# How to divide a fraction by another fraction or by an integer

• Dividing Fractions and Mixed Numbers
• Learning Objective(s)
• ·         Find the reciprocal of a number.
• ·         Divide two fractions.
• ·         Divide two mixed numbers.
• ·         Divide fractions, mixed numbers, and whole numbers.
• ·         Solve application problems that require division of fractions or mixed numbers.

There are times when you need to use division to solve a problem.

For example, if painting one coat of paint on the walls of a room requires 3 quarts of paint and there are 6 quarts of paint, how many coats of paint can you paint on the walls? You divide 6 by 3 for an answer of 2 coats. There will also be times when you need to divide by a fraction.

Suppose painting a closet with one coat only required  quart of paint. How many coats could be painted with the 6 quarts of paint? To find the answer, you need to divide 6 by the fraction, .

If the product of two numbers is 1, the two numbers are reciprocals of each other. Here are some examples:

 Original number Reciprocal Product

In each case, the original number, when multiplied by its reciprocal, equals 1.

To create two numbers that multiply together to give an answer of one, the numerator of one is the denominator of the other. You sometimes say one number is the “flip” of the other number: flip  to get the reciprocal . In order to find the reciprocal of a mixed number, write it first as an improper fraction so that it can be “flipped.”

 Example Problem Find the reciprocal of . Rewrite  as an improper fraction. The numerator is 4 • 5 + 1 = 21. Answer Find the reciprocal by interchanging (“flipping”) the numerator and denominator.
 What is the reciprocal of ? A) B) C) D) Show/Hide Answer A) Incorrect. The fractional parts of this answer and the original mixed number are reciprocals, but in order to find the reciprocal of the entire number, you must write the mixed number as an improper fraction before interchanging numerator and denominator. The correct answer is . B) Incorrect. You found the correct improper fraction that represents  but did not find the reciprocal. The reciprocal of  is . C) Correct. First, write  as  an improper fraction ,. The reciprocal of  is found by interchanging (“flipping”) the numerator and denominator. D) Incorrect. This is the correct reciprocal for the fractional part of the mixed number, but with a mixed number, you first need to write it as an improper fraction. The mixed number,  can be written as the improper fraction, . The reciprocal of  is found by interchanging (“flipping”) the numerator and denominator.

Dividing a Fraction or a Mixed Number by a Whole Number

When you divide by a whole number, you multiply by the reciprocal of the divisor. In the painting example where you need 3 quarts of paint for a coat and have 6 quarts of paint, you can find the total number of coats that can be painted by dividing 6 by 3, 6 ÷3 = 2. You can also multiply 6 by the reciprocal of 3, which is , so the multiplication problem becomes .

The same idea will work when the divisor is a fraction. If you have  of a candy bar and need to divide it among 5 people, each person gets  of the available candy:  of  is , so each person gets  of a whole candy bar.

If you have a recipe that needs to be divided in half, you can divide each ingredient by 2, or you can multiply each ingredient by  to find the new amount.

Similarly, with a mixed number, you can either divide by the whole number or you can multiply by the reciprocal. Suppose you have  pizzas that you want to divide evenly among 6 people.

Dividing by 6 is the same as multiplying by the reciprocal of 6, which is . Cut the available pizza into six equal-sized pieces.

• Each person gets one piece, so each person gets  of a pizza.
• Dividing a fraction by a whole number is the same as multiplying by the reciprocal, so you can always use multiplication of fractions to solve such division problems.
 Example Problem Find . Write your answer as a mixed number with any fraction part in lowest terms. Rewrite  as an improper fraction. The numerator is  2 • 3 + 2. The denominator is still 3. Dividing by 4 or  is the same as multiplying by the reciprocal of 4, which is . Multiply numerators and multiply denominators. Simplify to lowest terms by dividing numerator and denominator by the common factor 4. Answer =
 Find Simplify the answer and write as a mixed number. A) B) C) D). Show/Hide Answer A) Correct. Write  as the improper fraction . Then multiply by , the reciprocal of 2. This gives the improper fraction , and the mixed number is , . B) Incorrect. After changing the mixed number to an improper fraction, you may have inverted  instead of 2. Keep , and multiply by the reciprocal of 2, giving you . Finally, write  as a mixed number, , . C). Incorrect. This is the correct improper fraction, but you still need to write the final answer as a mixed number, , which is . D) Incorrect. You may have forgotten to find the reciprocal of 2 before multiplying. Once you have the improper fraction for , which is , multiply by the reciprocal of 2, which is , giving you . Finally, write  as a mixed number, , .

Sometimes you need to solve a problem that requires dividing by a fraction. Suppose you have a pizza that is already cut into 4 slices. How many  slices are there?

There are 8 slices. You can see that dividing 4 by  gives the same result as multiplying 4 by 2. What would happen if you needed to divide each slice into thirds?

You would have 12 slices, which is the same as multiplying 4 by 3.

 Dividing with Fractions 1.      Find the reciprocal of the number that follows the division symbol. 2.      Multiply the first number (the one before the division symbol) by the reciprocal of the second number (the one after the division symbol). Examples:  and

Any easy way to remember how to divide fractions is the phrase “keep, change, flip”. This means to KEEP the first number, CHANGE the division sign to multiplication, and then FLIP (use the reciprocal) of the second number.

 Example Problem Divide. Multiply by the reciprocal: Keep , change  to •, and flip . Multiply numerators and multiply denominators. Simplify. Answer
 Example Problem Divide. Multiply by the reciprocal: Keep , change  to •, and flip . Multiply numerators and multiply denominators. Answer

When solving a division problem by multiplying by the reciprocal, remember to write all whole numbers and mixed numbers as improper fractions. The final answer should be simplified and written as a mixed number.

 Example Problem Divide. Write   as an improper fraction. Multiply by the reciprocal: Keep , change  to •, and flip . Multiply numerators and multiply denominators. Simplify. Answer
 Example Problem Divide. Simplify the answer and write as a mixed number. Write  as improper fractions. Multiply by the reciprocal of . Multiply numerators, multiply denominators. Regroup. Simplify: . Multiply. Rewrite as a mixed number.
 Find . Simplify the answer and write as a mixed number. A) B) C) D) 8 Show/Hide Answer A) Incorrect. You may have incorrectly written  as the improper fraction . To write  as an improper fraction, multiply 5 times 3 and add 1. The denominator is 3.  = Then change division to multiplication and multiply by the reciprocal of , which is , giving you . B) Incorrect. You may have forgotten to use the reciprocal of . After finding the improper fraction  that represents , change division to multiplication and use the reciprocal of , giving you . C) Incorrect. This is the correct improper fraction, but the final answer needs to be a mixed number. Divide 16 by 2, which is 8 and no remainder. There is no fractional part, so the answer is a whole number, 8. D) 8 Correct. Write  as an improper fraction, . Then multiply by the reciprocal of , which is , giving you .

Dividing Fractions or Mixed Numbers to Solve Problems

Using multiplication by the reciprocal instead of division can be very useful to solve problems that require division and fractions.

 Example Problem A cook has  pounds of ground beef. How many quarter- pound burgers can he make? You need to find how many quarter pounds there are in , so use division. Write  as an improper fraction. Multiply by the reciprocal. Multiply numerators and multiply denominators. Regroup and simplify , which is 1. Answer 75 burgers
 Example Problem A child needs to take  tablespoons of medicine per day in 4 equal doses. How much medicine is in each dose? You need to make 4 equal doses, so you can use division. Write  as an improper fraction. Multiply by the reciprocal. Multiply numerators and multiply denominators. Simplify, if possible. Answer tablespoon in each dose.
 How many -cup salt shakers can be filled from 12 cups of salt? A) B) C) 30 D) Show/Hide Answer A) Incorrect. You probably forgot to find the reciprocal of . Find . Keep the 12, change division to multiplication, and use the reciprocal (“flip”) of , giving you . B) Incorrect. You still need to find the mixed number that represents . In this case, 60 divided by 2 is 30 R0, so the answer is a whole number, 30. C) 30 Correct.  will show how many salt shakers can be filled. Write 12 as and multiply by the reciprocal (“flip”) of , giving you . D) Incorrect. You incorrectly used the expression . This will show how many groups of 12 there are in . You need to find how many groups of  are in 12, which is . Then write 12 as and multiply by the reciprocal (“flip”) of , giving you .

Division is the same as multiplying by the reciprocal. When working with fractions, this is the easiest way to divide. Whether you divide by a number or multiply by the reciprocal of the number, the result will be the same. You can use these techniques to help you solve problems that involve division, fractions, and/or mixed numbers.

## Dividing Fractions by Whole Numbers

• Welcome to our Dividing Fractions by Whole Numbers page.
• Take a look at our worked examples or have a go at our practice sheets!
• We also have a calculator which will not only give you the answer, but also show you all the working out along the way!
• If you need support to divide fractions by other fractions, use the link below.
• How to Divide fractions page
1. On this page, you will learn how to divide a fraction by a whole number, and also some practice worksheets designed to help your child master this skill.
2. The sheets are carefully graded so that the easiest sheets come first, and the most difficult sheet is the last one.
3. Before your child tackles dividing fractions, they should be confident with multiplying fractions, and also converting mixed fractions to improper fractions and reducing fractions to simplest form.
• divide a fraction by a whole number;
• apply their understanding of simplest form;
• convert an improper fraction to a mixed number.
• If you want to divide fractions, you can use our Free Divide Fractions calculator.
• The calculator will help you dividing fractions by whole numbers, or fractions by other fractions or mixed numbers.
• The best thing about the calculator is that it also shows you all the working out along the way.

• Divide Fractions Calculator

Frazer says “To divide a fraction by an integer, or whole numbers, follow these 4 easy steps.”

1. Step 1
2. Change the whole number to a fraction by putting it over a denominator of 1.
3. If any of the fractions are mixed fractions (or mixed numbers), then first convert them into improper fractions.
4. Step 2
5. Swap the numerator and denominator of the dividend fraction (the fraction after the ÷ sign) and change the operator to a 'x' instead of a '÷ '
6. Step 3

Multiply the numerators of the fractions together, and the denominators of the fractions together. This will give you the answer.

Step 4 (Optional)

You may want to convert the fraction into its simplest form or convert it back to a mixed fraction (if it is an improper fraction).

• Step 1)
• Put the integer over a denominator of 1.
• So this gives us: [{3 over 4} ÷ 6; = ; {3 over 4} ÷ {6 over 1} ]
• Step 2)
• Invert the dividend fraction and change the operation to multiplication.
• So we now have [{3 over 4} ÷ {6 over 1} ; = ; {3 over 4} imes {1 over 6}]
• Now multiply the fractions:
• [{3 over 4} imes {1 over 6} ; = ; {3 imes 1 over 4 imes 6} ; = ; {3 over 24} ]
• Step 3)
• Simplify the answer by dividing the numerator and denominator by 3.
• [ {3 over 24} ; = ; {3 ÷ 3 over 24 ÷ 3} ; = ; {1 over 8}]
• Final answer: [{3 over 4} ÷ 6 ; = ; {1 over 8} ]

### Example 2) Work out: [{5 over 9} ÷ 4 ]

1. Step 1)
2. Put the integer over a denominator of 1.
3. So this gives us: [{5 over 9} ÷ 4 ; = ; {5 over 9} ÷ {4 over 1} ]
4. Step 2)
5. Invert the dividend fraction and change the operation to multiplication.

6. So we now have [{5 over 9} ÷ {4 over 1} ; = ; {5 over 9} imes {1 over 4}]
7. Now multiply the fractions:
8. [{5 over 9} imes {1 over 4} ; = ; {5 imes 1 over 9 imes 4} ; = ; {5 over 36} ]
9. Step 3)
10. This fraction is already in simplest form.

11. Final answer: [{5 over 9} ÷ 4 ; = ; {5 over 36} ]

### Example 3) Work out: [ 2 {2 over 3} ÷ 7 ]

• Step 1)
• Convert the mixed number to an improper fraction and put the integer over a denominator of 1.
• As an improper fraction: [ 2 {2 over 3} ; = ; {8 over 3} ]
• So this gives us: [2 {2 over 3} ÷ 7 ; = ; {8 over 3} ÷ {7 over 1} ]
• Step 2)
• Invert the dividend fraction and change the operation to multiplication.
• So we now have [{8 over 3} ÷ {7 over 1} ; = ; {8 over 3} imes {1 over 7}]
• Now multiply the fractions:
• [{8 over 3} imes {1 over 7} ; = ; {8 imes 1 over 3 imes 7} ; = ; {8 over 21} ]
• Step 3)
• This fraction is already in simplest form.
• Final answer: [2 {2 over 3} ÷ 7 ; = ; {8 over 21} ]

Here is our support page to help your child understand how to divide a fraction by another fraction.

