T class type: ONZ (discovery of new knowledge - according to the technology of the activity method of teaching).

Basic goals:

1. Deduce methods of dividing a fraction by a natural number;
2. To form the ability to perform the division of a fraction by a natural number;
3. Repeat and consolidate the division of fractions;
4. Train the ability to reduce fractions, analyze and solve problems.

Equipment demo material:

1. Tasks for updating knowledge:

Compare expressions:

Reference:

2. Trial (individual) task.

1. Perform division:

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

References:

• When dividing a fraction by a natural number, you can multiply the denominator by this number, and 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) for learning activities.

Purpose of the stage:

1. Organize the actualization of the requirements for the student on the part of educational activities (“must”);
2. Organize the activities of students to establish a thematic framework (“I can”);
3. To create conditions for the student to have an internal need for inclusion in educational activities (“I want”).

Organization of the educational process at stage I.

Hello! I'm glad to see you all in math class. I hope it's mutual.

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

Right. What helps you divide fractions? (Rule, properties).

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

Well done! You did well in the last lesson. Would you like to discover new knowledge yourself today? (Yes).

Then - go! And the motto of the lesson is the statement “Mathematics cannot be learned by watching how your neighbor does it!”.

II. Actualization of knowledge and fixation of an individual difficulty in a trial action.

Purpose of the stage:

1. To organize the actualization of the studied methods of action, sufficient to build new knowledge. Fix these methods verbally (in speech) and symbolically (standard) and generalize them;
2. Organize the actualization of mental operations and cognitive processes sufficient to build 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 the fixation of the educational goal and the topic of the lesson;
6. Organize the implementation of a trial action and fixing 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 the tablet. (2)

How to write this number as a fraction?

How did you perform the division action? (Children pronounce the rule, the teacher hangs letters on the board)

2. Calculate and record only the results:

3. Add up your results and write down your 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 will try)

Try this.

4. Individual (trial) task.

Do the division: (example a only)

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

And now divide the fraction by a natural number in a simpler way, without performing the entire chain of calculations: (example b). I give you 3 seconds for this.

Who failed to complete the task in 3 seconds?

Who made 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. Identification of the location and cause of the difficulty.

Purpose of the stage:

1. Organize the restoration of completed operations and fix (verbal and symbolic) place - step, operation, where the difficulty arose;
2. To organize the correlation of students' actions with the method (algorithm) used and the fixation in external speech of the cause of the difficulty - those specific knowledge, skills or abilities that are not enough 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 doing the whole chain of calculations)

What caused you difficulty? (Could not solve in a short time in a fast way)

What is the purpose of our lesson? (Find a quick way to divide a fraction by a natural number)

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

IV. Construction of the project of an exit from difficulty.

Purpose of the stage:

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

Organization of the educational process at stage IV.

Let's go back to the test case. Did you say that you divided by the rule of dividing fractions? (Yes)

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

What step(s) do you think you can skip?

(The solution chain is open on the board:

Analyze and draw a conclusion. (Step 1)

If there is no answer, then we summarize through the questions:

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

Has the numerator changed? (Not)

So what step can be "omitted"? (Step 1)

Action plan:

• Multiply the denominator of a fraction by a natural number.
• The numerator does not change.
• 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 fixation of the constructed method of action in speech and signs (with the help of a standard);
3. Organize the solution of the original problem and record the overcoming of the difficulty;
4. Organize a clarification of the general nature of the new knowledge.

Organization of the educational process at stage V.

Now run the test case in the new way quickly.

Are you able to complete the task quickly now? (Yes)

Explain how you did it? (Children speak)

This means that we have received 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, pronouncing the rule: when dividing a fraction by a natural number, you can multiply the denominator by this number, and leave the numerator the same.

(Everyone writes the formula in notebooks).

And now once again analyze the chain of solving the trial task, paying special attention to the answer. What did they 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 into the numerator of the new fraction, and leave the denominator the same)

Write this method in the form of a formula. (The student writes down the rule on the board. Everyone writes down the formula in notebooks.)

Let's go back to the first method. Can it be used if a:n? (Yes, this is the general way)

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

VI. Primary consolidation with pronunciation in external speech.

Purpose of the stage:

1. To organize the assimilation by children of a new method of action when solving typical 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) - perform at the blackboard, pronouncing the rule.
• No. 363 (d; f) - in pairs with a check on the sample.

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

Purpose of the stage:

1. To organize the students' independent fulfillment of tasks for a new mode of action;
2. Organize self-test based on comparison with the standard;
3. Based on the results of independent work, organize a reflection on the assimilation of a new mode of action.

Organization of the educational process at stage VII.

Calculate in a new way:

• No. 363 (b; c)

Students check the standard, note the correctness of the performance. 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 check their work.

VIII. Inclusion in the system of knowledge and repetition.

Purpose of the stage:

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

Organization of the educational process at stage VIII.

Organize the fixation of unresolved difficulties in the lesson as a direction for future learning activities;

Organize discussion and recording of homework.

Organization of the educational process at stage IX.

1. Dialog:

Guys, what new knowledge did you discover today? (We learned to divide a fraction by a natural number in a simple way)

Formulate a general way. (They say)

In what way, and in what cases can you still use it? (They say)

What is the advantage of the new method?

