Learn on PengiOpenstax Prealgebre 2EChapter 5: Decimals

Lesson 3: Decimals and Fractions

In this prealgebra lesson from OpenStax Prealgebra 2E, Chapter 5, students learn how to convert fractions to decimals by dividing the numerator by the denominator, including cases that produce repeating decimals. The lesson also covers ordering decimals and fractions, applying the order of operations to expressions with decimals, and finding the circumference and area of circles. Real-world money applications, such as calculating earnings from multiple jobs, reinforce decimal arithmetic skills.

Section 1

📘 Decimals and Fractions

New Concept

Master the connection between fractions and decimals. You will learn to convert fractions to decimals, a key skill for ordering numbers, simplifying complex expressions, and calculating the circumference and area of circles.

What’s next

Get ready to apply this concept! Next, you’ll work through interactive examples of converting, ordering, and calculating with fractions and decimals.

Section 2

Convert fractions to decimals

Property

To convert a fraction to a decimal, divide the numerator of the fraction by the denominator of the fraction. The fraction bar indicates division, so a fraction like ab\frac{a}{b} can be written as a÷ba \div b.

Examples

  • To write 34\frac{3}{4} as a decimal, we divide 3 by 4. The calculation 3.00÷43.00 \div 4 gives us 0.750.75. So, 34=0.75\frac{3}{4} = 0.75.
  • To convert the improper fraction 95\frac{9}{5} to a decimal, we divide 9 by 5. The calculation 9.0÷59.0 \div 5 results in 1.81.8. So, 95=1.8\frac{9}{5} = 1.8.

Section 3

Repeating decimals

Property

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

Examples

  • To convert 23\frac{2}{3} to a decimal, dividing 2 by 3 gives 0.666...0.666.... The digit 6 repeats endlessly, so we write it as 0.60.\overline{6}.
  • For the fraction 511\frac{5}{11}, dividing 5 by 11 gives 0.454545...0.454545.... The block of digits 45 repeats, so we write it as 0.450.\overline{45}.

Section 4

Order decimals and fractions

Property

To compare a decimal to a fraction, first convert the fraction to a decimal. Then, compare the two decimal numbers. When ordering negative numbers, remember that the number closer to zero on the number line is the larger number.

Examples

  • To order 38\frac{3}{8} and 0.40.4, first convert 38\frac{3}{8} to a decimal, which is 0.3750.375. Since 0.375<0.40.375 < 0.4, we know that 38<0.4\frac{3}{8} < 0.4.
  • To order 12-\frac{1}{2} and 0.55-0.55, convert 12-\frac{1}{2} to 0.5-0.5. On a number line, 0.5-0.5 is to the right of 0.55-0.55, so 0.5>0.55-0.5 > -0.55. Therefore, 12>0.55-\frac{1}{2} > -0.55.

Section 5

Simplify expressions using the order of operations

Property

The order of operations (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right) applies to expressions with decimals and fractions.

Examples

  • To simplify 8(4.512)8(4.5 - \frac{1}{2}), first solve the parenthesis: 4.50.5=44.5 - 0.5 = 4. Then multiply: 8(4)=328(4) = 32.
  • In the expression (15)2+(0.5)(1.2)(\frac{1}{5})^2 + (0.5)(1.2), first handle the exponent: (15)2=0.22=0.04(\frac{1}{5})^2 = 0.2^2 = 0.04. Then multiply: (0.5)(1.2)=0.6(0.5)(1.2) = 0.6. Finally, add: 0.04+0.6=0.640.04 + 0.6 = 0.64.

Section 6

Properties of circles

Property

For a circle with radius rr and diameter dd:
The circumference is C=2πrC = 2\pi r or C=πdC = \pi d.
The area is A=πr2A = \pi r^2.
The number π\pi (pi) is a constant used for all circles. For exact answers, leave the symbol π\pi in the solution. For approximate answers, use π3.14\pi \approx 3.14 or π227\pi \approx \frac{22}{7}.

Examples

  • For a circle with a radius of 5 cm, the approximate circumference is C2(3.14)(5)=31.4C \approx 2(3.14)(5) = 31.4 cm, and the approximate area is A3.14(52)=78.5A \approx 3.14(5^2) = 78.5 square cm.
  • A circle has a diameter of 20 feet, which means its radius is 10 feet. The exact area is A=π(102)=100πA = \pi (10^2) = 100\pi square feet.

