Multiplication and Division of Fractions

Multiplication and division in math are two of the basic four operations. Multiplication is nothing but the same as repetitive addition which means adding a quantity specific number of times. Similarly, the division is repetitive subtraction which means equal grouping or dividing a quantity into a specific number of groups.

You might be aware of multiplication and division operations and their rules with whole numbers and integers. Similarly, the multiplication and division of fractions have their own process and rules. Let’s understand how the multiplication and division of proper and improper fractions are performed. The process for mixed fractions is a bit different from that of proper and improper fractions.

Multiplication of Fractions

The essential condition in the case of addition and subtraction of fractions is that the denominators of the fractions should be the same or the fractions should be like fractions. 

But the multiplication of fractions is not like the addition or subtraction of fractions Here, you can multiply any two fractions with different denominators. 

The only thing to be kept in mind is that the fractions should not be mixed fractions, they should either be proper fractions or improper fractions. Let’s learn how to multiply proper or improper fractions.

Steps for Multiplication of Fractions

The steps involved in the multiplication of proper or improper fractions are

Step 1: Multiply the numerators.

Step 2: Multiply the denominators.

Step 3: Reduce (if necessary) the resultant fraction to its lowest terms or simplest form.

Examples

Ex 1: $\frac {3}{5} \times  \frac {7}{11}$ 

Multiply numerators: $3 \times 7 = 21$

Multiply denominators: $5 \times 11 = 55$

Therefore, $ \frac {3}{5} \times \frac {7}{11} = \frac {3 \times 7}{5 \times 11} = \frac {21}{55}$

Ex 2: $\frac {2}{3} \times \frac {9}{14}$

Multiply numerators: $2 \times 9 = 18$

Multiply denominators: $3 \times 14 = 42$

Therefore, $ \frac {2}{3} \times \frac {9}{14} = \frac {2 \times 9}{3 \times 14} = \frac {18}{42}$

Note: $ \frac {18}{42}$ is not in its lowest or simples form. ($18$ and $42$ have common factors)

Now, reduce $\frac {18}{42}$ to its lowest form

$ \frac {18}{42} = \frac {9}{21}$ (Dividing numerator and denominator by $2$)

$ \frac {9}{21} = \frac {3}{7}$ (Dividing numerator and denominator by $3$)

Since, $\frac {3}{7}$ is in its lowest form, therefore, $ \frac {2}{3} \times \frac {9}{14} = \frac {3}{7}$

Alternate method

$\frac {2}{3} \times \frac {9}{14} = \frac {2 \times 9}{3 \times 14} = \frac {1 \times 9}{3 \times 7}  =   \frac {1 \times 3}{1 \times 7} = \frac {3}{7}$

Multiplication of Fractions Using Visual Models

Now, let us see how we can visualize the multiplication of fractions by first taking an example of two simple fractions. 

$ \frac {1}{2} \times \frac {1}{3}$

To visualize the multiplication of $\frac {1}{2}$ and $\frac {1}{3}$ draw a rectangle and divide its length into $2$ equal parts. Each part (column) will represent $\frac {1}{2}$ of the whole rectangle. And then divide its width into $3$ equal parts. Each part (row) will represent $\frac {1}{3}$ of the whole rectangle.

Now, visualize these two figures together and observe the common portion of both $\frac {1}{2}$ and $\frac {1}{3}$ of the rectangle is the product of $\frac {1}{2}$ and $\frac {1}{3}$.

Therefore, $\frac {1}{2} \times \frac {1}{3} = \frac {1}{2}$.

Let’s consider one more example of $\frac {2}{3}$ and $\frac {3}{4}$.

Portion shaded blue is $2$ parts  (rows) of $3$ parts (rows) of a rectangle.

Portion shaded yellow is $3$ parts  (columns) of $4$ parts (columns) of a rectangle.

Now, visualizing these two figures together and observing the common portion of both $\frac {2}{3}$ and $\frac {3}{4}$  of the rectangle is the product of $\frac {2}{3}$  and $\frac {3}{4}$.

A common portion (shaded green) is $1$ part of $2$ parts of a rectangle.

(There is a total of $12$ square boxes. Out of these $12$ square boxes, $6$ are green, so the fraction of green is $\frac {6}{12} = \frac {1}{2}$)

Multiplying Fractions with Whole Numbers

In order to multiply a fraction with a whole number is to write the whole number in the form of a fraction. The numerator of a fraction will be the whole number and the denominator of the fraction will be $1$.

$5 = \frac {5}{1}$, $7 = \frac {7}{1}$, $29 = \frac {29}{1}$, …

Note: Every whole number can be represented in the form of a fraction where

• the numerator will be the whole number 
• the denominator is always $1$ (one)

Steps for Multiplying Fractions With Whole Numbers

Step 1: Write the whole number in the form of a fraction 

Step 2: Multiply the numerators.

Step 3: Multiply the denominators.

