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Contemporary Mathematics

3.4 Rational Numbers

Contemporary Mathematics3.4 Rational Numbers

Table of contents
  1. Preface
  2. 1 Sets
    1. Introduction
    2. 1.1 Basic Set Concepts
    3. 1.2 Subsets
    4. 1.3 Understanding Venn Diagrams
    5. 1.4 Set Operations with Two Sets
    6. 1.5 Set Operations with Three Sets
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  3. 2 Logic
    1. Introduction
    2. 2.1 Statements and Quantifiers
    3. 2.2 Compound Statements
    4. 2.3 Constructing Truth Tables
    5. 2.4 Truth Tables for the Conditional and Biconditional
    6. 2.5 Equivalent Statements
    7. 2.6 De Morgan’s Laws
    8. 2.7 Logical Arguments
    9. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  4. 3 Real Number Systems and Number Theory
    1. Introduction
    2. 3.1 Prime and Composite Numbers
    3. 3.2 The Integers
    4. 3.3 Order of Operations
    5. 3.4 Rational Numbers
    6. 3.5 Irrational Numbers
    7. 3.6 Real Numbers
    8. 3.7 Clock Arithmetic
    9. 3.8 Exponents
    10. 3.9 Scientific Notation
    11. 3.10 Arithmetic Sequences
    12. 3.11 Geometric Sequences
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  5. 4 Number Representation and Calculation
    1. Introduction
    2. 4.1 Hindu-Arabic Positional System
    3. 4.2 Early Numeration Systems
    4. 4.3 Converting with Base Systems
    5. 4.4 Addition and Subtraction in Base Systems
    6. 4.5 Multiplication and Division in Base Systems
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  6. 5 Algebra
    1. Introduction
    2. 5.1 Algebraic Expressions
    3. 5.2 Linear Equations in One Variable with Applications
    4. 5.3 Linear Inequalities in One Variable with Applications
    5. 5.4 Ratios and Proportions
    6. 5.5 Graphing Linear Equations and Inequalities
    7. 5.6 Quadratic Equations with Two Variables with Applications
    8. 5.7 Functions
    9. 5.8 Graphing Functions
    10. 5.9 Systems of Linear Equations in Two Variables
    11. 5.10 Systems of Linear Inequalities in Two Variables
    12. 5.11 Linear Programming
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  7. 6 Money Management
    1. Introduction
    2. 6.1 Understanding Percent
    3. 6.2 Discounts, Markups, and Sales Tax
    4. 6.3 Simple Interest
    5. 6.4 Compound Interest
    6. 6.5 Making a Personal Budget
    7. 6.6 Methods of Savings
    8. 6.7 Investments
    9. 6.8 The Basics of Loans
    10. 6.9 Understanding Student Loans
    11. 6.10 Credit Cards
    12. 6.11 Buying or Leasing a Car
    13. 6.12 Renting and Homeownership
    14. 6.13 Income Tax
    15. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  8. 7 Probability
    1. Introduction
    2. 7.1 The Multiplication Rule for Counting
    3. 7.2 Permutations
    4. 7.3 Combinations
    5. 7.4 Tree Diagrams, Tables, and Outcomes
    6. 7.5 Basic Concepts of Probability
    7. 7.6 Probability with Permutations and Combinations
    8. 7.7 What Are the Odds?
    9. 7.8 The Addition Rule for Probability
    10. 7.9 Conditional Probability and the Multiplication Rule
    11. 7.10 The Binomial Distribution
    12. 7.11 Expected Value
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  9. 8 Statistics
    1. Introduction
    2. 8.1 Gathering and Organizing Data
    3. 8.2 Visualizing Data
    4. 8.3 Mean, Median and Mode
    5. 8.4 Range and Standard Deviation
    6. 8.5 Percentiles
    7. 8.6 The Normal Distribution
    8. 8.7 Applications of the Normal Distribution
    9. 8.8 Scatter Plots, Correlation, and Regression Lines
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  10. 9 Metric Measurement
    1. Introduction
    2. 9.1 The Metric System
    3. 9.2 Measuring Area
    4. 9.3 Measuring Volume
    5. 9.4 Measuring Weight
    6. 9.5 Measuring Temperature
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  11. 10 Geometry
    1. Introduction
    2. 10.1 Points, Lines, and Planes
    3. 10.2 Angles
    4. 10.3 Triangles
    5. 10.4 Polygons, Perimeter, and Circumference
    6. 10.5 Tessellations
    7. 10.6 Area
    8. 10.7 Volume and Surface Area
    9. 10.8 Right Triangle Trigonometry
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  12. 11 Voting and Apportionment
    1. Introduction
    2. 11.1 Voting Methods
    3. 11.2 Fairness in Voting Methods
    4. 11.3 Standard Divisors, Standard Quotas, and the Apportionment Problem
    5. 11.4 Apportionment Methods
    6. 11.5 Fairness in Apportionment Methods
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  13. 12 Graph Theory
    1. Introduction
    2. 12.1 Graph Basics
    3. 12.2 Graph Structures
    4. 12.3 Comparing Graphs
    5. 12.4 Navigating Graphs
    6. 12.5 Euler Circuits
    7. 12.6 Euler Trails
    8. 12.7 Hamilton Cycles
    9. 12.8 Hamilton Paths
    10. 12.9 Traveling Salesperson Problem
    11. 12.10 Trees
    12. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  14. 13 Math and...
    1. Introduction
    2. 13.1 Math and Art
    3. 13.2 Math and the Environment
    4. 13.3 Math and Medicine
    5. 13.4 Math and Music
    6. 13.5 Math and Sports
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  15. A | Co-Req Appendix: Integer Powers of 10
  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
    12. Chapter 12
    13. Chapter 13
  17. Index
A close up view of a newspaper showing stock price changes.
Figure 3.20 Stock gains and losses are often represented as percentages.(credit: "stock market quotes in newspaper" by Andreas Poike/Flickr, CC BY 2.0)

