*}x\end{align*}.

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

First, notice that \begin{align*}x\end{align*} is being multiplied by 2.3 on one side of the equation. Divide both sides by 2.3 to isolate \begin{align*}x\end{align*}.

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

Next, simplify the left side of the equation. \begin{align*}\frac{2.3}{2.3}\end{align*} is equal to 1, which leaves you with \begin{align*}1x\end{align*} or \begin{align*}x\end{align*} on the left side of the equation.

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

Now, simplify the right side of the equation. Use what you have learned about decimal division. Move the decimal points 1 place to the right to change 2.3 into 23 and 5.06 into 50.6. Then, divide using long division.

\begin{align*}\begin{array}{rcl} & \ \ \overset{ \ \ \ \ \ \ \ \ \ 2.2}{23 \overline{ ) {50.6 \;}}}\\ & \ \ \quad \underline{-46\;\;\;\;\;}\\ & \qquad 46 \\ & \qquad \underline{-46\;\;}\\ & \qquad 0 \end{array}\end{align*}

\begin{align*}x=2.2\end{align*}

The answer is \begin{align*}x=2.2\end{align*}.

Now, check the answer.

\begin{align*}\begin{array}{rcl} 2.3x&=&5.06 \\ 2.3(2.2)&=&5.06 \\ 5.06&=&5.06 \end{array}\end{align*}

The answer is correct.

#### Example 4

Solve the following equation for \begin{align*}x\end{align*}.

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

First, notice that \begin{align*}x\end{align*} is being multiplied by 1.6 on one side of the equation. Divide both sides by 1.6 to isolate \begin{align*}x\end{align*}.

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

Next, simplify the left side of the equation. \begin{align*}\frac{1.6}{1.6}\end{align*} is equal to 1, which leaves you with \begin{align*}1x\end{align*} or \begin{align*}x\end{align*} on the left side of the equation.

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

Now, simplify the right side of the equation. Use what you have learned about decimal division. Move the decimal points 1 place to the right to change 1.6 into 16 and 5.76 into 57.6. Then, divide using long division.

\begin{align*}\begin{array}{rcl} & \ \ \overset{ \ \ \ \ \ \ \ \ \ \ \ 3.6}{16 \overline{ ) {57.6 \;}}}\\ & \ \ \quad \underline{-48\;\;\;\;\;}\\ & \qquad 96 \\ & \qquad \underline{-96\;\;}\\ & \qquad 0 \end{array}\end{align*}

\begin{align*}x=3.6\end{align*}

The answer is \begin{align*}x=3.6\end{align*}.

Now, check the answer.

\begin{align*}\begin{array}{rcl} 1.6x&=&5.76 \\ 1.6(3.6)&=&5.76 \\ 5.76&=&5.76 \end{array}\end{align*}

The answer is correct.

#### Example 5

Solve the following equation for \begin{align*}x\end{align*}.

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

First, notice that \begin{align*}x\end{align*} is being multiplied by 4.7 on one side of the equation. Divide both sides by 4.7 to isolate \begin{align*}x\end{align*}.

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

Next, simplify the left side of the equation. \begin{align*}\frac{4.7}{4.7}\end{align*} is equal to 1, which leaves you with \begin{align*}1x\end{align*} or \begin{align*}x\end{align*} on the left side of the equation.

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

Now, simplify the right side of the equation. Use what you have learned about decimal division. Move the decimal points 1 place to the right to change 4.7 into 47 and 10.81 into 108.1. Then, divide using long division.

\begin{align*}\begin{array}{rcl} & \ \ \overset{ \ \ \ \ \ \ \ \ \ \ \ 2.3}{47 \overline{ ) {108.1 \;}}}\\ & \ \ \quad \underline{-94\;\;\;\;\;}\\ & \qquad 141 \\ & \qquad \underline{-141\;\;}\\ & \qquad 0 \end{array}\end{align*}

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

The answer is \begin{align*}x=2.3\end{align*}.

Now, check the answer.

\begin{align*}\begin{array}{rcl} 4.7x&=&10.81 \\ 4.7(2.3)&=&10.81 \\ 10.81&=&10.81 \end{array}\end{align*}

The answer is correct.

### Review

Solve the following equations for \begin{align*}x\end{align*}.

- \begin{align*}3.7x = 7.77\end{align*}
- \begin{align*}3.1x = 10.23\end{align*}
- \begin{align*}7.2x = 29.52\end{align*}
- \begin{align*}2.7x = 11.34\end{align*}
- \begin{align*}1.2x = 6.72\end{align*}
- \begin{align*}11x = 27.5\end{align*}
- \begin{align*}6.7x = 42.21\end{align*}
- \begin{align*}8.2x = 51.66\end{align*}
- \begin{align*}1.9x = 12.92\end{align*}
- \begin{align*}5.7x = 54.72\end{align*}
- \begin{align*}0.55x = .31955\end{align*}
- \begin{align*}9.8x = 114.66\end{align*}
- 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.
- 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.
- 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.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 2.15.

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Dividend

In a division problem, the dividend is the number or expression that is being divided.divisor

In a division problem, the divisor is the number or expression that is being divided into the dividend. For example: In the expression , 6 is the divisor and 152 is the dividend.Estimation

Estimation is the process of finding an approximate answer to a problem.Inverse Operation

Inverse operations are operations that "undo" each other. Multiplication is the inverse operation of division. Addition is the inverse operation of subtraction.Inverse Property of Multiplication

The inverse property of multiplication states that the product of any real number and its multiplicative inverse (reciprocal) is one. If is a nonzero real number, then .Quotient

The quotient is the result after two amounts have been divided.### Image Attributions

In this concept, you will learn how to apply the inverse property of multiplication to solve decimal equations.

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