Have we reached our goal of the lesson? (Yes)

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

Have you succeeded?

What were the difficulties?

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

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

Multiplication and division of fractions.

Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")

This operation is much nicer than addition-subtraction! Because it's easier. I remind you: 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). I.e:

For example:

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

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

For example:

If multiplication or division with integers and fractions is caught, it's okay. As with addition, we make a fraction from a whole number with a unit in the denominator - and go! For example:

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

How to bring this fraction to a decent form? Yes, very easy! Use division through two points:

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

In the first case (expression on the left):

In the second (expression on the right):

Feel the difference? 4 and 1/9!

What is the order of division? Or brackets, or (as here) the length of horizontal dashes. Develop an eye. And if there are no brackets or dashes, like:

then divide-multiply in order, left to right!

And another very simple and important trick. In actions with degrees, it will come in handy for you! Let's divide the unit by any fraction, for example, by 13/15:

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

That's all the actions with fractions. The thing is quite simple, but gives more than enough errors. Take note of practical advice, and there will be fewer of them (mistakes)!

Practical Tips:

1. The most important thing when working with fractional expressions is accuracy and attentiveness! These are not common words, not good wishes! This is a severe need! Do all the calculations on the exam as a full-fledged task, with concentration and clarity. It is better to write two extra lines in a draft than to mess up when calculating in your head.

2. In examples with different types of fractions - go to ordinary fractions.

3. We reduce all fractions to the stop.

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

5. We divide the unit into a fraction in our mind, simply by turning the fraction over.

Here are the tasks you need to complete. Answers are given after all tasks. Use the materials of this topic and practical advice. Estimate how many examples you could solve correctly. The first time! Without a calculator! And draw the right conclusions...

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

So, solve in exam mode ! This is preparation for the exam, by the way. We solve an example, we check, we solve the following. We decided everything - we checked again from the first to the last. Only after look at the answers.

Calculate:

Did you decide?

Looking for answers that match yours. I specifically wrote them down in a mess, away from the temptation, so to speak ... Here they are, the answers, written down with a semicolon.

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

And now we draw conclusions. If everything worked out - happy for you! Elementary 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. Learning - with interest!)

you can get acquainted with functions and derivatives.

Is division. 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 dividing an ordinary fraction by a natural number and a number by a fraction. Finally, consider how the division of an ordinary fraction by a mixed number is carried out.

Page navigation.

Division of a common fraction by a common fraction

It is known that division is the inverse 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 sense of division is preserved when dividing ordinary fractions.

Consider examples of dividing ordinary fractions.

Note that we should not forget about the reduction of fractions and about the selection of the integer part from an improper fraction.

Division of a common 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 division rule for ordinary fractions. Indeed, the representation of a natural number as a fraction leads to the following equalities .

Consider an example of dividing a fraction by a number.

Example.

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

Decision.

By the rule of dividing a fraction by a number, we have . Let's do the reduction: . This division is completed.

Answer:

.

Division of a natural number by a common fraction

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

According to the voiced rule, , and the rule of multiplying a natural number by an ordinary fraction allows you to rewrite it in the form.

Consider an example.

Example.

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

Decision.

Let's move from division to multiplication, we have . After reduction and selection of the integer part, we get .

Answer:

.

Division of a common fraction by a mixed number

Division of a common fraction by a mixed number easily reduced to the division of ordinary fractions. To do this, it is enough to

Last time we learned how to add and subtract fractions (see the lesson "Addition and subtraction of fractions"). The most difficult moment in those actions was bringing fractions to a common denominator.

Now it's time to deal with multiplication and division. The good news is that these operations are even easier than addition and subtraction. To begin with, consider the simplest case, when there are two positive fractions without a distinguished integer part.

To multiply two fractions, you need to multiply their numerators and denominators separately. The first number will be the numerator of the new fraction, and the second will be the denominator.

To divide two fractions, you need to multiply the first fraction by the "inverted" second.

Designation:

From the definition it follows that the division of fractions is reduced to multiplication. To flip a fraction, just swap the numerator and denominator. Therefore, the entire lesson we will consider mainly multiplication.

As a result of multiplication, a reduced fraction can arise (and often does arise) - of course, it must be reduced. If, after all the reductions, the fraction turned out to be incorrect, the whole part should be distinguished in it. But what definitely won’t happen with multiplication is reduction to a common denominator: no crosswise methods, maximum factors and least common multiples.

By definition we have:

Multiplication of fractions with an integer part and negative fractions

If there is an integer part in the fractions, they must be converted to improper ones - and only then multiplied according to the schemes outlined above.

If there is a minus in the numerator of a fraction, in the denominator or in front of it, it can be taken out of the limits of multiplication or removed altogether according to the following rules:

1. Plus times minus gives minus;
2. Two negatives make an affirmative.

Until now, these rules have only been encountered when adding and subtracting negative fractions, when it was required to get rid of the whole part. For a product, they can be generalized in order to “burn” several minuses at once:

1. We cross out the minuses in pairs until they completely disappear. In an extreme case, one minus can survive - the one that did not find a match;
2. If there are no minuses left, the operation is completed - you can start multiplying. If the last minus is not crossed out, since it did not find a pair, we take it out of the limits of multiplication. You get a negative fraction.

