Multiplying Fractions with Mixed Numbers – Steps, Examples, FAQs

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What Is Multiplying Fractions with Mixed Numbers?

Multiplying fractions with mixed numbers refers to finding the product of a fraction and a mixed number (mixed fraction).

To multiply fractions and mixed numbers, the first step is to convert the mixed number into an improper fraction. After that, you proceed with the regular multiplication of two fractions.

A fraction represents parts of a whole. If a cake is divided into eight equal pieces, and three pieces of the cake are placed on a plate, then we can say that the plate has $\frac{3}{8}$ of the cake.

There are three types of fractions:

How To Multiply Fractions

To multiply two fractions $\frac{a}{b}$ and $\frac{c}{d}$, we first multiply the numerators and write the product as the numerator of the answer. Next, we multiply the denominators and write the result in the denominator. This is how the multiplication of fractions work. The fractions can be proper or improper.

$\frac{a}{b} \times\frac{c}{d} = \frac{a \times c}{b \times d}$

Example: $\frac{2}{3}\times\frac{7}{9} = \frac{2\times7}{3\times9} = \frac{14}{27}$

To multiply a fraction with a mixed number, we first convert the mixed number into an improper fraction. How do we convert a mixed number into an improper fraction? Let’s just quickly understand this conversion with examples.

•  $2\frac{1}{3} = \frac{(3\times2) + 1}{3} = \frac{7}{3}$
•  $9\frac{5}{6} = \frac{(6\times9) + 5}{6} = \frac{59}{6}$

How to Multiply Fractions with Mixed Numbers

Multiplication of two fractions is pretty straightforward. As mentioned earlier, there’s only one extra step that we have to complete in order to multiply a fraction and a mixed number, which is converting the given mixed number into an improper fraction. After that, we multiply the two fractions following the regular steps.

Note that multiplying mixed numbers with fractions follows the same procedure since the order does not matter. Multiplication of fractions is commutative.

Steps of Multiplying Fractions with Mixed Numbers

Let’s understand the steps with the help of an example.

Example: Multiply $\frac{2}{3}$ with $2\frac{1}{5}$.

Step 1: Convert all the mixed numbers from the given fractions into improper fractions.

Here, $2\frac{1}{5}$ is a mixed number. Converting it into an improper fraction, we get

$2\frac{1}{5} = \frac{(5\times2) + 1}{5} = \frac{11}{5}$

Step 2: Rewrite the problem using the new improper fractions.

Here, the problem becomes $\frac{2}{3}\times\frac{11}{5}$.

Step 3: Multiply the numerators. Write the result as the numerator of the answer.

Multiply the denominators. Write the result as the denominator of the answer.

$\frac{2}{3}\times\frac{11}{5} = \frac{2\times11}{3\times5} = \frac{22}{15}$

Step 4: Write the fraction in its simplest form. Rewrite the answer as a mixed number if required.

Here, $\frac{22}{15} = 1\frac{7}{15}$

We might come across three cases when we have to multiply a fraction with a mixed number.

• Multiplying a proper fraction with a mixed number
• Multiplying a proper fraction with a mixed number
• Multiplying a mixed fraction with a mixed fraction

In each case, we follow the same steps that we discussed earlier. Let’s see examples in every category.

Example 1: Multiplying proper fractions with mixed numbers

Multiply $\frac{1}{5}$ by $2\frac{1}{2}$.

Convert the mixed number into an improper fraction.

$2\frac{1}{2} = \frac{(2\times2) +1}{2} = \frac{5}{2}$

Next, we multiply the fractions $\frac{1}{5}$ and $\frac{5}{2}$.

$\frac{1}{5}\times\frac{5}{2} = \frac{1\times5}{5\times2} = \frac{1}{2}$

Example 2: Multiplying improper fractions with mixed numbers

Multiply $\frac{9}{8}$ by $6\frac{1}{3}$.

Convert the mixed number into an improper fraction.

$6\frac{1}{3} = \frac{(6\times3) + 1}{3} = \frac{19}{3}$

Next, we multiply the fractions $\frac{9}{8}$ and $\frac{19}{3}$.

$\frac{9\times19}{8\times3} = \frac{57}{8}$

Change the improper fraction $\frac{57}{8}$ into a mixed number.

$\frac{57}{8} = 7\frac{1}{8}$

So, $\frac{9}{8}\times6\frac{1}{3} = 7\frac{1}{8}$

Example 3: Multiplying mixed fractions

Multiply $3\frac{1}{5}$ by $6\frac{2}{3}$

Convert both the mixed numbers to improper fractions.

$3\frac{1}{5} = \frac{(3\times5) + 1}{5} = \frac{16}{5}$

$6\frac{2}{3} = \frac{(6\times3) + 2}{3} = \frac{20}{3}$

Multiply the fractions $\frac{16}{5}$ and $\frac{20}{3}$.

