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Dividing Fractions

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Dividing Fractions

Study Guide

Key Definition

To divide one fraction by another, multiply the first fraction by the reciprocal (multiplicative inverse) of the second: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

Important Notes

To divide $\frac{a}{b}$ by $\frac{c}{d}$, multiply by its reciprocal: $\frac{a}{b} \times \frac{d}{c}$
Always convert mixed numbers to improper fractions before dividing
Simplify the resulting fraction whenever possible
The reciprocal of a fraction is obtained by swapping its numerator and denominator
Multiplying by a reciprocal is equivalent to dividing by the original fraction

Mathematical Notation

$\div$ or $/$ represents division

$\times$ or $*$ represents multiplication

$\frac{a}{b}$ is a fraction

GCF stands for greatest common factor

Improper Fraction is a fraction whose numerator is greater than its denominator

Use proper mathematical notation when solving problems.

Why It Works

Dividing by a fraction is the same as multiplying by its reciprocal. This relies on the property of multiplicative inverses: $\frac{c}{d} \times \frac{d}{c}=1$.

Remember

Dividing by any non-zero number is the same as multiplying by its reciprocal.

Quick Reference

Basic Division:$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

Mixed Number Conversion:$3 \tfrac{3}{5} = \tfrac{18}{5}$

Understanding Dividing Fractions

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Step 1: Flip (find the reciprocal of) the second fraction. Step 2: Multiply. Example: $\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4}$

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is $\frac{2}{3} \div \frac{7}{9}$?

Real-World Problem

Question Exercise

Intermediate

Pizza Sharing

You have $\frac{3}{4}$ of a pizza. If one serving size is $\frac{5}{6}$ of a whole pizza, how many full servings can you make?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Divide $2 \tfrac{1}{2}$ by $1 \tfrac{3}{4}$.

Challenge Quiz

Single Choice Quiz

Advanced

Simplify $\frac{18}{5} \div \frac{21}{8}$.

Recap

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Review key concepts and takeaways