Word problem for multiplying fractions: Multiplying fractions word problem worksheets for grade 5

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Multiplying Fractions Word Problems | 10 Real Life Examples

Multiplying Fractions Word Problems

I guess maybe some students get it from their parents. After all, I know that I have been known to grumble about things I learned in school, but never used in real life. I’m sure when it comes to word problems for multiplying fractions, that phrase may have passed your lips too.

But, we do multiply fractions in real life and the real world sometimes. For example, recipes! Recipes and adjusting, maybe cutting a recipe in half or doubling it, those are real world examples of multiplying fractions.

Table of Contents

Most schools begin introducing fractions around 3rd grade, with multiplying fractions word problems beginning in 6th grade.

Here are 10 multiplying fractions word problems to work on multiplying fractions using real-life examples.

Have you ever heard, “I will never use this”?

It is important for children to see how math is used in everyday life. And word problems are one way to do this.

Multiplying Fractions

Multiplying Fractions seems like a foreign concept to many students, but it is a concept we use.

Have you ever wanted to only make 1/2 of a recipe? If so, you have probably multiplying fractions…..especially if the recipe called for 3/4 a cup of flour.

So today, we are going to finish up our multiplying fraction unit with some word problems. I am doing the activity in the post as well as some of the fraction games and activities below.

Grab my other free multiplying fractions by fractions activities.

  • How to Multiply Fractions by Fractions – Step by Step Instructions with Free Printable
  • Here is a Multiplying Fractions BINGO Game That’s Perfect for Extra Practice
  • 3 Cut and Paste Worksheets For Multiplying Fractions Practice
  • Free Printable Fraction Game For Multiplying Fractions
  • Many More on my TeachersPayTeachers page for Multiplying Fractions

Preparing the Fractions Real World Problems

These task cards are easy to prepare.

  1. Print off on copy paper.
  2. Have children cut them out and glue them in their math journal.
  3. Finally, provide them pencils, glue, and colored pencils. and you are ready to go.

Multiplying Fractions Word Problems Example

Rachel’s  Famous  Cookies sold  2/3 as many sugar cookies as peanut butter cookies. If they sold 1/4  of a box of peanut butter cookies, how many boxes of sugar cookies did they sell?

The first step is to look at we know. We know that they sold 1/4 a box of peanut butter cookies. But they only sold 2/3 of the 1/4 when it came to the sugar cookies.

So if we want to start with a diagram, we can begin by drawing a box of cookies and coloring in 1/4.

Many students require a multisensory approach to learning. It’s not just enough to talk about something and hear it. They have to see it. This activity works as a visual graphic organizer of sorts, so that the student can visualize what is going on.

Next, we can color in 2/3.

Finally, we look and see what part overlaps, and this is the answer! Two out of the 12 squares overlap, so our answer is 2/12 or 1/6 when the fraction is simplified. But using the squares on the paper and coloring with overlap, we have completed our first multiplying fractions word problem.

Using Fractions in Real Life

I’m a firm believer that children should not be given multiplying fractions word problems that require the same operation each time.

That is not real life, and that does teach them to learn what needs to be done to solve the problem. We need to give them the confidence that they can apply this knowledge to different situations and word problems.

That’s why I am including more than that in the multiplying fractions word problems worksheets.

So even though this printable is for practicing multiplying fractions with word problems there are a few word problems where multiplication is not used. Here is one example.

Luciana gave 1/4 of the cake to one friend, and 1/3 another friend. How much of the cake is left if she started with a whole cake?

This word problem has the children subtracting to figure out the answer.

  1. We know that Luciana started with a whole cake, which equals one. So we want to begin by drawing a square.
  2. The next thing we have to figure out is how many parts we need to divide it up into. Since we now we will be subtracting 1/4 and 1/3 we will need to get the Least Common Multiple. The multiples of three and four are…..

4: 4, 8, 12

3: 3, 6, 9, 12

As you can see 12 is the least common multiple. So our square needs to have 12 equal parts.

3. Next, we will need to find equivalent fractions for both of our fractions.

1/4 x 3/3 = 3/12 and 1/3 x 4/4 = 4/12.

4. Now all that is left is to subtract 3 part and then 4 parts.

5. Finally, we can see that there 5/12 of the cake left.

Word Problems are an important part of math instruction, and how we do math in everyday life. Enjoy working through these real life problems with your children.

The printable that accompanies this activity is below. Here you go, your free multiplying fractions word problems PDF. This free printable includes multiplying fractions word problems with answers.

You’ve Got This!

Multiplying-Fractions-Word-Problems

Get more fractions activities by ordering a workbook (but hey, nothing beats a free printable!)

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Solving Problems by Multiplying and Dividing Fractions and Mixed Numbers

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Fraction Word Problems With Interactive Exercises

Example 1: If it takes 5/6 yards of fabric to make a dress, then how many yards will it take to make 8 dresses?

Analysis: To solve this problem, we will convert the whole number to an improper fraction. Then we will multiply the two fractions.

Solution:

Answer: It will take 6 and 2/3 yards of fabric to make 8 dresses.


Example 2: Renee had a box of cupcakes, of which she gave 1/2 to her friend Juan. Juan gave 3/4 of his share to his friend Elena. What fractional part of the original box of cupcakes did Elena get?

Analysis: To solve this problem, we will multiply these two fractions.

Solution:

Answer: Elena got 3/8 of the original box of cupcakes.


Example 3: Nina’s math classroom is 6 and 4/5 meters long and 1 and 3/8 meters wide. What is the area of the classroom?

Analysis: To solve this problem, we will multiply these mixed numbers. But first we must convert each mixed number to an improper fraction.

Solution:

Answer: The area of the classroom is 9 and 7/20 square meters.


Example 4: A chocolate bar is 3/4 of an inch long. If it is divided into pieces that are 3/8 of an inch long, then how many pieces is that?

Analysis: To solve this problem, we will divide the first fraction by the second.

Solution:

Answer: 2 pieces


Example 5: An electrician has a piece of wire that is 4 and 3/8 centimeters long. She divides the wire into pieces that are 1 and 2/3 centimeters long. How many pieces does she have?

Analysis: To solve this problem, we will divide the first mixed number by the second.

Solution:

Answer: The electrician has 2 and 5/8 pieces of wire.


Example 6: A warehouse has 1 and 3/10 meters of tape. If they divide the tape onto pieces that are 5/8 meters long, then how many pieces will they have?

Analysis: To solve this problem, we will divide the first mixed number by the second. First, we will convert each mixed number into an improper fraction.

Solution:

Answer: The warehouse will have 2 and 2/25 pieces of tape.


Summary: In this lesson we learned how to solve word problems involving multiplication and division of fractions and mixed numbers. 


Exercises 

Directions: Subtract the mixed numbers in each exercise below. Be sure to simplify your result, if necessary. Click once in an ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR.

Note: To write the mixed number four and two-thirds, enter 4, a space, and then 2/3 into the form.

1. One batch of cookies contains 1 and 3/4 cups of melted chocolate. How many cups of melted chocolate are needed to make 8 batches of cookies?
 
ANSWER BOX:  cups   

RESULTS BOX: 

2. Todd drank 5/8 of a 24-ounce can of juice. Lila drank 1/3 as much juice as Todd did. How many ounces did Lila drink?
 
ANSWER BOX:   oz.   

RESULTS BOX: 

3. A rectangular area rug has a length of 3 and 2/3 feet and a width of 2 and 3/4 feet. What is the area of the rug?
 
ANSWER BOX:   sq ft   

RESULTS BOX: 

4. Janet has 5 and 3/4 centimeters of licorice. She divides the licorice into pieces that are 1 and 7/8 centimeters long. How many pieces of licorice will she have?
 