• How to Divide Fractions Support
1. Here you will find a selection of Fraction worksheets designed to help your child understand how to divide mixed fractions.
2. Once your child has mastered how to divide fractions, they are ready to learn to divide mixed fractions.
• divide a fraction by a mixed number;
• divide a mixed number by a fraction;
• divide two mixed fractions.
• How to Divide Mixed Numbers

Here you will find a selection of Fraction worksheets designed to help your child understand how to convert an improper fraction to a mixed number.

• convert an improper fraction to a mixed number;
• convert a mixed number to an improper fraction.

• Convert Improper Fractions

Here you will find a selection of Fraction worksheets designed to help your child understand how to convert a fraction to its simplest form.

• develop an understanding of equivalent fractions;
• know when a fraction is in its simplest form;
• convert a fraction to its simplest form.
• How to Simplify Fractions support page

How to Print or Save these sheets

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How to Print or Save these sheets

Need help with printing or saving? Follow these 3 easy steps to get your worksheets printed out perfectly!

• The Math Salamanders hope you enjoy using these free printable Math worksheets and all our other Math games and resources.

## Fractions – Dividing mixed numbers – In Depth

Dividing mixed numbers is very similar to multiplying mixed numbers. You just add one step—after changing the divisor into an improper fraction, you then find its reciprocal and multiply. Let's work through a “word problem” example.

The SuperQuik Market has just installed new scanners for its check-out lanes. They claim the average time to check out a customer is 2 ½ minutes. How many customers, on average, can they check out in half an hour?

To solve this problem, we have to know that half an hour is the same as 30 minutes. Then we can divide 30 by 2 ½.

• First step: Write the whole number and the mixed number as improper fractions.
• Second step: Write the reciprocal of the divisor, 2/5, and multiply.

Third step: Simplify, if possible. Notice that we can simplify our problem at this step, to make our calculations easier. Five goes evenly into 30, so we can divide both 5 and 30 by 5, to give 1 and 6.

Fourth step: Perform the simple multiplication of the numerators and the denominators. We find that the market can check out 12 customers in 30 minutes with its new scanners.

Fifth step: Put the answer in lowest terms, and check the answer. Our answer is already in lowest terms, so there is nothing left to do but check the answer, to be sure it makes sense. We can use estimation and rounding to do our check.

If we round 2 ½ minutes to 3 minutes and divide 3 into 30, we get 10 customers in 30 minutes.

So it is reasonable that 2 more customers per half hour, or 12 customers, can be checked, since 2 ½ minutes per customer is less than 3 minutes per customer.

## How to divide fractions: 3 easy steps to solve hard problems

Teaching your students how to divide fractions can be just as simple as teaching multiplication… once you know all of the little tricks to get the right answer.

But — as with any math concept — when you teach division, you don’t want your students to just solve a problem. You want them to understand what’s happening in each question.

But, that’s the thing. It’s tough to get them to understand dividing fractions if you don’t quite get it yourself.

We were a bit confused on the subject, too. That’s why we looked into the best tools and easiest ways to make sure your class understands the key concepts for dividing fractions. Pay close attention, and by the end of this article you’ll be a fully equipped, ultra confident fraction division master.

### How dividing fractions works

Teaching students how to divide fractions is part of the Common Core State Standards for Mathematical Practice. One of the most valuable things to teach your students when dividing fractions is what the answer means. Take a look at the example below:

½ ÷ ⅙ = 3

Why is the solution a bigger number than the fractions involved?

When you divide a fraction, you’re asking how many groups of the divisor (second number) can be found in the dividend (first number). For the above equation, we’re asking how many ⅙ appear in ½.

