Section 5.3 The Rational Numbers

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Section 5.3The Rational Numbers

• Objectives
• Define the rational numbers.
• Reduce rational numbers.
• Convert between mixed numbers and improper
  fractions.
• Express rational numbers as decimals.
• Express decimals in the form a / b.
• Multiply and divide rational numbers.
• Add and subtract rational numbers.
• Apply the density property of rational numbers.
• Solve problems involving rational numbers.

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Defining the Rational Numbers

• The set of rational numbers is the set of all
  numbers which can be expressed in the form ,
  where a and b are integers and b is not equal to
  0.
• The integer a is called the numerator.
• The integer b is called the denominator.
• Examples The following are examples of rational
  numbers
• ¼, -½, ¾, 5, 0

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Reducing a Rational Number

• If is a rational number and c is any number
  other than 0,
• The rational numbers and are called
  equivalent fractions.
• To reduce a rational number to its lowest terms,
  divide both the numerator and denominator by
  their greatest common divisor.

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Reducing a Rational Number

• Example Reduce to lowest terms.
• Solution Begin by finding the greatest common
  divisor of 130 and 455.
• Thus, 130 2 5 13, and 455 5 7 13. The
  greatest common divisor is 5 13 or 65.

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Reducing a Rational NumberExample Continued

• Divide the numerator and the denominator of the
  given rational number by 5 13 or 65.
• There are no common divisors of 2 and 7 other
  than 1. Thus, the rational number is in its
  lowest terms.

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Mixed Numbers and Improper Fractions

• A mixed number consists of the sum of an integer
  and a rational number, expressed without the use
  of an addition sign.
• Example
• An improper fraction is a rational number whose
  numerator is greater than its denominator.
• Example

19 is larger than 5
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Mixed Numbers and Improper FractionsConverting
from Mixed Number to an Improper Fraction

• Multiply the denominator of the rational number
  by the integer and add the numerator to this
  product.
• Place the sum in step 1 over the denominator on
  the mixed number.
• Example Convert to an improper fraction.
• Solution

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Mixed Numbers and Improper FractionsConverting
from an Improper Fraction to a Mixed Number

1. Divide the denominator into the numerator. Record
   the quotient and the remainder.
2. Write the mixed number using the following form

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Mixed Numbers and Improper FractionsConverting
from an Improper Fraction to a Mixed Number

• Example Convert to a mixed number.
• Solution Step 1. Divide the denominator into the
  numerator.
• Step 2. Write the mixed number with the
• Thus,

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Rational Numbers and Decimals

• Any rational number can be expresses as a decimal
  by dividing the denominator in to the numerator.
• Example Express each rational number as a
  decimal.
• a. b.
• Solution In each case, divide the denominator
  into the numerator.

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Rational Numbers and DecimalsExample Continued

1. b.

Notice the digits 63 repeat over and over
indefinitely. This is called a repeating decimal.
Notice the decimal stops. This is called a
terminating decimal.
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Expressing Decimals as a Quotient of Two Integers

• Terminating decimals can be expressed with
  denominators of 10, 100, 1000, 10,000, and so on.
• Using the chart, the digits to the right of the
  decimal point are the numerator of the rational
  number.

Example Express each terminating decimal as a
quotient of integers a. 0.7 b. 0.49 c.
0.048.
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Expressing Decimals as a Quotient of Two
IntegersExample Continued

• Solution
• 0.7 because the 7 is in the tenths
  position.
• 0.49 because the digit on the right, 9,
  is in the
• hundredths position.
• 0.048 because the digit on the right,
  8, is in the
• thousandths position. Reducing to lowest terms,

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Expressing Decimals as a Quotient of Two
IntegersRepeating Decimals

• Example Express as a quotient of
  integers.
• Solution Step1. Let n equal the repeating
  decimal such that n , or 0.6666
• Step 2. If there is one repeating digit, multiply
  both sides of the equation in step 1 by 10.
• n 0.66666
• 10n 10(0.66666)
• 10n 6.66666

Multiplying by 10 moves the decimal point one
place to the right.
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Expressing Decimals as a Quotient of Two
IntegersRepeating DecimalsExample Continued

• Step 3. Subtract the equation in step 1 from the
  equation in step 2.
• Step 4. Divide both sides of the equation in step
  3 by the number in front of n and solve for n.
• We solve 9n 6 for n

Thus, .
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Multiplying Rational Numbers

• The product of two rational numbers is the
  product of their numerators divided by the
  product of their denominator.
• If and are rational numbers, then
  .
• Example Multiply. If possible, reduce the
  product to its lowest terms

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Multiplying Rational NumbersExample Continued

• Solution

Multiply across.
Simplify to lowest terms.
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Dividing Rational Numbers

• The quotient of two rational numbers is a product
  of the first number and the reciprocal of the
  second number.
• If and are rational numbers, then
• Example Divide. If possible, reduce the quotient
  to its lowest terms

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Dividing Rational NumbersExample Continued

• Solution

Change to multiplication by using the reciprocal.
Multiply across.
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Adding and Subtracting Rational NumbersIdentical
Denominators

• The sum or difference of two rational numbers
  with identical denominators is the sum or
  difference of their numerators over the common
  denominator.
• If and are rational numbers, then
• and

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Adding and Subtracting Rational NumbersIdentical
Denominators

• Example Perform the indicated operations
• a. b. c.
• Solution

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Adding and Subtracting Rational NumbersUnlike
Denominators

• If the rational numbers to be added or subtracted
  have different denominators, we use the least
  common multiple of their denominators to rewrite
  the rational numbers.
• The least common multiple of their denominators
  is called the least common denominator or LCD.

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Adding and Subtracting Rational NumbersUnlike
Denominators

• Example Find the sum of .
• Solution Find the least common multiple of 4
  and 6 so that the denominators will be identical.
  LCM of 4 and 6 is 12. Hence, 12 is the LCD.

We multiply the first rational number by 3/3 and
the second one by 2/2 to obtain 12 in the
denominator for each number.
Notice, we have 12 in the denominator for each
number.
Thus, we add across to obtain the answer.
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Density of Rational Numbers

• If r and t represent rational numbers, with r lt
  t, then there is a rational number s such that s
  is between r and t
• r lt s lt t.
• Example Find a rational number halfway between ½
  and ¾.
• Solution First add ½ and ¾.

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Density of Rational NumbersExample Continued

• Next, divide this sum by 2.
• The number is halfway between ½ and ¾. Thus,

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Problem Solving with Rational Numbers

• A common application of rational numbers involves
  preparing food for a different number of servings
  than what the recipe gives.
• The amount of each ingredient can be found as
  follows

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Problem Solving with Rational NumbersChanging
the Size of a Recipe

• A chocolate-chip recipe for five dozen cookies
  requires ¾ cup of sugar. If you want to make
  eight dozen cookies, how much sugar is needed?
• Solution

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Problem Solving with Rational NumbersChanging
the Size of a Recipe

• The amount of sugar, in cups, needed is
  determined by multiplying the rational numbers
• Thus, cups of sugar is needed.