Learning Objectives

After completing this section, you should be able to:

  1. Define and identify numbers that are rational.
  2. Simplify rational numbers and express in lowest terms.
  3. Add and subtract rational numbers.
  4. Convert between improper fractions and mixed numbers.
  5. Convert rational numbers between decimal and fraction form.
  6. Multiply and divide rational numbers.
  7. Apply the order of operations to rational numbers to simplify expressions.
  8. Apply density property of rational numbers.
  9. Solve problems involving rational numbers.
  10. Use fractions to convert between units.
  11. Define and apply percent.
  12. Solve problems using percent.

We are often presented with percentages or fractions to explain how much of a population has a certain feature. For example, the 6-year graduation rate of college students at public institutions is 57.6%, or 72/125. That fraction may be unsettling. But without the context, the percentage is hard to judge. So how does that compare to private institutions? There, the 6-year graduation rate is 65.4%, or 327/500. Comparing the percentages is straightforward, but the fractions are harder to interpret due to different denominators. For more context, historical data could be found. One study reported that the 6-year graduation rate in 1995 was 56.4%. Comparing that historical number to the recent 6-year graduation rate at public institutions of 57.6% shows that there hasn't been much change in that rate.

Defining and Identifying Numbers That Are Rational

A rational number (called rational since it is a ratio) is just a fraction where the numerator is an integer and the denominator is a non-zero integer. As simple as that is, they can be represented in many ways. It should be noted here that any integer is a rational number. An integer, nn, written as a fraction of two integers is n1n1.

In its most basic representation, a rational number is an integer divided by a non-zero integer, such as 312312. Fractions may be used to represent parts of a whole. The denominator is the total number of parts to the object, and the numerator is how many of those parts are being used or selected. So, if a pizza is cut into 8 equal pieces, each piece is 1818 of the pizza. If you take three slices, you have 3838 of the pizza (Figure 3.21). Similarly, if in a group of 20 people, 5 are wearing hats, then 520520 of the people are wearing hats (Figure 3.22).

8 slices of pizza. Three slices are highlighted.
Figure 3.21 Pizza cut in 8 slices, with 3 slices highlighted
A group of 20 people. Five people are wearing hats.
Figure 3.22 Group of 20 people, with 5 people wearing hats

Another representation of rational numbers is as a mixed number, such as 258258 (Figure 3.23). This represents a whole number (2 in this case), plus a fraction (the 5858).

Two whole pizzas and 5 slices of a third pizza. The whole pizzas have 8 slices each.
Figure 3.23 Two whole pizzas and one partial pizza

Rational numbers may also be expressed in decimal form; for instance, as 1.34. When 1.34 is written, the decimal part, 0.34, represents the fraction 3410034100, and the number 1.34 is equal to 134100134100. However, not all decimal representations are rational numbers.

A number written in decimal form where there is a last decimal digit (after a given decimal digit, all following decimal digits are 0) is a terminating decimal, as in 1.34 above. Alternately, any decimal numeral that, after a finite number of decimal digits, has digits equal to 0 for all digits following the last non-zero digit.

All numbers that can be expressed as a terminating decimal are rational. This comes from what the decimal represents. The decimal part is the fraction of the decimal part divided by the appropriate power of 10. That power of 10 is the number of decimal digits present, as for 0.34, with two decimal digits, being equal to 3410034100.

Another form that is a rational number is a decimal that repeats a pattern, such as 67.1313… When a rational number is expressed in decimal form and the decimal is a repeated pattern, we use special notation to designate the part that repeats. For example, if we have the repeating decimal 4.3636…, we write this as 4.36¯4.36¯. The bar over the 36 indicates that the 36 repeats forever.

If the decimal representation of a number does not terminate or form a repeating decimal, that number is not a rational number.

One class of numbers that is not rational is the square roots of integers or rational numbers that are not perfect squares, such as 1010 and 256256. More generally, the number bb is the square root of the number aa if a=b2a=b2. The notation for this is b=ab=a, where the symbol is the square root sign. An integer perfect square is any integer that can be written as the square of another integer. A rational perfect square is any rational number that can be written as a fraction of two integers that are perfect squares.

Sometimes you may be able to identify a perfect square from memory. Another process that may be used is to factor the number into the product of an integer with itself. Or a calculator (such as Desmos) may be used to find the square root of the number. If the calculator yields an integer, the original number was a perfect square.

Tech Check

Using Desmos to Find the Square Root of a Number

When Desmos is used, there is a tab at the bottom of the screen that opens the keyboard for Desmos. The keyboard is shown below. On the keyboard (Figure 3.24) is the square root symbol ( )( ). To find the square root of a number, click the square root key, and then type the number. Desmos will automatically display the value of the square root as you enter the number.

Desmos keyboard is displayed. Three sets of keys are displayed. The first set has 4 rows of 4 keys, each. Row 1: x, y, a squared, and a to the power b. Row 2: open parenthesis, close parenthesis, lesser than, and greater than. Row 3: modulus of a, comma, lesser than or equal to, and greater than or equal to. Row 4: A B C, sound, square root, and pi. The second set has 4 rows of 4 keys, each. Row 1: 7, 8, 9, and division symbol. Row 2: 4, 5, 6, and multiplication symbol. Row 3: 1, 2, 3, and minus symbol. Row 4: 0, dot, equals sign, and plus sign. The third set has the following keys: functions, left arrow, right arrow, close, and back.
Figure 3.24 Desmos keyboard with square root key

Example 3.51

Identifying Perfect Squares

Which of the following are perfect squares?

