2.5: Multiplication of Rational Numbers
What if you had two numbers like \begin{align*}\frac{5}{6}\end{align*}
Watch This
CK12 Foundation: 0205S Multiplying Rationals
Try This
For more practice multiplying fractions, try playing the fraction game at http://www.aaamath.com/fra66mx2.htm, or the one at http://www.mathplayground.com/fractions_mult.html.
Guidance
Whenever we multiply a number by negative one, the sign of the number changes. In more mathematical terms, multiplying by negative one maps a number onto its opposite. The number line below shows two examples: \begin{align*}3 \cdot 1 = 3\end{align*}
When we multiply a number by negative one, the absolute value of the new number is the same as the absolute value of the old number, since both numbers are the same distance from zero.
The product of a number “\begin{align*}x\end{align*}
When you multiply an expression by negative one, remember to multiply the entire expression by negative one.
Example A
Multiply the following by negative one.
a) 79.5
b) \begin{align*}\pi\end{align*}
c) \begin{align*}(x + 1)\end{align*}
d) \begin{align*} x \end{align*}
Solution
a) 79.5
b) \begin{align*}\pi\end{align*}
c) \begin{align*}(x + 1) \ \text{or} \ x  1\end{align*}
d) \begin{align*} x \end{align*}
Note that in the last case the negative sign outside the absolute value symbol applies after the absolute value. Multiplying the argument of an absolute value equation (the term inside the absolute value symbol) does not change the absolute value. \begin{align*} x \end{align*}
Whenever you are working with expressions, you can check your answers by substituting in numbers for the variables. For example, you could check part \begin{align*}d\end{align*}
Careful, though—plugging in numbers can tell you if your answer is wrong, but it won’t always tell you for sure if your answer is right!
Multiply Rational Numbers
Example B
Simplify \begin{align*}\frac{1}{3} \cdot \frac{2}{5}\end{align*}
One way to solve this is to think of money. For example, we know that one third of sixty dollars is written as \begin{align*}\frac{1}{3} \cdot \$ 60\end{align*}
If we divide our rectangle into thirds one way and fifths the other way, here’s what we get:
Here is the intersection of the two shaded regions. The whole has been divided into five pieces widthwise and three pieces heightwise. We get two pieces out of a total of fifteen pieces.
Solution
\begin{align*}\frac{1}{3} \cdot \frac{2}{5} = \frac{2}{15}\end{align*}
Notice that \begin{align*}1 \cdot 2 = 2\end{align*}
When multiplying fractions: \begin{align*}\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}\end{align*}
This rule doesn’t just hold for the product of two fractions, but for any number of fractions.
Example C
Evaluate and simplify \begin{align*}\frac{12}{25} \cdot \frac{35}{42}\end{align*}
Solution
We can see that 12 and 42 are both multiples of six, 25 and 35 are both multiples of five, and 35 and 42 are both multiples of 7. That means we can write the whole product as \begin{align*}\frac{6 \cdot 2}{5 \cdot 5} \cdot \frac{5 \cdot 7}{6 \cdot 7} = \frac{6 \cdot 2 \cdot 5 \cdot 7}{5 \cdot 5 \cdot 6 \cdot 7}\end{align*}
Identify and Apply Properties of Multiplication
The four mathematical properties which involve multiplication are the Commutative, Associative, Multiplicative Identity and Distributive Properties.
Commutative property: When two numbers are multiplied together, the product is the same regardless of the order in which they are written.
Example: \begin{align*}4 \cdot 2 = 2 \cdot 4\end{align*}
We can see a geometrical interpretation of The Commutative Property of Multiplication to the right. The Area of the shape \begin{align*}(length \times width)\end{align*}
Associative Property: When three or more numbers are multiplied, the product is the same regardless of their grouping.
Example: \begin{align*}2 \cdot (3 \cdot 4) = (2 \cdot 3) \cdot 4\end{align*}
Multiplicative Identity Property: The product of one and any number is that number.
