Actions with fractions. Addition and subtraction of fractions Examples of addition and subtraction of fractions

Lesson content

Adding fractions with the same denominator

There are two types of addition of fractions:

Adding fractions with the same denominator
Adding fractions with different denominators

First, let's study the addition of fractions with the same denominators. Everything is simple here. To add fractions with the same denominator, add their numerators and leave the denominator unchanged. For example, add the fractions and. Add the numerators and leave the denominator unchanged:

This example can be easily understood if you think about the pizza, which is divided into four parts. If you add pizzas to pizza, you get pizzas:

Example 2. Add fractions and.

The answer is an incorrect fraction. If the end of the problem comes, then it is customary to get rid of incorrect fractions. To get rid of the incorrect fraction, you need to select the whole part in it. In our case, the whole part is easily distinguished - two divided by two is equal to one:

This example can be easily understood if you think about the pizza, which is divided into two parts. If you add pizza to the pizza, you get one whole pizza:

Example 3... Add fractions and.

Again, add up the numerators, and leave the denominator unchanged:

This example can be easily understood if you think about the pizza, which is divided into three parts. If you add pizza to the pizza, you get pizza:

Example 4. Find the value of an expression

This example is solved in the same way as the previous ones. The numerators must be added, and the denominator must be left unchanged:

Let's try to depict our solution using a picture. If you add pizzas to pizza and add pizzas to the pizza, you get 1 whole and more pizza.

As you can see, there is nothing difficult in adding fractions with the same denominators. It is enough to understand the following rules:

To add fractions with the same denominator, you need to add their numerators, and leave the denominator unchanged;

Adding fractions with different denominators

Now let's learn how to add fractions with different denominators. When adding up fractions, the denominators of those fractions should be the same. But they are not always the same.

For example, you can add and fractions because they have the same denominators.

But fractions cannot be added immediately, since these fractions have different denominators. In such cases, the fractions must be reduced to the same (common) denominator.

There are several ways to bring fractions to the same denominator. Today we will consider only one of them, since the rest of the methods may seem difficult for a beginner.

The essence of this method is that first the (LCM) is sought for the denominators of both fractions. Then the LCM is divided by the denominator of the first fraction and the first additional factor is obtained. Do the same with the second fraction - the LCM is divided by the denominator of the second fraction and a second additional factor is obtained.

Then the numerators and denominators of the fractions are multiplied by their additional factors. As a result of these actions, fractions with different denominators are converted into fractions with the same denominators. And we already know how to add such fractions.

Example 1... Add the fractions and

First of all, we find the least common multiple of the denominators of both fractions. The denominator of the first fraction is 3, and the denominator of the second fraction is 2. The least common multiple of these numbers is 6

LCM (2 and 3) = 6

Now we return to fractions and. First, divide the LCM by the denominator of the first fraction and get the first additional factor. LCM is the number 6, and the denominator of the first fraction is the number 3. Divide 6 by 3, we get 2.

The resulting number 2 is the first additional factor. We write it down to the first fraction. To do this, make a small oblique line above the fraction and write the additional factor found above it:

We do the same with the second fraction. We divide the LCM by the denominator of the second fraction and get the second additional factor. The LCM is the number 6, and the denominator of the second fraction is the number 2. Divide 6 by 2, we get 3.

The resulting number 3 is the second additional factor. We write it down to the second fraction. Again, we draw a small oblique line above the second fraction and write the additional factor found above it:

We are now ready to add. It remains to multiply the numerators and denominators of fractions by your additional factors:

Look closely at what we have arrived at. We came to the conclusion that fractions with different denominators turned into fractions with the same denominators. And we already know how to add such fractions. Let's finish this example to the end:

Thus, the example ends. It turns out to add.

Let's try to depict our solution using a picture. If you add pizza to the pizza, you get one whole pizza and another sixth pizza:

The reduction of fractions to the same (common) denominator can also be depicted using a picture. Reducing fractions and to a common denominator, we got fractions and. These two fractions will be represented by the same slices of pizza. The only difference is that this time they will be divided into equal shares (reduced to the same denominator).

The first picture depicts a fraction (four out of six pieces), and the second picture depicts a fraction (three out of six pieces). Putting these pieces together we get (seven pieces out of six). This fraction is incorrect, so we selected the whole part in it. As a result, we got (one whole pizza and another sixth pizza).

Note that we have described this example in too much detail. V educational institutions it is not customary to write so extensively. You need to be able to quickly find the LCM of both denominators and additional factors to them, as well as quickly multiply the found additional factors by your numerators and denominators. While in school, we would have to write this example as follows:

But there is also back side medals. If at the first stages of studying mathematics you do not make detailed notes, then questions of the kind begin to appear “Where's that figure come from?” “Why do the fractions suddenly turn into completely different fractions? «.

To make it easier to add fractions with different denominators, you can use the following step-by-step instructions:

Find the LCM of the denominators of fractions;
Divide the LCM by the denominator of each fraction and get an additional factor for each fraction;
Multiply the numerators and denominators of fractions by your additional factors;
Add fractions that have the same denominator;
If the answer turns out to be an incorrect fraction, then select its whole part;

Example 2. Find the value of an expression .

