Dividing Fractions

➗ Dividing Fractions

Part 1 of 5 — Concept Introduction

Welcome to dividing fractions! In Grade 5 you met multiplying and dividing fractions. Now we'll deepen your understanding and tackle trickier problems — including whole numbers and mixed numbers — with confidence.

The most reliable strategy is a three-step trick called Keep–Change–Flip (KCF), also known as "multiply by the reciprocal." Once you learn it, dividing fractions becomes just as easy as multiplying them.

The Three Steps

1. Keep the first fraction exactly as it is.
2. Change the division sign $\div$ into a multiplication sign $\times$.
3. Flip the second fraction (use its reciprocal).

In symbols:

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

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

What Is a Reciprocal? 🔄

The reciprocal of a fraction is what you get when you flip the numerator (top) and denominator (bottom) — they trade places.

Important: Every number except zero has a reciprocal. When you multiply a number by its reciprocal, you always get 1.

$\frac{3}{4} \times \frac{4}{3} = \frac{12}{12} = 1 \checkmark$

So "Flip" in Keep–Change–Flip really means "use the reciprocal of the second fraction." A whole number like $7$ secretly equals $\frac{7}{1}$, so its reciprocal is $\frac{1}{7}$.

A Worked Example

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

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

Now multiply straight across:

$\frac{3}{4} \times \frac{5}{2} = \frac{3 \times 5}{4 \times 2} = \frac{15}{8}$

Since $\frac{15}{8}$ is an improper fraction (top bigger than bottom), we can write it as a mixed number:

$\frac{15}{8} = 1\,\frac{7}{8}$

So $\frac{3}{4} \div \frac{2}{5} = 1\,\frac{7}{8}$. Keep, change, flip, multiply, simplify!

Concept Check 🎯

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

📝 Worked Examples

Part 2 of 5 — Worked Examples

Let's slow down and walk through problems one step at a time, including a whole number.

Example 1: Whole Number ÷ Fraction → $6 \div \frac{2}{3}$

Step 1 — Write the whole number as a fraction: $6 = \frac{6}{1}$

Step 2 — Keep–Change–Flip: $\frac{6}{1} \div \frac{2}{3} = \frac{6}{1} \times \frac{3}{2}$

Step 3 — Multiply:

$\frac{6}{1} \times \frac{3}{2} = \frac{6 \times 3}{1 \times 2} = \frac{18}{2} = 9$

Think about it: How many $\frac{2}{3}$ pieces fit into $6$ wholes? Nine pieces! ✅

Example 2: Fraction ÷ Whole Number → $\frac{3}{5} \div 4$

Step 1 — Write the whole number as a fraction: $4 = \frac{4}{1}$

Step 2 — Keep–Change–Flip: $\frac{3}{5} \div \frac{4}{1} = \frac{3}{5} \times \frac{1}{4}$

Step 3 — Multiply straight across:

$\frac{3}{5} \times \frac{1}{4} = \frac{3 \times 1}{5 \times 4} = \frac{3}{20}$

So $\frac{3}{5} \div 4 = \frac{3}{20}$. Notice that dividing by a whole number makes the piece smaller. 🎉

Your Turn 🧮

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

1. After you flip $\frac{2}{3}$, what is the new second fraction? (Type it like $3/2$.)

2. Multiply $\frac{3}{5} \times \frac{3}{2}$ straight across, before simplifying. What is the result? (Type it like $9/10$.)

3. The answer $\frac{9}{10}$ is already in lowest terms. Is it less than 1? Type yes or no.

🧭 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 $\frac{7}{8} \div \frac{1}{2}$.

🌍 Application & Word Problems

Part 4 of 5 — Real-World Problems

Dividing fractions answers questions like "How many small pieces fit into a larger amount?" That comes up everywhere — cooking, building, sewing, and sharing.

Trail Mix Example 🥜

You have $\frac{3}{4}$ pound of trail mix. Each snack bag holds $\frac{1}{8}$ pound. How many bags can you fill?

This is the 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 fill 6 bags! 🎉

Sharing With a Whole Number

A whole number becomes a fraction over $1$. For example, sharing $\frac{4}{5}$ of a cake among $3$ people:

$\frac{4}{5} \div 3 = \frac{4}{5} \div \frac{3}{1} = \frac{4}{5} \times \frac{1}{3} = \frac{4}{15}$

Word Problem Practice 🧮

A board is $\frac{9}{10}$ meter long. You need to cut it into pieces that are each $\frac{3}{10}$ meter long.

1. After you flip $\frac{3}{10}$, what fraction do you multiply by? (Type it like $10/3$.)

2. Multiply $\frac{9}{10} \times \frac{10}{3}$ straight across, before simplifying. (Type it like $90/30$.)

3. How many pieces can you cut? Simplify your answer. (Type a single whole number.)

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, and a number times its reciprocal equals 1.
• A whole number $n$ becomes $\frac{n}{1}$ before you flip.
• A mixed number must be changed to an improper fraction first. Example: $2\,\frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3}$.
• 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 — including a mixed number. Take your time!