Example: \begin{align*}5 \cdot 1 = 5\end{align*}
Distributive property: The multiplication of a number and the sum of two numbers is equal to the first number times the second number plus the first number times the third number.
Example: \begin{align*}4(6 + 3) = 4 \cdot 6 + 4 \cdot 3\end{align*}
Example D
A gardener is planting vegetables for the coming growing season. He wishes to plant potatoes and has a choice of a single \begin{align*}8 \times 7\end{align*}
Solution
In the first option, the gardener has a total area of \begin{align*}(8 \times 7)\end{align*}
In the second option, the gardener has \begin{align*}(3 \times 7)\end{align*}
Watch this video for help with the Examples above.
CK12 Foundation: Multiplying Rational Numbers
Vocabulary
When multiplying an expression by negative one, remember to multiply the entire expression by negative one.
To multiply fractions, multiply the numerators and multiply the denominators: \begin{align*}\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}\end{align*}
The multiplicative properties are:

Commutative Property: The product of two numbers is the same whichever order the items to be multiplied are written. Example: \begin{align*}2 \cdot 3 = 3 \cdot 2 \end{align*}
2⋅3=3⋅2 
Associative Property: When three or more numbers are multiplied, the sum is the same regardless of how they are grouped. Example: \begin{align*} 2 \cdot (3 \cdot 4) = (2 \cdot 3) \cdot 4\end{align*}
2⋅(3⋅4)=(2⋅3)⋅4 
Multiplicative Identity Property: The product of any number and one is the original number. Example: \begin{align*}2 \cdot 1 = 2\end{align*}
2⋅1=2 
Distributive property: The multiplication of a number and the sum of two numbers is equal to the first number times the second number plus the first number times the third number. Example: \begin{align*}4(2 + 3) = 4(2) + 4(3)\end{align*}
4(2+3)=4(2)+4(3)
Guided Practice
Multiply the following rational numbers:
a) \begin{align*}\frac{2}{5} \cdot \frac{5}{9}\end{align*}
b) \begin{align*}\frac{1}{3} \cdot \frac{2}{7} \cdot \frac{2}{5}\end{align*}
c) \begin{align*}\frac{1}{2} \cdot \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{4}{5}\end{align*}
Solution
a) With this problem, we can cancel the fives: \begin{align*}\frac{2}{5} \cdot \frac{5}{9} = \frac{2 \cdot 5}{5 \cdot 9} = \frac{2}{9}\end{align*}.
b) With this problem, we multiply all the numerators and all the denominators:
\begin{align*}\frac{1}{3} \cdot \frac{2}{7} \cdot \frac{2}{5} = \frac{1 \cdot 2 \cdot 2}{3 \cdot 7 \cdot 5} = \frac{4}{105}\end{align*}
c) With this problem, we multiply all the numerators and all the denominators, and then we can cancel most of them. The 2’s, 3’s, and 4’s all cancel out, leaving \begin{align*}\frac{1}{5}\end{align*}.
With multiplication of fractions, we can simplify before or after we multiply. The next example uses factors to help simplify before we multiply.
Practice
In 14, multiply the following expressions by negative one.
 25
 105
 \begin{align*}x^2\end{align*}
 \begin{align*}( 3 + x)\end{align*}
In 510, multiply the following rational numbers. Write your answer in the simplest form.
 \begin{align*}\frac{5}{12} \times \frac{9}{10}\end{align*}
 \begin{align*}\frac{2}{3} \times \frac{1}{4}\end{align*}
 \begin{align*}\frac{3}{4} \times \frac{1}{3}\end{align*}
 \begin{align*}\frac{15}{11} \times \frac{9}{7}\end{align*}
 \begin{align*}\frac{1}{13} \times \frac{1}{11}\end{align*}
 \begin{align*}\frac{12}{15} \times \frac{35}{13} \times \frac{10}{2} \times \frac{26}{36}\end{align*}
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 .multiplicative identity property
The product of any number and one is the number itself.Image Attributions
Description
Learning Objectives
Here you'll learn how to evaluate and simplify rational expressions involving multiplication. You'll also learn how to identify and apply the properties of multiplication.