Task. Find the value of the expression:

We translate all fractions into improper ones, and then we take out the minuses outside the limits of multiplication. What remains is multiplied according to the usual rules. We get:

Let me remind you once again that the minus that comes before a fraction with a highlighted integer part refers specifically to the entire fraction, and not just to its integer part (this applies to the last two examples).

Also pay attention to negative numbers: when multiplied, they are enclosed in brackets. This is done in order to separate the minuses from the multiplication signs and make the whole notation more accurate.

Reducing fractions on the fly

Multiplication is a very laborious operation. The numbers here are quite large, and to simplify the task, you can try to reduce the fraction even more before multiplication. Indeed, in essence, the numerators and denominators of fractions are ordinary factors, and, therefore, they can be reduced using the basic property of a fraction. Take a look at the examples:

Task. Find the value of the expression:

By definition we have:

In all examples, the numbers that have been reduced and what is left of them are marked in red.

Please note: in the first case, the multipliers were reduced completely. Units remained in their place, which, generally speaking, can be omitted. In the second example, it was not possible to achieve a complete reduction, but the total amount of calculations still decreased.

However, in no case do not use this technique when adding and subtracting fractions! Yes, sometimes there are similar numbers that you just want to reduce. Here, look:

You can't do that!

The error occurs due to the fact that when adding a fraction, the sum appears in the numerator of a fraction, and not the product of numbers. Therefore, it is impossible to apply the main property of a fraction, since this property deals specifically with the multiplication of numbers.

There is simply no other reason to reduce fractions, so the correct solution to the previous problem looks like this:

Correct solution:

As you can see, the correct answer turned out to be not so beautiful. In general, be careful.

§ 87. Addition of fractions.

Adding fractions has many similarities to adding whole numbers. Addition of fractions is an action consisting in the fact that several given numbers (terms) are combined into one number (sum), which contains all units and fractions of units of terms.

We will consider three cases in turn:

1. Addition of fractions with the same denominators.
2. Addition of fractions with different denominators.
3. Addition of mixed numbers.

1. Addition of fractions with the same denominators.

Consider an example: 1 / 5 + 2 / 5 .

Take the segment AB (Fig. 17), take it as a unit and divide it into 5 equal parts, then the part AC of this segment will be equal to 1/5 of the segment AB, and the part of the same segment CD will be equal to 2/5 AB.

It can be seen from the drawing that if we take the segment AD, then it will be equal to 3/5 AB; but segment AD is precisely the sum of segments AC and CD. So, we can write:

1 / 5 + 2 / 5 = 3 / 5

Considering these terms and the resulting amount, we see that the numerator of the sum was obtained by adding the numerators of the terms, and the denominator remained unchanged.

From this we get the following rule: To add fractions with the same denominators, you must add their numerators and leave the same denominator.

Consider an example:

2. Addition of fractions with different denominators.

Let's add fractions: 3/4 + 3/8 First they need to be reduced to the lowest common denominator:

The intermediate link 6/8 + 3/8 could not have been written; we have written it here for greater clarity.

Thus, to add fractions with different denominators, you must first bring them to the lowest common denominator, add their numerators and sign the common denominator.

Consider an example (we will write additional factors over the corresponding fractions):

3. Addition of mixed numbers.

Let's add the numbers: 2 3 / 8 + 3 5 / 6.

Let us first bring the fractional parts of our numbers to a common denominator and rewrite them again:

Now add the integer and fractional parts in sequence:

§ 88. Subtraction of fractions.

Subtraction of fractions is defined in the same way as subtraction of whole numbers. This is an action by which, given the sum of two terms and one of them, another term is found. Let's consider three cases in turn:

1. Subtraction of fractions with the same denominators.
2. Subtraction of fractions with different denominators.
3. Subtraction of mixed numbers.

1. Subtraction of fractions with the same denominators.

Consider an example:

13 / 15 - 4 / 15

Let's take the segment AB (Fig. 18), take it as a unit and divide it into 15 equal parts; then the AC part of this segment will be 1/15 of AB, and the AD part of the same segment will correspond to 13/15 AB. Let's set aside another segment ED, equal to 4/15 AB.

We need to subtract 4/15 from 13/15. In the drawing, this means that the segment ED must be subtracted from the segment AD. As a result, segment AE will remain, which is 9/15 of segment AB. So we can write:

The example we made shows that the numerator of the difference was obtained by subtracting the numerators, and the denominator remained the same.

Therefore, to subtract fractions with the same denominators, you need to subtract the numerator of the subtrahend from the numerator of the minuend and leave the same denominator.

2. Subtraction of fractions with different denominators.

Example. 3/4 - 5/8

First, let's reduce these fractions to the smallest common denominator:

The intermediate link 6 / 8 - 5 / 8 is written here for clarity, but it can be skipped in the future.

Thus, in order to subtract a fraction from a fraction, you must first bring them to the smallest common denominator, then subtract the numerator of the subtrahend from the numerator of the minuend and sign the common denominator under their difference.

Consider an example:

3. Subtraction of mixed numbers.

Example. 10 3 / 4 - 7 2 / 3 .

