Multiplying and Dividing Fractions

✖️➗ Multiplying and Dividing Fractions

Part 1 of 5 — Concept Introduction

In Grade 7 you put fractions to work. Two of the most powerful skills are multiplying and dividing them. The best part? Both come down to multiplying — once you learn one little trick for division, you are done.

Multiplying Fractions

To multiply two fractions, you do not need a common denominator. Just follow three steps:

1. Multiply the numerators (the tops).
2. Multiply the denominators (the bottoms).
3. Simplify the result.

In symbols:

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

Example:

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

Since $8$ and $15$ share no common factors, $\frac{8}{15}$ is already in simplest form. 🎉

The Reciprocal 🔄

The reciprocal of a fraction is what you get when you flip it upside down — the top and bottom numbers swap places.

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

The reciprocal is the secret ingredient for dividing fractions, which you will use next.

Dividing Fractions

Here is the one trick you need: to divide by a fraction, multiply by its reciprocal (flip the second fraction).

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

Example:

$\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8} = 1\frac{7}{8}$

Notice that the division turned into a multiplication the instant we flipped the second fraction. Many people remember this as "Keep, Change, Flip": keep the first fraction, change $\div$ to $\times$, and flip the second fraction.

Concept Check 🎯

Make sure you understand the division rule.

📝 Worked Examples

Part 2 of 5 — Worked Examples

Let's work two problems slowly, one step at a time.

Example 1: Multiplying Mixed Numbers

Solve $2\frac{1}{3} \times 1\frac{1}{2}$.

Step 1 — Convert to improper fractions. Always do this first with mixed numbers!

• $2\frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3}$
• $1\frac{1}{2} = \frac{1 \times 2 + 1}{2} = \frac{3}{2}$

Step 2 — Multiply straight across:

$\frac{7}{3} \times \frac{3}{2} = \frac{7 \times 3}{3 \times 2} = \frac{21}{6}$

Step 3 — Simplify by dividing top and bottom by $3$:

$\frac{21}{6} = \frac{7}{2} = 3\frac{1}{2}$

So $2\frac{1}{3} \times 1\frac{1}{2} = 3\frac{1}{2}$. ✅

Example 2: Dividing Fractions

Solve $\frac{3}{4} \div \frac{2}{5}$ using Keep, Change, Flip.

Keep the first fraction: $\frac{3}{4}$

Change the sign: $\div$ becomes $\times$

Flip the second fraction: $\frac{2}{5}$ becomes $\frac{5}{2}$

Multiply:

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

Since $\frac{15}{8}$ is an improper fraction, rewrite it as a mixed number. $8$ goes into $15$ once with $7$ left over:

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

So $\frac{3}{4} \div \frac{2}{5} = 1\frac{7}{8}$. 🎉

Your Turn 🧮

Solve $\frac{2}{3} \div \frac{4}{9}$ step by step. Type each answer in the box.

1. After you flip $\frac{4}{9}$, what is the new second fraction? (Type it like $9/4$.)

2. Multiply $\frac{2}{3} \times \frac{9}{4}$ before simplifying. What is the result? (Type it like $18/12$.)

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

🧭 Guided Practice

Part 3 of 5 — Guided Practice

Work each problem carefully. Multiply straight across, or Keep, Change, Flip to divide. Simplify when you can.

Fill In the Steps 🔍

Complete the two steps for solving $\frac{4}{5} \div \frac{2}{3}$.

🌍 Application & Word Problems

Part 4 of 5 — Real-World Practice

Fractions show up everywhere — in recipes, in building projects, and when sharing things fairly. The tricky part of a word problem is deciding whether to multiply or divide.

Two Helpful Clues

• Multiply when you want a fraction of an amount. Example: "How much is $\frac{2}{3}$ of $\frac{3}{4}$ cup?"
• Divide when you are splitting an amount into equal-size pieces and asking "how many fit?" Example: "How many $\frac{1}{4}$-cup scoops are in $2$ cups?"

Worked Word Problem

A recipe needs $\frac{3}{4}$ cup of sugar, but Maya is making only half the recipe. How much sugar does she need?

"Half of $\frac{3}{4}$" means multiply:

$\frac{1}{2} \times \frac{3}{4} = \frac{3}{8} \text{ cup of sugar.}$

Recipe Problem 🍪

A baker has $\frac{3}{4}$ pound of chocolate. Each cookie needs $\frac{1}{8}$ pound of chocolate.

To find how many cookies the baker can make, divide: $\frac{3}{4} \div \frac{1}{8}$.

1. After you flip $\frac{1}{8}$, what is the new second fraction? (Type it like $8/1$.)

2. Multiply $\frac{3}{4} \times \frac{8}{1}$ before simplifying. What is the result? (Type it like $24/4$.)

3. How many whole cookies can the baker make? (Type a whole number.)

Word Problem Check 🎯

🏆 Review & Challenge

Part 5 of 5 — Putting It All Together

You now have every tool for multiplying and dividing fractions. Here is the whole toolkit on one page.

Quick Reference Table

Remember: You never need a common denominator to multiply or divide — that rule is only for adding and subtracting! Now try the challenge questions below. 💪

Mixed Challenge 🎯

These questions blend everything you have learned.