Mixed Numbers: Multiplication

As you know, a mixed number is a number expressed as the sum of a whole number and a fraction. Examples include $3\frac{1}{4}$ , $2\frac{1}{2}$ , and $5\frac{7}{8}$ .

If you want to multiply mixed numbers, the first step is converting them to improper fractions. If the fractions have any common factors in the numerators and denominators, you can simplify them before you multiply to make the math a little easier. You will generally be asked to write your answer as a mixed number with the fraction component in simplest form when you are finished.

That might sound a little abstract, but working through a few practice problems should make things clearer.

An example of mixed numbers: Multiplication

Let's find the product of $1\frac{2}{5}$ and $2\frac{1}{2}$ to get started. The first step is converting both mixed numbers to improper fractions, giving us:

$1\frac{2}{5}=\frac{2+1\times 5}{5}=\frac{7}{5}$

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

Thus, our expression becomes:

$\frac{7}{5}\times \frac{5}{2}$

We have a 5 in both the numerator and denominator, allowing us to cancel them out and simplify the expression:

$\frac{7}{1}\times \frac{1}{2}$

Next, we multiply the numerators and denominators separately:

$7\times 1=7$ (numerator)

$1\times 2=2$ (denominator)

Our answer is $\frac{7}{2}$ in improper fraction form which becomes $3\frac{1}{2}$ as a mixed number. The $\frac{1}{2}$ is already in simplest form, so we've found the product!

A harder example of mixed numbers: Multiplication

It was easy to find the common factor of 5 in the example above because both fractions had a 5, but it won't always be that simple. Consider the following example:

$1\frac{1}{4}\times 3\frac{5}{9}$

Rewriting both mixed numbers as improper fractions, we get:

$\frac{1+1\times 4}{4}\times \frac{5+3\times 9}{9}=\frac{5}{4}\times \frac{32}{9}$

We know that 4 and 32 share at least one common factor since they're both even, so we look for the greatest common factor (GCF) to make the math as easy as possible. The GCF of 4 and 32 is 4, allowing us to factor out a 4 from both the numerator and denominator:

$\frac{5}{1}\times \frac{8}{9}$

Now, we multiply the numerators and denominators separately:

$5\times 8=40$ (numerator)

$1\times 9=9$ (denominator)

Our answer is $\frac{40}{9}$ in improper fraction form, which becomes $4\frac{4}{9}$ as a mixed number. We're finished!

Mixed numbers: Multiplication practice questions

a. $1\frac{3}{4}\times 2\frac{1}{2}$

First, convert each mixed number to an improper fraction:

$1\frac{3}{4}=\frac{1\times 4+3}{4}=\frac{7}{4}$

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

Now, multiply the fractions:

$\frac{7}{4}\times \frac{5}{2}=\frac{7\times 5}{4\times 2}=\frac{35}{8}$

Convert the improper fraction back to a mixed number:

$\frac{35}{8}=4+\frac{3}{8}$