Dividing Fractions

➗ Dividing Fractions

Part 1 of 5 — Concept Introduction

You already know how to multiply fractions. Great news: dividing fractions uses that exact skill, plus one tiny twist!

The secret is a three-step trick called "Keep, Change, Flip." Once you learn it, dividing fractions is just as easy as multiplying them.

What Does "Keep, Change, Flip" Mean?

To divide one fraction by another:

1. Keep the first fraction exactly the same.
2. Change the division sign $\div$ into a multiplication sign $\times$.
3. Flip the second fraction upside down.

In symbols, it looks like this:

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

After you keep, change, and flip, you just multiply like normal!

What Is a Reciprocal? 🔄

The flipped-over fraction has a special name: the reciprocal.

A reciprocal is just a fraction turned upside down — the top and bottom numbers trade places.

Key idea: "Flip" in Keep, Change, Flip really means "use the reciprocal of the second fraction."

A whole number like $5$ secretly equals $\frac{5}{1}$, so its reciprocal is $\frac{1}{5}$.

A Worked Example

Let's solve $\frac{2}{3} \div \frac{4}{5}$ together.

• Keep the first fraction: $\frac{2}{3}$
• Change $\div$ to $\times$
• Flip the second fraction: $\frac{4}{5}$ becomes $\frac{5}{4}$

Now multiply straight across:

$\frac{2}{3} \times \frac{5}{4} = \frac{2 \times 5}{3 \times 4} = \frac{10}{12}$

Finally, simplify by dividing top and bottom by $2$:

$\frac{10}{12} = \frac{5}{6}$

So $\frac{2}{3} \div \frac{4}{5} = \frac{5}{6}$. Keep, change, flip, multiply, simplify!

Concept Check 🎯

Make sure you know what the "Flip" step does.

📝 Worked Examples

Part 2 of 5 — Worked Examples

Let's slow down and walk through two problems one step at a time.

Example 1: $\frac{1}{2} \div \frac{1}{4}$

Step 1 — Keep the first fraction: $\frac{1}{2}$

Step 2 — Change the sign: $\div$ becomes $\times$

Step 3 — Flip the second fraction: $\frac{1}{4}$ becomes $\frac{4}{1}$

Step 4 — Multiply:

$\frac{1}{2} \times \frac{4}{1} = \frac{1 \times 4}{2 \times 1} = \frac{4}{2} = 2$

This makes sense! "How many $\frac{1}{4}$ pieces fit inside $\frac{1}{2}$?" The answer is 2 — two quarters make one half. ✅

Example 2: $\frac{4}{9} \div \frac{2}{3}$

Keep: $\frac{4}{9}$

Change: $\div \to \times$

Flip: $\frac{2}{3} \to \frac{3}{2}$

Multiply:

$\frac{4}{9} \times \frac{3}{2} = \frac{4 \times 3}{9 \times 2} = \frac{12}{18}$

Simplify by dividing top and bottom by $6$:

$\frac{12}{18} = \frac{2}{3}$

So $\frac{4}{9} \div \frac{2}{3} = \frac{2}{3}$. 🎉

Your Turn 🧮

Solve $\frac{3}{5} \div \frac{2}{5}$ step by step. Type each answer in the box.

1. After you flip $\frac{2}{5}$, what is the new second fraction? (Type it like $5/2$.)

2. Multiply $\frac{3}{5} \times \frac{5}{2}$ before simplifying. What is the result? (Type it like $15/10$.)

3. Write the final simplified answer. (Type it like $3/2$.)

🧭 Guided Practice

Part 3 of 5 — Guided Practice

Use Keep, Change, Flip on each problem. Simplify your answer when you can.

Fill In the Steps 🔍

Complete the two steps for solving $\frac{3}{4} \div \frac{1}{2}$.

🌍 Application & Word Problems

Part 4 of 5 — Real-World Problems

Dividing fractions answers questions like "How many small pieces fit into a bigger amount?" That comes up all the time in real life — cooking, sharing, building, and more.

Cookie Recipe Example 🍪

A recipe uses $\frac{1}{8}$ cup of sugar per batch. You have $\frac{3}{4}$ cup of sugar. How many batches can you make?

This is a division problem: $\frac{3}{4} \div \frac{1}{8}$.

• Keep: $\frac{3}{4}$
• Change: $\div \to \times$
• Flip: $\frac{1}{8} \to \frac{8}{1}$

$\frac{3}{4} \times \frac{8}{1} = \frac{24}{4} = 6$

You can make 6 batches of cookies! 🎉

Dividing by a Whole Number

Remember: a whole number becomes a fraction over $1$. For example, $\frac{3}{4} \div 2 = \frac{3}{4} \div \frac{2}{1} = \frac{3}{4} \times \frac{1}{2} = \frac{3}{8}$.

Word Problem Practice 🧮

Maria has $\frac{7}{8}$ of a pizza left. She wants to share it equally among 3 friends.

1. Rewrite $3$ as a fraction over $1$. (Type it like $3/1$.)

2. After you flip, what do you multiply $\frac{7}{8}$ by? (Type it like $1/3$.)

3. How much pizza does each friend get? (Type it like $7/24$.)

Word Problem Check 📋

🏆 Review & Challenge

Part 5 of 5 — Review & Challenge

You made it! Here is everything you learned, all in one place.

Quick Summary Table

Remember:

• The reciprocal flips a fraction upside down.
• A whole number $n$ becomes $\frac{n}{1}$ before you flip.
• Dividing by a fraction gives the same answer as multiplying by its reciprocal.

Now try these mixed challenge problems!

Challenge Round 🎯

These mix everything together. Take your time!