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2.4: Multiplication of Rational Numbers

Difficulty Level: At Grade Created by: CK-12
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When you began learning how to multiply whole numbers, you replaced repeated addition with the multiplication sign \begin{align*}(\times)\end{align*}. For example,

\begin{align*}6 + 6 + 6 + 6 + 6 = 5 \times 6 = 30\end{align*}

Multiplying rational numbers is performed the same way. We will start with the Multiplication Property of –1.

The Multiplication Property of –1: For any real numbers \begin{align*}a, (-1) \times a = -a\end{align*}.

This can be summarized by saying, "A number times a negative is the opposite of the number."

Example 1: Evaluate \begin{align*}-1 \cdot 9,876\end{align*}.

Solution: Using the Multiplication Property of \begin{align*}-1\end{align*}: \begin{align*} \ -1 \cdot 9,876 = -9,876\end{align*}.

This property can also be used when the values are negative, as shown in Example 2.

Example 2: Evaluate \begin{align*}-1 \cdot -322\end{align*}.

Solution: Using the Multiplication Property of \begin{align*}-1\end{align*}: \begin{align*} \ -1 \cdot -322 = 322\end{align*}.

A basic algebraic property is the Multiplicative Identity. Similar to the Additive Identity, this property states that any value multiplied by 1 will result in the original value.

The Multiplicative Identity Property: For any real numbers \begin{align*}a, \ (1) \times a = a\end{align*}.

A third property of multiplication is the Multiplication Property of Zero. This property states that any value multiplied by zero will result in zero.

The Zero Property of Multiplication: For any real numbers \begin{align*}a, \ (0) \times a = 0\end{align*}.

Multiplying Rational Numbers

You’ve decided to make cookies for a party. The recipe you’ve chosen makes 6 dozen cookies, but you only need 2 dozen. How do you reduce the recipe?

In this case, you should not use subtraction to find the new values. Subtraction means to make less by taking away. You haven’t made any cookies; therefore, you cannot take any away. Instead, you need to make \begin{align*}\frac{2}{6}\end{align*} or \begin{align*}\frac{1}{3}\end{align*} of the original recipe. This process involves multiplying fractions.

For any real numbers \begin{align*}a, b, c,\end{align*} and \begin{align*}d\end{align*}, where \begin{align*}\ b \neq 0\end{align*} and \begin{align*}\ d \neq 0\end{align*},

\begin{align*}\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}\end{align*}

Example 3: The original cookie recipe calls for 8 cups flour. How much is needed for the reduced recipe?

Solution: Begin by writing the multiplication situation. \begin{align*}8 \cdot \frac{1}{3}\end{align*}. You need to rewrite this product in the form of the property above. In order to perform this multiplication, you need to rewrite 8 as the fraction \begin{align*}\frac{8}{1}\end{align*}.

\begin{align*}8 \times \frac{1}{3} = \frac{8}{1} \times \frac{1}{3} = \frac{8 \cdot 1}{1 \cdot 3} = \frac{8}{3} = 2 \frac{2}{3}\end{align*}

You will need \begin{align*}2 \ \frac{2}{3}\end{align*} cups flour.

Multiplication of fractions can also be shown visually. For example, to multiply \begin{align*}\frac{1}{3} \cdot \frac{2}{5}\end{align*}, draw one model to represent the first fraction and a second model to represent the second fraction.

By placing one model (divided in thirds horizontally) on top of the other (divided in fifths vertically), you divide one whole rectangle into \begin{align*}bd\end{align*} smaller parts. Shade \begin{align*}ac\end{align*} smaller regions.

The product of the two fractions is the \begin{align*}\frac{shaded \ regions}{total \ regions}.\end{align*}

\begin{align*}\frac{1}{3} \cdot \frac{2}{5} = \frac{2}{15}\end{align*}

Example 4: Simplify \begin{align*}\frac{3}{7} \cdot \frac{4}{5}.\end{align*}

Solution: By drawing visual representations, you can see that

\begin{align*}\frac{3}{7} \cdot \frac{4}{5} = \frac{12}{35}\end{align*}

Multiplication Properties

Properties that hold true for addition such as the Associative Property and Commutative Property also hold true for multiplication. They are summarized below.

The Associative Property of Multiplication: For any real numbers \begin{align*}a, \ b,\end{align*} and \begin{align*}c,\end{align*}

\begin{align*}(a \cdot b)\cdot c = a \cdot (b \cdot c)\end{align*}

The Commutative Property of Multiplication: For any real numbers \begin{align*}a\end{align*} and \begin{align*}b,\end{align*}

\begin{align*}a(b) = b(a)\end{align*}

The Same Sign Multiplication Rule: The product of two positive or two negative numbers is positive.

The Different Sign Multiplication Rule: The product of a positive number and a negative number is a negative number.

Solving Real-World Problems Using Multiplication

Example 5: Anne has a bar of chocolate and she offers Bill a piece. Bill quickly breaks off \begin{align*}\frac{1}{4}\end{align*} of the bar and eats it. Another friend, Cindy, takes \begin{align*}\frac{1}{3}\end{align*} of what was left. Anne splits the remaining candy bar into two equal pieces, which she shares with a third friend, Dora. How much of the candy bar does each person get?

