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Prealgebra 2e

5.3 Decimals and Fractions

Prealgebra 2e5.3 Decimals and Fractions
  1. Preface
  2. 1 Whole Numbers
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Add Whole Numbers
    4. 1.3 Subtract Whole Numbers
    5. 1.4 Multiply Whole Numbers
    6. 1.5 Divide Whole Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 The Language of Algebra
    1. Introduction to the Language of Algebra
    2. 2.1 Use the Language of Algebra
    3. 2.2 Evaluate, Simplify, and Translate Expressions
    4. 2.3 Solving Equations Using the Subtraction and Addition Properties of Equality
    5. 2.4 Find Multiples and Factors
    6. 2.5 Prime Factorization and the Least Common Multiple
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Integers
    1. Introduction to Integers
    2. 3.1 Introduction to Integers
    3. 3.2 Add Integers
    4. 3.3 Subtract Integers
    5. 3.4 Multiply and Divide Integers
    6. 3.5 Solve Equations Using Integers; The Division Property of Equality
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Fractions
    1. Introduction to Fractions
    2. 4.1 Visualize Fractions
    3. 4.2 Multiply and Divide Fractions
    4. 4.3 Multiply and Divide Mixed Numbers and Complex Fractions
    5. 4.4 Add and Subtract Fractions with Common Denominators
    6. 4.5 Add and Subtract Fractions with Different Denominators
    7. 4.6 Add and Subtract Mixed Numbers
    8. 4.7 Solve Equations with Fractions
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Decimals
    1. Introduction to Decimals
    2. 5.1 Decimals
    3. 5.2 Decimal Operations
    4. 5.3 Decimals and Fractions
    5. 5.4 Solve Equations with Decimals
    6. 5.5 Averages and Probability
    7. 5.6 Ratios and Rate
    8. 5.7 Simplify and Use Square Roots
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Percents
    1. Introduction to Percents
    2. 6.1 Understand Percent
    3. 6.2 Solve General Applications of Percent
    4. 6.3 Solve Sales Tax, Commission, and Discount Applications
    5. 6.4 Solve Simple Interest Applications
    6. 6.5 Solve Proportions and their Applications
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 The Properties of Real Numbers
    1. Introduction to the Properties of Real Numbers
    2. 7.1 Rational and Irrational Numbers
    3. 7.2 Commutative and Associative Properties
    4. 7.3 Distributive Property
    5. 7.4 Properties of Identity, Inverses, and Zero
    6. 7.5 Systems of Measurement
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Solving Linear Equations
    1. Introduction to Solving Linear Equations
    2. 8.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 8.2 Solve Equations Using the Division and Multiplication Properties of Equality
    4. 8.3 Solve Equations with Variables and Constants on Both Sides
    5. 8.4 Solve Equations with Fraction or Decimal Coefficients
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Math Models and Geometry
    1. Introduction
    2. 9.1 Use a Problem Solving Strategy
    3. 9.2 Solve Money Applications
    4. 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem
    5. 9.4 Use Properties of Rectangles, Triangles, and Trapezoids
    6. 9.5 Solve Geometry Applications: Circles and Irregular Figures
    7. 9.6 Solve Geometry Applications: Volume and Surface Area
    8. 9.7 Solve a Formula for a Specific Variable
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Polynomials
    1. Introduction to Polynomials
    2. 10.1 Add and Subtract Polynomials
    3. 10.2 Use Multiplication Properties of Exponents
    4. 10.3 Multiply Polynomials
    5. 10.4 Divide Monomials
    6. 10.5 Integer Exponents and Scientific Notation
    7. 10.6 Introduction to Factoring Polynomials
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Graphs
    1. Graphs
    2. 11.1 Use the Rectangular Coordinate System
    3. 11.2 Graphing Linear Equations
    4. 11.3 Graphing with Intercepts
    5. 11.4 Understand Slope of a Line
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  13. A | Cumulative Review
  14. B | Powers and Roots Tables
  15. C | Geometric Formulas
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
  17. Index

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: 0.24÷8.0.24÷8.
If you missed this problem, review Example 5.19.

Be Prepared 5.8

Order 0.64__0.60.64__0.6 using << or >.>.
If you missed this problem, review Example 5.7.

Be Prepared 5.9

Order −0.2__−0.1−0.2__−0.1 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 4545 can be written 4÷54÷5 or 54.54. 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 3434 as a decimal.

Try It 5.55

Write the fraction as a decimal: 14.14.

Try It 5.56

Write the fraction as a decimal: 38.38.

Example 5.29

Write the fraction 7272 as a decimal.

Try It 5.57

Write the fraction as a decimal: 94.94.

Try It 5.58

Write the fraction as a decimal: 112.112.