Step 4: Reduce (if necessary) the resultant fraction to its lowest terms or simplest form.

Examples

Ex 1: $9 \times \frac {7}{11}$ 

Here $9$ is a whole number, so write it in the fraction form.

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

$9 \times \frac {7}{11} = \frac {9}{1} \times \frac {7}{11} = \frac {9 \times 7}{1 \times 11} = \frac {63}{11}$

$9 \times \frac {7}{11} = \frac {63}{11}$

Ex 2: $ \frac {2}{3} \times 7$

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

$ \frac {2}{3} \times 7 = \frac {2}{3} \times \frac {7}{1} = \frac {2 \times 7}{3 \times 1} = \frac {14}{3}$

$ \frac {2}{3} \times 7 = \frac {14}{3}$

Ex 3: $5 \times \frac {21}{4}$

$5 = \frac {5}{1}$

$5 \times \frac {21}{4} = \frac {5}{1} \times \frac {21}{4} = \frac {105}{4}$.

Note: 

$ \frac {105}{4}$ is an improper fraction and it can be further converted into a mixed fraction.

Multiplying Fractions with Whole Numbers Visual Model

Consider this example: $5 \times \frac {2}{3}$. 

This means $ \frac {2}{3}$ is added $5$ times. 

Let us represent this example using a visual model. Five times two-thirds is represented as:

Therefore, $5 \times \frac {2}{3}  = \frac {10}{3}$.

Multiplication of Fractions With Decimal Numbers

You might be knowing that decimal numbers can be converted to fractions with a denominator as a power of $10$. The first step in the multiplication of a fraction with a decimal number is to convert the decimal number into a fraction and then follow the steps of the multiplication of fractions.

Steps For Multiplication Of A Fraction With A Decimal Number

Step 1: Convert a decimal number to a fraction

Step 2: Multiply the numerators.

Step 3: Multiply the denominators.

Step 4: Reduce (if necessary) the resultant fraction to its lowest terms or simplest form

Examples

Ex 1: $2.3 \times \frac {4}{5}$

$2.3 = \frac {23}{10}$

$2.3 \times \frac {4}{5} = \frac {23}{10} \times \frac {4}{5} = \frac {23\times4}{10\times5} = \frac {92}{50} = \frac {46}{25} = 1 \frac {21}{25}$

Ex 2: $ \frac {6}{11} \times 1.56$

$1.56 = \frac {156}{100} = \frac {78}{50} = \frac {39}{25}$

$\frac {6}{11} \times 1.56 = \frac {6}{11} \times \frac {39}{25} = \frac {6 \times 39}{11 \times 25} = \frac {234}{275}$

General Rules of Multiplying Fractions

While multiplying fractions, the following rules have to be kept in mind:

Rule 1: The first rule is to convert mixed fractions to improper fractions if any. Then, multiply the numerators of the given fractions.

Rule 2: Multiply the denominators separately.

Rule 3: Simplify the value obtained to its lowest term, if needed.

Division of Fractions

You might have done division with whole numbers. Division means sharing an item equally. The division of fractions is quite similar to the multiplication of fractions with a slight difference. For the division of fractions, we multiply the first fraction by the reciprocal (multiplicative inverse) of the second fraction.

Consider the two fractions $\frac {a}{b}$ and $\frac {c}{d}$ and follow the steps to divide $\frac {a}{b}$ by $\frac {c}{d}$, i.e., $\frac {a}{b} \div \frac {c}{d}$.

Steps For Division Of A Fraction With A Fraction

Step 1: Write the reciprocal of the second fraction, i.e., reciprocal of $\frac {c}{d}$ which is $\frac {d}{c}$

Step 2: Multiply $\frac {a}{b}$ by $\frac {d}{c}$ using the steps mentioned below (These are the same as that of multiplying fractions)

Step 3: Multiply the numerators.

Step 4: Multiply the denominators.

Step 5: Reduce (if necessary) the resultant fraction to its lowest terms or simplest form.

Examples

Ex 1: Divide $\frac {6}{7}$ by $\frac {3}{4}$, i.e., $ \frac {6}{7} \div \frac {3}{4}$.

Reciprocal of $\frac {3}{4}$ is $\frac {4}{3}$

$ \frac {6}{7} \div \frac {3}{4} = \frac {6}{7} \times \frac {4}{3} = \frac {6 \times 4}{7 \times 3} = \frac {24}{21}$

$\frac {6}{7} \div \frac {3}{4} = \frac {24}{21}$

Ex 2: Divide $\frac {1}{2}$ by $\frac {1}{3}$, i.e., $\frac {1}{2} \div \frac {1}{3}$

Reciprocal of $\frac {1}{3}$ is $\frac {3}{1}$

$ \frac {1}{2} \div \frac {1}{3} = \frac {1}{2} \times \frac {3}{1} = \frac {1 \times 3}{2 \times 1} = \frac {3}{2}$

Division of Fractions With Whole Numbers

For the division of fractions with the whole number, the first step is to convert the whole number to a fraction and then follow the steps of the division of fractions.