Solution: Think of the bar as one whole.

\begin{align*}1- \frac{1}{4} = \frac{3}{4}\end{align*}. This is the amount remaining after Bill takes his piece.

\begin{align*}\frac{1}{3} \times \frac{3}{4} = \frac{1}{4}\end{align*}. This is the fraction Cindy receives.

\begin{align*}\frac{3}{4} - \frac{1}{4} = \frac{2}{4} = \frac{1}{2}\end{align*}. This is the amount remaining after Cindy takes her piece.

Anne divides the remaining bar into two equal pieces. Every person receives \begin{align*}\frac{1}{4}\end{align*} of the bar.

Example 6: Doris’s truck gets \begin{align*}10 \frac{2}{3}\end{align*} miles per gallon. Her tank is empty so she puts in \begin{align*}5 \frac{1}{2}\end{align*} gallons of gas.

How far can she travel?

Solution: Begin by writing each mixed number as an improper fraction.

\begin{align*}10 \frac{2}{3} = \frac{32}{3} && 5 \frac{1}{2} = \frac{11}{2}\end{align*}

Now multiply the two values together.

\begin{align*}\frac{32}{3} \cdot \frac{11}{2} = \frac{352}{6} = 58\frac{4}{6} = 58\frac{2}{3}\end{align*}

Doris can travel \begin{align*}58 \ \frac{2}{3}\end{align*} miles on 5.5 gallons of gas.

Practice Set

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Multiplication of Rational Numbers (8:56)

Multiply the following rational numbers.

  1. \begin{align*}\frac{1}{2} \cdot \frac{3}{4}\end{align*}
  2. \begin{align*}-7.85 \cdot -2.3\end{align*}
  3. \begin{align*}\frac{2}{5} \cdot \frac{5}{9}\end{align*}
  4. \begin{align*}\frac{1}{3} \cdot \frac{2}{7} \cdot \frac{2}{5}\end{align*}
  5. \begin{align*}4.5 \cdot -3\end{align*}
  6. \begin{align*}\frac{1}{2} \cdot \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{4}{5}\end{align*}
  7. \begin{align*}\frac{5}{12} \times \frac{9}{10}\end{align*}
  8. \begin{align*}\frac{27}{5} \cdot 0 \end{align*}
  9. \begin{align*}\frac{2}{3} \times \frac{1}{4}\end{align*}
  10. \begin{align*}-11.1 (4.1)\end{align*}
  11. \begin{align*}\frac{3}{4} \times \frac{1}{3}\end{align*}
  12. \begin{align*}\frac{15}{11} \times \frac{9}{7}\end{align*}
  13. \begin{align*}\frac{2}{7} \cdot -3.5\end{align*}
  14. \begin{align*}\frac{1}{13} \times \frac{1}{11}\end{align*}
  15. \begin{align*}\frac{7}{27} \times \frac{9}{14}\end{align*}
  16. \begin{align*}\left (\frac{3}{5} \right )^2\end{align*}
  17. \begin{align*}\frac{1}{11} \times \frac{22}{21} \times \frac{7}{10}\end{align*}
  18. \begin{align*}5.75 \cdot 0\end{align*}

Multiply the following by negative one.

  1. 79.5
  2. \begin{align*}\pi\end{align*}
  3. \begin{align*}(x + 1)\end{align*}
  4. \begin{align*}|x|\end{align*}
  5. 25
  6. –105
  7. \begin{align*}x^2\end{align*}
  8. \begin{align*}(3 + x)\end{align*}
  9. \begin{align*}(3 - x)\end{align*}

In 28 – 30, state the property that applies to each of the following situations.

  1. A gardener is planting vegetables for the coming growing season. He wishes to plant potatoes and has a choice of a single 8 by 7 meter plot, or two smaller plots of 3 by 7 meters and 5 by 7 meters. Which option gives him the largest area for his potatoes?
  2. Andrew is counting his money. He puts all his money into $10 piles. He has one pile. How much money does Andrew have?
  3. Nadia and Peter are raising money by washing cars. Nadia is charging $3 per car, and she washes five cars in the first morning. Peter charges $5 per car (including a wax). In the first morning, he washes and waxes three cars. Who has raised the most money?

Mixed Review

  1. Compare these rational numbers: \begin{align*}\frac{16}{27}\end{align*} and \begin{align*}\frac{2}{3}\end{align*}.
  2. Define rational numbers.
  3. Give an example of a proper fraction. How is this different from an improper fraction?
  4. Which property is being applied? \begin{align*}16 - (-14) = 16 + 14 = 30\end{align*}
  5. Simplify \begin{align*}11 \frac{1}{2} + \frac{2}{9}\end{align*}.

Quick Quiz

  1. Order from least to greatest: \begin{align*}\left (\frac{5}{6}, \ \frac{23}{26}, \ \frac{31}{32}, \ \frac{3}{14} \right )\end{align*}.
  2. Simplify \begin{align*}\frac{5}{9} \times \frac{27}{4}.\end{align*}
  3. Simplify \begin{align*}|-5 + 11| - |9 - 37|\end{align*}.
  4. Add \begin{align*}\frac{21}{5} + \frac{7}{8}.\end{align*}

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