Multiplying Mixed Numbers

How do you multiply mixed numbers?

Multiplying mixed numbers requires three essential steps.

1. Convert all mixed numbers and whole numbers into improper fractions.
2. Multiply the numerators together and the denominators together.
3. Simplify the resulting fraction and convert it back into a mixed number if necessary.

This process ensures that the fractional parts and whole parts are multiplied together accurately. Once the numbers are written as improper fractions, this uses the exact same rules as multiplying fractions.

Convert mixed numbers first

We cannot simply multiply the whole numbers together and the fractions together. Instead, we convert mixed numbers to improper fractions first.

To convert a mixed number, multiply the whole number by the denominator, add the numerator, and place that sum over the original denominator.

For example, to convert $1 \dfrac{3}{4}$, multiply $1 \times 4 = 4$, then add $3$ to get $7$. The improper fraction is $\dfrac{7}{4}$.

Multiply and simplify

Once you have improper fractions, multiply the top numbers (numerators) together to get the new numerator, and multiply the bottom numbers (denominators) together to get the new denominator.

By simplifying fractions before multiplying, we can often make the calculation easier. Look for common factors between any numerator and any denominator, and divide them out before finding the product.

Cross-simplify factors before multiplying to avoid working with large numbers.

Convert the product back

After multiplying, your result will usually be an improper fraction. To convert it back to a mixed number, divide the numerator by the denominator.

The quotient becomes the whole number part, the remainder becomes the new numerator, and the denominator stays exactly the same.

Visual interpretation

An area model is an excellent way to see why mixed number multiplication works. When we multiply $2 \dfrac{1}{2}$ by $1 \dfrac{1}{2}$, we are finding the total area of a rectangle with those side lengths.

The entire rectangle consists of all the whole parts and fractional pieces combined, proving that the product is $3 \dfrac{3}{4}$.

Worked examples

Applying the method to different multiplication scenarios builds confidence, such as in fraction word problems.

Example 1: Multiplying a mixed number by a whole number

Question: What is $3 \dfrac{1}{4} \times 2$?

Method:

1. Write the whole number as a fraction: $2 = \dfrac{2}{1}$.
2. Convert the mixed number to an improper fraction: $3 \dfrac{1}{4} = \dfrac{13}{4}$.
3. Multiply the numerators and denominators: $\dfrac{13}{4} \times \dfrac{2}{1} = \dfrac{26}{4}$.
4. Simplify the fraction: $\dfrac{26}{4} = \dfrac{13}{2}$.
5. Convert back to a mixed number: $\dfrac{13}{2} = 6 \dfrac{1}{2}$.

Answer: $6 \dfrac{1}{2}$.

Check: Using addition, $3 \dfrac{1}{4} + 3 \dfrac{1}{4} = 6 \dfrac{2}{4} = 6 \dfrac{1}{2}$.

Example 2: Multiplying two mixed numbers

Question: Evaluate $1 \dfrac{2}{3} \times 2 \dfrac{1}{5}$.

Method:

1. Convert both mixed numbers to improper fractions: $1 \dfrac{2}{3} = \dfrac{5}{3}$ and $2 \dfrac{1}{5} = \dfrac{11}{5}$.
2. Multiply the numerators: $5 \times 11 = 55$.
3. Multiply the denominators: $3 \times 5 = 15$.
4. Simplify the result: $\dfrac{55}{15} = \dfrac{11}{3}$.
5. Convert back to a mixed number: $\dfrac{11}{3} = 3 \dfrac{2}{3}$.

Answer: $3 \dfrac{2}{3}$.

Check: Estimate the product: $1.5 \times 2 = 3$. The exact result $3 \dfrac{2}{3}$ is reasonable.

Example 3: Solving a geometric word problem

Question: A rectangular garden is $4 \dfrac{1}{2}$ meters long and $2 \dfrac{2}{3}$ meters wide. What is the area of the garden?

Method:

1. Set up the multiplication for area: $4 \dfrac{1}{2} \times 2 \dfrac{2}{3}$.
2. Convert to improper fractions: $4 \dfrac{1}{2} = \dfrac{9}{2}$ and $2 \dfrac{2}{3} = \dfrac{8}{3}$.
3. Cross-simplify before multiplying: divide $9$ and $3$ by $3$, and divide $8$ and $2$ by $2$.
4. Multiply the simplified fractions: $\dfrac{3}{1} \times \dfrac{4}{1} = \dfrac{12}{1}$.

Answer: The area is $12$ square meters.

Check: Converting $12$ to a mixed number confirms it is exactly $12$ wholes.

Common mistakes

A frequent error is multiplying the whole numbers together and then multiplying the fractions together separately. This ignores the cross-multiplication of the whole parts with the fractional parts.

Always remember to convert to improper fractions first. Bypassing this step guarantees an incorrect mathematical result.

Frequently asked questions

Can you cross-simplify mixed numbers before converting them?

No. You must convert mixed numbers into improper fractions before you can search for common factors to cross-simplify.

Does the order of multiplication matter?

No. Multiplication is commutative, meaning $1 \dfrac{1}{2} \times 2 \dfrac{1}{3}$ yields the exact same product as $2 \dfrac{1}{3} \times 1 \dfrac{1}{2}$.