  1. 45
  2. 144

Your Turn 3.51

Determine if the following are perfect squares:
1.
94
2.
441

Example 3.52

Identifying Rational Numbers

Determine which of the following are rational numbers:

  1. 7373
  2. 4.5564.556
  3. 315315
  4. 41174117
  5. 5.64¯5.64¯

Your Turn 3.52

Determine which of the following are rational numbers:
1.
13
2.
13. 2 ¯ 1 ¯
3.
48 16
4.
4 18 19
5.
14.1131

Simplifying Rational Numbers and Expressing in Lowest Terms

A rational number is one way to express the division of two integers. As such, there may be multiple ways to express the same value with different rational numbers. For instance, 4545 and 12151215 are the same value. If we enter them into a calculator, they both equal 0.8. Another way to understand this is to consider what it looks like in a figure when two fractions are equal.

In Figure 3.25, we see that 3535 of the rectangle and 915915 of the rectangle are equal areas.

Two rectangles are plotted on a rectangular grid. The grid is made up of 15 rows of 20 unit squares, each. The first rectangle has 5 rows of 3 unit squares, each. The rectangle is divided into 5 equal pieces. Each piece has 3 unit squares. 3 pieces are shaded and labeled three-fifths of the rectangle is shaded. The second rectangle has 5 rows of 3 unit squares, each. The rectangle is divided into 15 equal pieces. Each piece has a unit square. 9 pieces are shaded and labeled 9 over 15 of the rectangle is shaded.
Figure 3.25 Two Rectangles with Equal Areas

They are the same proportion of the area of the rectangle. The left rectangle has 5 pieces, three of which are shaded. The right rectangle has 15 pieces, 9 of which are shaded. Each of the pieces of the left rectangle was divided equally into three pieces. This was a multiplication. The numerator describing the left rectangle was 3 but it becomes 3×33×3, or 9, as each piece was divided into three. Similarly, the denominator describing the left rectangle was 5, but became 5×35×3, or 15, as each piece was divided into 3. The fractions 3535 and 915915 are equivalent because they represent the same portion (often loosely referred to as equal).

This understanding of equivalent fractions is very useful for conceptualization, but it isn’t practical, in general, for determining when two fractions are equivalent. Generally, to determine if the two fractions abab and cdcd are equivalent, we check to see that a×d=b×ca×d=b×c. If those two products are equal, then the fractions are equal also.

Example 3.53

Determining If Two Fractions Are Equivalent

Determine if 12301230 and 14351435 are equivalent fractions.

Your Turn 3.53

1.
Determine if 8 14 and 12 26 are equivalent fractions.

That a×d=b×ca×d=b×c indicates the fractions abab and cdcd are equivalent is due to some algebra. One property of natural numbers, integers, and rational numbers (also irrational numbers) is that for any three numbers a,b,a,b, and cc with c0c0, if a=ba=b, then a/c=b/ca/c=b/c. In other words, when two numbers are equal, then dividing both numbers by the same non-zero number, the two newly obtained numbers are also equal. We can apply that to a×da×d and b×cb×c, to show that abab and cdcd are equivalent if a×d=b×ca×d=b×c.

If a×d=b×ca×d=b×c, and c0,d 0c0,d 0, we can divide both sides by and obtain the following: a×dc=b×cca×dc=b×cc. We can divide out the cc on the right-hand side of the equation, resulting in a×dc=ba×dc=b. Similarly, we can divide both sides of the equation by dd and obtain the following: a×dc×d=bda×dc×d=bd. We can divide out dd the on the left-hand side of the equation, resulting in ac=bdac=bd. So, the rational numbers acac and bdbd are equivalent when a×d=b×ca×d=b×c.

Recall that a common divisor or common factor of a set of integers is one that divides all the numbers of the set of numbers being considered. In a fraction, when the numerator and denominator have a common divisor, that common divisor can be divided out. This is often called canceling the common factors or, more colloquially, as canceling.

To show this, consider the fraction 36633663. The numerator and denominator have the common factor 3. We can rewrite the fraction as 3663=12×321×33663=12×321×3. The common divisor 3 is then divided out, or canceled, and we can write the fraction as 12×321×3=122112×321×3=1221. The 3s have been crossed out to indicate they have been divided out. The process of dividing out two factors is also referred to as reducing the fraction.

If the numerator and denominator have no common positive divisors other than 1, then the rational number is in lowest terms.

The process of dividing out common divisors of the numerator and denominator of a fraction is called reducing the fraction.

One way to reduce a fraction to lowest terms is to determine the GCD of the numerator and denominator and divide out the GCD. Another way is to divide out common divisors until the numerator and denominator have no more common factors.

Example 3.54

Reducing Fractions to Lowest Terms

Express the following rational numbers in lowest terms:

  1. 36483648
  2. 100250100250
  3. 5113651136

Your Turn 3.54

1.
Express 252 840 in lowest terms.

Tech Check

Using Desmos to Find Lowest Terms

Desmos is a free online calculator. Desmos supports reducing fractions to lowest terms. When a fraction is entered, Desmos immediately calculates the decimal representation of the fraction. However, to the left of the fraction, there is a button that, when clicked, shows the fraction in reduced form.

Adding and Subtracting Rational Numbers

Adding or subtracting rational numbers can be done with a calculator, which often returns a decimal representation, or by finding a common denominator for the rational numbers being added or subtracted.