ANSWER BOX:   pieces   

RESULTS BOX: 

5. A piece of wood is 15 feet long. How many 3/4-foot sections can be cut from it?
 
ANSWER BOX:   sections    

RESULTS BOX: 

 

Lessons on Multiplying and Dividing Fractions and Mixed Numbers
1. Multiplying Fractions
2. Multiplying Fractions by Cancelling Common Factors
3. Multiplying Mixed Numbers
4. Reciprocals
5. Dividing Fractions
6. Dividing Mixed Numbers
6. Solving Word Problems
7. Practice Exercises
8. Challenge Exercises
9. Solutions

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Dividing Fractions | Math Goodies

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Division indicates how many times one quantity is contained in another quantity. For example, in the illustration below, you can see that the whole number 2 contains 6 thirds.

Since 2 contains 6 one-thirds, we can say that 2 divided by one-third is 6.

so 2 ÷ = 6      
                 
and 2 x 3 = 6      
               
     

As you can see, dividing a first fraction by a second, nonzero fraction is the same as multiplying the first fraction by the reciprocal of the second fraction. This leads us to the following procedure.

Procedure: To divide a first fraction by a second, nonzero fraction, multiply the first traction by the reciprocal of the second fraction.

This method of dividing fractions is also referred to as invert and multiply since we are inverting the divisor and then multiplying. Basically, we are changing the division problem to a multiplication problem after inverting the divisor. This allows us to multiply the first fraction by the reciprocal of the second fraction.

Example 1: Divide.

Analysis:

Solution:


Example 2: Divide.

Analysis:

Solution:

In example 2, it was necessary to divide out common factors.


Example 3: Divide.

Analysis:

Solution:

In example 3, it was necessary to simplify the result.


Example 4: Divide

Analysis:

Solution:


Example 5: A candy bar is 3/4 of an inch long. If it is divided into pieces that are 1/8 of an inch long, then how many pieces is that?

Analysis: To solve this problem, we will divide 3/4 by 1/8.

Solution:

Answer: There will be 6 pieces of candy.


Summary: To divide a first fraction by a second, nonzero fraction, multiply the first fraction by the reciprocal of the second fraction. Simplify the result, if necessary.


Exercises

Directions: Divide the fractions in each exercise below. Be sure to simplify your result, if necessary. Click once in an ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR.

Note: To write the fraction three-fourths, enter 3/4 into the form. To write the mixed number four and two-thirds, enter 4, a space, and then 2/3 into the form.

1.
 
  ANSWER BOX:   

RESULTS BOX: 

2.
 
  ANSWER BOX:   

RESULTS BOX: 

3.
 
  ANSWER BOX:   

RESULTS BOX: 

4.
 
  ANSWER BOX:   

RESULTS BOX: 

5. Justin gave 2/3 of his pizza to 4 friends who shared the pizza equally. What fraction of the original pizza did each friend get?
 
  ANSWER BOX:     

RESULTS BOX: 

 

Lessons on Multiplying and Dividing Fractions and Mixed Numbers
1. Multiplying Fractions
2. Multiplying Fractions by Cancelling Common Factors
3. Multiplying Mixed Numbers
4. Reciprocals
5. Dividing Fractions
6. Dividing Mixed Numbers
6. Solving Word Problems
7. Practice Exercises
8. Challenge Exercises
9. Solutions

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Free Math Help on Multiplying Mixed Numbers

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Example 1: Nina’s garden is 4 and 2/3 feet long and 1 and 1/8 feet wide. What is the area of the garden?

Analysis: We will multiply these mixed numbers in order to solve this problem.

Solution: First we will convert each mixed number to an improper fraction. Then we can multiply.

Step 1:

Step 2:

Answer: The area of Nina’s garden is 5 and 1/4 sq ft.


Example 2: Multiply.

Analysis: First convert each mixed number to an improper fraction. Then multiply.

Step 1: 

Step 2:


The following is the procedure for multiplying mixed numbers..

Procedure: To multiply mixed numbers, first convert each mixed number to an improper fraction, then multiply. Simplify your result, if necessary.

Let’s look at some examples using this procedure.

Example 3: Multiply.

Step 1:

Step 2:

There are no common factors to divide out.


Example 4: Multiply

Analysis: First, convert the whole number and the mixed number to an improper fraction. Then multiply.

Step 1:

Step 2:

Now that we know how to multiply fractions, we do not need to show each and every part of the process below in our examples.


Example 5: Multiply

Step 1:

Step 2:


Example 6: Multiply.

Solution:

There are no common factors to divide out.


Example 7: Multiply

Solution: 


Summary: To multiply mixed numbers, first convert each mixed number to an improper fraction, then multiply. Simplify your result, if necessary.


Exercises

Directions: In each exercise below, multiply the fractions by dividing out common factors. Be sure to simplify your result, if necessary. Click once in an ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR.

Note: To write the mixed number four and two-thirds, enter 4, a space, and then 2/3 into the form.

1.
 
  ANSWER BOX:   

RESULTS BOX: 

2.
 
  ANSWER BOX:   

RESULTS BOX: 

3.
 
  ANSWER BOX:   

RESULTS BOX: 

4.
 
  ANSWER BOX:   

RESULTS BOX: 

5.
 
  ANSWER BOX:     

RESULTS BOX: 

Lessons on Multiplying and Dividing Fractions and Mixed Numbers
1. Multiplying Fractions
2. Multiplying Fractions by Cancelling Common Factors
3. Multiplying Mixed Numbers
4. Reciprocals
5. Dividing Fractions
6. Dividing Mixed Numbers
6. Solving Word Problems
7. Practice Exercises
8. Challenge Exercises
9. Solutions

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Multiplying and Dividing Fractions and Mixed Numbers |

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Directions: Perform the necessary operation in each exercise below. Be sure to simplify your result, if necessary.  Click once in an ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR.

Note: To write the fraction three-fourths, enter 3/4 into the form. To write the mixed number four and two-thirds, enter 4, a space, and then 2/3 into the form.

1. A candy bar is 9/15 of an inch long. If it is divided into pieces that are 1/5 of an inch long, then how many pieces is that?
 
  ANSWER BOX:   pieces   

RESULTS BOX: 

2. Justin gave 2/3 of his pizza to 8 friends who shared the pizza equally. What fraction of the original pizza did each friend get?
 
  ANSWER BOX:   

RESULTS BOX: 

3.
 
  ANSWER BOX:   

RESULTS BOX: 

4. Marcy’s garden is 3 and 4/5 meters long and 2 and 7/8 meters wide. What is the area of the garden?
 
  ANSWER BOX:  m2   

RESULTS BOX: 

5.  
 
  ANSWER BOX:   

RESULTS BOX: 

6. A chocolate bar is 11/12 of an inch long. If it is divided into pieces that are 5/8 of an inch long, then how many pieces is that?
 
  ANSWER BOX:   pieces   

RESULTS BOX: 

7. A painter had a trough with 22 liters of paint. If each bucket holds 2 and 3/4 liters, then how many buckets of paint can be poured from the trough?
 
  ANSWER BOX:   buckets   

RESULTS BOX: 

8.   
 
  ANSWER BOX:   

RESULTS BOX: 

9.   Frank has 8 and 2/3 feet of rope. If he divides the rope into pieces that are 1 and 4/9 feet long, then how many pieces of rope will he have?
 
  ANSWER BOX:  pieces    

RESULTS BOX: 

10. Cathy’s classroom has an area of 15 and 5/6 square feet. If the length is 8 and 3/4 feet, then what is the width of  her classroom?
 
