Multiplying Mixed Numbers

Multiplying mixed numbers might look a bit intimidating at first, but there is one golden rule that makes it incredibly easy: always convert mixed numbers into improper fractions first!

Once you do that, you are just multiplying regular fractions.

The 3-Step Method

To multiply a mixed number by a whole number, a fraction, or another mixed number, just follow these three steps:

1. Convert every mixed number into an improper fraction.
2. Multiply the numerators (top numbers) together, and multiply the denominators (bottom numbers) together.
3. Simplify the resulting fraction and convert it back into a mixed number if needed.

Step-by-Step Examples

Let's walk through some examples to see how this works in practice.

Example 1: Multiplying Two Mixed Numbers

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

• Step 1: Convert to improper fractions. For $2\frac{1}{3}$, multiply the whole number by the denominator and add the numerator: $(2 \times 3) + 1 = 7$. Keep the denominator the same to get $\frac{7}{3}$. For $1\frac{2}{5}$: $(1 \times 5) + 2 = 7$. Keep the denominator the same to get $\frac{7}{5}$.
• Step 2: Multiply. $\frac{7}{3} \times \frac{7}{5} = \frac{7 \times 7}{3 \times 5} = \frac{49}{15}$
• Step 3: Convert back to a mixed number. Divide $49$ by $15$. $15$ goes into $49$ three times with a remainder of $4$. Answer: $3\frac{4}{15}$

Example 2: Multiplying a Mixed Number by a Fraction

Problem: $3\frac{1}{2} \times \frac{4}{7}$

• Step 1: Convert. $3\frac{1}{2}$ becomes $\frac{7}{2}$. The second number is already a fraction: $\frac{4}{7}$.
• Step 2: Multiply. $\frac{7}{2} \times \frac{4}{7} = \frac{28}{14}$
• Step 3: Simplify. Divide $28$ by $14$. It divides evenly! Answer: $2$

Example 3: Simplifying Before You Finish

Problem: $1\frac{3}{4} \times 2\frac{2}{3}$

• Step 1: Convert. $1\frac{3}{4}$ becomes $\frac{7}{4}$. $2\frac{2}{3}$ becomes $\frac{8}{3}$.
• Step 2: Multiply. $\frac{7}{4} \times \frac{8}{3} = \frac{56}{12}$
• Step 3: Simplify and Convert. First, simplify $\frac{56}{12}$ by dividing the top and bottom by their greatest common factor, which is $4$. $\frac{56 \div 4}{12 \div 4} = \frac{14}{3}$ Now, convert $\frac{14}{3}$ to a mixed number. $14 \div 3 = 4$ with a remainder of $2$. Answer: $4\frac{2}{3}$

Quick Tip for Success

Always double-check your conversions! The most common mistake is adding incorrectly when turning a mixed number into an improper fraction. Remember the pattern: (Whole Number $\times$ Denominator) + Numerator.