5.5: Solve Equations with Decimals

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    Learning Objectives

    By the end of this section, you will be able to:

    • Determine whether a decimal is a solution of an equation
    • Solve equations with decimals
    • Translate to an equation and solve

    Be Prepared 5.10

    Before you get started, take this readiness quiz.

    Evaluate
    If you missed this problem, review Example 4.77.

    Be Prepared 5.11

    Evaluate when
    If you missed this problem, review Example 3.41.

    Be Prepared 5.12

    Solve
    If you missed this problem, review Example 4.99.

    Determine Whether a Decimal is a Solution of an Equation

    Solving equations with decimals is important in our everyday lives because money is usually written with decimals. When applications involve money, such as shopping for yourself, making your family’s budget, or planning for the future of your business, you’ll be solving equations with decimals.

    Now that we’ve worked with decimals, we are ready to find solutions to equations involving decimals. The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number, an integer, a fraction, or a decimal. We’ll list these steps here again for easy reference.

    How To

    Determine whether a number is a solution to an equation.

    1. Step 1. Substitute the number for the variable in the equation.
    2. Step 2. Simplify the expressions on both sides of the equation.
    3. Step 3. Determine whether the resulting equation is true.
      • If so, the number is a solution.
      • If not, the number is not a solution.

    Example 5.40

    Determine whether each of the following is a solution of

    ⓐⓑⓒ

    Answer

    ⓐ
    .
    . .
    Subtract. .

    Since does not result in a true equation, is not a solution to the equation.

    ⓑ
    .
    . .
    Subtract. .

    Since does not result in a true equation, is not a solution to the equation.

    ⓒ
    .
    . .
    Subtract. .

    Since results in a true equation, is a solution to the equation.

    Try It 5.79

    Determine whether each value is a solution of the given equation.

    ⓐⓑⓒ

    Try It 5.80

    Determine whether each value is a solution of the given equation.

    ⓐⓑⓒ

    Solve Equations with Decimals

    In previous chapters, we solved equations using the Properties of Equality. We will use these same properties to solve equations with decimals.

    Properties of Equality

    Subtraction Property of Equality
    For any numbers
    If then     		Addition Property of Equality
    For any numbers
    If then
    The Division Property of Equality
    For any numbers
    If then     		The Multiplication Property of Equality
    For any numbers
    If then

    When you add, subtract, multiply or divide the same quantity from both sides of an equation, you still have equality.

    Example 5.41

    Solve:

    Answer

    We will use the Subtraction Property of Equality to isolate the variable.

      .
    . .
    Simplify. .
    Check: .
    . .
    Simplify. .

    Since makes a true statement, we know we have found a solution to this equation.

    Try It 5.81

    Solve:

    Try It 5.82

    Solve:

    Example 5.42

    Solve:

    Answer

    We will use the Addition Property of Equality.

      .
    Add 4.75 to each side, to undo the subtraction. .
    Simplify. .
    Check: .
    . .
    .

    Since the result is a true statement, is a solution to the equation.

    Try It 5.83

    Solve:

    Try It 5.84

    Solve:

    Example 5.43

    Solve:

    Answer

    We will use the Division Property of Equality.

    Use the Properties of Equality to find a value for

      .
    We must divide both sides by 0.8 to isolate n. .
    Simplify. .
    Check: .
    . .
    .

    Since makes a true statement, we know we have a solution.

    Try It 5.85

    Solve:

    Try It 5.86

    Solve:

    Example 5.44

    Solve:

    Answer

    We will use the Multiplication Property of Equality.

      .
    Here, p is divided by −1.8. We must multiply by −1.8 to isolate p .
    Multiply. .
    Check: .
    . .
    .

    A solution to is

    Try It 5.87

    Solve:

    Try It 5.88

    Solve:

    Translate to an Equation and Solve

    Now that we have solved equations with decimals, we are ready to translate word sentences to equations and solve. Remember to look for words and phrases that indicate the operations to use.

    Example 5.45

    Translate and solve: The difference of and is

    Answer

    Translate. .
    Add to both sides of the equation. .
    Simplify. .
    Check: Is the difference of and 4.3 equal to 2.1?
    Let : Is the difference of 6.4 and 4.3 equal to 2.1?
    Translate. .
    Simplify. .

    Try It 5.89

    Translate and solve: The difference of and is

    Try It 5.90

    Translate and solve: The difference of and is

    Example 5.46

    Translate and solve: The product of and is

    Answer

    Translate. .
    Divide both sides by . .
    Simplify. .
    Check: Is the product of −3.1 and equal to ?
    Let : Is the product of and equal to ?
    Translate. .
    Simplify. .

    Try It 5.91

    Translate and solve: The product of and is

    Try It 5.92

    Translate and solve: The product of and is

    Example 5.47

    Translate and solve: The quotient of and is

    Answer

    Translate. .
    Multiply both sides by . .
    Simplify. .
    Check: Is the quotient of and equal to ?
    Let Is the quotient of and equal to ?
    Translate. .
    Simplify. .

    Try It 5.93

    Translate and solve: The quotient of and is

    Try It 5.94

    Translate and solve: The quotient of and is

    Example 5.48

    Translate and solve: The sum of and is

    Answer

    Translate. .
    Subtract from each side. .
    Simplify. .
    Check: Is the sum and equal to ?
    Let Is the sum and equal to ?
    Translate. .
    Simplify. .

    Try It 5.95

    Translate and solve: The sum of and is

    Try It 5.96

    Translate and solve: The sum of and is

    Media

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    Section 5.4 Exercises

    Practice Makes Perfect

    Determine Whether a Decimal is a Solution of an Equation

    In the following exercises, determine whether each number is a solution of the given equation.

    285.


    ⓐ ⓑ ⓒ

    286.


    ⓐⓑⓒ

    287.


    ⓐⓑⓒ

    288.


    ⓐⓑⓒ

    Solve Equations with Decimals

    In the following exercises, solve the equation.

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    Mixed Practice

    In the following exercises, solve the equation. Then check your solution.

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    Translate to an Equation and Solve

    In the following exercises, translate and solve.

    345.

    The difference of and is

    346.

    The difference and is

    347.

    The product of and is

    348.

    The product of and is

    349.

    The quotient of and is

    350.

    The quotient of and is

    351.

    The sum of and is

    352.

    The sum of and is

    Everyday Math

    353.

    Shawn bought a pair of shoes on sale for . Solve the equation to find the original price of the shoes,

    354.

    Mary bought a new refrigerator. The total price including sales tax was Find the retail price, of the refrigerator before tax by solving the equation

    Writing Exercises

    355.

    Think about solving the equation but do not actually solve it. Do you think the solution should be greater than or less than Explain your reasoning. Then solve the equation to see if your thinking was correct.

    356.

    Think about solving the equation but do not actually solve it. Do you think the solution should be greater than or less than Explain your reasoning. Then solve the equation to see if your thinking was correct.

    Self Check

    ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    .

    ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

    This page titled 5.5: Solve Equations with Decimals was last modified on Sun, 28 May 2023 07:42:58 GMT and is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax.