Dividing Fractions

Gaining facility in dividing fractions is of vital importance for mastering all the fractional operations: addition, subtraction, multiplication and division.

Probably starting in fourth grade math, and then receiving strong emphasis in fifth grade math and sixth grade math, your child learned the operations for fractions.

That is, how to do the four arithmetic operations of addition, subtraction, multiplication and division with fractions, or with combinations of fractions, mixed numbers and whole numbers.

Like dividing fractions, expertise in calculating percentages is a core skill that helps your child earn top scores in math and science classes, and on the ACT and SAT tests. Check out our exclusive resource to help your child become a Percents Expert.

The next step, and this topic is addressed throughout the middle school years, is Applications of Fractions. That means understanding when to use the different operations in the context of real life fractions situations.

This page is focusing solely on the real-life application of dividing fractions.

If your upper elementary or middle school child really doesn't know to do dividing fractions confidently, and doesn't know how or when to get a common denominator, you really need to work with him first on learning these basic steps. (Scroll down to the end of this article to see my TOP recommended fractions practice materials!)

OK, so assuming your child knows how to do operations with fractions pretty well, let's look at the applications. The real trick for this is recognizing the English words that point to the given operation. Another way to look at it is, recognizing the typical situations that relate to a given operation.

Here's a simple problem involving dividing fractions to see if your child knows how to compute:

• A small medicine bottle contains 6 ½ ounces. Each dose of the medicine is ½ ounce. How many doses of medicine are in the bottle?

13/2 ÷ 1/2 (make the mixed number into an improper fraction)

13/2 x 2/1 (invert and change to multiplication)

13/1 x 1/1 (cancel common factor of 2 and multiply)

13

Division is the trickiest fraction word problem, by far.

Fortunately, it is also the rarest by far.

You can think of this division problem as literally dividing that medicine bottle into ½ ounce portions. As though you had a large number of little ½ ounce plastic cups and poured out the whole bottle into ½ ounce doses, and then counted how many little cups you had.

The confusion is really created by the fraction itself. One trick to seeing that dividing fractions is the needed operation is to rephrase the problem without fractions.

• “A small medicine bottle contains 8 ounces. Each dose of the medicine is 2 ounces. How many doses of medicine are in the bottle?”

Now isn’t it obvious there are 4 doses? And where does 4 come from? 8 ÷ 2. So the operation is division.

We are dividing our bottle into smaller units and the question is how many of these smaller units will we have?

Fraction division problems are the hardest to visualize!

What I suggest to students is the following:

• If you see the concept of grouping or putting together, it’s addition.
• If you see the concept of taking away, it’s subtraction.
• If you see the wording “a fraction of” it’s multiplication.
• If it’s not one of those, and you think it might be division. Try substituting easy whole numbers in for the numbers of the problem and see if it makes sense as division.

Here is a second example:

• Mr. Van Gogh, the 7th grade art teacher, gave each student 1 1/2 pounds of clay for their pottery projects. If he used up a whole 30 pound bag of clay on this project, how many students were in Mr. Van Gogh’s art class?

Let’s see: ...It’s certainly not an adding fractions problem, because we’re not putting parts together to make a whole.

...It’s definitely not multiplying fractions, because there’s no “fraction of” wording in the problems.

It seems a tiny bit like subtracting fractions, because we did say the clay got used up, but it’s not that we started with 30 pounds and took away, or subtracted 1 1/2 pounds, but rather, we started with 30 pounds of clay and divided it into 1 1/2 pound units.

Let’s try the trick of substituting in easier numbers.

Let’s say he started with 10 pounds of clay, and gave each student 2 pounds. Then how many students would there be?

Obviously 5. Because 10 ÷ 2 is 5. So it must be that we want to divide.

Now let’s do the problem: 30 ÷ 1 1/2

30/1 ÷ 3/2 (make the mixed number an improper fraction)

30/1 x 2/3 (invert and multiply)

10/1 x 2/1 (cancel common factor of 3)

20/1 or 20 (multiply across the fractions)

Now, let me speak as a concerned teacher, to you, a concerned parent...

If your child can't, unassisted, accurately compute the simple dividing fractions problems I've given above, it's time for you to take action!

Your child needs help understanding the basics of dividing fractions.

This is not necessarily a sign of a learning difficulty, but more likely that your child's teacher never spent enough time teaching and practicing dividing fractions in the classroom. (I consider this a math education crime! Your child was robbed of the opportunity to gain mastery of a vital math education skill he'll need throughout middle school and high school math courses.)

It takes weeks of incremental skills development and practice to gain facility with dividing fractions!!! Did your child's teacher give him that? (You may be interested to read about some of the problems in today's American Math education to learn more about this phenomenon.)

To help your child bridge his or her knowledge gap, here are my TOP RECOMMENDED practice books for you to use with your child.

I only recommend products that I have used successfully in my classrooms, and in my own home with my children. (Cost matters too. We can buy good math practice books and tools without breaking the bank!)

For 4th Grade Math Practice:

• Kumon 4th grade math practice books are well presented and organized. At $6.95, its a great value compared to other math workbooks!
• For homeschoolers and those interested in more intense math practice at home: I would recommend this Saxon Publishing 4th grade math practice book. I have not only successfully used the Saxon materials in the classroom, but I have contributed and edited a text book for them. Their math materials are superior!
• And, I'm positive your children will love these as much as mine do. These fun fractions tiles will help visually demonstrate the operation of adding fractions. Use them as you're working out fractions problems together with your child.

• Kumon 5th grade math practice books are tops for zeroing in on one skill and working on incremental fractions skill development. At only $6.95 (and eligible for FREE Shipping from amazon.com) they are a real bargain!
• For Homeschoolers and parents who want more intense at-home math practice, you should consider the practice books and products in Saxon Math's excellent series of math instruction books.

• Kumon 6th grade math practice booksare tops for zeroing in on one skill and working on incremental fractions skill development. At only $6.95 (and eligible for FREE Shipping from amazon.com) they are a real bargain!
• For Homeschoolers and parents who want more intense at-home math practice, you should consider the practice books and products in Saxon Math's excellent series of math instruction books.

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