John was invited to Amy's birthday, and she has invited in total 7 friends to celebrate her birthday. To have equal cake pieces, each of the attendees would have \(\frac{1}{8}\) of the cake. Accidentally, Amy dropped her piece of cake, so John decided to give her a part of his. He divided his piece of cake by 2 and gave Amy half.
Can we calculate what fraction of the cake Amy got in the end? The answer is dividing the fraction of John by 2, that is, \(\dfrac{\dfrac{1}{8}}{2}=\dfrac{1}{16}\) of the cake.
In this article, we will learn to do the operations of multiplication and division with fractions.
Multiplication and Division of Fractions Step by Step
We are interested in looking at the operations of multiplication and division on fractions. Foremost, let's recall our knowledge on fractions.
A fraction represents a part of a whole. It has two parts – the numerator and the denominator. The numerator is written above the line and the denominator is written below the line. The denominator cannot be zero.
\(\dfrac{2}{3}, \dfrac{1}{2}, \dfrac{7}{8}, \cdots\) are examples of fractions.
We are familiar with multiplying and dividing two numbers. Now the question is how to perform these operations on fractions instead of whole numbers.
Suppose you are given two fractions, say \(\dfrac{a}{b}\) and \(\dfrac{c}{d}\), we want to know what do we mean by \(\dfrac{a}{b}\times \dfrac{c}{d}\) and \(\dfrac{\dfrac{a}{b}}{\dfrac{c}{d}}.\)
Multiplication and division of fractions rules
Multiplication of fractions rules
To multiply two fractions \(\dfrac{a}{b}\) and \(\dfrac{c}{d}\), you essentially multiply the numerators together and the denominators together. Thus. we have
\[\dfrac{a}{b}\times \dfrac{c}{d}=\dfrac{a\times b}{c\times d}.\]
We, in fact, follow the following steps to multiply fractions together.
Step 1. Multiply the numerators of the two fractions together and the denominators together.
Step 2. Divide the resultant numbers to get the new fraction.
We can stop at this point. However, if the numerator and denominator of the new fraction have common factors, we proceed with another step to obtain the simplest form of the fraction.
Step 3. Find the common factor of the numerator and denominator of the new fraction. Divide the numerator and denominator by this common factor. This gives the simplest form of the fraction.
Multiply the fractions \(\dfrac{3}{7}\) and \(\dfrac{5}{11}\).
Solution
Step 1. Multiplying the numerators of the fractions together, we get \[3\times 5=15.\]
Multiplying the denominators of the fractions together, we get
\[7\times 11=77.\]
Step 2. Dividing the resultant numbers gives the new fraction \(\dfrac{15}{77}.\)
Since the numerator and denominator of the new fraction do not have any common factors, this is the simplest form.
Multiply \(\dfrac{2}{5}\) and \(\dfrac{7}{9}\).
Solution
Multiplying the numerators and denominators, we get
\[\dfrac{2}{5}\times \dfrac{7}{9}=\dfrac{2\times 7}{5 \times 9}=\dfrac{14}{45}.\]
Multiply \(\dfrac{5}{8}\) and \(\dfrac{2}{3}.\)
Solution
Step 1. Multiplying the numerators of the two fractions together, we get
\(5 \times 2=10.\) Similarly, doing the same with the denominators gives \(8\times 3=24.\)
Step 2. Dividing the resultant numbers gives us the new fraction \(\dfrac{10}{24}.\)
We notice that the numerator and denominator of the new fraction have a common factor of 2.
Step 3. We get the simplest form of this fraction by dividing out the common factor 2 from the numerator 10 and the denominator 24. This gives us, \(10 \divsymbol 2=5\) and \(24\divsymbol 2=12\).
The simplest fraction is therefore \(\dfrac{5}{12}.\)
Division of fractions rules
To divide two fractions, you essentially invert the fraction you are dividing with and then multiply it with the former. So the division of two fractions of the form
\[\frac{a}{b}\divsymbol\frac{c}{d}=\frac{\frac{a}{b}}{\frac{c}{d}}\]
is the same as multiplying the fractions
\[\frac{a}{b}\times \frac{d}{c}.\]
And thus we have
\[\frac{a}{b}\divsymbol\frac{c}{d} =\frac{a}{b}\times\frac{d}{c}.\]
Since we saw already how to multiply two fractions, you just follow those steps from here.
In summary, we follow the following steps to perform division on fractions,
Step 1. Invert the divisor fraction – the numerator becomes the denominator and the denominator becomes the numerator.
Step 2. After inversion, multiply the resultant fractions together using the steps described for the multiplication of fractions.
Divide \(\dfrac{5}{8}\) by \(\dfrac{2}{3}.\)
Solution
Step 1. Inverting the divisor, we get \(\dfrac{3}{2}\).
Step 2. Now we perform the multiplication of the obtained fractions,
\(\dfrac{5}{8}\) and \(\dfrac{3}{2}\) to get,
\[\dfrac{5}{8}\times \dfrac{3}{2}=\dfrac{5\times 3}{8\times 2}=\dfrac{15}{16}.\]
Since the numerator and denominator have no common factors, this is the simplest form.