Tech Check

Using Desmos to Add Rational Numbers in Fractional Form

To create a fraction in Desmos, enter the numerator, then use the division key (/) on your keyboard, and then enter the denominator. The fraction is then entered. Then click the right arrow key to exit the denominator of the fraction. Next, enter the arithmetic operation (+ or –). Then enter the next fraction. The answer is displayed dynamically (calculates as you enter). To change the Desmos result from decimal form to fractional form, use the fraction button (Figure 3.26) on the left of the line that contains the calculation:

A fraction symbol is shown with empty boxes in the numerator and denominator.
Figure 3.26 Fraction button on the Desmos keyboard

Example 3.55

Adding Rational Numbers Using Desmos

Calculate 2342+9562342+956 using Desmos.

Your Turn 3.55

1.
Calculate 124 297 + 3 125 .

Performing addition and subtraction without a calculator may be more involved. When the two rational numbers have a common denominator, then adding or subtracting the two numbers is straightforward. Add or subtract the numerators, and then place that value in the numerator and the common denominator in the denominator. Symbolically, we write this as ac±bc=a±bcac±bc=a±bc. This can be seen in the Figure 3.27, which shows 320+420=720320+420=720.

Two rectangles are plotted on a rectangular grid. The grid is made up of 15 rows of 20 unit squares, each. The first rectangle has 5 rows of 4 unit squares, each. The rectangle is divided into 20 equal pieces. Each piece has a unit square. The second rectangle has 5 rows of 4 unit squares, each. The rectangle is divided into 20 equal pieces. 3 pieces are shaded in pink and 4 pieces are shaded in green. Text reads, 3 over 20 in pink, 4 over 20 in green. 3 over 20 plus 4 over 20 equals 7 over 20.
Figure 3.27 Partially Shaded Rectangle

It is customary to then write the result in lowest terms.

FORMULA

If cc is a non-zero integer, then ac±bc=a±bcac±bc=a±bc.

Example 3.56

Adding Rational Numbers with the Same Denominator

Calculate 1328+7281328+728.

Your Turn 3.56

1.
Calculate 38 73 + 7 73 .

Example 3.57

Subtracting Rational Numbers with the Same Denominator

Calculate 45136171364513617136.

Your Turn 3.57

1.
Calculate 21 40 8 40 .

When the rational numbers do not have common denominators, then we have to transform the rational numbers so that they do have common denominators. The common denominator that reduces work later in the problem is the LCM of the numerator and denominator. When adding or subtracting the rational numbers abab and cdcd, we perform the following steps.

Step 1: Find LCM(b,d)LCM(b,d).

Step 2: Calculate n=LCM(b,d)bn=LCM(b,d)b and m=LCM(b,d)dm=LCM(b,d)d.

Step 3: Multiply the numerator and denominator of abab by nn, yielding a×nb×na×nb×n.

Step 4: Multiply the numerator and denominator of cdcd by mm, yielding c×md×m c×md×m.

Step 5: Add or subtract the rational numbers from Steps 3 and 4, since they now have the common denominators.

You should be aware that the common denominator is LCM(b,d)LCM(b,d). For the first denominator, we have b×n=b×LCM(b,d)b=LCM(b,d)b×n=b×LCM(b,d)b=LCM(b,d), since we multiply and divide LCM(b,d)LCM(b,d) by the same number. For the same reason, d×m=d×LCM(b,d)b=LCM(b,d)d×m=d×LCM(b,d)b=LCM(b,d).

Example 3.58

Adding Rational Numbers with Unequal Denominators

Calculate 1118+2151118+215.

Your Turn 3.58

1.
Calculate 4 9 + 7 12 .

Example 3.59

Subtracting Rational Numbers with Unequal Denominators

Calculate 14259701425970.

Your Turn 3.59

1.
Calculate 10 99 17 300 .

Converting Between Improper Fractions and Mixed Numbers

One way to visualize a fraction is as parts of a whole, as in 512512 of a pizza. But when the numerator is larger than the denominator, as in 23122312, then the idea of parts of a whole seems not to make sense. Such a fraction is an improper fraction. That kind of fraction could be written as an integer plus a fraction, which is a mixed number. The fraction 23122312 rewritten as a mixed number would be 1111211112. Arithmetically, 1111211112 is equivalent to 1+11121+1112, which is read as “one and 11 twelfths.”

Improper fractions can be rewritten as mixed numbers using division and remainders. To find the mixed number representation of an improper fraction, divide the numerator by the denominator. The quotient is the integer part, and the remainder becomes the numerator of the remaining fraction.

Example 3.60

Rewriting an Improper Fraction as a Mixed Number

Rewrite 48134813 as a mixed number.

Your Turn 3.60

1.
Rewrite 95 26 as a mixed number.

Similarly, we can convert a mixed number into an improper fraction. To do so, first convert the whole number part to a fraction by writing the whole number as itself divided by 1, and then add the two fractions.

Alternately, we can multiply the whole number part and the denominator of the fractional part. Next, add that product to the numerator. Finally, express the number as that product divided by the denominator.

Example 3.61

Rewriting a Mixed Number as an Improper Fraction

Rewrite 549549 as an improper fraction.

Your Turn 3.61

1.
Rewrite 9 5 14 as an improper fraction.

Tech Check

Using Desmos to Rewrite a Mixed Number as an Improper Fraction

Desmos can be used to convert from a mixed number to an improper fraction. To do so, we use the idea that a mixed number, such as 56115611, is another way to represent 5+6115+611. If 5+6115+611 is entered in Desmos, the result is the decimal form of the number. However, clicking the fraction button to the left will convert the decimal to an improper fraction, 61116111. As an added bonus, Desmos will automatically reduce the fraction to lowest terms.