Let's bring the fractional parts of the minuend and the subtrahend to the lowest common denominator:

We subtracted a whole from a whole and a fraction from a fraction. But there are cases when the fractional part of the subtrahend is greater than the fractional part of the minuend. In such cases, you need to take one unit from the integer part of the minuend, split it into those parts in which the fractional part is expressed, and add to the fractional part of the minuend. And then the subtraction will be performed in the same way as in the previous example:

§ 89. Multiplication of fractions.

When studying the multiplication of fractions, we will consider the following questions:

1. Multiplying a fraction by an integer.
2. Finding a fraction of a given number.
3. Multiplication of a whole number by a fraction.
4. Multiplying a fraction by a fraction.
5. Multiplication of mixed numbers.
6. The concept of interest.
7. Finding percentages of a given number. Let's consider them sequentially.

1. Multiplying a fraction by an integer.

Multiplying a fraction by an integer has the same meaning as multiplying an integer by an integer. Multiplying a fraction (multiplicand) by an integer (multiplier) means composing the sum of identical terms, in which each term is equal to the multiplicand, and the number of terms is equal to the multiplier.

So, if you need to multiply 1/9 by 7, then this can be done like this:

We easily got the result, since the action was reduced to adding fractions with the same denominators. Hence,

Consideration of this action shows that multiplying a fraction by an integer is equivalent to increasing this fraction as many times as there are units in the integer. And since the increase in the fraction is achieved either by increasing its numerator

or by decreasing its denominator , then we can either multiply the numerator by the integer, or divide the denominator by it, if such a division is possible.

From here we get the rule:

To multiply a fraction by an integer, you need to multiply the numerator by this integer and leave the same denominator or, if possible, divide the denominator by this number, leaving the numerator unchanged.

When multiplying, abbreviations are possible, for example:

2. Finding a fraction of a given number. There are many problems in which you have to find, or calculate, a part of a given number. The difference between these tasks and others is that they give the number of some objects or units of measurement and you need to find a part of this number, which is also indicated here by a certain fraction. To facilitate understanding, we will first give examples of such problems, and then introduce the method of solving them.

Task 1. I had 60 rubles; 1 / 3 of this money I spent on the purchase of books. How much did the books cost?

Task 2. The train must cover the distance between cities A and B, equal to 300 km. He has already covered 2/3 of that distance. How many kilometers is this?

Task 3. There are 400 houses in the village, 3/4 of them are brick, the rest are wooden. How many brick houses are there?

Here are some of the many problems that we have to deal with to find a fraction of a given number. They are usually called problems for finding a fraction of a given number.

Solution of problem 1. From 60 rubles. I spent 1 / 3 on books; So, to find the cost of books, you need to divide the number 60 by 3:

Problem 2 solution. The meaning of the problem is that you need to find 2 / 3 of 300 km. Calculate first 1/3 of 300; this is achieved by dividing 300 km by 3:

300: 3 = 100 (that's 1/3 of 300).

To find two-thirds of 300, you need to double the resulting quotient, that is, multiply by 2:

100 x 2 = 200 (that's 2/3 of 300).

Solution of problem 3. Here you need to determine the number of brick houses, which are 3/4 of 400. Let's first find 1/4 of 400,

400: 4 = 100 (that's 1/4 of 400).

To calculate three quarters of 400, the resulting quotient must be tripled, that is, multiplied by 3:

100 x 3 = 300 (that's 3/4 of 400).

Based on the solution of these problems, we can derive the following rule:

To find the value of a fraction from a given number, you need to divide this number by the denominator of the fraction and multiply the resulting quotient by its numerator.

3. Multiplication of a whole number by a fraction.

Earlier (§ 26) it was established that the multiplication of integers should be understood as the addition of identical terms (5 x 4 \u003d 5 + 5 + 5 + 5 \u003d 20). In this paragraph (paragraph 1) it was established that multiplying a fraction by an integer means finding the sum of identical terms equal to this fraction.

In both cases, the multiplication consisted in finding the sum of identical terms.

Now we move on to multiplying a whole number by a fraction. Here we will meet with such, for example, multiplication: 9 2 / 3. It is quite obvious that the previous definition of multiplication does not apply to this case. This is evident from the fact that we cannot replace such multiplication by adding equal numbers.

Because of this, we will have to give a new definition of multiplication, i.e., in other words, to answer the question of what should be understood by multiplication by a fraction, how this action should be understood.

The meaning of multiplying an integer by a fraction is clear from the following definition: to multiply an integer (multiplier) by a fraction (multiplier) means to find this fraction of the multiplier.

Namely, multiplying 9 by 2/3 means finding 2/3 of nine units. In the previous paragraph, such problems were solved; so it's easy to figure out that we end up with 6.

But now an interesting and important question arises: why such seemingly different actions as finding the sum of equal numbers and finding the fraction of a number are called the same word “multiplication” in arithmetic?

This happens because the previous action (repeating the number with terms several times) and the new action (finding the fraction of a number) give an answer to homogeneous questions. This means that we proceed here from the considerations that homogeneous questions or tasks are solved by one and the same action.

To understand this, consider the following problem: “1 m of cloth costs 50 rubles. How much will 4 m of such cloth cost?

This problem is solved by multiplying the number of rubles (50) by the number of meters (4), i.e. 50 x 4 = 200 (rubles).

Let's take the same problem, but in it the amount of cloth will be expressed as a fractional number: “1 m of cloth costs 50 rubles. How much will 3/4 m of such cloth cost?

