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*}

### Video Review

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### 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*}

### Explore More

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*}
59×274. - Simplify \begin{align*}|-5 + 11| - |9 - 37|\end{align*}
|−5+11|−|9−37| . - Add \begin{align*}\frac{21}{5}\end{align*}
215 and \begin{align*}\frac{7}{8}.\end{align*}78.

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 2.6.