Division appears. In this article we will talk about division of ordinary fractions. First, we will give a rule for dividing ordinary fractions and look at examples of dividing fractions. Next we will focus on division common fraction on natural number and numbers into fractions. Finally, let's look at how to divide a common fraction by a mixed number.

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Dividing a common fraction by a common fraction

It is known that division is the inverse action of multiplication (see the connection between division and multiplication). That is, division involves finding an unknown factor when the product and another factor are known. The same meaning of division is preserved when dividing ordinary fractions.

Let's look at examples of dividing ordinary fractions.

Note that we should not forget about reducing fractions and separating the whole part from an improper fraction.

Dividing a fraction by a natural number

We'll give it right away rule for dividing a fraction by a natural number: to divide the fraction a/b by a natural number n, you need to leave the numerator the same and multiply the denominator by n, that is, .

This division rule follows directly from the rule for dividing ordinary fractions. Indeed, representing a natural number as a fraction leads to the following equalities .

Let's look at the example of dividing a fraction by a number.

Example.

Divide the fraction 16/45 by the natural number 12.

Solution.

According to the rule for dividing a fraction by a number, we have . Let's do the abbreviation: . This division is complete.

Answer:

.

Dividing a natural number by a fraction

The rule for dividing fractions is similar rule for dividing a natural number by a fraction: to divide a natural number n by a common fraction a/b, you need to multiply the number n by the reciprocal of the fraction a/b.

According to the stated rule, , and the rule for multiplying a natural number by an ordinary fraction allows it to be rewritten in the form .

Let's look at an example.

Example.

Divide the natural number 25 by the fraction 15/28.

Solution.

Let's move from division to multiplication, we have . After reducing and selecting the whole part, we get .

Answer:

.

Dividing a fraction by a mixed number

Dividing a fraction by a mixed number easily reduces to dividing ordinary fractions. To do this, it is enough to carry out

Ordinary fractional numbers first meet schoolchildren in the 5th grade and accompany them throughout their lives, since in everyday life it is often necessary to consider or use an object not as a whole, but in separate pieces. Start studying this topic - shares. Shares are equal parts, into which this or that object is divided. After all, it is not always possible to express, for example, the length or price of a product as a whole number; parts or fractions of some measure should be taken into account. Formed from the verb “to split” - to divide into parts, and having Arabic roots, the word “fraction” itself arose in the Russian language in the 8th century.

Fractional expressions have long been considered the most difficult branch of mathematics. In the 17th century, when first textbooks on mathematics appeared, they were called “broken numbers,” which was very difficult for people to understand.

The modern form of simple fractional remainders, the parts of which are separated by a horizontal line, was first promoted by Fibonacci - Leonardo of Pisa. His works are dated to 1202. But the purpose of this article is to simply and clearly explain to the reader how mixed fractions are multiplied with different denominators.

Multiplying fractions with different denominators

Initially it is worth determining types of fractions:

  • correct;
  • incorrect;
  • mixed.

Next, you need to remember how fractional numbers with the same denominators are multiplied. The very rule of this process is easy to formulate independently: the result of multiplication simple fractions with the same denominators is a fractional expression, the numerator of which is the product of the numerators, and the denominator is the product of the denominators of these fractions. That is, in fact, the new denominator is the square of one of the initially existing ones.

When multiplying simple fractions with different denominators for two or more factors the rule does not change:

a/b * c/d = a*c / b*d.

The only difference is that formed number under the fractional line will be the product of different numbers and, naturally, the square of one numerical expression it is impossible to name it.

It is worth considering the multiplication of fractions with different denominators using examples:

  • 8/ 9 * 6/ 7 = 8*6 / 9*7 = 48/ 63 = 16/2 1 ;
  • 4/ 6 * 3/ 7 = 2/ 3 * 3/7 <> 2*3 / 3*7 = 6/ 21 .

The examples use methods for reducing fractional expressions. You can only reduce numerator numbers with denominator numbers; adjacent factors above or below the fraction line cannot be reduced.

Along with simple fractions, there is the concept of mixed fractions. A mixed number consists of an integer and a fractional part, that is, it is the sum of these numbers:

1 4/ 11 =1 + 4/ 11.

How does multiplication work?

Several examples are provided for consideration.

