5.4: Decimals and Fractions

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    Learning Objectives

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

    • Convert fractions to decimals
    • Order decimals and fractions
    • Simplify expressions using the order of operations
    • Find the circumference and area of circles

    Be Prepared 5.7

    Before you get started, take this readiness quiz.

    Divide:
    If you missed this problem, review Example 5.19.

    Be Prepared 5.8

    Order using or
    If you missed this problem, review Example 5.7.

    Be Prepared 5.9

    Order using or
    If you missed this problem, review Example 5.8.

    Convert Fractions to Decimals

    In Decimals, we learned to convert decimals to fractions. Now we will do the reverse—convert fractions to decimals. Remember that the fraction bar indicates division. So can be written or This means that we can convert a fraction to a decimal by treating it as a division problem.

    Convert a Fraction to a Decimal

    To convert a fraction to a decimal, divide the numerator of the fraction by the denominator of the fraction.

    Example 5.28

    Write the fraction as a decimal.

    Answer

    A fraction bar means division, so we can write the fraction using division. A division problem is shown. 3 is on the inside of the division sign and 4 is on the outside.
    Divide. A division problem is shown. 3.00 is on the inside of the division sign and 4 is on the outside. Below the 3.00 is a 28 with a line below it. Below the line is a 20. Below the 20 is another 20 with a line below it. Below the line is a 0. Above the division sign is 0.75.
    So the fraction is equal to

    Try It 5.55

    Write the fraction as a decimal:

    Try It 5.56

    Write the fraction as a decimal:

    Example 5.29

    Write the fraction as a decimal.

    Answer

    The value of this fraction is negative. After dividing, the value of the decimal will be negative. We do the division ignoring the sign, and then write the negative sign in the answer.
    Divide by A division problem is shown. 7.0 is on the inside of the division sign and 2 is on the outside. Below the 7 is a 6 with a line below it. Below the line is a 10. Below the 10 is another 10 with a line below it. Below the line is a 0. 3.5 is written above the division sign.
    So,

    Try It 5.57

    Write the fraction as a decimal:

    Try It 5.58

    Write the fraction as a decimal:

    Repeating Decimals

    So far, in all the examples converting fractions to decimals the division resulted in a remainder of zero. This is not always the case. Let’s see what happens when we convert the fraction to a decimal. First, notice that is an improper fraction. Its value is greater than The equivalent decimal will also be greater than

    We divide by

    A division problem is shown. 4.000 is on the inside of the division sign and 3 is on the outside. Below the 4 is a 3 with a line below it. Below the line is a 10. Below the 10 is a 9 with a line below it. Below the line is another 10, followed by another 9 with a line, followed by another 10, followed by another 9 with a line, followed by a 1. Above the division sign is 1.333...

    No matter how many more zeros we write, there will always be a remainder of and the threes in the quotient will go on forever. The number is called a repeating decimal. Remember that the “…” means that the pattern repeats.

    Repeating Decimal

    A repeating decimal is a decimal in which the last digit or group of digits repeats endlessly.

    How do you know how many ‘repeats’ to write? Instead of writing we use a shorthand notation by placing a line over the digits that repeat. The repeating decimal is written The line above the tells you that the repeats endlessly. So

    For other decimals, two or more digits might repeat. Table 5.5 shows some more examples of repeating decimals.

    is the repeating digit
    is the repeating digit
    is the repeating block
    is the repeating block
    Table 5.5

    Example 5.30

    Write as a decimal.

    Answer

    Divide by

    A division problem is shown. 43.00000 is on the inside of the division sign and 22 is on the outside. Below the 43 is a 22 with a line below it. Below the line is a 210 with a 198 with a line below it. Below the line is a 120 with 110 and a line below it. Below the line is 100 with 88 and a line below it. Below the line is 120 with 110 and a line below it. Below the line is 100 with 88 and a line below it. Below the line is an ellipses. There are arrows pointing to the 120s saying 120 repeats. There are arrows pointing to the 100s saying 100 repeats. There are arrows pointing to the 88s saying, in red, “The pattern repeats, so the numbers in the quotient will repeat as well.” The quotient is shown above the division sign. It is 1.95454.