Converting Rational Numbers Between Decimal and Fraction Forms

Understanding what decimals represent is needed before addressing conversions between the fractional form of a number and its decimal form, or writing a number in decimal notation. The decimal number 4.557 is equal to 45571,00045571,000. The decimal portion, .557, is 557 divided by 1,000. To write any decimal portion of a number expressed as a terminating decimal, divide the decimal number by 10 raised to the power equal to the number of decimal digits. Since there were three decimal digits in 4.557, we divided 557 by 103=1000103=1000.

Decimal representations may be very long. It is convenient to round off the decimal form of the number to a certain number of decimal digits. To round off the decimal form of a number to nn (decimal) digits, examine the (n+1n+1)st decimal digit. If that digit is 0, 1, 2, 3, or 4, the number is rounded off by writing the number to the nnth decimal digit and no further. If the (n+1n+1)st decimal digit is 5, 6, 7, 8, or 9, the number is rounded off by writing the number to the nnth digit, then replacing the nnth digit by one more than the nnth digit.

Example 3.62

Rounding Off a Number in Decimal Form to Three Digits

Round 5.67849 to three decimal digits.

Your Turn 3.62

1.
Round 5.1082 to three decimal places.

Example 3.63

Rounding Off a Number in Decimal Form to Four Digits

Round 45.11475 to four decimal digits.

Your Turn 3.63

1.
Round 18.6298 to two decimal places.

To convert a rational number in fraction form to decimal form, use your calculator to perform the division.

Example 3.64

Converting a Rational Number in Fraction Form into Decimal Form

Convert 47254725 into decimal form.

Your Turn 3.64

1.
Convert 48 30 into decimal form.

Converting a terminating decimal to the fractional form may be done in the following way:

Step 1: Count the number of digits in the decimal part of the number, labeled nn.

Step 2: Raise 10 to the nnth power.

Step 3: Rewrite the number without the decimal.

Step 4: The fractional form is the number from Step 3 divided by the result from Step 2.

This process works due to what decimals represent and how we work with mixed numbers. For example, we could convert the number 7.4536 to fractional from. The decimal part of the number, the .4536 part of 7.4536, has four digits. By the definition of decimal notation, the decimal portion represents 4,536104=4,53610,0004,536104=4,53610,000. The decimal number 7.4536 is equal to the improper fraction 74,53610,00074,53610,000. Adding those to fractions yields 74,53610,00074,53610,000.

Example 3.65

Converting from Decimal Form to Fraction Form with Terminating Decimals

Convert 3.2117 to fraction form.

Your Turn 3.65

1.
Convert 17.03347 to fraction form.

The process is different when converting from the decimal form of a rational number into fraction form when the decimal form is a repeating decimal. This process is not covered in this text.

Multiplying and Dividing Rational Numbers

Multiplying rational numbers is less complicated than adding or subtracting rational numbers, as there is no need to find common denominators. To multiply rational numbers, multiply the numerators, then multiply the denominators, and write the numerator product divided by the denominator product. Symbolically, ab×cd=a×cb×dab×cd=a×cb×d. As always, rational numbers should be reduced to lowest terms.

FORMULA

If bb and dd are non-zero integers, then ab×cd=a×cb×dab×cd=a×cb×d.

Example 3.66

Multiplying Rational Numbers

Calculate 1225×10211225×1021.

Your Turn 3.66

1.
Calculate 45 88 × 28 75 .

As with multiplication, division of rational numbers can be done using a calculator.

Example 3.67

Dividing Decimals with a Calculator

Calculate 3.45 ÷ 2.341 using a calculator. Round to three decimal places if necessary.

Your Turn 3.67

1.
Calculate 45.63 ÷ 17.13 using a calculator. Round to three decimal places, if necessary.

Before discussing division of fractions without a calculator, we should look at the reciprocal of a number. The reciprocal of a number is 1 divided by the number. For a fraction, the reciprocal is the fraction formed by switching the numerator and denominator. For the fraction abab, the reciprocal is baba. An important feature for a number and its reciprocal is that their product is 1.

When dividing two fractions by hand, find the reciprocal of the divisor (the number that is being divided into the other number). Next, replace the divisor by its reciprocal and change the division into multiplication. Then, perform the multiplication. Symbolically,ba÷cd=ab×dc=a×db×cba÷cd=ab×dc=a×db×c. As before, reduce to lowest terms.

FORMULA

If b,cb,c and dd are non-zero integers, then ba÷cd=ab×dc=a×db×cba÷cd=ab×dc=a×db×c.

Example 3.68

Dividing Rational Numbers

  1. Calculate 421÷635421÷635.
  2. Calculate 18÷52818÷528.

Your Turn 3.68

1.
Calculate 46 175 ÷ 69 285 .
2.
Calculate 3 40 ÷ 42 55 .

Applying the Order of Operations to Simplify Expressions

The order of operations for rational numbers is the same as for integers, as discussed in Order of Operations. The order of operations makes it easier for anyone to correctly calculate and represent. The order follows the well-known acronym PEMDAS:

P Parentheses
E Exponents
M/D Multiplication and division
A/S Addition and subtraction

The first step in calculating using the order of operations is to perform operations inside the parentheses. Moving down the list, next perform all exponent operations moving from left to right. Next (left to right once more), perform all multiplications and divisions. Finally, perform the additions and subtractions.

Example 3.69

Applying the Order of Operations with Rational Numbers

Correctly apply the rules for the order of operations to accurately compute (5727)×23(5727)×23.

Your Turn 3.69

1.
Correctly apply the rules for the order of operations to accurately compute ( 3 16 + 7 16 ) 2 + 1 5 ÷ 3 10 .

Example 3.70

Applying the Order of Operations with Rational Numbers

Correctly apply the rules for the order of operations to accurately compute 4+23÷((59)2(23+5))24+23÷((59)2(23+5))2.