2 1/ 2 * 7 3/ 5 = 2 + 1/ 2 * 7 + 3/ 5 = 2*7 + 2* 3/ 5 + 1/ 2 * 7 + 1/ 2 * 3/ 5 = 14 + 6/5 + 7/ 2 + 3/ 10 = 14 + 12/ 10 + 35/ 10 + 3/ 10 = 14 + 50/ 10 = 14 + 5=19.

The example uses multiplication of a number by ordinary fractional part, the rule for this action can be written as:

a* b/c = a*b /c.

In fact, such a product is the sum of identical fractional remainders, and the number of terms indicates this natural number. Special case:

4 * 12/ 15 = 12/ 15 + 12/ 15 + 12/ 15 + 12/ 15 = 48/ 15 = 3 1/ 5.

There is another solution to multiplying a number by a fractional remainder. You just need to divide the denominator by this number:

d* e/f = e/f: d.

This technique is useful to use when the denominator is divided by a natural number without a remainder or, as they say, by a whole number.

Convert mixed numbers to improper fractions and obtain the product in the previously described way:

1 2/ 3 * 4 1/ 5 = 5/ 3 * 21/ 5 = 5*21 / 3*5 =7.

This example involves a way of representing a mixed fraction as an improper fraction, it can also be represented as general formula:

a bc = a*b+ c / c, where the denominator of the new fraction is formed by multiplying the whole part with the denominator and adding it with the numerator of the original fractional remainder, and the denominator remains the same.

This process also works in the opposite direction. To separate the whole part and the fractional remainder, you need to divide the numerator of an improper fraction by its denominator using a “corner”.

Multiplying improper fractions produced in a generally accepted way. When writing under a single fraction line, you need to reduce fractions as necessary in order to reduce numbers using this method and make it easier to calculate the result.

There are many helpers on the Internet to solve even complex mathematical problems in various variations of programs. A sufficient number of such services offer their assistance in counting multiplication of fractions with different numbers in denominators - so-called online calculators for calculating fractions. They are able not only to multiply, but also to perform all other simple arithmetic operations with ordinary fractions and mixed numbers. It’s not difficult to work with; you fill in the appropriate fields on the website page, select the sign of the mathematical operation, and click “calculate.” The program calculates automatically.

The topic of arithmetic operations with fractions is relevant throughout the education of middle and high school students. In high school, they no longer consider the simplest species, but integer fractional expressions, but the knowledge of the rules for transformation and calculations obtained earlier is applied in its original form. Well-mastered basic knowledge gives complete confidence in successfully solving the most complex problems.

In conclusion, it makes sense to quote the words of Lev Nikolaevich Tolstoy, who wrote: “Man is a fraction. It is not in the power of man to increase his numerator - his merits - but anyone can reduce his denominator - his opinion about himself, and with this decrease come closer to his perfection.

Multiplying and dividing fractions.

Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)

This operation is much nicer than addition-subtraction! Because it's easier. As a reminder, to multiply a fraction by a fraction, you need to multiply the numerators (this will be the numerator of the result) and the denominators (this will be the denominator). That is:

For example:

Everything is extremely simple. And please don't look common denominator! Don't need him here...

To divide a fraction by a fraction, you need to reverse second(this is important!) fraction and multiply them, i.e.:

For example:

If you come across multiplication or division with integers and fractions, it’s okay. As with addition, we make a fraction from a whole number with one in the denominator - and go ahead! For example:

In high school, you often have to deal with three-story (or even four-story!) fractions. For example:

How can I make this fraction look decent? Yes, very simple! Use two-point division:

But don't forget about the order of division! Unlike multiplication, this is very important here! Of course, we will not confuse 4:2 or 2:4. But it’s easy to make a mistake in a three-story fraction. Please note for example:

In the first case (expression on the left):

In the second (expression on the right):

Do you feel the difference? 4 and 1/9!

What determines the order of division? Either with brackets, or (as here) with the length of horizontal lines. Develop your eye. And if there are no brackets or dashes, like:

then divide and multiply in order, from left to right!

And another very simple and important technique. In actions with degrees, it will be so useful to you! Let's divide one by any fraction, for example, by 13/15:

The shot has turned over! And this always happens. When dividing 1 by any fraction, the result is the same fraction, only upside down.