  ANSWER BOX:  feet   

RESULTS BOX: 

Lessons on Multiplying and Dividing Fractions and Mixed Numbers
1. Multiplying Fractions
2. Multiplying Fractions by Cancelling Common Factors
3. Multiplying Mixed Numbers
4. Reciprocals
5. Dividing Fractions
6. Dividing Mixed Numbers
6. Solving Word Problems
7. Practice Exercises
8. Challenge Exercises
9. Solutions

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Multiply Fractions by Whole Numbers Word Problems — Math Worksheets

Multiply Fractions by Whole Numbers Word Problems — Math Worksheets — SplashLearn

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Multiply Fractions by Whole Numbers Word Problems

Develop math skills by practicing word problems on multiplying fractions by whole numbers.

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Know more about Multiply Fractions by Whole Numbers Word Problems Worksheets

Building a strong foundation in multiplication of fractions is an important step in helping your child become proficient and confident. Students will apply multiplication of fractions to real-world situations. They will work with multiply by scenarios. The worksheet requires them to make sense of each story situation and find the unknown quantity. Designed for your child’s grade level, this worksheet involves multiplication of fractions by whole numbers.

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Solving problems on the topic «Multiplication of fractions»

Lesson in the 6th grade on the topic «Solving problems on the topic» Multiplication of fractions»

Purpose: Generalizing repetition and systematization of knowledge and methods of action.

Tasks:

  • to develop computational skills and abilities in solving problems;

  • consolidate knowledge of the basic rules;

  • develop independence;

  • interest in the subject.

Planned results:

Subject: application of knowledge in practical tasks.

Meta-subject:

Cognitive: be able to extract information from different types of texts. Arbitrarily and consciously master the general method of solving tasks.

Regulatory: be aware of the level and quality of mastering knowledge and skills. Make a plan and sequence of work.

Communication: organize and plan educational cooperation with the teacher and classmates.

Lesson type: Lesson — travel

Lesson progress:

Hello guys! Write down the number in your notebooks, class work.

Let’s get ready for work, we have a lot to do today, do you think we can handle it?

— Yes!

— What will help us cope?

— Knowledge.

What have we learned? Remember the topics that we have been studying since the first day of the second term. (multiplying fractions, multiplying mixed numbers, multiplying a fraction by a natural number, finding a fraction from a number, etc. ) (You can open the textbook and see the topics covered)

What should I do in today’s lesson?

— By repetition. And what will we do during the repetition?

— solve examples, problems.

So, the theme of our lesson “Solving problems on the topic “Faction multiplication”

— And what mark will you put yourself at this moment on the topics covered? Please rate the fields.

And we have one more guest! (enable the quest, click on the Gorodovichka picture, a window appears, press ok, a video appears. After watching it, close the window and click on the slide field to go to the next one)

Teacher: What kind of transport does Gorodovichok suggest us to travel with? Let’s go by train. Boarding in the cars will be carried out after completing the tasks. (click on the slide field and go to slide 3)

Car 1, 2

Reduce 22/66=1/3; 4/10=2/5; 9/12=3/4

Do the following: answer 1/7;

(click on the slide field and go to slide 4. If the student finds it difficult to complete the action, then by clicking on the big daisy, we go to the slide where the desired rule is formulated. Attention: if the example is in line 1, then click on the top daisy, if in the second , then on the lower one. We return to the example by clicking on the large daisy.)

Car 3, 4

9/10*5/6 = 3/4;; 40/7*14/5=16

3/16 *4=3/4; 23*5/46=5/2=2 ½;

3 3/5 * 1 1/9= 4; 5*2 1/5=11; 4 2/7 * 2=8 4/7

(teacher writes down how many examples each student computed) Put marks for mental arithmetic in the margins. (by clicking on the slide field, go to slide 9 — map)

U: We have a map and Gorodovichok offers to choose a route. (children can suggest any route, it’s okay, you can move in any order)

  1. Metro bridge (click on the route number, first we get to the task, solve it, click on the slide field, go to slide 11, then turn on the video by clicking on the word metro bridge): The answer in the problem says Gorodovichok.

Evaluate your decision. Field mark. (by clicking on Gorodovichka we get to a slide with a map)

  1. Theaters (click on the route number — first the slide, then the video by clicking on Gorodovika, and then the task about Dunno) Answer: no, Dunno 9 will not be able to complete0003

Evaluate your decision. (by clicking on Dunno we return to the map)

  1. Museum (click on the route number — the video turns on, close, click on the arrow to get to the task) Answer on the slide by clicking. Rate your decision. Field score. Return to the map. (by clicking on Gorodovichka)

Teacher: We still have one more route with a question mark, it is very short and the arrow points to a cute animal. Who is it? (Students make assumptions, if they know they call them, if not, then the teacher talks about Ligrenok — this is a cub of a lion and a tigress, who was born in the Novosibirsk Zoo. In nature, such an animal does not exist.)

We have completed the route proposed by Gorodoviček. Find the arithmetic mean of the marks and compare with the mark you gave yourself at the beginning of the lesson.

(click on Gorodovichka on the map, go to the next slide 15. Click on the picture of a flower in a flower bed on the right, turn on the final video. Click on the slide field, go to the final slide).

U: The townsman is happy with us. Are you guys satisfied with your knowledge? What about the results? The result is effort multiplied by ability. If efforts are equal to zero, then the result will also be equal to zero. Therefore, efforts should not be zero, but you all have the ability!

Therefore, thanks for the lesson!

methodology and its implementation, examples of problem solving

Mathematics

11/12/21

13 min.

Calculations are performed not only with natural integers, but also with fractional ones. In mathematics lessons in grade 6, examples of multiplication of ordinary fractions are studied in more detail. For a correct calculation, it is necessary to apply a certain technique that experts have developed for this purpose. They recommend that you first acquire basic knowledge, and then move on to their practical implementation.

Contents:

  • General
  • Types of common fractions
  • Working with mixed numbers
  • Reduction Rules
  • Multiplication algorithm

General

The process of finding the product of two ordinary fractional identities is very simple. However, there are «pitfalls» that can cause many errors. To prevent this from happening, it is necessary to be guided by a special algorithm that is offered by leading specialist teachers.

An ordinary fraction has two components — a numerator and a denominator. The first is at the top and is called divisible, and the second is at the bottom. The latter is called the divider. It should be noted that the fractional form is a representation of a private, i. e., the result of a division operation. This notation is used for readable form, because sometimes one number is not divisible by another.

For example, dividing 2 by 3 results in a decimal infinite repeating fraction. It can be written in the following form: 0,(6). The brackets mean that the number 6 repeats an infinite number of times, this is how periodicity is indicated.

However, there are cases when a decimal non-periodic value is formed, and it somehow needs to be written with an accuracy of a ten-thousandth part. This operation is not possible because the integer part will be followed by 10000 digits. Here for its record it is also necessary to use an ordinary fraction.

It should be noted that multiplying infinite non-periodic fractions is also problematic. They need to be converted to ordinary values, and then apply the appropriate algorithm. To use the technique, you need to gain basic knowledge. These include the following:

  1. Classification of ordinary fractional numbers.
  2. Work with mixed fractions of ordinary type.
  3. Abbreviation.

It should be noted that each component must be analyzed in detail, since the speed of learning depends on the qualitative study of the material. If the student did not understand the difference between proper and improper fractions, then it does not make sense to move on to the second paragraph. This will cause confusion and valuable time will be wasted.

Types of common fractions

Classification of fractional expressions allows you to understand their main properties, conversion methods and the main differences between them. They are of three types: correct, incorrect and mixed. For convenience, it is necessary to write the fraction in the mathematical representation «p / t», where p is the numerator and t is the denominator.