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 4343 to a decimal. First, notice that 4343 is an improper fraction. Its value is greater than 1.1. The equivalent decimal will also be greater than 1.1.

We divide 44 by 3.3.

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 1,1, and the threes in the quotient will go on forever. The number 1.333…1.333… 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 1.3331.333 we use a shorthand notation by placing a line over the digits that repeat. The repeating decimal 1.3331.333 is written 1.3.1.3. The line above the 33 tells you that the 33 repeats endlessly. So 1.333…=1.31.333…=1.3

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

1.333…=1.31.333…=1.3 33 is the repeating digit
4.1666…=4.164.1666…=4.16 66 is the repeating digit
4.161616…=4.164.161616…=4.16 1616 is the repeating block
0.271271271…=0.271–––0.271271271…=0.271––– 271271 is the repeating block
Table 5.5

Example 5.30

Write 43224322 as a decimal.

Try It 5.59

Write as a decimal: 2711.2711.

Try It 5.60

Write as a decimal: 5122.5122.

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: 78+6.4.78+6.4.

Try It 5.61

Simplify: 38+4.9.38+4.9.

Try It 5.62

Simplify: 5.7+1320.5.7+1320.

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 38__0.438__0.4 using << or >.>.

Try It 5.63

Order each of the following pairs of numbers, using << or >.>.

1720__0.821720__0.82

Try It 5.64

Order each of the following pairs of numbers, using << or >.>.

34__0.78534__0.785

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 −0.5___34−0.5___34 using << or >.>.

Try It 5.65

Order each of the following pairs of numbers, using << or >:>:

58__−0.5858__−0.58

Try It 5.66

Order each of the following pairs of numbers, using << or >:>:

−0.53__1120−0.53__1120

Example 5.34

Write the numbers 1320,0.61,11161320,0.61,1116 in order from smallest to largest.

Try It 5.67

Write each set of numbers in order from smallest to largest: 78,45,0.82.78,45,0.82.

Try It 5.68

Write each set of numbers in order from smallest to largest: 0.835,1316,34.0.835,1316,34.

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. 7(18.321.7)7(18.321.7)
  2. 23(8.33.8)23(8.33.8)
Try It 5.69

Simplify: 8(14.637.5)8(14.637.5) 35(9.62.1).35(9.62.1).

Try It 5.70

Simplify: 25(25.6956.74)25(25.6956.74) 27(11.94.2).27(11.94.2).

Example 5.36

Simplify each expression:

  1. 6÷0.6+(0.2)4(0.1)26÷0.6+(0.2)4(0.1)2
  2. (110)2+(3.5)(0.9)(110)2+(3.5)(0.9)
Try It 5.71

Simplify: 9÷0.9+(0.4)3(0.2)2.9÷0.9+(0.4)3(0.2)2.

Try It 5.72

Simplify: (12)2+(0.3)(4.2).(12)2+(0.3)(4.2).

Find the Circumference and Area of Circles

The properties of circles have been studied for over 2,0002,000 years. All circles have exactly the same shape, but their sizes are affected by the length of the radius, a line segment from the center to any point on the circle. A line segment that passes through a circle’s center connecting two points on the circle is called a diameter. The diameter is twice as long as the radius. See Figure 5.6.

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 ππ (pronounced pie). However, the exact value of ππ cannot be calculated since the decimal never ends or repeats (we will learn more about numbers like this in The Properties of Real Numbers.)

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 ππ in the answer. We can get an approximate answer by substituting 3.143.14 as the value of π.π. 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.
ris the length of the radius.dis the length of the diameter.ris the length of the radius.dis the length of the diameter.
The circumference is2πr.C=2πrThe area isπr2.A=πr2The circumference is2πr.C=2πrThe area isπr2.A=πr2

Since the diameter is twice the radius, another way to find the circumference is to use the formula C=πd.C=πd.

Suppose we want to find the exact area of a circle of radius 1010 inches. To calculate the area, we would evaluate the formula for the area when r=10r=10 inches and leave the answer in terms of π.π.

A=πr2A=π(102)A=π·100A=πr2A=π(102)A=π·100

We write ππ after the 100.100. So the exact value of the area is A=100πA=100π square inches.

To approximate the area, we would substitute π3.14.π3.14.

A = 100 π 100 · 3.14 314 square inches A = 100 π 100 · 3.14 314 square inches

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

Example 5.37

A circle has radius 1010 centimeters. Approximate its circumference and area.

Try It 5.73

A circle has radius 5050 inches. Approximate its circumference and area.

Try It 5.74

A circle has radius 100100 feet. Approximate its circumference and area.

Example 5.38

A circle has radius 42.542.5 centimeters. Approximate its circumference and area.

