dividing fractions by integers problem solving

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Course: 6th grade   >   Unit 2

• Understanding division of fractions
• Dividing fractions: 2/5 ÷ 7/3

Dividing fractions: 3/5 ÷ 1/2

• Dividing fractions
• Dividing mixed numbers
• Divide mixed numbers
• Writing fraction division story problems
• Interpret fraction division
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Video transcript

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Division of Fractions and Integers

A fraction is a division operation. 1 2 = 0 . 5 because 1 ÷ 2 = 0 . 5 . But you can also divide fractions by fractions! Below is the rule for what to do when you have one fraction divided by another fraction.

Keep, Change, Flip

What happens is that you flip the bottom fraction, and replace the division sign with a multiplication sign. The math terminology is that you are “multiplying by the inverse fraction ”. I call it “Keep, Change, Flip Formula”!

Want to watch animated videos and solve interactive exercises about fractions? Click here to try Video Crash Courses !

Using the flip formula, you get

Note! When you divide an integer or fraction by a proper fraction (which is less than 1 ), you’ll find that the answer you get is actually larger than the integer or fraction you started with. This is different than normal division with whole numbers where the answer is smaller.

If you divide a fraction by an integer, or an integer by a fraction, it may be easier to use the trick you apply on integers multiplied by fractions—that is, to write the integer as a fraction.

Since 5 = 5 1 , you get

Since 3 = 3 1 , you get

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11.7: Dividing Fractions- Problems

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• Michelle Manes
• University of Hawaii

We’ve spent the last couple of chapters talking about dividing fractions: how to make sense of the operation, how to picture what’s going on, and how to do the computations. But all of this kind of begs the question: When would you ever want to divide fractions, anyway? How does that even come up?

It’s important that teachers are able to come up with situations and problems that model particular operations, which means you have to really understand what the operations mean and when they are used.

Think / Pair / Share

• Use one of our methods (draw a picture, rectangles, common denominator, missing factor) to compute \(1 \frac{3}{4} \div \frac{1}{2}\).
• Come up with a situation where you would want to compute \(1 \frac{3}{4} \div \frac{1}{2}\). (That is, write a word problem that would require you to do this computation to solve it.)

When to Multiply, When to Divide?

A common answer to

Come up with a situation where you would want to compute \(1 \frac{3}{4} \div \frac{1}{2}\).

Is something like this:

My recipe calls for \(1 \frac{3}{4} \div \frac{1}{2}\) cups of flour, but I only want to make half a recipe. How much flour should I use?

But that problem doesn’t ask you to divide fractions. It asks you to cut your recipe in half, which means dividing by 2 or multiplying by \(\frac{1}{2}\).

Why is it so hard to come up with division problems that use fractions? Maybe it’s because fractions are already the answer to a division problem, so you’re dividing and then dividing some more. Maybe it’s because they just make it look so complicated. In any case, it’s worth spending some time thinking about division problems that involve fractions and how to recognize and solve them.

One handy trick: Write a problem that involves division of whole numbers, and then see if you can change the numbers to fractions in a sensible way.

Example \(\PageIndex{1}\):

Here are some division problems involving whole numbers:

• I have 10 feet of ribbon. How many 2-inch pieces can I cut from it?
• I have a fancy old clock that rings once every 15 minutes. How many times will it ring over the course of 2 hours (120 minutes)?
• My fish tank needs 6 gallons of water, and my bucket holds 3 gallons. How many times will I need to fill my bucket in order to fill the tank?
• A recipe calls for 6 cups of flour, and my largest scoop measures exactly 2 cups. How many times should I use it?
• I ran 12 miles and went around the the same route 3 times. How long was the route?

Here are some very similar problems, rewritten to use fractions instead:

• I have \(1 \frac{3}{4}\) feet of ribbon. How many 6-inch (that’s \(\frac{1}{2}\) a foot) pieces can I cut from it?
• My watch alarm goes off every half hour, and I don’t know how to shut it off. How many times will it go off during the \(1 \frac{3}{4}\) hour movie?
• My fish tank needs \(1 \frac{3}{4}\) gallons of water, and my bucket holds \(\frac{1}{2}\) gallon. How many times will I need to fill my bucket in order to fill the tank?
• I want to measure \(1 \frac{3}{4}\) cups of flour for a recipe, but I only have a \(\frac{1}{2}\) cup measuring cup. How many times should I fill it?
• I ran \(1 \frac{3}{4}\) miles before I twisted my ankle. I only finished half the race. How long was the race course?

For each one of the fraction division questions, we can understand why it’s a division problem:

• I have \(1 \frac{3}{4}\) feet of ribbon. How many 6-inch (that’s \(\frac{1}{2}\) a foot) pieces can I cut from it? This means making equal groups of \(\frac{1}{2}\) foot each and asking how many groups. That’s quotative division.
• My watch alarm goes off every half hour, and I don’t know how to shut it off. How many times will it go off during the \(1 \frac{3}{4}\) hour movie? Again, we’re making equal groups of \(\frac{1}{2}\) hour each, and asking how many groups. Quotative division.
• My fish tank needs \(1 \frac{3}{4}\) gallons of water, and my bucket holds \(\frac{1}{2}\) gallon. How many times will I need to fill my bucket in order to fill the tank? Once again: we’re making equal groups of \(\frac{1}{2}\) gallon each, and asking how many groups (buckets).
• I want to measure \(1 \frac{3}{4}\) cups of flour for a recipe, but I only have a \(\frac{1}{2}\) cup measuring cup. How many times should I fill it? This is making equal groups of \(\frac{1}{2}\) cup and asking how many groups.
• I ran \(1 \frac{3}{4}\) miles before I twisted my ankle. I only finished half the race. How long was the race course? This one is a little different. This one is a little different. It’s the fraction version of partitive division.

