A Guide to Dividing Fractions

Important Takeaways

The phrase 'dividing fractions' can be intimidating to many—there are terms like divisor, dividend, and reciprocal to contend with, and the process requires flipping fractions. It might seem difficult at first, but once you familiarize yourself with the rules and concepts, it becomes much easier. With a little practice, you'll find dividing fractions to be quite straightforward. Math is about mastering terms and techniques, and with that, dividing fractions becomes second nature.

Division is essentially the opposite of multiplication, meaning that when you divide fractions, the result will always be greater than either of the individual fractions in the problem. You're essentially determining how many times the divisor (the second number) fits into the dividend (the first number). If you know how to multiply fractions, you won’t struggle with dividing them.

What is the Process for Dividing Two Fractions?

Step 1: Retain the Numerators and Denominators

Before diving into the problem, take a moment to look at your fractions, take a deep breath, and remind yourself that if a sixth grader can master dividing fractions, so can you!

The first step in dividing fractions is as simple as that encouraging statement. For example, if you're solving 2/3 ÷ 1/6, don’t make any changes! Simply leave both the numerator and denominator of each fraction as they are.

Step 2: Switch the Division Symbol

The next step in dividing fractions is to turn the division sign (÷) into a multiplication sign (x). So, 2/3 ÷ 1/6 becomes 2/3 x 1/6.

Step 3: Invert the Second Fraction

The third step is to find the reciprocal of the second fraction, but don’t worry! Dividing fractions is just like multiplying the first fraction by the reciprocal of the second one.

This means you’ll flip the numerator (the top part) and the denominator (the bottom part) of the fraction on the right side of the division sign, which is called the divisor.

For example, if you divide 2/3 by 1/6, you begin by inverting the divisor: 2/3 x 6/1 = 12/3.

Now you're dealing with an improper fraction.

You may observe that the fraction is no longer in proper form, where the numerator is smaller than the denominator; instead, it has become an improper fraction.

Improper fractions are those where the value the fraction represents exceeds 1.

Step 4: Is your answer final? Not yet. Simplify the fraction.

You're almost there, but it's not the final answer yet.

Now, all you need to do is simplify the fraction 12/3. This involves finding the greatest common divisor of both the numerator and denominator, which in this case is 3. So, the fraction simplifies to 4/1, or simply 4. That’s your final result.

How Do You Divide Mixed Fractions?

Dividing fractions with mixed numbers requires an extra step. You need to convert the mixed fractions (fractions with whole numbers) into improper fractions first, and then proceed to divide them just like you would with two simple fractions. Here’s an example: 3/4 ÷ 1 1/2.

Step 1: Convert the Mixed Fraction into an Improper Fraction.

The first step is to transform 1 1/2 into an improper fraction. 1 1/2 is equivalent to 3/2. Now, the problem becomes: 3/4 ÷ 3/2.

Step 2: Change the Division Operator.

Next, change the division sign (÷) to a multiplication sign (x): 3/4 ÷ 3/2 becomes 3/4 x 3/2.

Step 3: Flip the Second Fraction.

Keep the first fraction unchanged, then invert the second fraction. For example, turning 3/4 x 3/2 into 3/4 x 2/3 = 6/12.

Step 4: Reduce the Fraction to its Simplest Form.

Next, simplify the fraction: 6/12 = 1/2.

Thus, the solution to the equation 3/4 ÷ 1 1/2 = 1/2 is found.

To divide a mixed number by a fraction, begin by converting the mixed number to an improper fraction and proceed with the steps outlined above.

How Can You Divide Fractions When Their Denominators Are Different?

Once we have grasped the basic concepts and worked through two examples, dividing fractions with differing denominators becomes straightforward.

Step 1: Determine the Reciprocal of the Second Fraction

All you need to do is swap the numerator and denominator.

Step 2: Multiply the Numerators Together

Step 3: Multiply the Denominators Together

Step 4: Reduce the Fraction to its Simplest Form