Your Turn 3.70

1.
Correctly apply the rules for the order of operations to accurately compute ( 3 5 + 2 ) × ( 4 5 1 2 ) 2 ÷ 11 15 .

Applying the Density Property of Rational Numbers

Between any two rational numbers, there is another rational number. This is called the density property of the rational numbers.

Finding a rational number between any two rational numbers is very straightforward.

Step 1: Add the two rational numbers.

Step 2: Divide that result by 2.

The result is always a rational number. This follows what we know about rational numbers. If two fractions are added, then the result is a fraction. Also, when a fraction is divided by a fraction (and 2 is a fraction), then we get another fraction. This two-step process will give a rational number, provided the first two numbers were rational.

Example 3.71

Applying the Density Property of Rational Numbers

Demonstrate the density property of rational numbers by finding a rational number between 411411 and 712712.

Your Turn 3.71

1.
Demonstrate the density property of rational numbers by finding a rational number between 27 13 and 21 10 .

Solving Problems Involving Rational Numbers

Rational numbers are used in many situations, sometimes to express a portion of a whole, other times as an expression of a ratio between two quantities. For the sciences, converting between units is done using rational numbers, as when converting between gallons and cubic inches. In chemistry, mixing a solution with a given concentration of a chemical per unit volume can be solved with rational numbers. In demographics, rational numbers are used to describe the distribution of the population. In dietetics, rational numbers are used to express the appropriate amount of a given ingredient to include in a recipe. As discussed, the application of rational numbers crosses many disciplines.

Example 3.72

Mixing Soil for Vegetables

James is mixing soil for a raised garden, in which he plans to grow a variety of vegetables. For the soil to be suitable, he determines that 2525 of the soil can be topsoil, but 2525 needs to be peat moss and 1515 has to be compost. To fill the raised garden bed with 60 cubic feet of soil, how much of each component does James need to use?

Your Turn 3.72

1.
Ashley wants to study for 10 hours over the weekend. She plans to spend half the time studying math, a quarter of the time studying history, an eighth of the time studying writing, and the remaining eighth of the time studying physics. How much time will Ashley spend on each of those subjects?

Example 3.73

Determining the Number of Specialty Pizzas

At Bella’s Pizza, one-third of the pizzas that are ordered are one of their specialty varieties. If there are 273 pizzas ordered, how many were specialty pizzas?

Your Turn 3.73

1.
Danny, a nutritionist, is designing a diet for her client, Callum. Danny determines that Callum’s diet should be 30% protein. If Callum consumes 2,400 calories per day, how many calories of protein should Danny tell Callum to consume?

Using Fractions to Convert Between Units

A common application of fractions is called unit conversion, or converting units, which is the process of changing from the units used in making a measurement to different units of measurement.

For instance, 1 inch is (approximately) equal to 2.54 cm. To convert between units, the two equivalent values are made into a fraction. To convert from the first type of unit to the second type, the fraction has the second unit as the numerator, and the first unit as the denominator.

From the inches and centimeters example, to change from inches to centimeters, we use the fraction 2.54cm1in2.54cm1in. If, on the other hand, we wanted to convert from centimeters to inches, we’d use the fraction 1in2.54cm1in2.54cm. This fraction is multiplied by the number of units of the type you are converting from, which means the units of the denominator are the same as the units being multiplied.

Example 3.74

Converting Liters to Gallons

It is known that 1 liter (L) is 0.264172 gallons (gal). Use this to convert 14 liters into gallons.

Your Turn 3.74

1.
One mile is equal to 1.60934 km. Convert 200 miles to kilometers. Round off the answer to three decimal places.

Example 3.75

Converting Centimeters to Inches

It is known that 1 inch is 2.54 centimeters. Use this to convert 100 centimeters into inches.

Your Turn 3.75

1.
It is known that 4 quarts equals 3.785 liters. If you have 25 quarts, how many liters do you have? Round off to three decimal places.

Defining and Applying Percent

A percent is a specific rational number and is literally per 100. nn percent, denoted nn%, is the fraction n100n100.

Example 3.76

Rewriting a Percentage as a Fraction

Rewrite the following as fractions:

  1. 31%
  2. 93%

Your Turn 3.76

Rewrite the following as fractions:
1.
4%
2.
50%

Example 3.77

Rewriting a Percentage as a Decimal

Rewrite the following percentages in decimal form:

  1. 54%
  2. 83%

Your Turn 3.77

Rewrite the following percentages in decimal form:
1.
14%
2.
7%

You should notice that you can simply move the decimal two places to the left without using the fractional definition of percent.

Percent is used to indicate a fraction of a total. If we want to find 30% of 90, we would perform a multiplication, with 30% written in either decimal form or fractional form. The 90 is the total, 30 is the percentage, and 27 (which is 0.30×900.30×90) is the percentage of the total.

FORMULA

n%n% of xx items is n100×xn100×x. The xx is referred to as the total, the nn is referred to as the percent or percentage, and the value obtained from n100×xn100×x is the part of the total and is also referred to as the percentage of the total.

Example 3.78

Finding a Percentage of a Total

  1. Determine 40% of 300.
  2. Determine 64% of 190.

Your Turn 3.78

1.
Determine 25% of 1,200.
2.
Determine 53% of 1,588.

In the previous situation, we knew the total and we found the percentage of the total. It may be that we know the percentage of the total, and we know the percent, but we don't know the total. To find the total if we know the percentage the percentage of the total, use the following formula.

FORMULA

If we know that nn% of the total is xx, then the total is 100×xn100×xn.