Steps For Division Of A Fraction With A Whole Number

Step 1: Convert a whole number to a fraction

Step 2: Write the reciprocal of the second fraction, i.e., reciprocal of $\frac {c}{d}$ which is $\frac {d}{c}$

Step 3: Multiply $\frac {a}{b}$ by $\frac {d}{c}$ using the steps mentioned below (These are the same as that of multiplying fractions)

Step 4: Multiply the numerators.

Step 5: Multiply the denominators.

Step 6: Reduce (if necessary) the resultant fraction to its lowest terms or simplest form.

Examples

Ex 1: Divide $\frac {2}{3}$ with $4$, i.e., $\frac {2}{3} \div 4$

$4$ in fraction form is $\frac {4}{1}$

$ \frac {2}{3} \div 4 = \frac {2}{3} \div \frac {4}{1}$

Reciprocal of $ \frac {4}{1}$ is $\frac {1}{4}$

$ \frac {2}{3} \div 4 = \frac {2}{3} \times \frac {1}{4} = \frac {2 \times 1}{3 \times 4} = \frac {2}{12} = \frac {1}{6}$

Therefore, $\frac {2}{3} \div 4 = \frac {1}{6}$. 

Ex 2: Divide $5$ with $\frac {6}{11}$, i.e., $5 \div \frac {6}{11}$.

$5$ in fraction is $\frac {5}{1}$

$5 \div \frac {6}{11} = \frac {5}{1} \div \frac {6}{11}$

Reciprocal of $ \frac {6}{11}$ is $ \frac {11}{6}$

$ \frac {5}{1} \div \frac {6}{11} = \frac {5}{1} \times \frac {11}{6} = \frac {5 \times 11}{1 \times 6} = \frac {55}{6}$

Division of Fractions With Decimal Numbers

You might be knowing that decimal numbers can be converted to fractions with a denominator as a power of $10$. The first step in the division of a fraction with a decimal number is to convert the decimal number into a fraction and then follow the steps of the division of fractions.

Steps For Division Of A Fraction With A Decimal Number

Step 1: Convert a decimal number to a fraction

Step 2: Write the reciprocal of the second fraction, i.e., reciprocal of $\frac {c}{d}$ which is $\frac {d}{c}$

Step 3: Multiply $\frac {a}{b}$ by $\frac {d}{c}$ using the steps mentioned below (These are the same as that of multiplying fractions)

Step 4: Multiply the numerators.

Step 5: Multiply the denominators.

Step 6: Reduce (if necessary) the resultant fraction to its lowest terms or simplest form.

Examples

Ex 1: Divide $2.5$ by $\frac {2}{3}$ i.e., $2.5 \div \frac {2}{3}$. 

$2.5 = \frac {25}{10} = \frac {5}{2}$

$2.5 \div \frac {2}{3} = \frac {5}{2} \div \frac {2}{3} = \frac {5}{2} \times \frac {3}{2} = \frac {5 \times 3}{2 \times 2} = \frac {15}{4} = 3 \frac {3}{4}$. 

Ex 2: Divide $\frac {8}{11}$ by $11.9$ i.e., $\frac {8}{11} \div 11.9$.

$11.9 = \frac {119}{10}$

$ \frac {8}{11} \div 11.9 = \frac {8}{11} \div \frac {119}{10}$

$ \frac {8}{11} \times \frac {10}{119} = \frac {8 \times 10}{11 \times 119} = \frac {80}{1309}$

Conclusion

We can multiply a fraction with any other fraction and it’s not necessary for fractions to be like fractions, as in the case of addition or subtraction of fractions. The division of fractions is quite similar to that of multiplication with the difference that the reciprocal of the second fraction is multiplied by the first fraction.

Practice Problems

1. Multiply the following
   1. $\frac{2}{5} \times \frac {1}{7}$
   2. $\frac{7}{8} \times \frac {4}{5}$
   3. $\frac{3}{11} \times \frac {6}{19}$
   4. $\frac {1}{3} \times \frac {1}{7}$
   5. $\frac{3}{5} \times \frac {5}{9}$
   6. $\frac{7}{11} \times 2.6$
   7. $7.8 \times \frac {1}{5}$
   8. $\frac{4}{7} \times 3.82$
2. Divide the following
   1. $\frac{5}{8} \div \frac {2}{3}$
   2. $\frac{1}{9} \div \frac {5}{18}$
   3. $\frac{2}{15} \div \frac {4}{5}$
   4. $\frac {1}{2} \div \frac {1}{4}$
   5. $\frac{3}{7} \div \frac {5}{12}$
   6. $\frac{8}{13} \div 6.5$
   7. $2.8 \div {2}{5}$
   8. $\frac{3}{7} \div 1.02$

Recommended Reading

• Mixed Fractions – Definition & Operations (With Examples)
• Addition and Subtraction of Fractions (With Pictures)