### 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:

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 “

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)

c)

d)

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.

Whenever you are working with expressions, you can check your answers by substituting in numbers for the variables. For example, you could check part

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

One way to solve this is to think of money. For example, we know that *one third of sixty dollars* is written 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.

Notice that

When multiplying fractions:

This rule doesn’t just hold for the product of two fractions, but for any number of fractions.

2. Evaluate and simplify

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

**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:**

*We can see a geometrical interpretation of* **The Commutative Property of Multiplication** *to the right. The Area of the shape (length×width) 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:**

**Multiplicative Identity Property:** The product of one and any number is that number.

**Example:**

**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

**Example:**

#### 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 8×7 meter plot, or two smaller plots of 3×7 and 5×7 meters. Which option gives him the largest area for his potatoes?*

In the first option, the gardener has a total area of

In the second option, the gardener has

### Examples

Multiply the following rational numbers:

#### Example 1

With this problem, we can cancel the fives:

#### Example 2

With this problem, we can multiply all the numerators and all the denominators:

#### Example 3

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.

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

In 5-10, 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*}

### Review (Answers)

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