Learn on PengiBig Ideas Math, Course 2Chapter 2: Rational Numbers

Lesson 4: Multiplying and Dividing Rational Numbers

In this Grade 7 lesson from Big Ideas Math, Course 2, Chapter 2, students learn how to multiply and divide rational numbers — including fractions, mixed numbers, and decimals with negative values — using the same sign rules that apply to integers. The lesson also explores why the product of two negative rational numbers is positive, using the Additive Inverse Property and properties of multiplication such as the Commutative and Associative properties to build a formal justification.

Section 1

Multiply Fractions

Property

If $a$ , $b$ , $c$ , and $d$ are numbers where $b \neq 0$ and $d \neq 0$ , then

\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}

To multiply fractions, we multiply the numerators and multiply the denominators. Then we write the fraction in simplified form.

Examples

To multiply $\frac{2}{3} \cdot \frac{4}{5}$ , multiply the numerators ( $2 \cdot 4 = 8$ ) and the denominators ( $3 \cdot 5 = 15$ ). The result is $\frac{8}{15}$ .
Multiply $-\frac{4}{7} \cdot \frac{14}{16}$ . The product will be negative. We can simplify before multiplying: $-\frac{4}{7} \cdot \frac{2 \cdot 7}{4 \cdot 4} = -\frac{1}{1} \cdot \frac{2}{4} = -\frac{2}{4}$ , which simplifies to $-\frac{1}{2}$ .
To multiply $8 \cdot \frac{3}{4}$ , first write 8 as $\frac{8}{1}$ . Then multiply: $\frac{8}{1} \cdot \frac{3}{4} = \frac{24}{4}$ . Simplifying this fraction gives 6.

Explanation

Multiplying fractions is straightforward: multiply the numerators to get the new numerator, and multiply the denominators to get the new denominator. Remember to simplify the resulting fraction by canceling any common factors for the final answer.

Section 2

Divide Fractions

Property

Reciprocal
The reciprocal of the fraction $\frac{a}{b}$ is $\frac{b}{a}$ .

Fraction Division
If $a, b, c, d$ are numbers where $b \neq 0, c \neq 0, d \neq 0$ , then

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}

Examples

To divide $\frac{2}{5} \div \frac{3}{4}$ , we keep $\frac{2}{5}$ , change $\div$ to $\cdot$ , and flip $\frac{3}{4}$ to $\frac{4}{3}$ . This gives $\frac{2}{5} \cdot \frac{4}{3} = \frac{8}{15}$ .
To divide $-\frac{5}{18} \div (-\frac{15}{24})$ , the result is positive. We calculate $\frac{5}{18} \cdot \frac{24}{15} = \frac{5 \cdot 6 \cdot 4}{3 \cdot 6 \cdot 3 \cdot 5} = \frac{4}{9}$ .
To solve $\frac{5x}{7} \div \frac{10x}{21}$ , we multiply by the reciprocal: $\frac{5x}{7} \cdot \frac{21}{10x}$ . After canceling common factors, we get $\frac{1}{1} \cdot \frac{3}{2} = \frac{3}{2}$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Multiply Fractions

Property

If $a$ , $b$ , $c$ , and $d$ are numbers where $b \neq 0$ and $d \neq 0$ , then

\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}

To multiply fractions, we multiply the numerators and multiply the denominators. Then we write the fraction in simplified form.

Examples

To multiply $\frac{2}{3} \cdot \frac{4}{5}$ , multiply the numerators ( $2 \cdot 4 = 8$ ) and the denominators ( $3 \cdot 5 = 15$ ). The result is $\frac{8}{15}$ .
Multiply $-\frac{4}{7} \cdot \frac{14}{16}$ . The product will be negative. We can simplify before multiplying: $-\frac{4}{7} \cdot \frac{2 \cdot 7}{4 \cdot 4} = -\frac{1}{1} \cdot \frac{2}{4} = -\frac{2}{4}$ , which simplifies to $-\frac{1}{2}$ .
To multiply $8 \cdot \frac{3}{4}$ , first write 8 as $\frac{8}{1}$ . Then multiply: $\frac{8}{1} \cdot \frac{3}{4} = \frac{24}{4}$ . Simplifying this fraction gives 6.