This problem also needs to be solved by multiplying the number of rubles (50) by the number of meters (3/4).

You can also change the numbers in it several times without changing the meaning of the problem, for example, take 9/10 m or 2 3/10 m, etc.

Since these problems have the same content and differ only in numbers, we call the actions used in solving them the same word - multiplication.

How is a whole number multiplied by a fraction?

Let's take the numbers encountered in the last problem:

According to the definition, we must find 3 / 4 of 50. First we find 1 / 4 of 50, and then 3 / 4.

1/4 of 50 is 50/4;

3/4 of 50 is .

Hence.

Consider another example: 12 5 / 8 = ?

1/8 of 12 is 12/8,

5/8 of the number 12 is .

Hence,

From here we get the rule:

To multiply an integer by a fraction, you need to multiply the integer by the numerator of the fraction and make this product the numerator, and sign the denominator of the given fraction as the denominator.

We write this rule using letters:

To make this rule perfectly clear, it should be remembered that a fraction can be considered as a quotient. Therefore, it is useful to compare the found rule with the rule for multiplying a number by a quotient, which was set out in § 38

It must be remembered that before performing multiplication, you should do (if possible) cuts, For example:

4. Multiplying a fraction by a fraction. Multiplying a fraction by a fraction has the same meaning as multiplying an integer by a fraction, that is, when multiplying a fraction by a fraction, you need to find the fraction in the multiplier from the first fraction (multiplier).

Namely, multiplying 3/4 by 1/2 (half) means finding half of 3/4.

How do you multiply a fraction by a fraction?

Let's take an example: 3/4 times 5/7. This means that you need to find 5 / 7 from 3 / 4 . Find first 1/7 of 3/4 and then 5/7

1/7 of 3/4 would be expressed like this:

5 / 7 numbers 3 / 4 will be expressed as follows:

Thus,

Another example: 5/8 times 4/9.

1/9 of 5/8 is ,

4/9 numbers 5/8 are .

Thus,

From these examples, the following rule can be deduced:

To multiply a fraction by a fraction, you need to multiply the numerator by the numerator, and the denominator by the denominator and make the first product the numerator and the second product the denominator of the product.

This rule can be written in general as follows:

When multiplying, it is necessary to make (if possible) reductions. Consider examples:

5. Multiplication of mixed numbers. Since mixed numbers can easily be replaced by improper fractions, this circumstance is usually used when multiplying mixed numbers. This means that in those cases where the multiplicand, or the multiplier, or both factors are expressed as mixed numbers, then they are replaced by improper fractions. Multiply, for example, mixed numbers: 2 1/2 and 3 1/5. We turn each of them into an improper fraction and then we will multiply the resulting fractions according to the rule of multiplying a fraction by a fraction:

Rule. To multiply mixed numbers, you must first convert them to improper fractions and then multiply according to the rule of multiplying a fraction by a fraction.

Note. If one of the factors is an integer, then the multiplication can be performed based on the distribution law as follows:

6. The concept of interest. When solving problems and when performing various practical calculations, we use all kinds of fractions. But one must keep in mind that many quantities admit not any, but natural subdivisions for them. For example, you can take one hundredth (1/100) of a ruble, it will be a penny, two hundredths is 2 kopecks, three hundredths is 3 kopecks. You can take 1/10 of the ruble, it will be "10 kopecks, or a dime. You can take a quarter of the ruble, i.e. 25 kopecks, half a ruble, i.e. 50 kopecks (fifty kopecks). But they practically don’t take, for example , 2/7 rubles because the ruble is not divided into sevenths.

The unit of measurement for weight, i.e., the kilogram, allows, first of all, decimal subdivisions, for example, 1/10 kg, or 100 g. And such fractions of a kilogram as 1/6, 1/11, 1/13 are uncommon.

In general our (metric) measures are decimal and allow decimal subdivisions.

However, it should be noted that it is extremely useful and convenient in a wide variety of cases to use the same (uniform) method of subdividing quantities. Many years of experience have shown that such a well-justified division is the "hundredths" division. Let's consider a few examples related to the most diverse areas of human practice.

1. The price of books has decreased by 12/100 of the previous price.

Example. The previous price of the book is 10 rubles. She went down by 1 ruble. 20 kop.

2. Savings banks pay out during the year to depositors 2/100 of the amount that is put into savings.

Example. 500 rubles are put into the cash desk, the income from this amount for the year is 10 rubles.

3. The number of graduates of one school was 5/100 of the total number of students.

EXAMPLE Only 1,200 students studied at the school, 60 of them graduated from school.

The hundredth of a number is called a percentage..

The word "percent" is borrowed from the Latin language and its root "cent" means one hundred. Together with the preposition (pro centum), this word means "for a hundred." The meaning of this expression follows from the fact that initially in ancient Rome interest was the money that the debtor paid to the lender “for every hundred”. The word "cent" is heard in such familiar words: centner (one hundred kilograms), centimeter (they say centimeter).

For example, instead of saying that the plant produced 1/100 of all the products produced by it during the past month, we will say this: the plant produced one percent of the rejects during the past month. Instead of saying: the plant produced 4/100 more products than the established plan, we will say: the plant exceeded the plan by 4 percent.