A proper fraction is an expression in which the numerator is less than the denominator, i. e. the condition p

However, when calculating, one can see a mixed representation in textbooks (for example, Vilenkina N. Ya.). For example, 6[2/3]. The latter consists of an integer and a fractional part, the latter being represented as an ordinary fractional value. This form of notation is used for the final display of the result obtained during the calculations.

Mathematicians recommend always transforming the answer into a readable form so that other people can use it later. Next, you need to analyze in detail the work with mixed numerical representations, since in this case it is problematic to multiply ordinary fractions. Lack of conversion can lead to many errors in calculations.

Working with mixed numbers

There is also a certain algorithm for working with mixed numbers. It has two directions: direct and inverse transformation. In the first case, the mixed fractional identity is converted into an improper fraction of ordinary type. It looks like this:

  1. Write value: M[p/t].
  2. Calculate the value of the numerator «P» using the following formula: P=Mt+p.
  3. Write an improper fraction: P/t.

It should be noted that the algorithm for converting an incorrect ordinary value is performed strictly in the reverse order . The technique is as follows :

  1. Write an incorrect identity of an ordinary fractional form: Р/t.
  2. Select an integer constant by dividing the numerator by the denominator: Р/t=M.
  3. Calculate a new numerator that must be less than the denominator: p=P-M*t.
  4. Write search value: M[p/t].

It should be noted that in the last step it is recommended to reduce the fractional part. This operation must be done constantly in order to optimize further calculations. Next, you need to understand the method of reducing the numerator and denominator.

Abbreviation rules

Reducing the numerator and denominator is necessary to reduce the amount of calculations. For example, you want to perform a multiplication operation on two fractional values ​​44/55 and 90/100. If you leave the expressions in this form, then to calculate the product you need to operate with large numbers, and this is very inconvenient. Therefore, the fractions must be reduced. For this purpose, a special technique is used. It looks like this:

  1. Write fractional value.
  2. Find the common factor for the numerator and denominator.
  3. Take out the value obtained in the first paragraph.
  4. Reduce the fraction by writing the result.

However, the algorithm needs to be worked out in practice. Its implementation looks like this:

  1. 44/55 and 90/100.
  2. 11 and 10 are common factors for two fractional values.
  3. (11*4)/(11*5) and (10*9)/(10*10).
  4. 4/5 and 9/10.

It should be noted that it is more convenient to perform any arithmetic operations with ordinary fractions obtained at the fourth step of the algorithm than with their initial values. Based on this, we can conclude that reduction is a necessary measure used all over the world to optimize calculations. Then you can move on to the method of multiplying fractions in grade 6.

Multiplication algorithm

The technique for multiplying fractional ordinary values ​​is quite simple. However, in mathematics there are only three cases that are not always amenable to explanation in the classroom (very often teachers do not pay students’ attention to them):

  1. Same denominators.
  2. Equal numerators but different denominators.
  3. Each element is equal to the component of the same type, i.e. the numerator of the first fraction is equivalent to the numerator of the second, and the denominators are also equal to each other.

In fact, the multiplication of simple fractions with different denominators is one and the same operation, that is, the search for a solution is carried out according to the same principle. To explain it, you need to disassemble the implementation methodology. It looks like this:

  1. Write two fractions.
  2. Convert mixed numbers to improper fractional numbers.
  3. Bring them back to normal using the reduce operation.
  4. Reduce the numerator and denominator of one value by the improper fraction elements of another value.
  5. Multiply numerators and denominators.
  6. Write down the desired result, reducing it if necessary and converting it to a proper fraction.

To understand the algorithm, you need to learn how to solve problems for multiplying fractions with different denominators for grade 6. For example, you need to multiply 6[4/8] and 3[20/35]. Their product is found by the following method:

  1. 6[4/8] and 3[20/35].
  2. The conversion should be performed only after reducing the fractional values ​​to the optimal form: 6[4/8]=6[½] and 3[20/35]=3[4/7].
  3. Conversion to wrong fractional identities: 13/2 and 25/7.
  4. Reducing between values ​​is not possible because 25 is not divisible by 2, but 13 by 7.
  5. Multiplication: (13*25)/(2*7)=325/14.
  6. For reduction, you need to find a common factor for the numbers 325 and 14 (minus is not divisible, but plus is divisible): 2 (-), 3 (-), 4 (-), 5 (-), 6 (-), 7 (-), 8 (-), 9 (-). It is not possible to shorten a fractional expression.
  7. Mixed notation, following the method of converting an incorrect fractional value into a mixed number: 23[(325−23*14)/14]=23[3/14].

It should be noted that each step of the method needs to be optimized. To do this, you need to get rid of unnecessary calculations, constantly reducing fractional values. However, some may not understand how the multiplication technique affects the result. To do this, you need to solve the example with a different method:

  1. For convenience, reduce the fractional values: 6[½] and 3[4/7].
  2. Multiply integer and fractional parts: 18[4/14].
  3. Shorten: 18[2/7].

It should be noted that the results do not match, since the latter method is incorrect. Based on this, we can conclude that it is required to solve problems according to the methodology. If you do not follow the rules, then errors may appear in the calculations.

Thus, to perform the operation of the product of two ordinary fractions, it is necessary to use a certain algorithm, as well as be able to reduce fractional values ​​and convert mixed numbers.

Lesson summary on the topic «Multiplication and division of mixed fractions» | Lesson plan in mathematics (grade 5):

Outline of an open lesson in mathematics in grade 6

Topic: «Multiplication and division of mixed fractions»

Type of lesson: Lesson of generalization and systematization of knowledge, skills.

Lesson objectives:

1. Repeat the rules for multiplication and division of ordinary fractions

2. Fix the solution of equations, division problems and multiplication of fractions

3. To develop students’ logical thinking

Lesson objectives:

— to create conditions for systematizing students’ knowledge on the topic «Multiplication and division of mixed fractions»

Student-oriented lesson objectives:

— to promote the formation of cognitive interest in mathematics , ways of generalizing and systematizing knowledge, skills of cooperation with adults and peers, skills of introspection and self-control

Metasubject goals:

communicative:

— to be able to accurately and competently express one’s thoughts

— to perceive the text taking into account the set educational task

— to form communicative actions aimed at structuring information on the topic

regulatory:

— to determine the sequence of intermediate actions taking into account final result

— adjust activities: make changes to the process, taking into account the difficulties and errors that have arisen, outline ways to eliminate them

— Aware of the level and quality of assimilation of material

Cognitive:

— Focus on a variety of solutions to

— Create and transform the solutions to problems

— make the most effective solutions to

Technical support for the lesson: multimedia projector and computer at the teacher.

Lesson plan.

I. Organizational moment. Motivation for learning activities (2 min.)

II. Update of basic knowledge (7 min)

Oral exercises (9 min)

III. Fizminutka (2 min)

IV. Inclusion in the knowledge system and repetition (20 min)

V. Information about homework (2 min)

VI. Summing up. Reflection of educational activity and assessment of students (3 min.)

Lesson progress

I. Organizational moment (setting goals and objectives of the lesson)

Before you is a rebus. What word is encoded here? slide 2

— That’s right, guys! The word is «fraction».

— What do you think we will do at the lesson today? (children’s answers)

— Today we will continue to work on the topic «Multiplication and division of mixed fractions», we will prepare for the test ..

