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# 2.5: Multiplication of Rational Numbers

Difficulty Level: At Grade Created by: CK-12
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### Multiplication of Rational Numbers

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*} and \begin{align*}-1 \cdot -1 = 1\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*}” and negative one is \begin{align*}-x\end{align*}. This does not mean that \begin{align*}-x\end{align*} is necessarily less than zero! If \begin{align*}x\end{align*} itself is negative, then \begin{align*}-x\end{align*} will be positive because a negative times a negative (negative one) is a positive.

When you multiply an expression by negative one, remember to multiply the entire expression by negative one.

#### Multiplying Rational Numbers

Multiply the following by negative one.

a) 79.5

-79.5

b) \begin{align*}\pi\end{align*}

\begin{align*}-\pi\end{align*}

c) \begin{align*}(x + 1)\end{align*}

\begin{align*}-(x + 1) \ \text{or} \ -x - 1\end{align*}

d) \begin{align*}| x |\end{align*}

\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*} is always positive. \begin{align*}| -x |\end{align*} is always positive. \begin{align*}-| x |\end{align*} is always negative.

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*} of Example 1 by letting \begin{align*}x = -3\end{align*}. Then you’d see that \begin{align*}| -3 | \neq -| 3 |\end{align*}, because \begin{align*}| -3 | = 3\end{align*} and \begin{align*}-| 3 | = -3\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!

#### Multiplying Fractions

1. 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*}. We can read the above problem as one-third of two-fifths. Here is a visual picture of the fractions one-third and two-fifths.

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 width-wise and three pieces height-wise. We get two pieces out of a total of fifteen pieces.

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

Notice that \begin{align*}1 \cdot 2 = 2\end{align*} and \begin{align*}3 \cdot 5 = 15\end{align*}. This turns out to be true in general: when you multiply rational numbers, the numerators multiply together and the denominators multiply together. Or, to put it more formally:

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.

2. Evaluate and simplify \begin{align*}\frac{12}{25} \cdot \frac{35}{42}\end{align*}.

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*}. Then we can cancel out the 5, the 6, and the 7, leaving \begin{align*}\frac{2}{5}\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*} is the same no matter which way we draw it.

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 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 expressions \begin{align*}a, b, \end{align*} and \begin{align*}c\end{align*}, \begin{align*}a(b+c)=ab+ac\end{align*}.

Example: \begin{align*}4(6 + 3) = 4 \cdot 6 + 4 \cdot 3\end{align*}

#### Real-World Application

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*} meter plot, or two smaller plots of \begin{align*}3 \times 7 \end{align*} and \begin{align*}5 \times 7 \end{align*} meters. Which option gives him the largest area for his potatoes?

In the first option, the gardener has a total area of \begin{align*}(8 \times 7)\end{align*} or 56 square meters.

In the second option, the gardener has \begin{align*}(3 \times 7)\end{align*} or 21 square meters, plus \begin{align*}(5 \times 7)\end{align*} or 35 square meters. \begin{align*}21 + 35 = 56\end{align*}, so the area is the same as in the first option.

### Examples

Multiply the following rational numbers:

#### Example 1

\begin{align*}\frac{2}{5} \cdot \frac{5}{9}\end{align*}

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*}

#### Example 2

\begin{align*}\frac{1}{3} \cdot \frac{2}{7} \cdot \frac{2}{5}\end{align*}

With this problem, we can 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*}

#### Example 3

\begin{align*}\frac{1}{2} \cdot \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{4}{5}\end{align*}

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.

### Review

In 1-4, multiply the following expressions by negative one.

1. 25
2. -105
3. \begin{align*}x^2\end{align*}
4. \begin{align*}( 3 + x)\end{align*}

In 5-10, multiply the following rational numbers. Write your answer in the simplest form.

1. \begin{align*}\frac{5}{12} \times \frac{9}{10}\end{align*}
2. \begin{align*}\frac{2}{3} \times \frac{1}{4}\end{align*}
3. \begin{align*}\frac{3}{4} \times \frac{1}{3}\end{align*}
4. \begin{align*}\frac{15}{11} \times \frac{9}{7}\end{align*}
5. \begin{align*}\frac{1}{13} \times \frac{1}{11}\end{align*}
6. \begin{align*}\frac{12}{15} \times \frac{35}{13} \times \frac{10}{2} \times \frac{26}{36}\end{align*}

To view the Review answers, open this PDF file and look for section 2.5.

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### Vocabulary Language: English

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 $a+b=b+a \text{ and\,} (a)(b)=(b)(a)$.

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, $a(b + c) = ab + ac$.

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 $4\frac{3}{5}$.

multiplicative identity property

The product of any number and one is the number itself.

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