Example 3.79

Finding the Total When the Percentage and Percentage of the Total Are Known

  1. What is the total if 28% of the total is 140?
  2. What is the total if 6% of the total is 91?

Your Turn 3.79

1.
What is the total if 25% of the total is 30?
2.
What is the total if 45% of the total is 360?

The percentage can be found if the total and the percentage of the total is known. If you know the total, and the percentage of the total, first divide the part by the total. Move the decimal two places to the right and append the symbol %. The percentage may be found using the following formula.

FORMULA

The percentage, nn, of bb that is aa is ab×100%ab×100%.

Example 3.80

Finding the Percentage When the Total and Percentage of the Total Are Known

Find the percentage in the following:

  1. Total is 300, percentage of the total is 60.
  2. Total is 440, percentage of the total is 176.

Your Turn 3.80

Find the percentage in the following:
1.
Total is 1,000, percentage of the total is 70.
2.
Total is 500, percentage of the total is 425.

Solve Problems Using Percent

In the media, in research, and in casual conversation percentages are used frequently to express proportions. Understanding how to use percent is vital to consuming media and understanding numbers. Solving problems using percentages comes down to identifying which of the three components of a percentage you are given, the total, the percentage, or the percentage of the total. If you have two of those components, you can find the third using the methods outlined previously.

Example 3.81

Percentage of Students Who Are Sleep Deprived

A study revealed that 70% of students suffer from sleep deprivation, defined to be sleeping less than 8 hours per night. If the survey had 400 participants, how many of those participants had less than 8 hours of sleep per night?

Your Turn 3.81

1.
Riley has a daily calorie intake of 2,200 calories and wants to take in 20% of their calories as protein. How many calories of protein should be in their daily diet?

Example 3.82

Amazon Prime Subscribers

There are 126 million users who are U.S. Amazon Prime subscribers. If there are 328.2 million residents in the United States, what percentage of U.S. residents are Amazon Prime subscribers?

Your Turn 3.82

1.
A small town has 450 registered voters. In the primaries, 54 voted. What percentage of registered voters in that town voted in the primaries?

Example 3.83

Finding the Percentage When the Total and Percentage of the Total Are Known

Evander plays on the basketball team at their university and 73% of the athletes at their university receive some sort of scholarship for attending. If they know 219 of the student-athletes receive some sort of scholarship, how many student-athletes are at the university?

Your Turn 3.83

1.
A store declares a deep discount of 40% for an item, which they say will save $30. What was the original price of the item?