Imagine the example equation as a cake. You’ve got half of the cake remaining. If each serving of the cake is ⅙ of the whole, how many servings do you have left?

As you can see, you’ve got three servings of cake remaining!

## 21 Multiply and Divide Fractions

Simplify Fractions

In the following exercises, simplify each fraction. Do not convert any improper fractions to mixed numbers.

Multiply Fractions

In the following exercises, use a diagram to model.

In the following exercises, multiply, and write the answer in simplified form.

Find Reciprocals

In the following exercises, find the reciprocal.

Fill in the chart.

Opposite
Absolute Value
Reciprocal

Fill in the chart.

Opposite
Absolute Value
Reciprocal

Divide Fractions

In the following exercises, model each fraction division.

In the following exercises, divide, and write the answer in simplified form.

## How to Divide a Whole Number by a Fraction

1. 1

Convert the whole number to a fraction. To do this, make the whole number the numerator of a fraction. Make the denominator 1.[1]

• For example, if you are calculating 7÷34{displaystyle 7div {frac {3}{4}}}, you would first change 7{displaystyle 7} to 71{displaystyle {frac {7}{1}}}.
2. 2

Find the reciprocal of the divisor. The reciprocal of a number is the inverse of the number. To find the reciprocal of a fraction, reverse the numerator and denominator.[2]

• For example, the inverse of 34{displaystyle {frac {3}{4}}} is 43{displaystyle {frac {4}{3}}}.
3. 3

Multiply the two fractions. To multiply fractions, first multiply the numerators together. Then, multiply the denominators together. The product of the two fractions equals the quotient of your original division problem.[3]

• For example, 71×43=283{displaystyle {frac {7}{1}} imes {frac {4}{3}}={frac {28}{3}}}
4. 4

Simplify, if necessary. If you have an improper fraction (a fraction with a larger numerator than denominator), your teacher may require you to change it to a mixed number. Usually your teacher will require you to reduce proper fractions to lowest terms.

• For example, 283{displaystyle {frac {28}{3}}} simplifies to the mixed number 913{displaystyle 9{frac {1}{3}}}.
1. 1

Draw shapes representing the whole number. Your shape should be one capable of dividing into equal groups, such as a square or circle. Draw the shapes large enough that you can divide them into smaller pieces.

• For example, if you are calculating 5÷34{displaystyle 5div {frac {3}{4}}}, you would draw 5 circles.
2. 2

Divide each whole shape according to the fraction’s denominator. The denominator of a fraction tells you how many pieces a whole is divided into. Divide each whole shape into its fractional parts.[4]

• For example, if you are dividing by 34{displaystyle {frac {3}{4}}}, the 4 in the denominator tells you to divide each whole shape into fourths.
3. 3

Shade in groups representing the fraction. Since you are dividing the whole number by the fraction, you are looking to see how many groups of the fraction are in the whole number.[5] So, first, you need to create your groups. It might be helpful to shade each group a different color, since some groups will have pieces in two different wholes. Leave any leftover pieces unshaded.

• For example, if you are dividing 5 by 34{displaystyle {frac {3}{4}}}, you would color 3 quarters a different color for each group. Note that many groups will contain 2 quarters in one whole, and 1 quarter in another whole.
4. 4

Count the number of whole groups. This will give you the whole number of your answer.

• For example, you should have created 6 groups of 34{displaystyle {frac {3}{4}}} among your 5 circles.
5. 5

Interpret leftover pieces. Compare the number of pieces you have left over to a complete group. The fraction of a group that you have left over will indicate the fraction of your answer. Make sure you do not compare the number of pieces you have to a whole shape, as this will give you the wrong fraction.

• For example, after dividing the 5 shapes into groups of 34{displaystyle {frac {3}{4}}}, you have 2 quarters, or 24{displaystyle {frac {2}{4}}} left. Since a whole group is 3 pieces, and you have 2 pieces, your fraction is 23{displaystyle {frac {2}{3}}}.
6. 6

Write the answer. Combine your whole number of groups with your fractional number of groups to find the quotient of your original division problem.

• For example, 5÷34=623{displaystyle 5div {frac {3}{4}}=6{frac {2}{3}}}.
1. 1

Solve this math problem: How many times does 12{displaystyle {frac {1}{2}}} go into 8{displaystyle 8}?