Find \(\dfrac{2}{5}\divsymbol \dfrac{3}{8}\).
Solution
Here \(\dfrac{2}{5}\)is the dividend fraction and \(\dfrac{3}{8}\)is the divisor fraction.
Step 1. Invert the divisor, we get \(\dfrac{8}{3}.\)
Step 2. Now multiply the fractions we get,
\[\frac{2}{5}\divsymbol\frac{3}{8}=\frac{2}{5}\times \frac{8}{3}=\frac{2\times 8}{3\times 5} =\frac{16}{15}.\]
Since the numerator and denominator has no common factors, this is the simplest form.
When multiplying or dividing a fraction with a whole number \(a\), \(a\) can be written as its equivalent form \(\dfrac{a}{1}\) and thus no change in procedure is required.
Find \(\dfrac{\dfrac{2}{5}}{3}.\)
Solution
Here \(\dfrac{2}{5}\)is the dividend fraction and \(3=\dfrac{3}{1}\) is the divisor fraction.
Step 1. Invert the divisor, we get \(\dfrac{1}{3}\).
Step 2. Now multiply the fractions to get,
\[\dfrac{2}{5}\times \dfrac{1}{3}=\dfrac{2\times 1}{5\times 3}=\dfrac{2}{15}.\]
Since the numerator and denominator have no common factors, this is the simplest form.
Simplify \(\dfrac{4}{\dfrac{7}{9}}\).
Solution
Here \(4=\dfrac{4}{1}\)is the dividend fraction and \(\dfrac{7}{9}\)is the divisor fraction.
Solution
Step 1. Invert the divisor, we get \(\dfrac{9}{7}\).
Step 2. Now multiply the fractions together to get,
\[\dfrac{4}{\dfrac{7}{9}}=\dfrac{4}{1}\times \dfrac{9}{7}=\dfrac{4\times 9}{1\times 7}=\dfrac{36}{7}.\]
Since the numerator and denominator have no common factors, this is the simplest form.
To simplify our work by avoiding giant multiplications, we can “cancel” common factors between the numerators and denominators in the beginning before we multiply the terms together. This would modify the steps for multiplying fractions together to the following,
Step 1. If any numerator and denominator have a common factor, divide the corresponding numerator and denominator by the common factor to “cancel out” the common factor. Do this until no common factors remain between numerators and denominators.
Step 2. Perform multiplication of the resultant fractions.
In the following examples, we have used the aforementioned method.
Examples of Multiplication and Division of Fractions
So far, we have looked at examples involving operations of multiplication and division between two fractions. You can multiply/divide multiple fractions together using the same rules as described above. If there is a chain of multiple multiplications and divisions, you must first invert the divisor terms.
Simplify \(\dfrac{5}{9}\times\dfrac{18}{13}\times\dfrac{21}{20}\)
Solution
Here we have three fractions under multiplication. The first step is to multiply the numerators of the fractions together \(5\times 18\times 21\) and the denominators together \(9\times 13\times 20.\)
We see here that we end up with a multiplication of huge numbers. To avoid this, we are going first to cancel the common factors, where possible.
Step 1 . The numerators are 5,18,21 and the denominators are 9,13,20. We see 9 and 18 has 9 as common factor and 5 and 20 has 5 as a factor, thus we have
\[\frac{5}{9}\times\dfrac{18}{13}\times\dfrac{21}{20}=\dfrac{1}{1}\times\dfrac{2}{13}\times\dfrac{21}{4}.\]
Further, we can simplify 2 and 4 by dividing by 2, to get
\[\dfrac{5}{9}\times\dfrac{18}{13}\times\dfrac{21}{20}=\dfrac{1}{13} \times\dfrac{21}{2}.\]
Step 2. And the final answer is,
\[\dfrac{5}{9}\times\dfrac{18}{13}\times\dfrac{21}{20}=\dfrac{21}{13\times 2}=\dfrac{21}{26}.\]
Simplify
\[\dfrac{14}{39}\times\dfrac{12}{35}\divsymbol\dfrac{8}{13}\times\dfrac{2}{9}\]
Solution
Step 1. Invert the divisor fraction to get,
\[\dfrac{14}{39}\times\dfrac{12}{35}\divsymbol\dfrac{8}{13}\times\dfrac{2}{9}=\dfrac{14}{39}\times\dfrac{12}{35}\times\dfrac{13}{8}\times\dfrac{2}{9}\]
Step 2. Now we try to bring the terms to the simplest form. Dividing 14 and 35 by 7, 13 and 39 by 13, 12 and 9 by 3, 2 and 8 by 2 we get,
\[\dfrac{14}{39}\times\frac{12}{35}\times\dfrac{13}{8}\times\dfrac{2}{9}=\dfrac{2}{3}\times\dfrac{4}{5}\times\dfrac{1}{4}\times\dfrac{1}{3}\]
Step 3. Cancel out the 4, we get\[\dfrac{2}{3}\times\dfrac{4}{5}\times\dfrac{1}{4}\times\dfrac{1}{3}=\dfrac{2}{5}\times\dfrac{1}{5}\times \dfrac{1}{3}=\dfrac{2}{45}.\]
In the next example, we perform multiplication and division of mixed fractions.
