Multiplying Fractions

The concept of fractions is useful in our day to day life. For example, if you divide an apple into 4 equal parts, the value of each part is ¼ and if you divide it into 8 equal parts, the value of each piece is 1/8th of the whole apple.


Multiplying fractions is as simple as the multiplication of numbers in the number system. A fraction is a representation of the division of the whole. Fractions, when multiplied by whole numbers, can either give a fraction or a whole number. However, when multiplied by a variable, it becomes a variable quantity. A fraction consists of a numerator and denominator. The different types of fractions are proper fractions, improper fractions and mixed fractions.


Multiplication of fraction also results in a fraction that has a numerator and a denominator as seen while dividing the fractions. There are various concepts that you can learn when it comes to fractions and operations performed on it. Multiplication and division of fractions are needed to be understood properly. However, addition and subtraction of fractions are very easy to solve. Multiplying fractions with like denominators or different ones require the same effort unlike dividing fractions for the same conditions.


Multiplication of Fractions

\[\frac{a}{b}\] × \[\frac{p}{q}\]

\[\frac{a × p}{b × q}\]


\[\frac{ap}{bq}\]


Multiplying Fractions Definition

Multiplication of fraction is just like the multiplication of two given real numbers. You can apply the simple method for multiplying fractions. You can write the formula for multiplication of fraction as:


Product of fraction = product of numerator ÷ product of denominator


Rule:


Whenever multiplying fractions together:


\[\frac{a}{b}\] x \[\frac{c}{d}\]   = \[\frac{a × c}{b × d}\]


Multiply the numerators together, then multiply the denominator together.

Types of Fraction Multiplication

There are different types of fraction multiplication that you must know about. Let us discuss multiplying fractions examples in detail.


  1. Multiplication of a fraction with whole numbers

  2. Multiplication of a fraction with other fraction

  3. Multiplication of a fraction with variables


Multiplication Of Fractions With Fractions

In this section, we will learn how to multiply fractions with fractions. Below are the basic steps that you need to follow for multiplying fractions:


  1. First, multiply the top numbers that are the numerators.

  2. Then, multiply the bottom numbers that are the denominators. 

  3. Lastly, simplify the fraction if it is needed.


Consider the following example


Multiply the two fractions \[\frac{1}{2}\] x \[\frac{2}{5}\]


The first step is to multiply the numerators. Doing so, you get,


\[\frac{1}{2}\] x \[\frac{2}{5}\] = \[\underline{1 × 2}\] = \[\underline{2}\]


The next step is to multiply the denominators.


\[\frac{1}{2}\] x \[\frac{2}{5}\] = \[\frac{1 × 2}{2 × 5}\] = \[\frac{2}{10}\]


The last step is to simplify the fraction.


\[\frac{2}{10}\] = \[\frac{1}{5}\]


Since the result is a proper fraction, it is an example of how to multiply proper fractions.

Multiplying Fractions With Whole Numbers

If you multiply a whole number or a real number with a fraction, it equals to the real number times the fraction added. Let us look at the example below on multiplying fractions with whole numbers.


Multiply 7 x \[\frac{1}{2}\].


7 x \[\frac{1}{2}\] refers to 7 times of \[\frac{1}{2}\]. This means that if you add \[\frac{1}{2}\] for 7 times, you would get the answer.


Hence, 


\[\frac{1}{2}\] + \[\frac{1}{2}\] + \[\frac{1}{2}\] + \[\frac{1}{2}\] + \[\frac{1}{2}\] + \[\frac{1}{2}\] + \[\frac{1}{2}\] + \[\frac{1}{2}\] = \[\frac{7}{2}\] = 3.5


If you perform multiplication of mixed fractions with a whole number, you would get a fraction itself. Consider the following example:


Multiply 5 \[\frac{1}{2}\] x 11. 


Simplifying the mixed fraction, you get,


5 \[\frac{1}{2}\] x 11 = \[\frac{11 × 11}{2}\]  = \[\frac{121}{2}\] = 60.5

Multiplying Fractions With Variables

When multiplying fractions with variables, the result you get is shown below. Consider the given example:


Multiply \[\frac{5x}{2y}\] x \[\frac{2x}{3z}\]


\[\frac{5x}{2y}\] x \[\frac{2x}{3z}\] = \[\frac{5x × 2x}{2y × 3z}\]


=  \[\frac{10x^{2}}{6yz}\]


FAQ (Frequently Asked Questions)

1. How to multiply fractions?

For multiplying fractions, the first step is to simplify the fraction to its lowest term. In the case of multiplication of mixed fractions, simplify it. Once you simplify the fraction, multiply the numerators together and then the denominators together. Then, the product of fractions would be obtained in a/b form.

2. How to multiply fractions with the same denominator?

Multiplying fractions with like denominators is very simple. First, multiply the numerators together and then the denominators. Once you get the product, reduce and simplify the fraction in case it is needed. Consider the following example:


Multiply  ¾  x 1/4 .


Follow the steps as mentioned above and you would get

¾ x ¼ = (3 x 1)/(4 x 4)


= 3/16


Since this cannot be reduced further, the answer is 3/16.