    Notice that the differences of and repeat, so there is a repeat in the digits of the quotient; will repeat endlessly. The first decimal place in the quotient, is not part of the pattern. So,

    Try It 5.59

    Write as a decimal:

    Try It 5.60

    Write as a decimal:

    It is useful to convert between fractions and decimals when we need to add or subtract numbers in different forms. To add a fraction and a decimal, for example, we would need to either convert the fraction to a decimal or the decimal to a fraction.

    Example 5.31

    Simplify:

    Answer

    Change to a decimal. .
    Add.

    Try It 5.61

    Simplify:

    Try It 5.62

    Simplify:

    Order Decimals and Fractions

    In Decimals, we compared two decimals and determined which was larger. To compare a decimal to a fraction, we will first convert the fraction to a decimal and then compare the decimals.

    Example 5.32

    Order using or

    Answer

    Convert to a decimal.
    Compare to
    Rewrite with the original fraction.

    Try It 5.63

    Order each of the following pairs of numbers, using or

    Try It 5.64

    Order each of the following pairs of numbers, using or

    When ordering negative numbers, remember that larger numbers are to the right on the number line and any positive number is greater than any negative number.

    Example 5.33

    Order using or

    Answer

    Convert to a decimal.
    Compare to .
    Rewrite the inequality with the original fraction.

    Try It 5.65

    Order each of the following pairs of numbers, using or

    Try It 5.66

    Order each of the following pairs of numbers, using or

    Example 5.34

    Write the numbers in order from smallest to largest.

    Answer

    Convert the fractions to decimals.
    Write the smallest decimal number first.
    Write the next larger decimal number in the middle place.
    Write the last decimal number (the larger) in the third place.
    Rewrite the list with the original fractions.

    Try It 5.67

    Write each set of numbers in order from smallest to largest:

    Try It 5.68

    Write each set of numbers in order from smallest to largest:

    Simplify Expressions Using the Order of Operations

    The order of operations introduced in Use the Language of Algebra also applies to decimals. Do you remember what the phrase “Please excuse my dear Aunt Sally” stands for?

    Example 5.35

    Simplify the expressions:

    1. ⓐ
    2. ⓑ
    Answer

    ⓐ
    Simplify inside parentheses.
    Multiply.
    ⓑ
    Simplify inside parentheses.
    Write as a fraction.
    Multiply.
    Simplify.

    Try It 5.69

    Simplify: ⓐ ⓑ

    Try It 5.70

    Simplify: ⓐ ⓑ

    Example 5.36

    Simplify each expression:

    1. ⓐ
    2. ⓑ
    Answer

    ⓐ
    Simplify exponents.
    Divide.
    Multiply.
    Add.
    Subtract.
    ⓑ
    Simplify exponents.
    Multiply.
    Convert to a decimal.
    Add.

    Try It 5.71

    Simplify:

    Try It 5.72

    Simplify:

    Find the Circumference and Area of Circles

    The properties of circles have been studied for over

    The size of a circle can be measured in two ways. The distance around a circle is called its circumference.

    A circle is shown. A dotted line running through the widest portion of the circle is labeled as a diameter. A dotted line from the center of the circle to a point on the circle is labeled as a radius. Along the edge of the circle is the circumference.
    Figure 5.6

    Archimedes discovered that for circles of all different sizes, dividing the circumference by the diameter always gives the same number. The value of this number is pi, symbolized by Greek letter

    Manipulative Mathematics

    Doing the Manipulative Mathematics activity Pi Lab will help you develop a better understanding of pi.

    If we want the exact circumference or area of a circle, we leave the symbol in the answer. We can get an approximate answer by substituting as the value of We use the symbol to show that the result is approximate, not exact.

    Properties of Circles

    A circle is shown. A line runs through the widest portion of the circle. There is a red dot at the center of the circle. The half of the line from the center of the circle to a point on the right of the circle is labeled with an r. The half of the line from the center of the circle to a point on the left of the circle is also labeled with an r. The two sections labeled r have a brace drawn underneath showing that the entire segment is labeled d.