— What tasks will we set for ourselves? (children’s answers)

— Today we will repeat the rules for multiplying and dividing mixed fractions. slide 3

II. Updating knowledge

1) Theory survey: slide 4

  • How to multiply a fraction by a natural number?
  • How to multiply two fractions?
  • How to multiply mixed numbers?
  • What do you know about multiplication of zero and one?
  • Divide by zero …
  • Formulate a rule for dividing fractions.
  • How do you divide mixed numbers?
  • If the given fraction is correct, then its reciprocal will be …
  • If the given fraction is improper, then its reciprocal will be …
  • To find a fraction of a number you need…
  • What numbers are called reciprocals?

2) Mental counting

Slide 5 Task for ingenuity: out of three numbers 4, 0.3 and , choose one that can be inserted into an empty circle. What is the pattern?

In the recording record Find errors and execute the solution correctly:

1) Slide 6

2) Slide 7

3) 25 = Slide 8

4) 6 (2 +) = 6 + Slide 9

5): 5 = Slide 10

6) 24: = Slide 11

7) 2: 5 =: = Slide 12

III. Eye Physics

IV. Inclusion in the knowledge system and repetition

1. Solve the equations orally, choose the correct answer, justify it slide 13

-What did you notice in all the equations? (the product is equal to 1)

— When multiplying which numbers, the product is equal to 1? (mutually inverse)

— So, to solve the equation it is enough …. (define the number reciprocal to a known factor)

a \u003d 1 Answer: 8

b \u003d 1 Answer: 1.5

1 s \u003d 1 Answer:

0.5 x \u003d 1 Answer: slide 14, 15 9002 slide 14, 9002 slide 9002 by options, to the board 2 students, then check with the class — who has it wrong)

(+ x): 25 \u003d 0.04 y + y \u003d 6.3

slide 17

3. Solution of examples ( work in rows, to the board 3 students in turn — decide with commenting, then check with the class — who has it wrong)

7: 3+ 0.2

1.75 + 3: 2

4: — 27

The first box contains 8 kg of pears, which is 1 times more than in the second, and 1 times less than in the third. How many kilograms of pears are in three boxes?

— How many boxes of pears are there in the problem?

— Who can make a short note to the problem?

(1 student makes a short note on the board) — slide 19

1 — 8 kg

2 — ? 1 time

3 — ? 1 time >

Action solution — 1 learner at the board :

  1. 8 :1= := = 7(kg) — in box 2
  2. 8 1= = = 9 (kg) — in box 3
  3. 8 + 7 + 9 \u003d 24 (kg) — in three boxes

Answer: 24.

5) A task for an independent solution. One student solves on the back of the board.

Slide check.

The first car traveled 60 km in an hour, the second traveled 54 km in an hour. Which car has the highest speed?

V. Information about homework

Homework on cards of three colors with ready-made multi-level tasks

slide 20

Students choose a task from the offered cards.

VI. Summarizing. Reflection of educational activity and assessment of students. slide 21

— What task did we set ourselves at the beginning of the lesson? (review the rules for multiplying and dividing mixed fractions.

— Lesson marks ….

— How we coped with the task, I will ask you to evaluate as follows. Choose a smiley corresponding to your mood at the moment:

1.

I understand everything, I am sure of my knowledge

2.

I am not entirely confident

3.

I don’t understand much

rules, examples, solutions, multiplication of fractions with different denominators

Another action that can be performed with ordinary fractions is multiplication. We will try to explain its basic rules when solving problems, show how an ordinary fraction is multiplied by a natural number and how to correctly multiply three or more ordinary fractions.

How to multiply one common fraction by another

First write down the basic rule:

Definition 1

If we multiply one ordinary fraction, then the numerator of the resulting fraction will be equal to the product of the numerators of the original fractions, and the denominator to the product of their denominators. In literal form, for two fractions a/b and c/d, this can be expressed as ab·cd=a·cb·d.

Let’s look at an example of how to correctly apply this rule. Let’s say we have a square whose side is equal to one numerical unit. Then the area of ​​the figure will be 1 square. unit. If we divide the square into equal rectangles with sides equal to 14 and 18 numerical units, we get that it now consists of 32 rectangles (because 8 4 = 32). Accordingly, the area of ​​each of them will be equal to 132 of the area of ​​the entire figure, i.e. 132 sq. units.

Next, we need to color a part of the original square as it is done in the figure:

We have a shaded fragment with sides equal to 58 numerical units and 34 numerical units. Accordingly, to calculate its area, it is necessary to multiply the first fraction by the second. It will be equal to 58 34 sq. units. But we can simply count how many rectangles are included in the fragment: there are 15 of them, which means that the total area is 1532 square units.

Since 5 3=15 and 8 4=32, we can write the following equation:

58 34=5 38 4=1532

It is a confirmation of our rule for multiplying ordinary fractions, which is expressed as ab cd=a cb d. It works the same for both proper and improper fractions; It can be used to multiply fractions with different and the same denominators.

Let’s analyze the solutions of several problems on the multiplication of ordinary fractions.

Example 1

Multiply 711 by 98.

Solution

To begin with, we calculate the product of the numerators of the indicated fractions, multiplying 7 by 9. We got 63. Then we calculate the product of the denominators and get: 11 8 = 88. Compose the answer from two numbers: 6388.

The whole solution can be written as follows:

711 98=7 911 8=6388

Answer: 711 98=6388.

If we get a reduced fraction in the answer, we need to complete the calculation and reduce it. If we get an improper fraction, we need to select the whole part from it.

Example 2

Calculate the product of fractions 415 and 556.

Solution

According to the rule studied above, we need to multiply the numerator by the numerator, and the denominator by the denominator. The solution entry will look like this:

415 556=4 5515 6=22090

one that has a sign of divisibility by 10.

Let’s reduce the fraction: 22090 GCD (220, 90)=10, 22090=220:1090:10=229. As a result, we got an improper fraction, from which we will select the whole part and get a mixed number: 229=249.

Answer: 415 556=249.

For the convenience of calculation, we can also reduce the original fractions before performing the multiplication operation, for which we need to reduce the fraction to the form a cb d. We decompose the values ​​of the variables into simple factors and cancel the same ones.

Let’s explain how this looks using the data of a specific task.

Example 3

Calculate the product 415 556.

Solution

Let’s write the calculations based on the multiplication rule. We will get:

415 556=4 5515 6

5 113 5 2 3.

Next, we can simply reduce some factors and get the following: .

It remains for us to calculate the simple products in the numerator and denominator and extract the integer part from the resulting improper fraction:

2 113 3=229=249

Answer : 415 556=249.

A numerical expression in which the multiplication of ordinary fractions takes place has a commutative property, that is, if necessary, we can change the order of the factors:

ab cd=cd ab=a cb d

How to multiply an ordinary fraction with a natural number

Let’s write down the basic rule right away, and then try to explain it in practice.

Definition 2

To multiply an ordinary fraction by a natural number, you need to multiply the numerator of this fraction by this number. In this case, the denominator of the final fraction will be equal to the denominator of the original ordinary fraction. Multiplication of some fraction ab by a natural number n can be written as a formula ab n=a nb.

It is easy to understand this formula if you remember that any natural number can be represented as an ordinary fraction with a denominator equal to one, that is:

ab n=ab n1=a nb 1=a nb

Let us explain our idea with concrete examples.

Example 4

Calculate the product of 227 times 5.

Solution :

227 5=2 527=1027

Answer: 227 5=1027

When we multiply a natural number with a common fraction, we often have to abbreviate the result or represent it as a mixed number.

Example 5

Condition: Calculate the product of 8 times 512.

Solution

According to the rule above, we multiply a natural number by the numerator. As a result, we get that 512 8=5 812=4012. The final fraction has signs of divisibility by 2, so we need to reduce it:

LCM(40, 12)=4, so 4012=40:412:4=103

Now we just have to select the whole part and write down the finished answer: 103=313.