That's it for operations with fractions. The thing is quite simple, but it gives more than enough errors. Note practical advice, and there will be fewer of them (errors)!

Practical tips:

1. The most important thing when working with fractional expressions is accuracy and attentiveness! Is not common words, not good wishes! This dire necessity! Do all calculations on the Unified State Exam as a full-fledged task, focused and clear. It’s better to write two extra lines in your draft than to mess up when doing mental calculations.

2. In examples with different types of fractions, we move on to ordinary fractions.

3. We reduce all fractions until they stop.

4. We reduce multi-level fractional expressions to ordinary ones using division through two points (we follow the order of division!).

5. Divide a unit by a fraction in your head, simply turning the fraction over.

Here are the tasks that you must definitely complete. Answers are given after all tasks. Use the materials on this topic and practical tips. Estimate how many examples you were able to solve correctly. The first time! Without a calculator! And draw the right conclusions...

Remember - the correct answer is received from the second (especially the third) time does not count! Such is the harsh life.

So, solve in exam mode ! This is already preparation for the Unified State Exam, by the way. We solve the example, check it, solve the next one. We decided everything - checked again from first to last. But only Then look at the answers.

Calculate:

Have you decided?

We are looking for answers that match yours. I deliberately wrote them down in disarray, away from temptation, so to speak... Here they are, the answers, written with semicolons.

0; 17/22; 3/4; 2/5; 1; 25.

Now we draw conclusions. If everything worked out, I’m happy for you! Basic calculations with fractions are not your problem! You can do more serious things. If not...

So you have one of two problems. Or both at once.) Lack of knowledge and (or) inattention. But this solvable Problems.

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)

You can get acquainted with functions and derivatives.

) and denominator by denominator (we get the denominator of the product).

Formula for multiplying fractions:

For example:

Before you begin multiplying numerators and denominators, you need to check whether the fraction can be reduced. If you can reduce the fraction, it will be easier for you to make further calculations.

Dividing a common fraction by a fraction.

Dividing fractions involving natural numbers.

It's not as scary as it seems. As in the case of addition, we convert the integer into a fraction with one in the denominator. For example:

Multiplying mixed fractions.

Rules for multiplying fractions (mixed):

  • convert mixed fractions to improper fractions;
  • multiplying the numerators and denominators of fractions;
  • reduce the fraction;
  • If you get an improper fraction, then we convert the improper fraction into a mixed fraction.

Note! To multiply a mixed fraction by another mixed fraction, you first need to convert them to the form of improper fractions, and then multiply according to the rule for multiplying ordinary fractions.

The second way to multiply a fraction by a natural number.

It may be more convenient to use the second method of multiplying a common fraction by a number.

Note! To multiply a fraction by a natural number, you must divide the denominator of the fraction by this number, and leave the numerator unchanged.

From the example given above, it is clear that this option is more convenient to use when the denominator of a fraction is divided without a remainder by a natural number.

Multistory fractions.

In high school, three-story (or more) fractions are often encountered. Example:

To bring such a fraction to its usual form, use division through 2 points:

Note! When dividing fractions, the order of division is very important. Be careful, it's easy to get confused here.

Note, For example:

When dividing one by any fraction, the result will be the same fraction, only inverted:

Practical tips for multiplying and dividing fractions:

1. The most important thing when working with fractional expressions is accuracy and attentiveness. Do all calculations carefully and accurately, concentratedly and clearly. It's better to write a few extra lines in your draft than to get lost in mental calculations.

2. In tasks with different types of fractions, go to the type of ordinary fractions.

3. We reduce all fractions until it is no longer possible to reduce.

4. We transform multi-level fractional expressions into ordinary ones using division through 2 points.

5. Divide a unit by a fraction in your head, simply turning the fraction over.

T lesson type: ONZ (discovery of new knowledge - using the technology of the activity-based teaching method).

Basic goals:

  1. Deduce methods for dividing a fraction by a natural number;
  2. Develop the ability to divide a fraction by a natural number;
  3. Repeat and reinforce division of fractions;
  4. Train the ability to reduce fractions, analyze and solve problems.

Equipment demonstration material:

1. Tasks for updating knowledge:

Compare expressions:

Reference:

2. Trial (individual) task.

1. Perform division:

2. Perform division without performing the entire chain of calculations: .

Standards:

  • When dividing a fraction by a natural number, you can multiply the denominator by that number, but leave the numerator the same.