The above examples can be expressed differently:

1. The price of books has decreased by 12 percent of the previous price.

2. Savings banks pay depositors 2 percent per year of the amount put into savings.

3. The number of graduates of one school was 5 percent of the number of all students in the school.

To shorten the letter, it is customary to write the% sign instead of the word "percentage".

However, it must be remembered that the % sign is usually not written in calculations, it can be written in the problem statement and in the final result. When performing calculations, you need to write a fraction with a denominator of 100 instead of an integer with this icon.

You need to be able to replace an integer with the specified icon with a fraction with a denominator of 100:

Conversely, you need to get used to writing an integer with the indicated icon instead of a fraction with a denominator of 100:

7. Finding percentages of a given number.

Task 1. The school received 200 cubic meters. m of firewood, with birch firewood accounting for 30%. How much birch wood was there?

The meaning of this problem is that birch firewood was only a part of the firewood that was delivered to the school, and this part is expressed as a fraction of 30 / 100. So, we are faced with the task of finding a fraction of a number. To solve it, we must multiply 200 by 30 / 100 (tasks for finding the fraction of a number are solved by multiplying a number by a fraction.).

So 30% of 200 equals 60.

The fraction 30 / 100 encountered in this problem can be reduced by 10. It would be possible to perform this reduction from the very beginning; the solution to the problem would not change.

Task 2. There were 300 children of various ages in the camp. Children aged 11 were 21%, children aged 12 were 61% and finally 13 year olds were 18%. How many children of each age were in the camp?

In this problem, you need to perform three calculations, that is, successively find the number of children 11 years old, then 12 years old, and finally 13 years old.

So, here it will be necessary to find a fraction of a number three times. Let's do it:

1) How many children were 11 years old?

2) How many children were 12 years old?

3) How many children were 13 years old?

After solving the problem, it is useful to add the numbers found; their sum should be 300:

63 + 183 + 54 = 300

You should also pay attention to the fact that the sum of the percentages given in the condition of the problem is 100:

21% + 61% + 18% = 100%

This suggests that the total number of children in the camp was taken as 100%.

3 a da cha 3. The worker received 1,200 rubles per month. Of these, he spent 65% on food, 6% on an apartment and heating, 4% on gas, electricity and radio, 10% on cultural needs and 15% he saved. How much money was spent on the needs indicated in the task?

To solve this problem, you need to find a fraction of the number 1,200 5 times. Let's do it.

1) How much money is spent on food? The task says that this expense is 65% of all earnings, i.e. 65/100 of the number 1,200. Let's do the calculation:

2) How much money was paid for an apartment with heating? Arguing like the previous one, we arrive at the following calculation:

3) How much money did you pay for gas, electricity and radio?

4) How much money is spent on cultural needs?

5) How much money did the worker save?

For verification, it is useful to add the numbers found in these 5 questions. The amount should be 1,200 rubles. All earnings are taken as 100%, which is easy to check by adding up the percentages given in the problem statement.

We have solved three problems. Despite the fact that these tasks were about different things (delivery of firewood for the school, the number of children of different ages, the expenses of the worker), they were solved in the same way. This happened because in all tasks it was necessary to find a few percent of the given numbers.

§ 90. Division of fractions.

When studying the division of fractions, we will consider the following questions:

1. Divide an integer by an integer.
2. Division of a fraction by an integer
3. Division of an integer by a fraction.
4. Division of a fraction by a fraction.
5. Division of mixed numbers.
6. Finding a number given its fraction.
7. Finding a number by its percentage.

Let's consider them sequentially.

1. Divide an integer by an integer.

As was indicated in the section on integers, division is the action consisting in the fact that, given the product of two factors (the dividend) and one of these factors (the divisor), another factor is found.

The division of an integer by an integer we considered in the department of integers. We met there two cases of division: division without a remainder, or "entirely" (150: 10 = 15), and division with a remainder (100: 9 = 11 and 1 in the remainder). We can therefore say that in the realm of integers, exact division is not always possible, because the dividend is not always the product of the divisor and the integer. After the introduction of multiplication by a fraction, we can consider any case of division of integers as possible (only division by zero is excluded).

For example, dividing 7 by 12 means finding a number whose product times 12 would be 7. This number is the fraction 7/12 because 7/12 12 = 7. Another example: 14: 25 = 14/25 because 14/25 25 = 14.

Thus, to divide an integer by an integer, you need to make a fraction, the numerator of which is equal to the dividend, and the denominator is the divisor.

2. Division of a fraction by an integer.

Divide the fraction 6 / 7 by 3. According to the definition of division given above, we have here the product (6 / 7) and one of the factors (3); it is required to find such a second factor that, when multiplied by 3, would give the given product 6 / 7. Obviously, it should be three times smaller than this product. This means that the task set before us was to reduce the fraction 6 / 7 by 3 times.

We already know that the reduction of a fraction can be done either by decreasing its numerator or by increasing its denominator. Therefore, you can write:

In this case, the numerator 6 is divisible by 3, so the numerator should be reduced by 3 times.

Let's take another example: 5 / 8 divided by 2. Here the numerator 5 is not divisible by 2, which means that the denominator will have to be multiplied by this number:

Based on this, we can state the rule: To divide a fraction by an integer, you need to divide the numerator of the fraction by that integer(if possible), leaving the same denominator, or multiply the denominator of the fraction by this number, leaving the same numerator.

