Dividing Fractions
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Search | Updated December 26th, 2017

    This lesson will inform you how to divide fractions. Here are the sections within this lesson page:

    Before progressing with this lesson, it is strongly recommended that you have a firm grasp of fraction multiplication. Dividing fractions relies on it. So, please review our lesson on fraction multiplication before moving on with the sections below.

    esson: Multiplying Fractions

    Say you found $20 and you wanted to equally share your finding with three friends. If the four of you were to take an equal portion of the $20, each of you would receive $20/4 or $5. Twenty divided by four… It is an easy problem.

    However, if you asked yourself ‘How many 50-cent pieces are there in $20?” it would lead to a harder problem, mathematically speaking.

    The problem demands that we divide the 20 by a 1/2 dollar.

    In the section Example 1, we will see how to mathematically perform this task.

    To divide fractions, carry out these two steps.
  • Step 1: Take the reciprocal of the second fraction.
  • Step 2: Multiply the two fractions.
    That is all there is to it. If you would like to know why these steps work, jump to our last section: How It Works.

    Here is our first example:

dividing fractions

    Following our steps, first we must first take the reciprocal of the second fraction.

multiplying fractions

    We can see that 2 divided by 1 is 2. So, our problem is now…

multiplying fractions

    This answers our question from our previous section. There are 40 half-dollars in $20.

    ideo: Dividing Fractions
    uiz: Dividing Fractions

    Here is our second exmaple.

dividing fractions

    The Steps tell us to: 1) take the reciprocal of the second fraction and 2) switch to multiplication.

multiplying fractions

    Now, we will reduce. Starting with the 12 and 32, we get this.

reducing multiplication fractions

    Now, we can reduce the 42 and the 35.

reducing multiplication fractions

    We can still reduce some more. Look at the 6 and 8. They can both be divided by 2 because they are even numbers.

reducing multiplication fractions

    Since no remaining numerator has any factor in common with any remaining denominator, we need to multiply across to obtain the solution.

multiplication fractions multiply across

    ideo: Dividing Fractions
    uiz: Dividing Fractions

    So far, we have seen examples that describe how to perform tasks that lead to correct answers. However, it is worth mentioning the mathematics behind this process. Why? Mathematics is more than about mindlessly following tasks. Knowing why it works is important for deep understanding of this topic and all of mathematics.

    Here is a problem.

dividing fractions

    This horizontal layout can be changed to a vertical layout, like so because a fraction is really a division problem.

compound fraction

    To simplify the fraction in the denominator of our compound fraction (the 1/6), we can multiply it by its reciprocal, which is 6/1. However, if we multiply the bottom of a fraction by 6/1, then we must also multiply the numerator by the same quantity. Doing so maintains the value of our complex fraction.

    So, let us now multiply both the numerator and the denominator of our compound fraction by 6/1.

simplifying compound fraction

    Cleaning up the denominator, we get…

simplifying compound fraction

    It is clear now that when we place our original problem next to this result, we can see that our Two-Step Process is accurate.

changing from division to multiplication fractions

    We can see that a division problem can be altered to a multiplication problem if we flip the second fraction.

    Since we only wanted to understand the underlying process behind the steps, we will stop here. [The final answer is 4.]

    Use this instructional video to learn how to divide fractions.

    ideo: Dividing Fractions

    Use this interactive quiz to determine if you can divide fractions.

    uiz: Dividing Fractions

    Try these lessons, which are related to the sections above.

    esson: Divisibility Check
    esson: Fractions, Decimals, Percents
    esson: Reducing Fractions
    esson: Multiplying Fractions
    esson: Adding and Subtracting Fractions
    esson: Percent Word Problems
    esson: Dividing Rational Expressions