Book overview

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Chapter 5: Decimals

  1. Lesson 1

    Lesson 1: Decimals

  2. Lesson 2

    Lesson 2: Decimal Operations

  3. Lesson 3Current

    Lesson 3: Decimals and Fractions

  4. Lesson 4

    Lesson 4: Solve Equations with Decimals

  5. Lesson 5

    Lesson 5: Averages and Probability

  6. Lesson 6

    Lesson 6: Ratios and Rate

  7. Lesson 7

    Lesson 7: Simplify and Use Square Roots

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

📘 Decimals and Fractions

New Concept

Master the connection between fractions and decimals. You will learn to convert fractions to decimals, a key skill for ordering numbers, simplifying complex expressions, and calculating the circumference and area of circles.

What’s next

Get ready to apply this concept! Next, you’ll work through interactive examples of converting, ordering, and calculating with fractions and decimals.

Section 2

Convert fractions to decimals

Property

To convert a fraction to a decimal, divide the numerator of the fraction by the denominator of the fraction. The fraction bar indicates division, so a fraction like ab\frac{a}{b} can be written as a÷ba \div b.

Examples

  • To write 34\frac{3}{4} as a decimal, we divide 3 by 4. The calculation 3.00÷43.00 \div 4 gives us 0.750.75. So, 34=0.75\frac{3}{4} = 0.75.
  • To convert the improper fraction 95\frac{9}{5} to a decimal, we divide 9 by 5. The calculation 9.0÷59.0 \div 5 results in 1.81.8. So, 95=1.8\frac{9}{5} = 1.8.

Section 3

Repeating decimals

Property

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

Examples

  • To convert 23\frac{2}{3} to a decimal, dividing 2 by 3 gives 0.666...0.666.... The digit 6 repeats endlessly, so we write it as 0.60.\overline{6}.
  • For the fraction 511\frac{5}{11}, dividing 5 by 11 gives 0.454545...0.454545.... The block of digits 45 repeats, so we write it as 0.450.\overline{45}.

Section 4

Order decimals and fractions

Property

To compare a decimal to a fraction, first convert the fraction to a decimal. Then, compare the two decimal numbers. When ordering negative numbers, remember that the number closer to zero on the number line is the larger number.

Examples

  • To order 38\frac{3}{8} and 0.40.4, first convert 38\frac{3}{8} to a decimal, which is 0.3750.375. Since 0.375<0.40.375 < 0.4, we know that 38<0.4\frac{3}{8} < 0.4.
  • To order 12-\frac{1}{2} and 0.55-0.55, convert 12-\frac{1}{2} to 0.5-0.5. On a number line, 0.5-0.5 is to the right of 0.55-0.55, so 0.5>0.55-0.5 > -0.55. Therefore, 12>0.55-\frac{1}{2} > -0.55.

Section 5

Simplify expressions using the order of operations

Property

The order of operations (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right) applies to expressions with decimals and fractions.

Examples

  • To simplify 8(4.512)8(4.5 - \frac{1}{2}), first solve the parenthesis: 4.50.5=44.5 - 0.5 = 4. Then multiply: 8(4)=328(4) = 32.
  • In the expression (15)2+(0.5)(1.2)(\frac{1}{5})^2 + (0.5)(1.2), first handle the exponent: (15)2=0.22=0.04(\frac{1}{5})^2 = 0.2^2 = 0.04. Then multiply: (0.5)(1.2)=0.6(0.5)(1.2) = 0.6. Finally, add: 0.04+0.6=0.640.04 + 0.6 = 0.64.

Section 6

Properties of circles

Property

For a circle with radius rr and diameter dd:
The circumference is C=2πrC = 2\pi r or C=πdC = \pi d.
The area is A=πr2A = \pi r^2.
The number π\pi (pi) is a constant used for all circles. For exact answers, leave the symbol π\pi in the solution. For approximate answers, use π3.14\pi \approx 3.14 or π227\pi \approx \frac{22}{7}.

Examples

  • For a circle with a radius of 5 cm, the approximate circumference is C2(3.14)(5)=31.4C \approx 2(3.14)(5) = 31.4 cm, and the approximate area is A3.14(52)=78.5A \approx 3.14(5^2) = 78.5 square cm.
  • A circle has a diameter of 20 feet, which means its radius is 10 feet. The exact area is A=π(102)=100πA = \pi (10^2) = 100\pi square feet.

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 5: Decimals

  1. Lesson 1

    Lesson 1: Decimals

  2. Lesson 2

    Lesson 2: Decimal Operations

  3. Lesson 3Current

    Lesson 3: Decimals and Fractions

  4. Lesson 4

    Lesson 4: Solve Equations with Decimals

  5. Lesson 5

    Lesson 5: Averages and Probability

  6. Lesson 6

    Lesson 6: Ratios and Rate

  7. Lesson 7

    Lesson 7: Simplify and Use Square Roots