Try It 5.75

A circle has radius 51.851.8 centimeters. Approximate its circumference and area.

Try It 5.76

A circle has radius 26.426.4 meters. Approximate its circumference and area.

Approximate ππ with a Fraction

Convert the fraction 227227 to a decimal. If you use your calculator, the decimal number will fill up the display and show 3.14285714.3.14285714. But if we round that number to two decimal places, we get 3.14,3.14, the decimal approximation of π.π. When we have a circle with radius given as a fraction, we can substitute 227227 for ππ instead of 3.14.3.14. And, since 227227 is also an approximation of π,π, we will use the symbol to show we have an approximate value.

Example 5.39

A circle has radius 14151415 meter. Approximate its circumference and area.

Try It 5.77

A circle has radius 521521 meters. Approximate its circumference and area.

Try It 5.78

A circle has radius 10331033 inches. Approximate its circumference and area.

Section 5.3 Exercises

Practice Makes Perfect

Convert Fractions to Decimals

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

201.

2525

202.

4545

203.

3838

204.

5858

205.

17201720

206.

13201320

207.

114114

208.

174174

209.

3102531025

210.

2842528425

211.

5959

212.

2929

213.

15111511

214.

18111811

215.

1511115111

216.

2511125111

In the following exercises, simplify the expression.

217.

12+6.512+6.5

218.

14+10.7514+10.75

219.

2.4+582.4+58

220.

3.9+9203.9+920

221.

9.73+17209.73+1720

222.

6.29+21406.29+2140

Order Decimals and Fractions

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

223.

18___0.818___0.8

224.

14___0.414___0.4

225.

25___0.2525___0.25

226.

35___0.3535___0.35

227.

0.725___340.725___34

228.

0.92___780.92___78

229.

0.66___230.66___23

230.

0.83___560.83___56

231.

−0.75___45−0.75___45

232.

−0.44___920−0.44___920

233.

34___−0.92534___−0.925

234.

23___−0.63223___−0.632

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

235.

35,916,0.5535,916,0.55

236.

38,720,0.3638,720,0.36

237.

0.702,1320,580.702,1320,58

238.

0.15,316,150.15,316,15

239.

−0.3,13,720−0.3,13,720

240.

−0.2,320,16−0.2,320,16

241.

34,79,−0.734,79,−0.7

242.

89,45,−0.989,45,−0.9

Simplify Expressions Using the Order of Operations

In the following exercises, simplify.

243.

10(25.143.8)10(25.143.8)

244.

30(18.132.5)30(18.132.5)

245.

62(9.754.99)62(9.754.99)

246.

42(8.455.97)42(8.455.97)

247.

34(12.44.2)34(12.44.2)

248.

45(8.6+3.9)45(8.6+3.9)

249.

512(30.58+17.9)512(30.58+17.9)

250.

916(21.969.8)916(21.969.8)

251.

10÷0.1+(1.8)4(0.3)210÷0.1+(1.8)4(0.3)2

252.

5÷0.5+(3.9)6(0.7)25÷0.5+(3.9)6(0.7)2

253.

(37.1+52.7)÷(12.5÷62.5)(37.1+52.7)÷(12.5÷62.5)

254.

(11.4+16.2)÷(18÷60)(11.4+16.2)÷(18÷60)

255.

(15)2+(1.4)(6.5)(15)2+(1.4)(6.5)

256.

(12)2+(2.1)(8.3)(12)2+(2.1)(8.3)

257.

910·815+0.25910·815+0.25

258.

38·1415+0.7238·1415+0.72

Mixed Practice

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

259.

3146.53146.5

260.

5258.755258.75

261.

10.86÷2310.86÷23

262.

5.79÷345.79÷34

263.

78(103.48)+112(361)78(103.48)+112(361)

264.

516(117.6)+213(699)516(117.6)+213(699)

265.

3.6(982.72)3.6(982.72)

266.

5.1(1253.91)5.1(1253.91)

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.

267.

radius=5 in.radius=5 in.

268.

radius=20 in.radius=20 in.

269.

radius=9 ft.radius=9 ft.

270.

radius=4 ft.radius=4 ft.

271.

radius=46 cmradius=46 cm

272.

radius=38 cmradius=38 cm

273.

radius=18.6 mradius=18.6 m

274.

radius=57.3 mradius=57.3 m

275.

radius=710mileradius=710mile

276.

radius=711mileradius=711mile

277.

radius=38yardradius=38yard

278.

radius=512yardradius=512yard

279.

diameter=56mdiameter=56m

280.

diameter=34mdiameter=34m

Everyday Math

281.

Kelly wants to buy a pair of boots that are on sale for 2323 of the original price. The original price of the boots is $84.99.$84.99. 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 66 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?

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