$\frac{16\times20}{5\times3} = \frac{320}{15}$

Change the improper fraction $\frac{320}{15}$ into a mixed number.

$\frac{320}{15} = 21\frac{5}{15}$.

So, $3\frac{1}{5}\times 6\frac{2}{3} = 19\frac{5}{15}$

Solved Examples on Multiplying Fractions with Mixed Numbers

1. Multiply fractions with mixed numbers and simplify.

i) $\frac{1}{9}\times2\frac{1}{4}$

ii) $\frac{2}{5} 1\frac{3}{7}$

Solution:

i) Convert the mixed number $2\frac{1}{4}$ into an improper fraction.

$2\frac{1}{4} = \frac{(2\times4) + 1}{4} = \frac{9}{4}$

Multiply $\frac{9}{4}$ with $\frac{1}{9}$.

$\frac{1}{9}\times\frac{9}{4} = \frac{1}{4}$

Thus, $\frac{1}{9}\times2\frac{1}{4} = \frac{1}{4}$

ii) Convert the mixed number $1\frac{3}{7}$ into an improper fraction.

$1\frac{3}{7} = \frac{10}{7}$

Multiply $\frac{2}{5}$ with $\frac{10}{7}$.

$\frac{2}{5}\times\frac{10}{7} = \frac{4}{7}$

Thus, $\frac{2}{5}\times1\frac{3}{7} = \frac{4}{7}$

2. What is the product of $\frac{5}{3}$ and $8\frac{2}{5}$.

Solution:

Convert the mixed number $8\frac{2}{5}$ into an improper fraction.

$8\frac{2}{5} = \frac{(8\times5) + 2}{5} = \frac{42}{5}$

Multiply $\frac{5}{3}$ with $4\frac{2}{5}$.

$\frac{5}{3}\times\frac{42}{5} = \frac{1}{4}$.

Thus, $\frac{5}{3}\times8\frac{2}{5} = \frac{1}{4}$

3. What is the result when $2\frac{2}{5}$ is multiplied by $1\frac{1}{3}$?

Solution:

Convert both the mixed numbers to improper fractions.

$2\frac{2}{5} = \frac{(2\times5) + 2}{5} = \frac{12}{5}$

$1\frac{1}{3} = \frac{(1\times3) + 1}{3} = \frac{4}{3}$

Multiply $\frac{12}{5}$ with $\frac{4}{3}$.

$\frac{12}{5}\times\frac{4}{3} = \frac{48}{15} = \frac{16}{5} = 3\frac{1}{5}$

Thus, $2\frac{2}{5}\times1\frac{1}{3} = 3\frac{1}{5}$

Practice Problems on Multiplying Fractions with Mixed Numbers

1

On multiplying $\frac{5}{6}$ by $2\frac{1}{2}$, we get ____.

$2\frac{1}{6}$
$2\frac{1}{12}$
$3\frac{1}{12}$
$1\frac{1}{6}$
CorrectIncorrect
Correct answer is: $2\frac{1}{12}$
$2\frac{1}{2} = \frac{5}{2}$
$\frac{5}{6}\times\frac{5}{2} = \frac{25}{12} = 2\frac{1}{12}$
2

$3\frac{8}{9}\times1\frac{2}{7} =$

$1\frac{2}{3}$
$2\frac{1}{2}$
$4$
$5$
CorrectIncorrect
Correct answer is: $5$
$3\frac{8}{9} = \frac{3\times9 + 8}{9} = \frac{35}{9}$ and $1\frac{2}{7} = \frac{(1\times7) + 2}{7} = \frac{9}{7}$
$\frac{35}{9}\times\frac{9}{7} = 5$
3

To multiply mixed numbers with fractions, we convert mixed numbers into

decimals
improper fractions
whole number
proper fractions
CorrectIncorrect
For multiplying mixed numbers with fraction, we convert the mixed number into an improper fraction.
4

$4\frac{2}{3}\times2\frac{1}{7} = ?$

10
12
15
20
CorrectIncorrect
$4\frac{2}{3} = \frac{(4\times3) + 2}{3} = \frac{14}{3}$
$2\frac{1}{7} = \frac{15}{7}$
$\frac{15}{7}\times\frac{14}{3} = \frac{14\times15}{7\times3} = 10$
To multiply a fraction $\frac{a}{b}$ with the whole number m, we express the whole number in fractional form as $\frac{m}{1}$. Next, we multiply the two fractions $\frac{a}{b}$ and $\frac{m}{1}$. For example, $\frac{2}{5} \times \frac{8}{1} = \frac{16}{5}$
For dividing fractions with mixed numbers, we first convert the mixed number to an improper fraction. Next, in order to divide the fraction with the improper fraction, we multiply the given fraction by the reciprocal of the improper fraction. $\frac{2}{3}\div2\frac{1}{7} = \frac{2}{3}2\frac{1}{7} = \frac{2}{3}\div\frac{15}{7} = \frac{2}{3}\times\frac{7}{15} = \frac{14}{45}$