    Since the diameter is twice the radius, another way to find the circumference is to use the formula

    Suppose we want to find the exact area of a circle of radius inches. To calculate the area, we would evaluate the formula for the area when inches and leave the answer in terms of

    We write after the So the exact value of the area is square inches.

    To approximate the area, we would substitute

    Remember to use square units, such as square inches, when you calculate the area.

    Example 5.37

    A circle has radius centimeters. Approximate its ⓐ circumference and ⓑ area.

    Answer

    ⓐ Find the circumference when
    Write the formula for circumference.
    Substitute 3.14 for and 10 for ,.
    Multiply.
    ⓑ Find the area when
    Write the formula for area.
    Substitute 3.14 for and 10 for .
    Multiply.

    Try It 5.73

    A circle has radius inches. Approximate its ⓐ circumference and ⓑ area.

    Try It 5.74

    A circle has radius feet. Approximate its ⓐ circumference and ⓑ area.

    Example 5.38

    A circle has radius centimeters. Approximate its ⓐ circumference and ⓑ area.

    Answer

    ⓐ Find the circumference when
    Write the formula for circumference.
    Substitute 3.14 for and 42.5 for
    Multiply.
    ⓑ Find the area when .
    Write the formula for area.
    Substitute 3.14 for and 42.5 for .
    Multiply.

    Try It 5.75

    A circle has radius centimeters. Approximate its ⓐ circumference and ⓑ area.

    Try It 5.76

    A circle has radius meters. Approximate its ⓐ circumference and ⓑ area.

    Approximate with a Fraction

    Convert the fraction to a decimal. If you use your calculator, the decimal number will fill up the display and show But if we round that number to two decimal places, we get the decimal approximation of When we have a circle with radius given as a fraction, we can substitute for instead of And, since is also an approximation of we will use the symbol to show we have an approximate value.

    Example 5.39

    A circle has radius meter. Approximate its ⓐ circumference and ⓑ area.

    Answer

    ⓐ Find the circumference when
    Write the formula for circumference.
    Substitute for and for .
    Multiply.
    ⓑ Find the area when
    Write the formula for area.
    Substitute for and for .
    Multiply.

    Try It 5.77

    A circle has radius meters. Approximate its ⓐ circumference and ⓑ area.

    Try It 5.78

    A circle has radius inches. Approximate its ⓐ circumference and ⓑ area.

    Media

    ACCESS ADDITIONAL ONLINE RESOURCES

    Section 5.3 Exercises

    Practice Makes Perfect

    Convert Fractions to Decimals

    In the following exercises, convert each fraction to a decimal.

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    In the following exercises, simplify the expression.

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    Order Decimals and Fractions

    In the following exercises, order each pair of numbers, using or

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    In the following exercises, write each set of numbers in order from least to greatest.

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    Simplify Expressions Using the Order of Operations

    In the following exercises, simplify.

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    Mixed Practice

    In the following exercises, simplify. Give the answer as a decimal.

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    Find the Circumference and Area of Circles

    In the following exercises, approximate the ⓐ circumference and ⓑ area of each circle. If measurements are given in fractions, leave answers in fraction form.

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    Everyday Math

    281.

    Kelly wants to buy a pair of boots that are on sale for of the original price. The original price of the boots is What is the sale price of the shoes?

    282.

    An architect is planning to put a circular mosaic in the entry of a new building. The mosaic will be in the shape of a circle with radius of feet. How many square feet of tile will be needed for the mosaic? (Round your answer up to the next whole number.)

    Writing Exercises

    283.

    Is it easier for you to convert a decimal to a fraction or a fraction to a decimal? Explain.

    284.

    Describe a situation in your life in which you might need to find the area or circumference of a circle.

    Self Check

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

    .

    ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

    This page titled 5.4: Decimals and Fractions was last modified on Sun, 28 May 2023 07:57:52 GMT and is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax.