Multiplying and Dividing Fractions

🍫 Multiplying Fractions

Part 1 of 5 — Multiply Across the Top, Across the Bottom

Topics in This Part

🔑 Key Concept: Multiplying fractions is the easy operation — there is no common denominator to find. You just multiply straight across: tops times tops, bottoms times bottoms.

The Multiply-Across Rule

To multiply two fractions, multiply the numerators together and the denominators together:

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

That's the whole rule. No matching denominators, no flipping — just multiply.

Examples

💡 Did you notice? When you multiply two fractions that are each less than 1, the answer is smaller than both of them. Taking a part of a part gives you less — that's the opposite of what adding does.

Concept Check 🎯

The Magic Word: "Of"

In math, the word "of" almost always means multiply when fractions are involved.

$\frac{1}{2} \text{ of } \frac{1}{3} \;=\; \frac{1}{2} \times \frac{1}{3} \;=\; \frac{1}{6}$

Picture a chocolate bar split into $3$ equal rows. One row is $\frac{1}{3}$ of the bar. Now take half of that row. You end up with a piece that is $\frac{1}{6}$ of the whole bar. 🍫

🔑 Translate first. Whenever you see "$\frac{2}{3}$ of something," rewrite it as "$\frac{2}{3}$ times something" and multiply across.

Multiply Across 🧮

Multiply straight across. Enter each answer as a fraction like 8/15 (you do not need to simplify yet).

1) $\frac{2}{3} \times \frac{4}{5} = \,?$ 2) $\frac{3}{4} \times \frac{1}{2} = \,?$ 3) $\frac{5}{6} \times \frac{2}{7} = \,?$

"Of" Means Multiply 🎯

🍫 Multiplying Fractions

Part 2 of 5 — Whole Numbers, Mixed Numbers & Simplifying

🔑 The Idea: Any whole number is secretly a fraction over $1$, and any mixed number can become an improper fraction. Once everything is a fraction, the multiply-across rule handles it all.

Multiplying a Fraction by a Whole Number

Every whole number can be written as a fraction over $1$:

$5 = \frac{5}{1}, \qquad 3 = \frac{3}{1}, \qquad 12 = \frac{12}{1}$

So to multiply a fraction by a whole number, just put the whole number over $1$ and multiply across:

$\frac{2}{3} \times 6 = \frac{2}{3} \times \frac{6}{1} = \frac{2\times 6}{3\times 1} = \frac{12}{3} = 4$

💡 A quick shortcut: $\frac{2}{3}$ of $6$ means "split $6$ into $3$ equal groups ($= 2$ each) and take $2$ groups" $= 4$. Same answer, no surprise.

Simplifying the Answer

A fraction is in simplest form when the top and bottom share no common factor except $1$. To simplify, divide the top and bottom by the same number.

$\frac{6}{12} = \frac{6 \div 6}{12 \div 6} = \frac{1}{2}$

Cross-Cancel Before You Multiply (the pro move)

You can simplify early by cancelling a top number with a bottom number that share a factor. This keeps the numbers small.

$\frac{3}{4} \times \frac{2}{9}$

The $3$ on top and the $9$ on the bottom share a factor of $3$ → becomes $1$ and $3$. The $2$ on top and the $4$ on the bottom share a factor of $2$ → becomes $1$ and $2$:

$\frac{\cancel{3}^{\,1}}{\cancel{4}_{\,2}} \times \frac{\cancel{2}^{\,1}}{\cancel{9}_{\,3}} = \frac{1\times 1}{2\times 3} = \frac{1}{6}$

💡 Cross-cancelling is optional but smart. Whether you cancel first or simplify at the end, you get the same answer. Cancelling early just keeps the multiplication easier.

Concept Check 🎯

Multiplying Mixed Numbers

You cannot multiply mixed numbers straight across. First turn each mixed number into an improper fraction.

To convert: multiply the whole number by the denominator, add the numerator, keep the same denominator.

$1\tfrac{1}{2} = \frac{(1\times 2) + 1}{2} = \frac{3}{2}, \qquad 2\tfrac{1}{3} = \frac{(2\times 3) + 1}{3} = \frac{7}{3}$

Worked Example: $1\tfrac{1}{2} \times \tfrac{2}{3}$

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

Worked Example: $2\tfrac{1}{4} \times \tfrac{2}{3}$

$2\tfrac{1}{4} = \frac{9}{4}, \quad\text{so}\quad \frac{9}{4}\times\frac{2}{3} = \frac{18}{12} = \frac{3}{2} = 1\tfrac{1}{2}$

⚠️ Never multiply the whole numbers and the fractions separately. Convert to improper fractions first, then multiply across.

Convert, Then Multiply 🔽

Work through $1\tfrac{2}{3} \times \tfrac{1}{2}$ one step at a time.

Multiply & Simplify 🧮

Find each product. Enter whole-number answers as whole numbers (like 4) and fraction answers in simplest form (like 1/6).

1) $\frac{3}{8} \times 4 = \,?$ (simplify) 2) $\frac{2}{3} \times \frac{3}{4} = \,?$ (simplify) 3) $1\tfrac{1}{2} \times \frac{4}{9} = \,?$ (simplify)

➗ Dividing Fractions

Part 3 of 5 — Reciprocals & What Division Really Asks

Topics in This Part

🔑 Key Concept: Dividing by a fraction asks "how many of that fit inside this?" The trick that answers it every time is the reciprocal — and that's what this part builds.

What Division Really Asks

The question $a \div b$ means: "How many groups of $b$ fit into $a$?"

That's true for fractions too. Look at $1 \div \frac{1}{4}$:

$1 \div \frac{1}{4} = \text{"How many quarters fit in one whole?"} = 4$

One whole pizza cut into quarters gives 4 slices. 🍕 So $1 \div \frac{1}{4} = 4$.