Check Your Understanding

17.
Identify which of the following are rational numbers.
41 , 13 , 4 3 , 2.75 , 0.2 13 ¯
18.
Express 18 30 in lowest terms.
19.
Calculate 3 8 + 5 12 and express in lowest terms.
20.
Convert 0.34 into fraction form.
21.
Convert 47 12 into a mixed number.
22.
Calculate 2 9 × 21 22 and express in lowest terms.
23.
Calculate 2 5 ÷ 3 10 + 1 6 .
24.
Identify a rational number between 7 8 and 20 21 .
25.
Rewrite 3 2 7 as an improper fraction.
26.
Lina decides to save 1 8 of her take-home pay every paycheck. Her most recent paycheck was for $882. How much will she save from that paycheck?
27.
Determine 38% of 600.
28.
A microchip factory has decided to increase its workforce by 10%. If it currently has 70 employees, how many new employees will the factory hire?
For the following exercises, identify which of the following are rational numbers.
1 .
4.598
2 .
144
3 .
131
For the following exercises, reduce the fraction to lowest terms
4 .
8 10
5 .
30 105
6 .
36 539
7 .
231 490
8 .
750 17 , 875
For the following exercises, do the indicated conversion. If it is a repeating decimal, use the correct notation.
9 .
Convert 25 6 to a mixed number.
10 .
Convert 240 53 to a mixed number.
11 .
Convert 2 3 8 to an improper fraction.
12 .
Convert 15 7 30 to an improper fraction.
13 .
Convert 4 9 to decimal form.
14 .
Convert 13 20 to decimal form.
15 .
Convert 27 625 to decimal form.
16 .
Convert 11 14 to decimal form.
17 .
Convert 0.23 to fraction form and reduce to lowest terms.
18 .
Convert 3.8874 to fraction form and reduce to lowest terms.
For the following exercises, perform the indicated operations. Reduce to lowest terms.
19 .
3 5 + 3 10
20 .
3 14 + 8 21
21 .
13 36 14 99
22 .
13 24 4 117
23 .
3 7 × 21 48
24 .
48 143 × 77 120
25 .
14 27 ÷ 7 12
26 .
44 75 ÷ 484 285
27 .
( 3 5 + 2 7 ) × 10 21
28 .
3 8 × ( 13 12 35 36 )
29 .
( 3 7 + 5 16 ) 2 5 12
30 .
3 8 × ( 4 9 1 8 ) 2
31 .
( 2 5 × ( 7 8 2 3 ) ) 2 ÷ ( 4 9 + 5 6 ) + 7 12
32 .
( 1 5 ÷ ( 3 10 + 11 15 ) ) × ( 2 21 + 5 9 ) ( 8 15 ÷ 4 33 ) 2
33 .
Find a rational number between 8 17 and 15 28
34 .
Find a rational number between 3 50 and 13 98 .
35 .
Find two rational numbers between 3 10 and 19 45 .
36 .
Find three rational numbers between 5 12 and 175 308 .
37 .
Convert 24% to fraction form and reduce completely.
38 .
Convert 95% to fraction form and reduce completely.
39 .
Convert 0.23 to a percentage.
40 .
Convert 1.22 to a percentage.
41 .
Determine 30% of 250.
42 .
Determine 75% of 600.
43 .
If 25% of a group is 41 members, how many members total are in the group?
44 .
If 80% of the total is 60, how much is in the total?
45 .
13 is what percent of 20?
46 .
80 is what percent of 320?
47 .
Professor Donalson’s history of film class has 60 students. Of those students, 2 5 say their favorite movie genre is comedy. How many of the students in Professor Donalson’s class name comedy as their favorite movie genre?
48 .
Naia’s dormitory floor has 80 residents. Of those, 3 8 play Fortnight for at least 15 hours per week. How many students on Naia’s floor play Fortnight at least 15 hours per week?
49 .
In Tara’s town there are 24,000 people. Of those, 13 100 are food insecure. How many people in Tara’s town are food insecure?
50 .
Roughly 4 5 of air is nitrogen. If an enclosure holds 2,000 liters of air, how many liters of nitrogen should be expected in the enclosure?
51 .
To make the dressing for coleslaw, Maddie needs to mix it with 3 5 mayonnaise and 2 5 apple cider vinegar. If Maddie wants to have 8 cups of dressing, how many cups of mayonnaise and how many cups of apple cider vinegar does Maddie need?
52 .
Malika is figuring out their schedule. They wish to spend 4 15 of their time sleeping, 1 3 of their time studying and going to class, 1 5 of their time at work, and 2 15 of their time doing other activities, such as entertainment or exercising. There are 168 hours in a week. How many hours in a week will Malika spend:
  1. Sleeping?
  2. Studying and going to class?
  3. Not sleeping?
53 .
Roughly 20.9% of air is oxygen. How much oxygen is there in 200 liters of air?
54 .
65% of college students graduate within 6 years of beginning college. A first-year cohort at a college contains 400 students. How many are expected to graduate within 6 years?
55 .
A 20% discount is offered on a new laptop. How much is the discount if the new laptop originally cost $700?
56 .
Leya helped at a neighborhood sale and was paid 5% of the proceeds. If Leya is paid $171.25, what were the total proceeds from the neighborhood sale?
57 .
Unit Conversion. 1 kilogram (kg) is equal to 2.20462 pounds. Convert 13 kg to pounds. Round to three decimal places, if necessary.
58 .
Unit Conversion. 1 kilogram (kg) is equal to 2.20462 pounds. Convert 200 pounds to kilograms. Round to three decimal places, if necessary.
59 .
Unit Conversion. There are 12 inches in a foot, 3 feet in a yard, and 1,760 yards in a mile. Convert 10 miles to inches. To do so, first convert miles to yards. Next, convert the yards to feet. Last, convert the feet to inches.
60 .
Unit Conversion. There are 1,000 meters (m) in a kilometer (km), and 100 centimeters (cm) in a meter. Convert 4 km to centimeters.
61 .
Markup. In this exercise, we introduce the concept of markup. The markup on an item is the difference between how much a store sells an item for and how much the store paid for the item. Suppose Wegmans (a northeastern U.S. grocery chain) buys cereal at $1.50 per box and sells the cereal for $2.29.
  1. Determine the markup in dollars.
  2. The markup is what percent of the original cost? Round the percentage to one decimal place.
62 .
In this exercise, we explore what happens when an item is marked up by a percentage, and then marked down using the same percentage.
Wegmans purchases an item for $5 per unit. The markup on the item is 25%.
  1. Calculate the markup on the item, in dollars.
  2. What is the price for which Wegmans sells the item? This is the price Wegmans paid, plus the markup.
  3. Suppose Wegmans then offers a 25% discount on the sale price of the item (found in part b). In dollars, how much is the discount?
  4. Determine the price of the item after the discount (this is the sales price of the item minus the discount). Round to two decimal places.
  5. Is the new price after the markup and discount equal to the price Wegmans paid for the item? Explain.
63 .
Repeated Discounts. In this exercise, we explore applying more than one discount to an item.
Suppose a store cuts the price on an item by 50%, and then offers a coupon for 25% off any sale item. We will find the price of the item after applying the sale price and the coupon discount.
  1. The original price was $150. After the 50% discount, what is the price of the item?
  2. The coupon is applied to the discount price. The coupon is for 25%. Find 25% of the sale price (found in part a).
  3. Find the price after applying the coupon (this is the value from part a minus the value from part b).
  4. The total amount saved on the item is the original price after all the discounts. Determine the total amount saved by subtracting the final price paid (part c) from the original price of the item.
  5. Determine the effective discount percentage, which is the total amount saved divided by the original price of the item.
  6. Was the effective discount percentage equal to 75%, which would be the 50% plus the 25%? Explain.
Converting Repeating Decimals to a Fraction
It was mentioned in the section that repeating decimals are rational numbers. To convert a repeating decimal to a rational number, perform the following steps:
Step 1: Label the original number S .
Step 2: Count the number of digits, n , in the repeating part of the number.
Step 3: Multiply S by 10 n , and label this as 10 n × S .
Step 4: Determine 10 n 1 .
Step 5: Calculate 10 n × S S . If done correctly, the repeating part of the number will cancel out.
Step 6: If the result from Step 5 has decimal digits, count the number of decimal digits in the number from Step 5. Label this m .
Step 7: Remove the decimal from the result of Step 5.
Step 8: Add m zeros to the end of the number from Step 4.
Step 9: Divide the result from Step 7 by the result from Step 8. This is the fraction form of the repeating decimal.
64 .
Convert 0. 7 ¯ to fraction form.
65 .
Convert 0. 45 ¯ to fraction form.
66 .
Convert 3.1 5 ¯ to fraction form.
67 .
Convert 2.71 94 ¯ to fraction form.
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