In this entry, you can see the entire solution: 512 8=5 812=4012=103=313.

We could also reduce the fraction by factoring the numerator and denominator into prime factors, and the result would be exactly the same.

Answer: 512 8=313.

A numeric expression in which a natural number is multiplied by a fraction also has the displacement property, that is, the order of the factors does not affect the result:

ab n=n ab=a nb

How to multiply three or more common fractions

We can extend to the operation of multiplying common fractions the same properties that are characteristic of multiplying natural numbers. This follows from the very definition of these concepts.

Thanks to the knowledge of the associative and commutative properties, it is possible to multiply three or more ordinary fractions. It is permissible to rearrange the factors in places for greater convenience or arrange the brackets in a way that will make it easier to count.

Let’s use an example to show how this is done.

Example 6

Multiply four common fractions 120, 125, 37 and 58.

Solution: First, let’s write down the product. We get 120 125 37 58. We need to multiply all the numerators and all the denominators together: 120 125 37 58=1 12 3 520 5 7 8.

Before we start multiplication, we can make things a little easier for ourselves and decompose some numbers into prime factors for further reduction. This will be easier than reducing the finished fraction resulting from it.

1 12 3 520 5 7 8=1 (2 2 3) 3 52 2 5 5 7(2 2 2)=3 35 7 2 2 2=9280

Answer: 1 12 3 520 5 7 8=9280.

Example 7

Multiply 5 numbers 78 12 8 536 10.

Solution

For convenience, we can group the fraction 78 with the number 8, and the number 12 with the fraction 536, since future reductions will be obvious to us. As a result, we get:
78 12 8 536 10=78 8 12 536 10=7 88 12 536 10=71 2 2 3 52 2 3 3 10==7 53 10=7 5 103=3503=11623

Answer: 78 12 8 536 10=11623.

Problem solving

from 1 day / from 150 rubles

Course work

from 5 days / from 1800 rubles

abstract

from 1 day / from 700 rubles

C 16 multiplication and division of algebraic fractions.

Solving problems on multiplication and division of algebraic fractions

In this lesson, we will consider the rules for multiplying and dividing algebraic fractions, as well as examples for applying these rules. Multiplication and division of algebraic fractions is no different from multiplication and division of ordinary fractions. However, the presence of variables leads to somewhat more complex ways of simplifying the resulting expressions. Despite the fact that multiplying and dividing fractions is easier than adding and subtracting them, the study of this topic must be approached very responsibly, since there are many «pitfalls» in it that are usually not paid attention to. As part of the lesson, we will not only study the rules for multiplying and dividing fractions, but also analyze the nuances that may arise when applying them.

Subject:
Algebraic fractions. Arithmetic operations on algebraic fractions

Lesson:
Multiplication and division of algebraic fractions

The rules of multiplication and division of algebraic fractions are absolutely similar to the rules of multiplication and division of ordinary fractions. Recall them:

That is, in order to multiply fractions, it is necessary to multiply their numerators (this will be the numerator of the product), and multiply their denominators (this will be the denominator of the product).

Division by a fraction is multiplication by an inverted fraction, that is, in order to divide two fractions, it is necessary to multiply the first of them (the dividend) by the inverted second (the divisor).

Despite the simplicity of these rules, many people make mistakes in a number of special cases when solving examples on this topic. Let’s consider these particular cases in more detail:

In all these rules we used the following fact: .

Let’s solve some examples of multiplication and division of ordinary fractions in order to remember how to use these rules.

Example 1

Note:
when reducing fractions, we used the decomposition of a number into prime factors. Recall that are prime numbers

are natural numbers that are divisible only by and by itself. The remaining numbers are called composite

. The number is neither prime nor composite. Examples of prime numbers: .

Example 2

Consider now one of the special cases with ordinary fractions.

Example 3

As you can see, the multiplication and division of ordinary fractions, if the rules are correctly applied, is not difficult.

Consider the multiplication and division of algebraic fractions.

Example 4

Example 5

Note that it is possible and even necessary to reduce fractions after multiplication according to the same rules that we previously considered in the lessons on the reduction of algebraic fractions. Let’s consider some simple examples for special cases.

Example 6

Example 7

Let’s now consider some more complex examples of multiplication and division of fractions.

Example 8

Example 9

Example 10

Example 11

Example 12

Example 13

Before that, we considered fractions in which both the numerator and denominator were monomials. However, in some cases it is necessary to multiply or divide fractions whose numerators and denominators are polynomials. In this case, the rules remain the same, and for reduction it is necessary to use the formulas of abbreviated multiplication and parentheses.

Example 14

Example 15

Example 16

Example 17

Example 18

In this article, we continue our study of the basic operations that can be performed with algebraic fractions. Here we will consider multiplication and division: first we derive the necessary rules, and then we illustrate them with problem solutions.

How to properly divide and multiply algebraic fractions

To multiply algebraic fractions or divide one fraction into another, we need to use the same rules as for ordinary fractions. Let’s take a look at their wording.

When we need to multiply one ordinary fraction by another, we perform the multiplication of numerators and denominators separately, after which we write down the final fraction, putting the corresponding products in their places. An example of such a calculation:

2 3 4 7 = 2 4 3 7 = 8 21

And when we need to divide ordinary fractions, we do this by multiplying by the reciprocal of the divisor, for example: 7 11 \u003d 2 3 11 7 \u003d 22 7 \u003d 1 1 21

Multiplication and division of algebraic fractions is performed in accordance with the same principles. Let’s formulate the rule:

Definition 1

To multiply two or more algebraic fractions, you need to multiply the numerators and denominators separately. The result will be a fraction, the numerator of which will be the product of the numerators, and the denominator will be the product of the denominators.

In literal form, the rule can be written as a b · c d = a · c b · d . Here a , b , c and d
will be certain polynomials, with b and d
cannot be null.

Definition 2

To divide one algebraic fraction by another, you need to multiply the first fraction by the reciprocal of the second.

This rule can also be written as a b: c d = a b d c = a d b c . Letters a , b , c and d
here means polynomials, of which a , b , c and d
cannot be null.

Let us dwell separately on what an inverse algebraic fraction is. It is a fraction that, when multiplied by the original, gives a unit as a result. That is, such fractions will be similar to mutually reciprocal numbers. Otherwise, we can say that the inverse algebraic fraction consists of the same values ​​as the original one, but the numerator and denominator are reversed. So, in relation to the fraction a b + 1 a 3, the fraction a 3 a b + 1 will be inverse.

Solving problems on multiplication and division of algebraic fractions

In this paragraph, we will see how to correctly apply the above rules in practice. Let’s start with a simple and illustrative example.

Example 1

Condition:
multiply the fraction 1 x + y by 3 x y x 2 + 5 and then divide one fraction by the other.

Solution

Let’s do the multiplication first. According to the rule, you need to multiply the numerators and denominators separately:

1 x + y 3 x y x 2 + 5 = 1 3 x y (x + y) (x 2 + 5)

We have a new polynomial that needs to be reduced to standard form. We finish the calculation:

1 3 x y (x + y) (x 2 + 5) = 3 x y x 3 + 5 x + x 2 y + 5 y

Now let’s see how correctly divide one fraction by another. As a rule, we need to replace this action by multiplying by the reciprocal x 2 + 5 3 x y:

1 x + y: 3 x y x 2 + 5 = 1 x + y x 2 + 5 3 x y

Let’s bring the resulting fraction to the standard form:

1 x + y x 2 + 5 3 x y = 1 x 2 + 5 (x + y) 3 x y = x 2 + 5 3 x 2 y + 3 x y 2

Answer:
1 x + y 3 x y x 2 + 5 = 3 x y x 3 + 5 x + x 2 y + 5 y ; 1 x + y: 3 x y x 2 + 5 = x 2 + 5 3 x 2 y + 3 x y 2 .