3. Division of an integer by a fraction.

Let it be required to divide 5 by 1 / 2, i.e. find a number that, after multiplying by 1 / 2, will give the product 5. Obviously, this number must be greater than 5, since 1 / 2 is a proper fraction, and when multiplying a number by a proper fraction, the product must be less than the multiplicand. To make it clearer, let's write our actions as follows: 5: 1 / 2 = X , so x 1 / 2 \u003d 5.

We must find such a number X , which, when multiplied by 1/2, would give 5. Since multiplying a certain number by 1/2 means finding 1/2 of this number, then, therefore, 1/2 of the unknown number X is 5, and the whole number X twice as much, i.e. 5 2 \u003d 10.

So 5: 1 / 2 = 5 2 = 10

Let's check:

Let's consider one more example. Let it be required to divide 6 by 2 / 3 . Let's first try to find the desired result using the drawing (Fig. 19).

Fig.19

Draw a segment AB, equal to 6 of some units, and divide each unit into 3 equal parts. In each unit, three-thirds (3 / 3) in the entire segment AB is 6 times larger, i.e. e. 18/3. We connect with the help of small brackets 18 obtained segments of 2; There will be only 9 segments. This means that the fraction 2/3 is contained in b units 9 times, or, in other words, the fraction 2/3 is 9 times less than 6 integer units. Hence,

How to get this result without a drawing using only calculations? We will argue as follows: it is required to divide 6 by 2 / 3, i.e., it is required to answer the question, how many times 2 / 3 is contained in 6. Let's find out first: how many times is 1 / 3 contained in 6? In a whole unit - 3 thirds, and in 6 units - 6 times more, i.e. 18 thirds; to find this number, we must multiply 6 by 3. Hence, 1/3 is contained in b units 18 times, and 2/3 is contained in b units not 18 times, but half as many times, i.e. 18: 2 = 9. Therefore , when dividing 6 by 2 / 3 we did the following:

From here we get the rule for dividing an integer by a fraction. To divide an integer by a fraction, you need to multiply this integer by the denominator of the given fraction and, making this product the numerator, divide it by the numerator of the given fraction.

We write the rule using letters:

To make this rule perfectly clear, it should be remembered that a fraction can be considered as a quotient. Therefore, it is useful to compare the found rule with the rule for dividing a number by a quotient, which was set out in § 38. Note that the same formula was obtained there.

When dividing, abbreviations are possible, for example:

4. Division of a fraction by a fraction.

Let it be required to divide 3/4 by 3/8. What will denote the number that will be obtained as a result of division? It will answer the question how many times the fraction 3/8 is contained in the fraction 3/4. To understand this issue, let's make a drawing (Fig. 20).

Take the segment AB, take it as a unit, divide it into 4 equal parts and mark 3 such parts. Segment AC will be equal to 3/4 of segment AB. Let us now divide each of the four initial segments in half, then the segment AB will be divided into 8 equal parts and each such part will be equal to 1/8 of the segment AB. We connect 3 such segments with arcs, then each of the segments AD and DC will be equal to 3/8 of the segment AB. The drawing shows that the segment equal to 3/8 is contained in the segment equal to 3/4 exactly 2 times; So the result of the division can be written like this:

3 / 4: 3 / 8 = 2

Let's consider one more example. Let it be required to divide 15/16 by 3/32:

We can reason like this: we need to find a number that, after being multiplied by 3 / 32, will give a product equal to 15 / 16. Let's write the calculations like this:

15 / 16: 3 / 32 = X

3 / 32 X = 15 / 16

3/32 unknown number X make up 15 / 16

1/32 unknown number X is ,

32 / 32 numbers X make up .

Hence,

Thus, to divide a fraction by a fraction, you need to multiply the numerator of the first fraction by the denominator of the second, and multiply the denominator of the first fraction by the numerator of the second and make the first product the numerator and the second the denominator.

Let's write the rule using letters:

When dividing, abbreviations are possible, for example:

5. Division of mixed numbers.

When dividing mixed numbers, they must first be converted into improper fractions, and then the resulting fractions should be divided according to the rules for dividing fractional numbers. Consider an example:

Convert mixed numbers to improper fractions:

Now let's split:

Thus, to divide mixed numbers, you need to convert them to improper fractions and then divide according to the rule for dividing fractions.

6. Finding a number given its fraction.

Among the various tasks on fractions, there are sometimes those in which the value of some fraction of an unknown number is given and it is required to find this number. This type of problem will be inverse to the problem of finding a fraction of a given number; there a number was given and it was required to find some fraction of this number, here a fraction of a number is given and it is required to find this number itself. This idea will become even clearer if we turn to the solution of this type of problem.

Task 1. On the first day, glaziers glazed 50 windows, which is 1 / 3 of all windows of the built house. How many windows are in this house?

Decision. The problem says that 50 glazed windows make up 1/3 of all the windows of the house, which means that there are 3 times more windows in total, i.e.

The house had 150 windows.

Task 2. The shop sold 1,500 kg of flour, which is 3/8 of the total stock of flour in the shop. What was the store's initial supply of flour?

