Learn on PengienVision, Mathematics, Grade 7Chapter 1: Integers and Rational Numbers

Lesson 9: Divide Rational Numbers

In this Grade 7 enVision Mathematics lesson from Chapter 1, students learn how to divide rational numbers — including fractions, mixed numbers, and decimals — using multiplicative inverses and reciprocals. The lesson extends integer division sign rules (same signs yield a positive quotient, different signs yield a negative quotient) to all rational numbers, with students rewriting division expressions as multiplication by the reciprocal to find quotients. Real-world contexts such as glacier retreat, water drainage rates, and submarine depth changes help students apply these skills across positive and negative rational number scenarios.

Section 1

Dividing Rational Numbers in Different Forms

Property

To divide rational numbers that are in different forms, first convert them to the same form (usually fractions) and then divide.

If the signs are the same (both positive or both negative), the quotient is positive
If the signs are different (one positive, one negative), the quotient is negative

Examples

Section 2

Simplify complex fractions

Property

A complex fraction is a fraction in which the numerator or the denominator contains a fraction. To simplify a complex fraction, remember that the fraction bar means division.
Step 1. Rewrite the complex fraction as a division problem.
Step 2. Follow the rules for dividing fractions.
Step 3. Simplify if possible.

Examples

To simplify $\frac{\frac{1}{3}}{\frac{4}{9}}$ , rewrite as $\frac{1}{3} \div \frac{4}{9}$ . This becomes $\frac{1}{3} \cdot \frac{9}{4} = \frac{9}{12}$ , which simplifies to $\frac{3}{4}$ .

To simplify $\frac{5}{\frac{2}{3}}$ , rewrite as $5 \div \frac{2}{3}$ . This becomes $\frac{5}{1} \cdot \frac{3}{2} = \frac{15}{2}$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Dividing Rational Numbers in Different Forms

Property

To divide rational numbers that are in different forms, first convert them to the same form (usually fractions) and then divide.

If the signs are the same (both positive or both negative), the quotient is positive
If the signs are different (one positive, one negative), the quotient is negative

Examples

Section 2

Simplify complex fractions

Property

Examples

To simplify $\frac{\frac{1}{3}}{\frac{4}{9}}$ , rewrite as $\frac{1}{3} \div \frac{4}{9}$ . This becomes $\frac{1}{3} \cdot \frac{9}{4} = \frac{9}{12}$ , which simplifies to $\frac{3}{4}$ .

To simplify $\frac{5}{\frac{2}{3}}$ , rewrite as $5 \div \frac{2}{3}$ . This becomes $\frac{5}{1} \cdot \frac{3}{2} = \frac{15}{2}$ .

Overview

Lesson overview

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

Overview

Section 1

Dividing Rational Numbers in Different Forms

Property

To divide rational numbers that are in different forms, first convert them to the same form (usually fractions) and then divide.

If the signs are the same (both positive or both negative), the quotient is positive
If the signs are different (one positive, one negative), the quotient is negative

Examples

Section 2

Simplify complex fractions

Property

Examples

To simplify $\frac{\frac{1}{3}}{\frac{4}{9}}$ , rewrite as $\frac{1}{3} \div \frac{4}{9}$ . This becomes $\frac{1}{3} \cdot \frac{9}{4} = \frac{9}{12}$ , which simplifies to $\frac{3}{4}$ .

To simplify $\frac{5}{\frac{2}{3}}$ , rewrite as $5 \div \frac{2}{3}$ . This becomes $\frac{5}{1} \cdot \frac{3}{2} = \frac{15}{2}$ .