A mixed fraction is a combination of a whole number and a fraction. To multiply or divide mixed fractions, first, convert them into improper fractions and then continue with the standard process.
Simplify
\[4\dfrac{2}{7}\times 2\dfrac{1}{3}\div \dfrac{3}{5}.\]
Solution
Converting the mixed fractions into improper fractions, we get,
\[4\dfrac{2}{7}\times 2\dfrac{1}{3}\div \frac{3}{5} = \dfrac{30}{7}\times \dfrac{7}{3} \div \dfrac{3}{5}.\]
Inverting the divisor, we get,
\[\dfrac{30}{7}\times\dfrac{7}{3}\div\dfrac{3}{5}= \dfrac{30}{7} \times \dfrac{7}{3} \times \dfrac{5}{3}\]
Dividing 30 and 3 by 3, cancelling the 7 in the numerator and denominator, we have
\[\dfrac{30}{7}\times\dfrac{7}{3}\times \dfrac{5}{3}= \dfrac{10}{1} \times \dfrac{1}{1} \times \dfrac{5}{3}.\]
Multiplying the above fractions gives,
\[\dfrac{10}{1}\times\dfrac{5}{3}= \dfrac{50}{3} = 16\dfrac{2}{3}.\]
You can express your answer as a mixed fraction or improper fraction as necessary.
Multiplication and Division of Algebraic fractions
You can perform multiplication and division on algebraic fractions containing variable in the numerator and/or denominator, following the same steps that we have been using so far.
Simplify \(\dfrac{4xy}{5} \times \dfrac{2y}{x^3}\div \dfrac{y}{x}\).
Solution
Inverting the divisor, we get
\[\dfrac{4xy}{5} \times \dfrac{2y}{x^3} \div \dfrac{y}{x} = \dfrac{4xy}{5} \times \dfrac{2y}{x^3} \times \dfrac{x}{y}.\]
Dividing \(4xy\) and \(x^{3}\) by \(x\) and \(2y\) and \(y\) by \(y\), we get
\[ \dfrac{4xy}{5}\times\dfrac{2y}{x^3}\times\dfrac{x}{y}= \dfrac{4y}{5} \times \dfrac{2}{x^2} \times \dfrac{x}{1}.\]
Dividing \(x^2\) and \(x\) by \(x\) we get,
\[ \dfrac{4y}{5}\times\dfrac{2}{x^2}\dfrac{x}{1}= \dfrac{4y}{5} \times \dfrac{2}{x} \times \dfrac{1}{1}\]
Multiplying the above fractions gives,
\[ \dfrac{4y}{5}\times\dfrac{2}{x}\times\dfrac{1}{1}= \dfrac{8y}{5x}.\]
Multiply \( 2y^3 + 3xy + 5x^2 + 7\) by \(4x^2\).
Solution
\[\begin{align} &(2y^3 + 3xy + 5x^2 + 7) \times 4x^2 \\ &= (2y^3 \times 4x^2) + (3xy\times 4x^2) + (5x^2\times 4x^2) + (7\times 4x^2)\\ &= 8x^2y^3 + 12x^3 y + 20x^4 + 28x^2.\end{align}\]
Simplify \(\dfrac{2x^2 y^3}{7} \times \dfrac{14}{xy} \times \dfrac{y}{x^3}\).
Solution
Dividing \(2x^2y^3\) and \(xy\) by \(xy\), and 7 and 14 by 7, we get
\[ \frac{2x^2y^3}{7} \times \frac{14}{xy} \times \frac{y}{x^3} = \frac{2xy^2}{1} \times \frac{2}{1} \times \frac{y}{x^3} .\]
Dividing \(2xy^2\) and \(x^3\) by \(x\), we get,
\[\frac{2xy^{2}}{1}\times\frac{2}{1}\times\frac{y}{x^3}= \frac{2y^2}{1} \times \frac{2}{1} \times \frac{y}{x^2} .\]
Multiplying the above fractions, we get
\[ \frac{2y^2}{1} \times \frac{2}{1} \times \frac{y}{x^2} = \frac{4y^3}{x^2}. \]
Multiplication and Division of Fractions - Key takeaways
- To multiply fractions, you essentially multiply the numerators together and the denominators together. So a multiplication of the form\( \dfrac{a}{b}\times \dfrac{c}{d}\) is equivalent to \(\dfrac{a\times c}{b\times d}.\)
- To divide a number(whole number or fraction) with a fraction, we have first to invert the divisor and to apply the multiplication process to the remaining of the expression.
- To multiply or divide mixed fractions, first convert them into improper fractions and then continue with the standard process.
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