Decision. It can be seen from the condition of the problem that the sold 1,500 kg of flour make up 3/8 of the total stock; this means that 1/8 of this stock will be 3 times less, i.e., to calculate it, you need to reduce 1500 by 3 times:

1,500: 3 = 500 (that's 1/8 of the stock).

Obviously, the entire stock will be 8 times larger. Hence,

500 8 \u003d 4,000 (kg).

The initial supply of flour in the store was 4,000 kg.

From the consideration of this problem, the following rule can be deduced.

To find a number by a given value of its fraction, it is enough to divide this value by the numerator of the fraction and multiply the result by the denominator of the fraction.

We solved two problems on finding a number given its fraction. Such problems, as it is especially well seen from the last one, are solved by two actions: division (when one part is found) and multiplication (when the whole number is found).

However, after we have studied the division of fractions, the above problems can be solved in one action, namely: division by a fraction.

For example, the last task can be solved in one action like this:

In the future, we will solve the problem of finding a number by its fraction in one action - division.

7. Finding a number by its percentage.

In these tasks, you will need to find a number, knowing a few percent of this number.

Task 1. At the beginning of this year, I received 60 rubles from the savings bank. income from the amount I put into savings a year ago. How much money did I put in the savings bank? (Cash offices give depositors 2% of income per year.)

The meaning of the problem is that a certain amount of money was put by me in a savings bank and lay there for a year. After a year, I received 60 rubles from her. income, which is 2/100 of the money I put in. How much money did I deposit?

Therefore, knowing the part of this money, expressed in two ways (in rubles and in fractions), we must find the entire, as yet unknown, amount. This is an ordinary problem of finding a number given its fraction. The following tasks are solved by division:

So, 3,000 rubles were put into the savings bank.

Task 2. In two weeks, fishermen fulfilled the monthly plan by 64%, having prepared 512 tons of fish. What was their plan?

From the condition of the problem, it is known that the fishermen completed part of the plan. This part is equal to 512 tons, which is 64% of the plan. How many tons of fish need to be harvested according to the plan, we do not know. The solution of the problem will consist in finding this number.

Such tasks are solved by dividing:

So, according to the plan, you need to prepare 800 tons of fish.

Task 3. The train went from Riga to Moscow. When he passed the 276th kilometer, one of the passengers asked the passing conductor how much of the way they had already traveled. To this the conductor replied: “We have already covered 30% of the entire journey.” What is the distance from Riga to Moscow?

It can be seen from the condition of the problem that 30% of the journey from Riga to Moscow is 276 km. We need to find the entire distance between these cities, i.e., for this part, find the whole:

§ 91. Reciprocal numbers. Replacing division with multiplication.

Take the fraction 2/3 and rearrange the numerator to the place of the denominator, we get 3/2. We got a fraction, the reciprocal of this one.

In order to get a fraction reciprocal of a given one, you need to put its numerator in the place of the denominator, and the denominator in the place of the numerator. In this way, we can get a fraction that is the reciprocal of any fraction. For example:

3 / 4 , reverse 4 / 3 ; 5 / 6 , reverse 6 / 5

Two fractions that have the property that the numerator of the first is the denominator of the second and the denominator of the first is the numerator of the second are called mutually inverse.

Now let's think about what fraction will be the reciprocal of 1/2. Obviously, it will be 2 / 1, or just 2. Looking for a fraction, the reciprocal of this, we got an integer. And this case is not isolated; on the contrary, for all fractions with a numerator of 1 (one), the reciprocals will be integers, for example:

1 / 3, inverse 3; 1 / 5, reverse 5

Since, when searching for reciprocals, we also met with integers, in the future we will not talk about reciprocals, but about reciprocals.

Let's figure out how to write the reciprocal of a whole number. For fractions, this is solved simply: you need to put the denominator in the place of the numerator. In the same way, you can get the reciprocal of an integer, since any integer can have a denominator of 1. So the reciprocal of 7 will be 1 / 7, because 7 \u003d 7 / 1; for the number 10 the reverse is 1 / 10 since 10 = 10 / 1

This idea can be expressed in another way: the reciprocal of a given number is obtained by dividing one by the given number. This statement is true not only for integers, but also for fractions. Indeed, if you want to write a number that is the reciprocal of 5 / 9, then we can take 1 and divide it by 5 / 9, i.e.

Now let's point out one property mutually reciprocal numbers, which will be useful to us: the product of mutually reciprocal numbers is equal to one. Indeed:

Using this property, we can find reciprocals in the following way. Let's find the reciprocal of 8.

Let's denote it with the letter X , then 8 X = 1, hence X = 1 / 8 . Let's find another number, the inverse of 7/12, denote it by a letter X , then 7 / 12 X = 1, hence X = 1:7 / 12 or X = 12 / 7 .

We introduced here the concept of mutually reciprocal numbers in order to slightly supplement information about the division of fractions.

When we divide the number 6 by 3 / 5, then we do the following:

Pay special attention to the expression and compare it with the given one: .

If we take the expression separately, without connection with the previous one, then it is impossible to solve the question of where it came from: from dividing 6 by 3/5 or from multiplying 6 by 5/3. In both cases the result is the same. So we can say that the division of one number by another can be replaced by multiplying the dividend by the reciprocal of the divisor.

The examples that we give below fully confirm this conclusion.