Multiplying and Dividing Fractions

✖️ Multiplying Fractions

Good news: multiplying fractions is one of the easiest things you can do with fractions. You do not need a common denominator like you do for adding or subtracting. You just go straight across.

The Rule

To multiply two fractions:

1. Multiply the numerators (the top numbers)
2. Multiply the denominators (the bottom numbers)
3. Simplify the answer if you can

Written as a formula:

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

The tops go across, and the bottoms go across. That's the whole trick!

A First Example

Let's multiply $\frac{1}{2} \times \frac{1}{3}$.

• Step 1 — Numerators: $1 \times 1 = 1$
• Step 2 — Denominators: $2 \times 3 = 6$
• Step 3 — Simplify: $\frac{1}{6}$ is already in lowest terms.

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

Picture it: if you take half of one-third of a pizza, you get one-sixth of the whole pizza. The answer is smaller than what you started with — that always happens when you take a "part of a part."

What "of" Means 🤔

When a problem says "of," that is a clue to multiply.

• $\frac{1}{2} \times 8 = 4$ → half of 8 is 4
• $\frac{1}{4} \times 12 = 3$ → one-fourth of 12 is 3

A Smart Shortcut: Cancel First ✂️

Sometimes you can cancel a common factor before you multiply. This keeps the numbers small.

Look at $\frac{2}{3} \times \frac{3}{4}$. The number 3 is on top in one fraction and on the bottom in the other, so we can cancel both 3s:

$\frac{2}{\cancel{3}} \times \frac{\cancel{3}}{4} = \frac{2}{4} = \frac{1}{2}$

You get the same answer either way — canceling first just makes the multiplication easier.

Concept Check 🎯

📝 Worked Examples: Multiplying

Example 1 — Multiply, then simplify

$\frac{2}{3} \times \frac{3}{4}$

• Numerators: $2 \times 3 = 6$
• Denominators: $3 \times 4 = 12$
• Simplify: $\frac{6}{12} = \frac{1}{2}$

$\frac{2}{3} \times \frac{3}{4} = \frac{1}{2}$

Example 2 — Whole number times a fraction

To multiply a whole number by a fraction, write the whole number over 1.

$3 \times \frac{2}{5} = \frac{3}{1} \times \frac{2}{5}$

• Numerators: $3 \times 2 = 6$
• Denominators: $1 \times 5 = 5$
• Result: $\frac{6}{5} = 1\frac{1}{5}$

Example 3 — Mixed numbers

First change each mixed number into an improper fraction, then multiply.

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

• Convert: $1\frac{1}{2} = \frac{3}{2}$ and $2\frac{1}{3} = \frac{7}{3}$
• Multiply: $\frac{3}{2} \times \frac{7}{3} = \frac{21}{6}$
• Simplify: $\frac{21}{6} = \frac{7}{2} = 3\frac{1}{2}$

Your Turn ✍️

Solve $\frac{3}{5} \times \frac{2}{7}$.

Fill in the numerator, the denominator, and the final fraction. (This one does not simplify, so the final answer is just the result.)

➗ Dividing Fractions: Keep, Change, Flip

Dividing fractions uses a trick called Keep–Change–Flip (KCF):

1. Keep the first fraction the same.
2. Change the $\div$ sign into a $\times$ sign.
3. Flip the second fraction upside down (its reciprocal).

Then just multiply!

Reciprocal means you swap the top and bottom:

Example: $\frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times \frac{4}{1} = \frac{4}{2} = 2$. (How many $\frac{1}{4}$s fit in $\frac{1}{2}$? Two!)

Practice ✅

Finish Each Problem 🔽

Choose the correct final answer for each one.

• $\frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times \frac{2}{1} = \frac{6}{4} = ?$
• $\frac{5}{6} \div \frac{2}{3} = \frac{5}{6} \times \frac{3}{2} = \frac{15}{12} = ?$

🌍 Fractions in Real Life

Fractions show up all the time in cooking, building, and sharing. Remember the two clues:

• "of" usually means multiply. Half of a recipe → multiply by $\frac{1}{2}$.
• "split into equal parts" usually means divide. Sharing a board into equal pieces → divide.

Recipe example: A recipe uses $\frac{3}{4}$ cup of sugar, but you only want to make half the recipe.

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

Sharing example: You have $\frac{3}{4}$ of a pizza and want to split it equally between 2 people.

$\frac{3}{4} \div 2 = \frac{3}{4} \times \frac{1}{2} = \frac{3}{8} \text{ each}$

Word Problem ✍️

A ribbon is $\frac{4}{5}$ of a meter long. You cut it into 4 equal pieces.

Use Keep–Change–Flip on $\frac{4}{5} \div 4$.

• Box 1: the result before simplifying, written as a fraction (numerator/denominator).
• Box 2: that same answer simplified, written as a fraction.

Apply It 🍪

🏆 Review & Challenge

You now know how to multiply and divide fractions. Here is the whole idea on one card:

Remember: multiplying by a fraction less than 1 makes things smaller, and dividing by a fraction less than 1 makes things bigger. Let's mix it all together!

Mixed Challenge 🎯