Quite often, in the process of dividing and multiplying ordinary fractions, results are obtained that can be reduced, for example, 2 9 3 8 = 6 72 = 1 12 . When we perform these operations on algebraic fractions, we can also get reducible results. To do this, it is useful to first decompose the numerator and denominator of the original polynomial into separate factors. If necessary, re-read the article on how to do it correctly. Let’s look at an example of a problem in which it will be necessary to perform the reduction of fractions.

Example 2

Condition:
multiply the fractions x 2 + 2 x + 1 18 x 3 and 6 x x 2 — 1 .

Solution

Before calculating the product, we decompose the numerator of the first initial fraction and the denominator of the second into separate factors. To do this, we need formulas for abbreviated multiplication. Calculate:

x 2 + 2 x + 1 18 x 3 6 x x 2 — 1 = x + 1 2 18 x 3 6 x (x — 1) (x + 1) = x + 1 2 6 x 18 x 3 x — 1 x + 1

We have a fraction that can be reduced:

x + 1 2 6 x 18 x 3 x — 1 x + 1 = x + 1 3 x 2 (x — 1)

O how this is done, we wrote in an article on the reduction of algebraic fractions.

Multiplying the monomial and the polynomial in the denominator, we get the result we need:

x + 1 3 x 2 (x — 1) = x + 1 3 x 3 — 3 x 2 explanation:

x 2 + 2 x + 1 18 x 3 6 x x 2 — 1 = x + 1 2 18 x 3 6 x (x — 1) (x + 1) = x + 1 2 6 x 18 x 3 x — 1 x + 1 = = x + 1 3 x 2 (x — 1) = x + 1 3 x 3 — 3 x 2

Answer:
x 2 + 2 x + 1 18 x 3 6 x x 2 — 1 = x + 1 3 x 3 — 3 x 2 .

In some cases, it is convenient to transform the original fractions before multiplying or dividing so that further calculations become faster and easier.

Example 3

Condition:
divide 2 1 7 x — 1 by 12 x 7 — x .

Solution: Let’s start by simplifying the algebraic fraction 2 1 7 · x — 1 to get rid of the fractional coefficient. To do this, we multiply both parts of the fraction by seven (this action is possible due to the main property of the algebraic fraction). As a result, we get the following:

2 1 7 x — 1 = 7 2 7 1 7 x — 1 = 14 x — 7

the resulting fractions are opposite expressions. By changing the signs of the numerator and denominator 12 x 7 — x, we get 12 x 7 — x \u003d — 12 x x — 7.

After all the transformations, we can finally go directly to the division of algebraic fractions:

2 1 7 x — 1: 12 x 7 — x = 14 x — 7: — 12 x x — 7 = 14 x — 7 x — 7 — 12 x = 14 x — 7 x — 7 — 12 x = = 14 — 12 x = 2 7 — 2 2 3 x = 7 — 6 x = — 7 6 x

Answer:
2 1 7 x — 1: 12 x 7 — x = — 7 6 x .

How to multiply or divide an algebraic fraction by a polynomial

To perform such an operation, we can use the same rules that we have given above. First you need to represent the polynomial as an algebraic fraction with a unit in the denominator. This action is similar to converting a natural number into an ordinary fraction. For example, you can replace the polynomial x 2 + x − 4
by x 2 + x − 4 1
. The resulting expressions will be identically equal.

Example 4

Condition:
Divide the algebraic fraction by the polynomial x + 4 5 · x · y: x 2 — 16 .

Solution

x + 4 5 x y: x 2 — 16 = x + 4 5 x y: x 2 — 16 1 = x + 4 5 x y 1 x 2 — 16 = = x + 4 5 x y 1 (x — 4) x + 4 = (x + 4) 1 5 x y (x — 4) (x + 4) = 1 5 x y x — 4 = = 1 5 x 2 y — 20 x y

Answer:
x + 4 5 x y: x 2 — 16 = 1 5 x 2 y — 20 x y .

If you notice a mistake in the text, please highlight it and press Ctrl+Enter

Subject: Multiplication and division of algebraic fractions

Education is what remains when everything learned has already been forgotten

Laue

Targets:

Educational:

fix ZUN on topic

conduct primary current control of knowledge

work on gaps

Developing:

promote the development of communicative competence, i.e. the ability to work effectively with others.

promote the development of cooperative competence, i.e. ability to work in pairs.

contribute to the development of problem competence, i. e. the ability to understand the inevitability of difficulties in the course of any activity.

Educational:

to instill the ability to adequately evaluate the work done by a friend;

when working in pairs, cultivate the qualities of mutual assistance and support.

Methodical:

creation of conditions for the manifestation of individuality, cognitive activity of students;

show the methodology for conducting a lesson with the design of the results of educational activities and methods for their research based on a competency-based approach.

Equipment:
board, colored chalk. Table «Multiplication and division of algebraic fractions»; cards for individual work, cards — «reminders». Free minute assignment.

Course of the lesson

Organizational moment

Lesson plan written on the board:

Oral warm-up.

Individual work.

Problem solving.

Pair work.

Summary of the lesson.

Homework.

Teacher:

In the old days in Russia it was believed that if a person was versed in mathematics, then this meant the highest degree of scholarship. And the ability to see and hear correctly is the first step to wisdom. I want all the students in your class today to show how wise they are and how well-versed people are in algebra of the 7th grade.

So, the topic of the lesson is «Multiplication and division of algebraic fractions» In the last lesson, you started studying this topic, and we discussed why we are studying it. Let’s remember where it will come in handy in a few lessons.

Students:

For joint actions with algebraic fractions, for solving equations, and hence problems.

Teacher:

Even in the old days in Russia they said that multiplication is torment, and division is trouble. Anyone who could quickly and accurately multiply and divide was considered a great mathematician.

What are your goals?

Students:

Continue to study the topic, learn to quickly and accurately multiply and divide.

Teacher:

To achieve our goals, we (opens the plan written on the board, pronounces it)

1. Oral warm-up: (at this time 3-4 people solve the fraction reduction simulator in pairs) factorize by filling in the gaps

1= (y-1) (…), 5a+5b=… (a+b), xy-x=x (…), 14-2x=…

reduce the fraction

.

find the mistake made when multiplying and dividing algebraic fractions

Teacher:

Where is the error? Why is the error made? What rule did the student not know? What did you know? How to do it right?

2. Work in a notebook, № from the textbook 488 (1) Analysis, solution, verification.

Teacher:

And now you will have the opportunity to show your knowledge when taking the test, and to inspire you to work, I will read the rhyme «So that the teacher writes» 5 «in your diary, manage to multiply the numerator by the numerator in an instant, and so that the teacher is pleased with you, you multiply the first denominator by second»

Self-check, mutual check. According to the criteria (posted on the board) B-1 (321), B-2 (132) according to the correct codes, assessment in pairs. initial result. Estimates.

Work on mistakes in pairs «student-teacher»

If there are no mistakes in pairs, do the task in a free minute.

Simplify the expression and find its value at

5. Lesson summary

In conclusion of the lesson, I would like to ask you what types of work caused you difficulties? Why do you think? What did you learn new? Which of you is satisfied with your work in the classroom? Do you think the goals set at the beginning of the lesson have been achieved?

Teacher: I would like to finish the lesson with the words of the French engineer-physicist Laue: «Education is what remains when everything learned is already forgotten» No. 486,487,488 are even.

