Multiplying Fractions by Whole Numbers

🍕 Multiplying Fractions by Whole Numbers

Part 1 of 5 — The Big Idea

You already know how to multiply whole numbers, like $3 \times 4 = 12$. Multiplying a fraction by a whole number works in a very similar way.

Multiplying a fraction by a whole number just means adding that fraction over and over again.

For example, $3 \times \frac{2}{5}$ means "three groups of $\frac{2}{5}$":

$3 \times \frac{2}{5} = \frac{2}{5} + \frac{2}{5} + \frac{2}{5} = \frac{6}{5} = 1\frac{1}{5}$

When we add fractions with the same denominator, we add the numerators (the top numbers) and keep the denominator (the bottom number) the same.

✨ The Shortcut

Adding the fraction many times works, but there is a faster way:

$n \times \frac{a}{b} = \frac{n \times a}{b}$

In plain words:

• Multiply the whole number by the numerator (the top number).
• Keep the denominator (the bottom number) exactly the same.

Notice the bottom number never changes. Only the top number gets multiplied.

🤔 Why Does the Shortcut Work?

Think about $4 \times \frac{1}{3}$. That is four copies of $\frac{1}{3}$:

$\frac{1}{3} + \frac{1}{3} + \frac{1}{3} + \frac{1}{3} = \frac{4}{3}$

Each piece is one-third, and there are 4 of them. So we have $\frac{4}{3}$. That is exactly $4 \times 1$ on top, with the $3$ staying on the bottom. The shortcut and the repeated addition give the same answer every time.

Concept Check 🎯

🧮 Worked Examples

Part 2 of 5 — Step by Step

Let's work through a few problems slowly so you can see every step.

Example 1: $4 \times \frac{3}{8}$

• Step 1 — Multiply the top: $4 \times 3 = 12$.
• Step 2 — Keep the bottom: the denominator stays $8$.
• Step 3 — Write it: $\frac{12}{8}$.
• Step 4 — Simplify: $\frac{12}{8} = \frac{3}{2} = 1\frac{1}{2}$.

$4 \times \frac{3}{8} = \frac{12}{8} = 1\frac{1}{2}$

Example 2: $5 \times \frac{1}{3}$

• Multiply the top: $5 \times 1 = 5$.
• Keep the bottom: $3$.
• Result: $\frac{5}{3}$, which equals $1\frac{2}{3}$.

Example 3: $6 \times \frac{2}{7}$

• Multiply the top: $6 \times 2 = 12$.
• Keep the bottom: $7$.
• Result: $\frac{12}{7} = 1\frac{5}{7}$.

🔁 Turning a Fraction Into a Mixed Number

When the top number is bigger than the bottom number, the fraction is greater than 1. We can write it as a mixed number (a whole number plus a fraction).

To do this, ask: how many times does the bottom go into the top?

• $\frac{12}{7}$: the $7$ goes into $12$ one time with $5$ left over, so $\frac{12}{7} = 1\frac{5}{7}$.
• $\frac{5}{3}$: the $3$ goes into $5$ one time with $2$ left over, so $\frac{5}{3} = 1\frac{2}{3}$.

Your Turn — Type Each Answer 🧮

Use the shortcut: multiply the top, keep the bottom.

1. $2 \times \frac{3}{8} = \frac{?}{8}$ — What is the numerator (top number)?

2. $5 \times \frac{2}{9} = \frac{?}{9}$ — What is the numerator (top number)?

3. $3 \times \frac{1}{4} = \frac{?}{4}$ — What is the numerator (top number)?

🏋️ Guided Practice

Part 3 of 5 — Practice Together

Pick the correct answer for each problem.

Choose the Correct Result 🔍

Use the shortcut, then simplify if you can.

🌎 Word Problems

Part 4 of 5 — Real-World Math

Multiplying fractions by whole numbers shows up all the time in real life — sharing pizza, measuring ingredients, or running laps.

Example: Sharing Pizza 🍕

Each person eats $\frac{3}{8}$ of a pizza. How much pizza do 4 people eat?

We need 4 groups of $\frac{3}{8}$, so we multiply:

$4 \times \frac{3}{8} = \frac{12}{8} = 1\frac{4}{8} = 1\frac{1}{2} \text{ pizzas}$

Tip: Look for the words "each" and "how many in total." That is a clue to multiply!

Word Problem Practice ✏️

Read carefully and type each answer.

1. A recipe uses $\frac{2}{3}$ cup of flour. You make 3 batches. Total cups of flour $= 3 \times \frac{2}{3}$. Type the answer as a whole number.

2. Maya runs $\frac{3}{10}$ of a mile each day. After 4 days she has run $4 \times \frac{3}{10} = \frac{?}{10}$ miles. Type the numerator (top number).

3. Each cup holds $\frac{1}{4}$ liter of juice. 5 cups hold $5 \times \frac{1}{4} = \frac{?}{4}$ liters. Type the numerator (top number).

Solve the Story Problem 📖

🏆 Review & Challenge

Part 5 of 5 — Putting It All Together

You made it! Here is everything in one place.

Remember the rule: $n \times \frac{a}{b} = \frac{n \times a}{b}$ — and always simplify your final answer!

Challenge Questions 🌟

Mix everything you learned!