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Multiplying and Dividing Fractions

This article covers how to multiply and divide fractions.

First, let’s start with a “how to” video on multiplying fractions:

 

Always remember: to multiply two fractions simply multiply the numerators and multiply the denominators. Your new numerator will be the product of the two numerators, and your new denominator will be the product of the two denominators.

Now, let’s go over some practice questions step-by-step.

What is the fraction of the color blue in the figure below?

If you said \(\frac{1}{4}\) then you’re correct! The whole figure is divided into four parts and one part of the whole can be written as the fraction \(\frac{1}{4}\). Now what if you are asked what is \(\frac{1}{4}\) of that? In other words, if the blue part is divided further into four parts what will be the fraction of one sub-part of that division, with respect to the whole figure? Let’s divide the blue portion and see what happens.

Notice that dividing the blue portion further into parts, has resulted into unequal partitions of the figure. We cannot write the shaded blue part as fraction unless the whole figure is divided into equal parts. In order to achieve equal portion size, all the colored boxes must be divided into four.

Now, there are 16 equal parts of the figure and the fraction of shaded blue part is \(\frac{1}{16}\). This means that \(\frac{1}{4}\text{ of } \frac{1}{4}\) is equal to \(\frac{1}{16}\)! Mathematically translating this statement, โ€˜ofโ€™ means multiplication.

This teaches us the important concept of multiplying fractions! When we have no figures available, and just plain old fractions, what do we do then? How do we multiply? It’s very easy! Just follow one simple step:

Multiply the numerator by the numerator and multiply the denominator by the denominator!

Here are some other examples: \(\frac{1}{5} \times \frac{1}{6}=\frac{1 \times 1}{5 \times 6}=\frac{1}{30}\),

\(\frac{2}{4} \times \frac{3}{7}=\frac{2 \times 3}{4 \times 7}=\frac{8}{21}\)

 

If your final answer needs to be simplified further, you can simplify the fraction. See our article on โ€˜simplifying fractionsโ€™ to learn how to do that!

Multiplying Fractions and Integers

Multiplying fractions and integers is very simple and you follow the same procedure as above. The important thing to remember is that integers are also fractions with a denominator equal to 1!

This is illustrated by some examples:

\(7 \times \frac{2}{9}=\frac{7}{1}\times \frac{2}{9}=\frac{14}{9}\),

\(12 \times \frac{4}{16}=\frac{12}{1}\times \frac{4}{16}=\frac{48}{16}\).

Multiplying Mixed Numbers

Mixed numbers consist of whole numbers and fractions like: \(1\frac{1}{4}, 5\frac{4}{7}\), etc. In order to multiply mixed numbers, you first need to convert mixed numbers into fractions. The steps to do that are explained below:

Convert to fraction: \(4\frac{4}{8}\)

Step 1: Multiply the whole number and the denominator of the fraction.

Step 2: Add the numerator to the answer of step 1.

Step 3: The final answer obtained in step two is the numerator of the fraction and the denominator remains the same as in the original mixed number. So, the final answer is:

\(\frac{36}{8}\)

 

Remember: Converting mixed numbers into a fraction always results in an improper fraction.

Once the mixed number is converted to a fraction, follow the same procedure as outlined above to multiply them.

Multiply: \(3\frac{1}{2} \times 6\frac{2}{3}\)

First, convert both mixed numbers into improper fractions. Remember, an improper fraction is when the numerator is greater than the denominator.

Now multiply: \(\frac{7}{2} \times \frac{20}{3}=\frac{7 \times 20}{2 \times 3}=\frac{140}{6}\)

The final answer is the form of the improper fraction \(\frac{140}{6}\)! If you want to convert the final answer to mixed number form, see our article on simplifying fractions. We have covered this conversion in that article.

Dividing Fractions

Multiplications sounds easy enough, but how do you divide two fractions? Before we do that, let’s learn about reciprocals. What is the reciprocal of a fraction?

When we replace the numerator by the denominator and the denominator by the numerator, we get the reciprocal of a fraction.

If the statement sounds confusing, see the example given below and re-read the statement.

\(\frac{5}{9} \div \frac{4}{16} = \frac{5}{9} \times \frac{16}{4}=\frac{80}{36}\)

 

When we divide one fraction by another, the first fraction is the dividend and the second is the divisor. In case of the example, \(\frac{5}{9}\) is the dividend and \(\frac{4}{16}\) is the divisor.

How to solve this? Follow this simple procedure:

Take the reciprocal of the divisor and multiply it by the dividend!

The same goes for dividing fractions and integers. See the examples below for understanding:

\(\frac{4}{20} \div 4=\frac{4}{20} \times \frac{1}{4} = \frac{4}{80}\)

 

For dividing mixed numbers, convert them to improper fractiona first and then take the reciprocal of the divisor and multiply it by the dividend.

Final Thoughts!

So, this was all about multiplying and dividing fractions. The main concept to learn and remember here is how to multiply fractions. As you have seen, division of fractions also involves a multiplication step, so if you know multiplication, you are all set! Click for more help with fractions, decimals, and percentages. If you want to learn how to convert an improper fraction to a mixed number, see our article on Simplifying Fractions. Explore our resources further, to enhance your mathematical learning in an easy and fun way!