  • If the numerator is divisible by a natural number, then when dividing a fraction by this number, you can divide the numerator by the number and leave the denominator the same.

During the classes

I. Motivation (self-determination) to educational activities.

Purpose of the stage:

  1. Organize the updating of requirements for the student in terms of educational activities (“must”);
  2. Organize student activities to establish thematic frameworks (“I can”);
  3. Create conditions for the student to develop an internal need for inclusion in educational activities (“I want”).

Organization educational process at stage I.

Hello! I'm glad to see you all at the math lesson. I hope it's mutual.

Guys, what new knowledge did you acquire in the last lesson? (Divide fractions).

Right. What helps you do division of fractions? (Rule, properties).

Where do we need this knowledge? (In examples, equations, problems).

Well done! You did well on the assignments in the last lesson. Do you want to discover new knowledge yourself today? (Yes).

Then - let's go! And the motto of the lesson will be the statement “You can’t learn mathematics by watching your neighbor do it!”

II. Updating knowledge and fixing individual difficulties in a trial action.

Purpose of the stage:

  1. Organize the updating of learned methods of action sufficient to build new knowledge. Record these methods verbally (in speech) and symbolically (standard) and generalize them;
  2. Organize the actualization of mental operations and cognitive processes, sufficient for the construction of new knowledge;
  3. Motivate for a trial action and its independent implementation and justification;
  4. Present an individual task for a trial action and analyze it in order to identify new educational content;
  5. Organize commit educational purpose and lesson topics;
  6. Organize the implementation of a trial action and fix the difficulty;
  7. Organize an analysis of the responses received and record individual difficulties in performing a trial action or justifying it.

Organization of the educational process at stage II.

Frontally, using tablets (individual boards).

1. Compare expressions:

(These expressions are equal)

What interesting things did you notice? (The numerator and denominator of the dividend, the numerator and denominator of the divisor in each expression increased by the same number of times. Thus, the dividends and divisors in the expressions are represented by fractions that are equal to each other).

Find the meaning of the expression and write it down on your tablet. (2)

How can I write this number as a fraction?

How did you perform the division action? (Children recite the rule, the teacher hangs it on the board letter designations)

2. Calculate and record the results only:

3. Add up the results and write down the answer. (2)

What is the name of the number obtained in task 3? (Natural)

Do you think you can divide a fraction by a natural number? (Yes, we'll try)

Try this.

4. Individual (trial) task.

Perform division: (example a only)

What rule did you use to divide? (According to the rule of dividing fractions by fractions)

Now divide the fraction by a natural number greater than in a simple way, without performing the entire chain of calculations: (example b). I'll give you 3 seconds for this.

Who couldn't complete the task in 3 seconds?

Who did it? (There are no such)

Why? (We don't know the way)

What did you get? (Difficulty)

What do you think we will do in class? (Divide fractions by natural numbers)

That's right, open your notebooks and write down the topic of the lesson: “Dividing a fraction by a natural number.”

Why does this topic sound new when you already know how to divide fractions? (Need a new way)

Right. Today we will establish a technique that simplifies the division of a fraction by a natural number.

III. Identifying the location and cause of the problem.

Purpose of the stage:

  1. Organize the restoration of completed operations and record (verbal and symbolic) the place - step, operation - where the difficulty arose;
  2. Organize the correlation of students’ actions with the method (algorithm) used and fixation in external speech of the cause of the difficulty - that specific knowledge, skills or abilities that are lacking to solve the initial problem of this type.

Organization of the educational process at stage III.

What task did you have to complete? (Divide a fraction by a natural number without going through the entire chain of calculations)

What caused you difficulty? (We couldn’t solve it in a short time using a quick method)

What goal do we set for ourselves in the lesson? (Find quick way dividing a fraction by a natural number)

What will help you? (Already known rule for dividing fractions)

IV. Building a project for getting out of a problem.

Purpose of the stage:

  1. Clarification of the project goal;
  2. Choice of method (clarification);
  3. Determination of means (algorithm);
  4. Building a plan to achieve the goal.

Organization of the educational process at stage IV.

