Rational Numbers and Algebra

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Questions and Answers

Match the following rational numbers with their equivalent forms:

  • = 10/10 0 = 10/10 -2 = -20/10 -5/6 = -20/24

Match the following statements with their explanations:

Rational numbers can be found between any two given rational numbers = Countless rational numbers can be found between two rational numbers -2 can be written as -20/10 = Conversion of rational numbers to equivalent forms 0 can be written as 0/10 = Conversion of rational numbers to equivalent forms Countless rational numbers can be found between -2 and 0 = Rational numbers can be found between any two given rational numbers

Match the following with their corresponding numerators:

  • = -10 0 = 0 -5/6 = -20 -2 = -20

Match the following rational numbers with their denominators:

<ul> <li>= 10 0 = 10 -5/6 = 24 -2 = 10</li> </ul>
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Match the following rational numbers with their equivalent forms:

<p>-5/6 = -20/24 -2 = -20/10 0 = 0/10</p> <ul> <li>= -10/10</li> </ul>
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Match the following with their corresponding explanations:

<p>Countless rational numbers can be found between -5/6 and 5/8 = Rational numbers can be found between any two given rational numbers -5/6 can be written as -20/24 = Conversion of rational numbers to equivalent forms 5/8 can be written as 15/24 = Conversion of rational numbers to equivalent forms -20/24 and 15/24 are equivalent forms = Conversion of rational numbers to equivalent forms</p>
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Match the following with their corresponding properties:

<p>Rational numbers = Can be found between any two given rational numbers -5/6 = Can be written in equivalent forms 0 = Can be written in equivalent forms</p> <ul> <li>= Can be written in equivalent forms</li> </ul>
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Match the following with their corresponding characteristics:

<p>Rational numbers = Can be written in the form a/b where a and b are integers -5/6 = Has a denominator of 6 0 = Has a numerator of 0</p> <ul> <li>= Has a numerator of -10</li> </ul>
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Match the following expressions with their simplified forms:

<p>2 3 5 3 1 = 1/14 − 3 2 3 1 2 = 2/35</p>
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Match the following expressions with the properties used to simplify them:

<p>2 3 5 3 1 = Commutativity − 3 2 3 1 2 = Distributivity</p>
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Match the following numbers with their additive inverses:

<p>8 = -8 -5 = 5 2 = -2 9 = -9</p>
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Match the following values of x with the verification of the equation -(-x) = x:

<p>11 = True -13 = True 15 = False -17 = True</p>
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Match the following numbers with their multiplicative inverses:

<p>8 = 1/8 -5 = -1/5 2 = 1/2 19 = 1/19</p>
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Match the following expressions with their equivalent forms:

<p>2 3 5 3 1 = 2 3/5 1 3/5</p> <ul> <li>3 2 3 1 2 = -3 2/5 1 3/5</li> </ul>
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Match the following equations with their solutions:

<ul> <li>3 2 3 1 2 = 2/35 2 3 5 3 1 = 1/14</li> </ul>
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Match the following expressions with their correct orders of operations:

<p>2 3 5 3 1 = Multiplication first, then division</p> <ul> <li>3 2 3 1 2 = Division first, then multiplication</li> </ul>
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Match the property with the correct operation for rational numbers:

<p>Commutative = Multiplication Not commutative = Subtraction</p>
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Match the operation with the correct statement for rational numbers:

<p>Addition = a + b = b + a Subtraction = a - b ≠ b - a Multiplication = a × b = b × a Division = a ÷ b ≠ b ÷ a</p>
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Match the expression with the correct calculation for rational numbers:

<p>$\frac{-3}{8} + \frac{1}{7}$ = $\frac{1}{7} + \frac{-3}{8}$ $\frac{2}{5} - \frac{3}{4}$ = $\frac{3}{4} - \frac{2}{5}$ $\frac{-7}{6} × \frac{-4}{5}$ = $\frac{-4}{5} × \frac{-7}{6}$ $\frac{-5}{3} ÷ \frac{4}{7}$ = $\frac{4}{7} ÷ \frac{-5}{3}$</p>
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Match the operation with the correct characteristic for rational numbers:

<p>Addition = Always commutative Subtraction = Never commutative Multiplication = Always commutative Division = Never commutative</p>
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Match the statement with the correct operation for rational numbers:

<p>a + b = b + a = Addition a - b ≠ b - a = Subtraction a × b = b × a = Multiplication a ÷ b ≠ b ÷ a = Division</p>
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Match the operation with the correct example for rational numbers:

<p>Addition = $\frac{-3}{8} + \frac{1}{7}$ Subtraction = $\frac{2}{5} - \frac{3}{4}$ Multiplication = $\frac{-7}{6} × \frac{-4}{5}$ Division = $\frac{-5}{3} ÷ \frac{4}{7}$</p>
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Match the operation with the correct reason for rational numbers:

<p>Addition = Commutative property holds Subtraction = Commutative property does not hold Multiplication = Commutative property holds Division = Commutative property does not hold</p>
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Match the operation with the correct result for rational numbers:

<p>Addition = Equal result Subtraction = Not equal result Multiplication = Equal result Division = Not equal result</p>
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Study Notes

Commutative Properties of Rational Numbers

  • Addition is commutative for rational numbers, meaning that the order of numbers does not change the result: a + b = b + a.
  • Subtraction is not commutative for rational numbers, meaning that the order of numbers changes the result: a - b ≠ b - a.
  • Multiplication is commutative for rational numbers, meaning that the order of numbers does not change the result: a × b = b × a.
  • Division is not commutative for rational numbers, meaning that the order of numbers changes the result: a ÷ b ≠ b ÷ a.

Rational Numbers Between Two Given Numbers

  • There are countless rational numbers between any two given rational numbers.
  • Example: Find three rational numbers between -2 and 0. Solution: Convert -2 and 0 to rational numbers with the same denominators, and then find rational numbers between them.
  • Example: Find ten rational numbers between -5/6 and 5/8. Solution: Convert -5/6 and 5/8 to rational numbers with the same denominators, and then find rational numbers between them.

Multiplication of Rational Numbers

  • The commutative property of multiplication can be used to simplify multiplication of rational numbers.
  • Example: Find the value of (2 -3 × (-1 × 3 - 3)) × (- 3/5).
  • Solution: Use the commutative property of multiplication to simplify the expression.

Exercise 1.1

  • Problems involve using properties of rational numbers to simplify expressions.
  • Examples: Find the value of expressions such as (-2 × 3/5 + 3/2 × -5/6) and (-3/7 × 2/5 - 1/2 × 3/7).
  • Problems also involve finding additive inverses and multiplicative inverses of rational numbers.

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