Explanation

Section 2

Divide Fractions

Property

Reciprocal
The reciprocal of the fraction $\frac{a}{b}$ is $\frac{b}{a}$ .

Fraction Division
If $a, b, c, d$ are numbers where $b \neq 0, c \neq 0, d \neq 0$ , then

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}

Examples

To divide $\frac{2}{5} \div \frac{3}{4}$ , we keep $\frac{2}{5}$ , change $\div$ to $\cdot$ , and flip $\frac{3}{4}$ to $\frac{4}{3}$ . This gives $\frac{2}{5} \cdot \frac{4}{3} = \frac{8}{15}$ .
To divide $-\frac{5}{18} \div (-\frac{15}{24})$ , the result is positive. We calculate $\frac{5}{18} \cdot \frac{24}{15} = \frac{5 \cdot 6 \cdot 4}{3 \cdot 6 \cdot 3 \cdot 5} = \frac{4}{9}$ .
To solve $\frac{5x}{7} \div \frac{10x}{21}$ , we multiply by the reciprocal: $\frac{5x}{7} \cdot \frac{21}{10x}$ . After canceling common factors, we get $\frac{1}{1} \cdot \frac{3}{2} = \frac{3}{2}$ .

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Multiply Fractions

Property

If $a$ , $b$ , $c$ , and $d$ are numbers where $b \neq 0$ and $d \neq 0$ , then

\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}

To multiply fractions, we multiply the numerators and multiply the denominators. Then we write the fraction in simplified form.

Examples

To multiply $\frac{2}{3} \cdot \frac{4}{5}$ , multiply the numerators ( $2 \cdot 4 = 8$ ) and the denominators ( $3 \cdot 5 = 15$ ). The result is $\frac{8}{15}$ .
Multiply $-\frac{4}{7} \cdot \frac{14}{16}$ . The product will be negative. We can simplify before multiplying: $-\frac{4}{7} \cdot \frac{2 \cdot 7}{4 \cdot 4} = -\frac{1}{1} \cdot \frac{2}{4} = -\frac{2}{4}$ , which simplifies to $-\frac{1}{2}$ .
To multiply $8 \cdot \frac{3}{4}$ , first write 8 as $\frac{8}{1}$ . Then multiply: $\frac{8}{1} \cdot \frac{3}{4} = \frac{24}{4}$ . Simplifying this fraction gives 6.

Explanation

Section 2

Divide Fractions

Property

Reciprocal
The reciprocal of the fraction $\frac{a}{b}$ is $\frac{b}{a}$ .

Fraction Division
If $a, b, c, d$ are numbers where $b \neq 0, c \neq 0, d \neq 0$ , then

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}

Examples

To divide $\frac{2}{5} \div \frac{3}{4}$ , we keep $\frac{2}{5}$ , change $\div$ to $\cdot$ , and flip $\frac{3}{4}$ to $\frac{4}{3}$ . This gives $\frac{2}{5} \cdot \frac{4}{3} = \frac{8}{15}$ .
To divide $-\frac{5}{18} \div (-\frac{15}{24})$ , the result is positive. We calculate $\frac{5}{18} \cdot \frac{24}{15} = \frac{5 \cdot 6 \cdot 4}{3 \cdot 6 \cdot 3 \cdot 5} = \frac{4}{9}$ .
To solve $\frac{5x}{7} \div \frac{10x}{21}$ , we multiply by the reciprocal: $\frac{5x}{7} \cdot \frac{21}{10x}$ . After canceling common factors, we get $\frac{1}{1} \cdot \frac{3}{2} = \frac{3}{2}$ .