Recall what partitive division asks: For \(20 \div 4\), we ask 20 is 4 groups of what size?

So for \(1 \frac{3}{4} \div \frac{1}{2}\), we ask: \(1 \frac{3}{4}\) is half a group of what size?

You try it.

• First write five different division word problems that use whole numbers. (Try to write at least a couple each of partitive and quotative division problems.)
• Then change the problems so that they are fraction division problems instead. You might need to rewrite the problem a bit so that it makes sense.
• Solve your problems!

Dividing Fractions Practice Questions

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Dividing Fractions Worksheets

Welcome to our Dividing Fractions Worksheets page. Here you will find a range of free printable sheets and support to help your child learn to divide fractions by other fractions.

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• Dividing Fractions Calculator
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Dividing Fractions Online Quiz

Divding fractions calculator.

We have a calculator to help you to divide a fraction by another fraction.

You can divide mixed fractions or divide a fraction by an integer too.

All the working out is shown step-by-step so you can see how it is worked out.

• Divide Fractions Calculator

How to Divide Fractions Support

This support page will help you find out all about how to divide fractions.

There are several worked examples and a printable support sheet.

Formula for dividing two fractions

\[{a \over b} \div {c \over d} = {ad \over bc} \]

• How to Divide Fractions

About our Dividing Fractions Worksheets

Here you will find a selection of Fraction worksheets designed to help your child understand how to divide a fraction by another fraction, or to divide a fraction by a whole number.

The sheets are carefully graded so that the easiest sheets come first, and the most difficult sheet is the last one.

Before your child tackles dividing fractions, they should first be confident with multiplying fractions .

Using these sheets will help your child to:

• divide a fraction by another fraction;
• apply their understanding of simplest form;
• convert between mixed fractions and improper fractions.

To divide mixed fractions, use the link below!

Sheet 1 is supported and has examples and instructions to follow.

Sheet 2 has less support - also the fractions do not need to be simplified.

Sheets 3 and 4 have no support and the fractions need to be simplified.

Sheet 5 is similar to sheets 3 & 4, but the fractions also need to be converted to mixed fraction answers.

• Dividing Fractions Sheet 1
• PDF version
• Dividing Fractions Sheet 2
• Dividing Fractions Sheet 3
• Dividing Fractions Sheet 4
• Dividing Fractions Sheet 5

Dividing Fractions Walkthrough Video

This short video walkthrough shows several problems from our Dividing Fractions Worksheet 1 being solved and has been produced by the West Explains Best math channel.

If you would like some support in solving the problems on these sheets, please check out the video below!

Multiplying and Dividing Fractions Worksheets

We also have some multiplying and dividing fractions worksheets.

Often when you have taught both of these concepts, it is important to muddle them up and practice them both to develop confidence.

The sheets below are practice sheets for students who are already fairly confident multiplying and dividing fractions.

Sheets 1 & 2 have 24 questions. The fractions need simplifying, but can be left as improper fractions.

Sheet 3 has just 12 questions. The fractions need simplifying and converting to mixed fractions.

• Multiplying and Dividing Fractions 1
• Multiplying and Dividing Fractions 2
• Multiplying and Dividing Fractions 3

More Recommended Math Worksheets

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Multiplying and Dividing Fractions Support

We have a number of support pages designed to help you understand how to multiply and divide fractions.

On each support page, there are some worked examples to show you what to do.

We also have some math videos on some support pages to help explain what to do.

• Dividing Fractions by whole numbers
• Divide Whole numbers by Fractions

• How do you Multiply Fractions support

• How to Multiply Mixed Fractions support page
• How to Divide Fractions support page
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We have more worksheets to help your child learn and practice multiplying and dividing fractions.

All the sheets are carefully graded with different levels of support.

Take a look at our full range below.

• Multiplying Fractions Worksheets
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How to Convert Improper Fractions

This support page shows how to convert improper fractions to mixed numbers, and how to convert mixed numbers to improper fractions.

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• Convert Improper Fractions Support

Simplifying Fractions

This is also sometimes called reducing fractions to their simplest form.

This involves dividing both the numerator and denominator by a common factor to reduce the fraction to the equivalent fraction with the smallest possible numerator and denominator.

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This quick quiz tests your understanding and skill at dividing fractions.

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This resource is suitable for the White Rose Maths Scheme of Learning, covering Autumn Term Block 4 (Fractions B), Step 3 (Divide a Fraction by an Integer).

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• Key Stage: Key Stage 2
• Subject: Maths
• Topic: White Rose Maths | Autumn Term: Fractions B | Small Step 3
• Topic Group: Fractions, Decimals and Percentages
• Year(s): Year 6
• Media Type: PDF
• Resource Type: Worksheet
• Last Updated: 24/10/2023
• Resource Code: M2WAT14323
• Curriculum Point(s): Divide proper fractions by whole numbers [for example, 1/3 divided by 2 = 1/6].

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