Lesson 66: Multiplying Mixed Numbers

New Concept

To multiply mixed numbers, you must first convert them into improper fractions. Then, multiply the numerators and denominators as you would with regular fractions.

What’s next

Now you're ready to apply this method. Soon, you'll solve practice problems involving mixed numbers, whole numbers, and fractions in different combinations.

Property

Step 1: Put the problem into the correct shape (if it is not already).
Step 2: Perform the operation indicated.
Step 3: Simplify the answer if possible.

Examples

$1\frac{1}{2} \times \frac{2}{3} \rightarrow \frac{3}{2} \times \frac{2}{3} = \frac{6}{6} = 1$
$2\frac{1}{2} \times 1\frac{1}{3} \rightarrow \frac{5}{2} \times \frac{4}{3} = \frac{20}{6} = 3\frac{1}{3}$
$4 \times 2\frac{1}{4} \rightarrow \frac{4}{1} \times \frac{9}{4} = \frac{36}{4} = 9$

Explanation

For multiplication, 'correct shape' means turning mixed numbers into improper fractions. It’s like a costume party for numbers where everyone must wear a fraction outfit, even whole numbers! This single step makes multiplying different types of numbers, like $3\frac{1}{2}$ and 5, super straightforward and prevents common errors. Get them dressed up first!

Property

To multiply mixed numbers, you must first write the mixed numbers and whole numbers as improper fractions. Then, multiply the numerators together and the denominators together.

Examples

$2\frac{2}{3} \times 4 = \frac{8}{3} \times \frac{4}{1} = \frac{32}{3} = 10\frac{2}{3}$
$2\frac{1}{2} \times 1\frac{1}{3} = \frac{5}{2} \times \frac{4}{3} = \frac{20}{6} = 3\frac{1}{3}$
$3\frac{1}{2} \times 1\frac{2}{3} = \frac{7}{2} \times \frac{5}{3} = \frac{35}{6} = 5\frac{5}{6}$

Explanation

Never multiply the whole numbers and fractions separately—it's a trap! Think of it like a recipe: you can't just throw ingredients next to each other. You must prepare them first by converting everything into improper fractions. This blends all the parts together correctly, ensuring you get the delicious, accurate final product every single time.

Property

After multiplying, the final answer must be simplified. This involves converting any improper fraction into a mixed number and reducing the fractional part to its lowest terms.

Examples

$\frac{32}{3} = 10\frac{2}{3}$
$\frac{44}{8} = 5\frac{4}{8} = 5\frac{1}{2}$
$\frac{50}{12} = 4\frac{2}{12} = 4\frac{1}{6}$

Explanation

Simplifying is like tidying up your final answer to make it look sharp and be easily understood. An improper fraction like $\frac{25}{4}$ is technically right, but it's clunky. Converting it to a mixed number like $6\frac{1}{4}$ makes it neat and practical. It’s the final flourish that turns a correct answer into a great one.