Let's return to the test task. You said you divided according to the rule for dividing fractions? (Yes)

To do this, replace the natural number with a fraction? (Yes)

What step (or steps) do you think can be skipped?

(The solution chain is open on the board:

Analyze and draw a conclusion. (Step 1)

If there is no answer, then we lead you through questions:

Where did the natural divisor go? (Into the denominator)

Has the numerator changed? (No)

So which step can you “omit”? (Step 1)

Action plan:

  • Multiply the denominator of a fraction by a natural number.
  • We do not change the numerator.
  • We get a new fraction.

V. Implementation of the constructed project.

Purpose of the stage:

  1. Organize communicative interaction in order to implement the constructed project aimed at acquiring the missing knowledge;
  2. Organize the recording of the constructed method of action in speech and signs (using a standard);
  3. Organize the solution to the initial problem and document how to overcome the difficulty;
  4. Organize clarification of the general nature of new knowledge.

Organization of the educational process at stage V.

Now run the test case in a new way quickly.

Now you were able to complete the task quickly? (Yes)

Explain how you did this? (Children talk)

This means that we have gained new knowledge: the rule for dividing a fraction by a natural number.

Well done! Say it in pairs.

Then one student speaks to the class. We fix the rule-algorithm verbally and in the form of a standard on the board.

Now enter the letter designations and write down the formula for our rule.

The student writes on the board, saying the rule: when dividing a fraction by a natural number, you can multiply the denominator by this number, but leave the numerator the same.

(Everyone writes the formula in their notebooks).

Now analyze the chain of solving the test task again, paying special attention to the answer. What did you do? (The numerator of the fraction 15 was divided (reduced) by the number 3)

What is this number? (Natural, divisor)

So how else can you divide a fraction by a natural number? (Check: if the numerator of a fraction is divisible by this natural number, then you can divide the numerator by this number, write the result in the numerator of the new fraction, and leave the denominator the same)

Write this method down as a formula. (The student writes the rule on the board while pronouncing it. Everyone writes the formula in their notebooks.)

Let's return to the first method. You can use it if a:n? (Yes, this is the general way)

And when is it convenient to use the second method? (When the numerator of a fraction is divided by a natural number without a remainder)

VI. Primary consolidation with pronunciation in external speech.

Purpose of the stage:

  1. Organize children’s assimilation of a new method of action when solving standard problems with their pronunciation in external speech (frontally, in pairs or groups).

Organization of the educational process at stage VI.

Calculate in a new way:

  • No. 363 (a; d) - performed at the board, pronouncing the rule.
  • No. 363 (e; f) - in pairs with checking according to the sample.

VII. Independent work with self-test according to the standard.

Purpose of the stage:

  1. Organize students’ independent completion of tasks for a new way of action;
  2. Organize self-test based on comparison with the standard;
  3. Based on the results of execution independent work organize reflection on the assimilation of a new way of action.

Organization of the educational process at stage VII.

Calculate in a new way:

  • No. 363 (b; c)

Students check against the standard and mark the correctness of execution. The causes of errors are analyzed and errors are corrected.

The teacher asks those students who made mistakes, what is the reason?

At this stage, it is important that each student independently checks their work.

VIII. Inclusion in the knowledge system and repetition.

Purpose of the stage:

  1. Organize the identification of the boundaries of application of new knowledge;
  2. Organize repetition of educational content necessary to ensure meaningful continuity.

Organization of the educational process at stage VIII.

  • Organize the recording of unresolved difficulties in the lesson as a direction for future educational activities;
  • Organize a discussion and recording of homework.

    • Organization of the educational process at stage IX.

    1. Dialogue:

    Guys, what new knowledge have you discovered today? (Learned how to divide a fraction by a natural number in a simple way)

    Formulate a general method. (They say)

    In what way and in what cases can it be used? (They say)

    What is the advantage of the new method?

    Have we achieved our lesson goal? (Yes)

    What knowledge did you use to achieve your goal? (They say)

    Did everything work out for you?

    What were the difficulties?

    2. Homework: clause 3.2.4.; No. 365(l, n, o, p); No. 370.

    3. Teacher: I’m glad that everyone was active today and managed to find a way out of the difficulty. And most importantly, they were not neighbors when opening a new one and establishing it. Thanks for the lesson, kids!