Video lesson “Multiplication and division of algebraic fractions. Raising an algebraic fraction to a power ”is an auxiliary tool for teaching a mathematics lesson on this topic. With the help of a video lesson, it is easier for a teacher to form students’ ability to perform multiplication and division of algebraic fractions. The visual aid contains a detailed, understandable description of examples in which the operations of multiplication and division are performed. The material can be demonstrated during the teacher’s explanation or become a separate part of the lesson.

In order to form the ability to solve tasks for multiplying and dividing algebraic fractions, important comments are given during the description of the solution, points that require memorization and deep understanding are highlighted using color, bold type, and pointers. With the help of a video lesson, the teacher can increase the effectiveness of the lesson. This visual aid will help you quickly and effectively achieve your learning goals.

The video lesson begins with the introduction of the topic. After that, it is indicated that the operations of multiplication and division with algebraic fractions are performed similarly to operations with ordinary fractions. The screen shows the rules for multiplication, division and exponentiation of fractions. The multiplication of fractions is demonstrated using literal parameters. It is noted that when multiplying fractions, numerators, as well as denominators, are multiplied. This is how the resulting fraction a/b c/d=ac/bd is obtained. The division of fractions is demonstrated using the expression a/b:c/d as an example. It is indicated that to perform the division operation, it is necessary to write the product of the numerator of the dividend and the denominator of the divisor into the numerator. The denominator of the quotient is the product of the denominator of the dividend and the numerator of the divisor. Thus, the operation of division turns into the operation of multiplying the fraction of the dividend and the fraction reciprocal of the divisor. Raising to the power of a fraction is equivalent to a fraction in which the numerator and denominator are raised to the designated power.

The following is an example solution. In example 1, you need to perform the actions (5x-5y) / (x-y) (x 2 -y 2) / 10x. To solve this example, the numerator of the second fraction included in the product is decomposed into factors. Using the formulas of abbreviated multiplication, a transformation is made x 2 -y 2 \u003d (x + y) (x-y). Then the numerators of the fractions and the denominators are multiplied. After carrying out the operations, it is clear that there are factors in the numerator and denominator that can be reduced using the main property of the fraction. As a result of transformations, a fraction (x + y) 2 / 2x is obtained. It also considers the execution of actions 7a 3 b 5 / (3a-3b) (6b 2 -12ab + 6a 2) / 49a 4 b 5 . All numerators and denominators are considered for the possibility of factorization, allocation of common factors. Then the numerators and denominators are multiplied. After multiplication, reductions are made. The result of the transformation is the fraction 2(a-b)/7a.

An example is considered in which it is necessary to perform the actions (x 3 -1) / 8y: (x 2 + x + 1) / 16y 2 . To solve the expression, it is proposed to convert the numerator of the first fraction using the abbreviated multiplication formula x 3 -1 \u003d (x-1) (x 2 + x + 1). According to the rule of division of fractions, the first fraction is multiplied by the reciprocal of the second. After multiplying the numerators and denominators, a fraction is obtained that contains the same factors in the numerator and denominator. They are shrinking. The result is a fraction (x-1) 2y. The solution of the example (a 4 -b 4)/(ab+2b-3a-6):(b-a)(a+2) is also described here. Similar to the previous example, the abbreviated multiplication formula is used to convert the numerator. The denominator of the fraction is also converted. Then the first fraction is multiplied with the reciprocal of the second fraction. After multiplication, transformations are performed, reductions of the numerator and denominator by common factors. The result is a fraction — (a + b) (a 2 + b 2) / (b-3). The attention of students is drawn to how the signs of the numerator and denominator change during multiplication.

In the third example, it is necessary to perform actions with fractions ((x+2)/(3x 2 -6x)) 3:((x 2 +4x+4)/(x 2 -4x+4)) 2 . In solving this example, the rule of raising a fraction to a power is applied. Both the first and second fractions are raised to a power. They are converted by raising the numerator and denominator to a power. In addition, to convert the denominators of fractions, the abbreviated multiplication formula is used, highlighting the common factor. To divide the first fraction by the second, you need to multiply the first fraction by the reciprocal of the second. The numerator and denominator form expressions that can be reduced. After the conversion, a fraction (x-2) / 27x 3 (x + 2) is obtained.

Video lesson “Multiplication and division of algebraic fractions. Raising an algebraic fraction to a power ”is used to increase the effectiveness of a traditional mathematics lesson. The material may be useful to a teacher who provides distance learning. A detailed clear description of the solution of examples will help students who independently master the subject or require additional classes.

In this article we will look at basic operations with algebraic fractions
:

  • fraction reduction
  • multiplication of fractions
  • division of fractions

Let’s start with reduction of algebraic fractions
.

It would seem that algorithm

obvious.

To reduce algebraic fractions
, need

1. Factorize the numerator and denominator of the fraction.

2. Reduce the same multipliers.

However, schoolchildren often make the mistake of «reducing» not factors, but terms. For example, there are amateurs who «reduce» by in fractions and get as a result, which, of course, is not true.

Consider the examples:

1.

Reduce the fraction:

1. Factorize the numerator using the sum square formula, and the denominator using the difference of squares formula

2. Divide the numerator and denominator by

2.
Reduce the fraction:

1. Factor the numerator. Since the numerator contains four terms, we apply the grouping.

2. Factor the denominator. The same applies to grouping.

3. Let’s write down the fraction that we got and reduce the same factors:

Multiplication of algebraic fractions.

When multiplying algebraic fractions, we multiply the numerator by the numerator, and we multiply the denominator by the denominator.

Important!

No need to rush to perform multiplication in the numerator and denominator of a fraction. After we have written the product of the numerators of fractions in the numerator, and the product of the denominators in the denominator, we need to factor each factor and reduce the fraction.

Consider the examples:

3.

Simplify the expression:

1. Let’s write the product of fractions: in the numerator the product of the numerators, and in the denominator the product of the denominators:

2. Let’s factorize each bracket:

Now we need to reduce the same factors. Note that the expressions and differ only in sign: and as a result of dividing the first expression by the second, we get -1.

So,

We perform the division of algebraic fractions according to this rule:

That is, to divide by a fraction, you need to multiply by the «inverted» one.

We see that the division of fractions is reduced to multiplication, and multiplication, in the end, is reduced to the reduction of fractions.

Consider the example:

4.
Simplify the expression:

Multiplication of fractions. Division of Fractions.

  • Alphaschool
  • Articles
  • org/ListItem»> Multiplication and division of fractions

Any natural number can be represented as an ordinary fraction.

In order to multiply two fractions, you need to:

  • convert fractions to improper;
  • multiply their numerators and write the result to the numerator;
  • multiply their denominators and write the result into the denominator;
  • if abbreviated;

Example 1 . Multiply \(\frac{7}{8}\) and \(\frac{5}{6}\):

then do the multiplication:

Two fractions are said to be mutually inverse \(\).

Example: 3/4 and 4/3 are mutually inverse, since the result is \(1\):

It is also worth remembering that you cannot divide by zero.


Task 1 . Multiply \(2\frac{5}{7} \) and \(2 \frac{8}{9}\).

Solution.

\(\frac{19}{7}*\frac{26}{9}\)=\(\frac{494}{63}\)\(=7\frac{53}{63}\)

Answer: \(7\frac{53}{63}\).

Task 2. Separate \(2\frac{5}{6}\) and \(\frac{3}{4} \).

Solution.

\(\frac{17}{6}:\frac{3}{4}\)\(=\frac{17*4}{6*3}=\frac{17*2}{3*3 }=\frac{34}{9}=3\frac{7}{9}\)

Answer: \(3\frac{7}{9}\).

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