2.6: Multiplication of Rational Numbers
Suppose you wrote a computer program that multiplies \begin{align*}\frac{1}{5}\end{align*}
Guidance
When you began learning how to multiply whole numbers, you replaced repeated addition with the multiplication sign \begin{align*}(\times)\end{align*}
\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 number \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 A
Evaluate \begin{align*}1 \cdot 9,876\end{align*}
Solution:
Using the Multiplication Property of \begin{align*}1\end{align*}
This property can also be used when the values are negative, as shown in Example B.
Example B
Evaluate \begin{align*}1 \cdot 322\end{align*}
Solution:
Using the Multiplication Property of \begin{align*}1\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 number \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 number \begin{align*}a, \ (0) \times a = 0\end{align*}
Multiplication of fractions can also be shown visually, as you can see in the example below.
Example C
Find \begin{align*}\frac{1}{3} \cdot \frac{2}{5}\end{align*}
Solution:
By placing one model (divided in thirds horizontally) on top of the other (divided in fifths vertically), you divide one whole rectangle into smaller parts.
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*}
Guided Practice
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*}
Practice
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. CK12 Basic Algebra: Multiplication of Rational Numbers (8:56)
Multiply the following rational numbers.

\begin{align*}\frac{1}{2} \cdot \frac{3}{4}\end{align*}
12⋅34 
\begin{align*}7.85 \cdot 2.3\end{align*}
−7.85⋅−2.3 
\begin{align*}\frac{2}{5} \cdot \frac{5}{9}\end{align*}
25⋅59 
\begin{align*}\frac{1}{3} \cdot \frac{2}{7} \cdot \frac{2}{5}\end{align*}
13⋅27⋅25 
\begin{align*}4.5 \cdot 3\end{align*}
4.5⋅−3 
\begin{align*}\frac{1}{2} \cdot \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{4}{5}\end{align*}
12⋅23⋅34⋅45 
\begin{align*}\frac{5}{12} \times \frac{9}{10}\end{align*}
512×910 
\begin{align*}\frac{27}{5} \cdot 0 \end{align*}
275⋅0 
\begin{align*}\frac{2}{3} \times \frac{1}{4}\end{align*}
23×14 
\begin{align*}11.1 (4.1)\end{align*}
−11.1(4.1)
Multiply the following by negative one.
 79.5

\begin{align*}\pi\end{align*}
π 
\begin{align*}(x + 1)\end{align*}
(x+1) 
\begin{align*}x\end{align*}
x  25
 –105

\begin{align*}x^2\end{align*}
x2 
\begin{align*}(3 + x)\end{align*}
(3+x) 
\begin{align*}(3  x)\end{align*}
(3−x)
Quick Quiz
 Order from least to greatest: \begin{align*}\left (\frac{5}{6}, \ \frac{23}{26}, \ \frac{31}{32}, \ \frac{3}{14} \right )\end{align*}
(56, 2326, 3132, 314) .  Simplify \begin{align*}\frac{5}{9} \times \frac{27}{4}.\end{align*}
 Simplify \begin{align*}5 + 11  9  37\end{align*}.
 Add \begin{align*}\frac{21}{5}\end{align*} and \begin{align*}\frac{7}{8}.\end{align*}
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Multiplication Property of –1
For any real number .multiplicative identity property
The product of any number and one is the number itself.Zero Property of Multiplication
For any real number .Associative Property
The associative property states that you can change the groupings of numbers being added or multiplied without changing the sum. For example: (2+3) + 4 = 2 + (3+4), and (2 X 3) X 4 = 2 X (3 X 4).Commutative Property
The commutative property states that the order in which two numbers are added or multiplied does not affect the sum or product. For example .distributive property
The distributive property states that the product of an expression and a sum is equal to the sum of the products of the expression and each term in the sum. For example, .Integer
The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., 3, 2, 1, 0, 1, 2, 3...Mixed Number
A mixed number is a number made up of a whole number and a fraction, such as .