• Since the problem is asking how many groups of 12{displaystyle {frac {1}{2}}} are in 8, the problem is one of division.
• Turn 8 into a fraction by placing it over 1: 8=81{displaystyle 8={frac {8}{1}}}.
• Find the reciprocal of the fraction by reversing the numerator and denominator: 12{displaystyle {frac {1}{2}}} becomes 21{displaystyle {frac {2}{1}}}.
• Multiply the two fractions together: 81×21=161{displaystyle {frac {8}{1}} imes {frac {2}{1}}={frac {16}{1}}}.
• Simplify, if necessary: 161=16{displaystyle {frac {16}{1}}=16}.
2. 2

Solve the following problem: 16÷58{displaystyle 16div {frac {5}{8}}}.

• Convert 16 into a fraction by placing it over 1: 16=161{displaystyle 16={frac {16}{1}}}.
• Take the fraction’s reciprocal by reversing the numerator and denominator: 58{displaystyle {frac {5}{8}}} becomes 85{displaystyle {frac {8}{5}}}.
• Multiply the two fractions together: 161×85=1285{displaystyle {frac {16}{1}} imes {frac {8}{5}}={frac {128}{5}}}.
• Simplify, if necessary: 1285=2535{displaystyle {frac {128}{5}}=25{frac {3}{5}}}.
3. 3

Solve the following problem by drawing a diagram. Rufus has 9 cans of food. She eats 23{displaystyle {frac {2}{3}}} of a can every day. How many days will her food last?

• Draw 9 circles representing the 9 cans.
• Since she eats 23{displaystyle {frac {2}{3}}} at a time, divide each circle into thirds.
• Color groups of 23{displaystyle {frac {2}{3}}}.
• Count the number of complete groups. There should be 13.
• Interpret the leftover pieces. There is 1 piece leftover, which is 13{displaystyle {frac {1}{3}}}. Since a whole group is 23{displaystyle {frac {2}{3}}}, you have half a group left over. So, your fraction is 12{displaystyle {frac {1}{2}}}.
• Combine the number of whole groups and fractional groups to find your final answer: 9÷23=1312{displaystyle 9div {frac {2}{3}}=13{frac {1}{2}}}.

• Question What is 12 3/7 divided by 9? First convert 12 3/7 to the improper fraction 87/7. Then divide by 9. The easiest way to do that is to multiply the denominator by 9: 7 x 9 = 63. So 87/7 ÷ 9 = 87/63, which reduces to 29/21. If you want the answer in mixed-number form, it's 1 8/21.
• Question How do I will simplify 24/1? 24/1 = 24.
• Question How do I divide 100,000 to 8 shares and 1/2 share? Divide 100,000 by 8.5.
• Question How do I divide 1/4 and 2x? If you're asking how to divide 2x into ¼, you would multiply the denominator (4) by 2x. ¼ ÷ 2x = 1/(8x).
• Question If given the fraction 4/9, how many pieces or groups is the whole divided in? The “whole” is 4, and it's divided into 9 equal pieces or groups.
• Question What is 40 divided by 1/16? Dividing by a fraction is the same thing as multiplying by the inverted fraction. In this case, 40 ÷ 1/16 is identical to 40 multiplied by 16/1, or (40)(16) = 640. (That means that 640 units of 1/16 will fit into 40.)
• Question How do I multiply a fraction by a fraction? Multiply the two numerators to get the numerator of the product. Then multiply the denominators to get the denominator of the product. For example, 2/5 multiplied by 3/8 equals (2 x 3) / (5 x 8) = 6 / 40, which reduces to 3 / 20.
• Question How do I divide 5/3 by 3? If you're asking how to divide 3 into 5/3, you would multiply the denominator (3) by 3. 5/3 ÷ 3 = 5/9.
• Question What is the correct answer to the following math equation? 9 – 31/3 + 1 9 + 1 = 10 = 30/3. 30/3 – 31/3 = -1/3.
• Question How do I solve the fraction 4/1 ÷ 38/3? 4/1 ÷ 38/3 = (4/1)(3/38) = 12/38 = 6/19.

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Categories: Fractions | Multiplication and Division

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