💡 Surprise! Dividing by a fraction less than $1$ makes the answer bigger, not smaller. That feels strange, but it makes sense: lots of tiny pieces fit inside a whole.

Concept Check 🎯

Reciprocals: Just Flip It

The reciprocal of a fraction is what you get when you flip it upside down — swap the numerator and denominator.

$\text{reciprocal of } \frac{3}{4} = \frac{4}{3}, \qquad \text{reciprocal of } \frac{2}{5} = \frac{5}{2}$

A whole number flips by putting it over $1$ first:

$\text{reciprocal of } 5 = \text{reciprocal of } \frac{5}{1} = \frac{1}{5}$

The defining property: a number times its reciprocal always equals $1$.

$\frac{3}{4} \times \frac{4}{3} = \frac{12}{12} = 1 \qquad ✓$

⚠️ A mixed number must become an improper fraction first, then flip. The reciprocal of $1\tfrac{1}{2} = \frac{3}{2}$ is $\frac{2}{3}$ — not $1\tfrac{2}{1}$.

Find the Reciprocal 🧮

Write the reciprocal (flip it). Enter fractions like 4/3; whole numbers like 5 become 1/5.

1) Reciprocal of $\frac{5}{8} = \,?$ 2) Reciprocal of $7 = \,?$ 3) Reciprocal of $1\tfrac{1}{4} = \,?$ (convert to improper first, then flip)

Why Dividing = Multiplying by the Reciprocal

Here is the connection that powers the whole method:

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

Dividing by a fraction is the same as multiplying by its reciprocal. Check it against something we already know:

$1 \div \frac{1}{4} = 1 \times \frac{4}{1} = 4 \qquad ✓$

We counted $4$ quarters in a whole earlier — and flipping-and-multiplying gives the exact same $4$. The trick isn't magic; it's just the counting question written as multiplication.

🔑 Remember this sentence: "Dividing by a fraction = multiplying by its flip." In Part 4 we turn it into a 3-step recipe you can run on autopilot.

Concept Check 🎯

➗ Dividing Fractions

Part 4 of 5 — Keep, Change, Flip

🔑 The Recipe: Keep the first fraction, Change $\div$ to $\times$, Flip the second fraction. Then multiply across just like Part 1. That's the entire method.

The Keep–Change–Flip Method

To compute $\frac{a}{b} \div \frac{c}{d}$:

1. Keep the first fraction exactly as it is: $\frac{a}{b}$
2. Change the division sign to multiplication: $\times$
3. Flip the second fraction (use its reciprocal): $\frac{d}{c}$

$\frac{a}{b} \div \frac{c}{d} \;\xrightarrow{\;\text{KCF}\;}\; \frac{a}{b} \times \frac{d}{c}$

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

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

✅ Check with meaning: "How many quarters fit in a half?" Two quarters make a half, so the answer is $2$. ✓

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

$\frac{3}{4} \div \frac{2}{3} = \frac{3}{4} \times \frac{3}{2} = \frac{9}{8} = 1\tfrac{1}{8}$

⚠️ Only flip the SECOND fraction. A very common slip is flipping the first one too. Keep the first one exactly as it is.

Run the Recipe 🔽

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

Dividing with Whole Numbers & Mixed Numbers

Same recipe — just turn every whole or mixed number into a fraction first.

Whole ÷ Fraction: $6 \div \frac{1}{2}$

$6 \div \frac{1}{2} = \frac{6}{1} \times \frac{2}{1} = \frac{12}{1} = 12$

"How many halves in $6$?" → $12$. ✓

Fraction ÷ Whole: $\frac{2}{3} \div 4$

Write $4 = \frac{4}{1}$, then Keep–Change–Flip:

$\frac{2}{3} \div \frac{4}{1} = \frac{2}{3} \times \frac{1}{4} = \frac{2}{12} = \frac{1}{6}$

Mixed ÷ Fraction: $1\tfrac{1}{2} \div \frac{3}{4}$

Convert $1\tfrac{1}{2} = \frac{3}{2}$, then KCF:

$\frac{3}{2} \div \frac{3}{4} = \frac{3}{2} \times \frac{4}{3} = \frac{12}{6} = 2$

💡 Two checkpoints before you flip: (1) every number is a fraction, and (2) you only flip the divisor (the one after the $\div$).

Concept Check 🎯

Keep–Change–Flip Drill 🧮

Divide using KCF. Enter whole-number answers as whole numbers (like 12) and fractions in simplest form (like 1/6).

1) $\frac{5}{6} \div \frac{1}{2} = \,?$ 2) $\frac{4}{9} \div \frac{2}{3} = \,?$ (simplify) 3) $3 \div \frac{1}{4} = \,?$

🏆 Multiplying & Dividing Fractions

Part 5 of 5 — Mixed Practice, Word Problems & Mastery Check

You can now (1) multiply fractions straight across, (2) handle whole and mixed numbers, (3) find reciprocals, and (4) divide with Keep–Change–Flip. Time to put it all together.

Quick Reference

⚠️ Two reminders: simplify your final answer, and when dividing, only flip the second fraction — never the first.

Multiply or Divide? 🔽

The hardest part of a word problem is choosing the operation. Decide for each story.

Word Problems 🧮

Solve each story. Enter whole numbers as whole numbers (like 8) and fractions in simplest form (like 3/8).

1) A recipe uses $\frac{2}{3}$ cup of flour. You make $\frac{1}{2}$ a batch. How many cups of flour? $\,?$ 2) A $4$-foot board is cut into $\frac{1}{2}$-foot pieces. How many pieces? $\,?$ 3) You have $\frac{3}{4}$ of a pizza and split it equally among $3$ friends. How much pizza per friend? $\,?$

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.