Let's use the instructions above.

Step 1. Find the LCM of the denominators of fractions

Find the LCM of the denominators of both fractions. The denominators of the fractions are the numbers 2, 3 and 4.

Step 2. Divide the LCM by the denominator of each fraction and get an additional factor for each fraction

We divide the LCM by the denominator of the first fraction. LCM is the number 12, and the denominator of the first fraction is the number 2. Divide 12 by 2, we get 6. We got the first additional factor 6. We write it over the first fraction:

Now we divide the LCM by the denominator of the second fraction. LCM is the number 12, and the denominator of the second fraction is the number 3. Divide 12 by 3, we get 4. We got the second additional factor 4. We write it over the second fraction:

Now we divide the LCM by the denominator of the third fraction. LCM is the number 12, and the denominator of the third fraction is the number 4. Divide 12 by 4, we get 3. We got the third additional factor 3. We write it over the third fraction:

Step 3. Multiply the numerators and denominators of fractions by your additional factors

We multiply the numerators and denominators by our additional factors:

Step 4. Add fractions with the same denominators

We came to the conclusion that fractions with different denominators turned into fractions with the same (common) denominators. It remains to add these fractions. We add:

The addition did not fit on one line, so we moved the remaining expression to the next line. This is allowed in mathematics. When an expression does not fit on one line, it is transferred to the next line, and it is necessary to put an equal sign (=) at the end of the first line and at the beginning new line... The equal sign on the second line indicates that this is a continuation of the expression that was on the first line.

Step 5. If the answer turns out to be an incorrect fraction, then select the whole part in it

We got the wrong fraction in our answer. We have to select the whole part from it. Highlight:

Received an answer

Subtracting fractions with the same denominator

There are two types of subtraction of fractions:

Subtracting fractions with the same denominator
Subtracting fractions with different denominators

First, let's study the subtraction of fractions with the same denominator. Everything is simple here. To subtract another from one fraction, you need to subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator the same.

For example, let's find the value of an expression. To solve this example, subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator unchanged. So let's do it:

This example can be easily understood if you think about the pizza, which is divided into four parts. If you cut pizzas from pizza, you get pizzas:

Example 2. Find the value of the expression.

Again, subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator unchanged:

This example can be easily understood if you think about the pizza, which is divided into three parts. If you cut pizzas from pizza, you get pizzas:

Example 3. Find the value of an expression

This example is solved in the same way as the previous ones. From the numerator of the first fraction, you need to subtract the numerators of the remaining fractions:

As you can see, there is nothing difficult in subtracting fractions with the same denominators. It is enough to understand the following rules:

To subtract another from one fraction, you need to subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator unchanged;
If the answer turns out to be an incorrect fraction, then you need to select the whole part in it.

Subtracting fractions with different denominators

For example, you can subtract a fraction from a fraction, since these fractions have the same denominator. But you cannot subtract a fraction from a fraction, since these fractions have different denominators. In such cases, the fractions must be reduced to the same (common) denominator.

The common denominator is found according to the same principle that we used when adding fractions with different denominators. First of all, find the LCM of the denominators of both fractions. Then the LCM is divided by the denominator of the first fraction and the first additional factor is obtained, which is written over the first fraction. Similarly, the LCM is divided by the denominator of the second fraction and a second additional factor is obtained, which is written over the second fraction.

The fractions are then multiplied by their additional factors. As a result of these operations, fractions with different denominators are converted into fractions with the same denominators. We already know how to subtract such fractions.

Example 1. Find the value of an expression:

These fractions have different denominators, so you need to bring them to the same (common) denominator.

First, we find the LCM of the denominators of both fractions. The denominator of the first fraction is 3, and the denominator of the second fraction is 4. The least common multiple of these numbers is 12

LCM (3 and 4) = 12

Now back to fractions and

Let's find an additional factor for the first fraction. To do this, we divide the LCM by the denominator of the first fraction. LCM is the number 12, and the denominator of the first fraction is the number 3. Divide 12 by 3, we get 4. Write the four over the first fraction:

We do the same with the second fraction. We divide the LCM by the denominator of the second fraction. LCM is the number 12, and the denominator of the second fraction is the number 4. Divide 12 by 4, we get 3. Write the three over the second fraction:

We are now ready to subtract. It remains to multiply the fractions by their additional factors:

We came to the conclusion that fractions with different denominators turned into fractions with the same denominators. We already know how to subtract such fractions. Let's finish this example to the end:

Received an answer

Let's try to depict our solution using a picture. If you cut pizzas from pizza, you get pizza

This is a detailed version of the solution. In school, we would have to solve this example in a shorter way. Such a solution would look like this:

The reduction of fractions and to a common denominator can also be depicted using the figure. Bringing these fractions to a common denominator, we got fractions and. These fractions will be represented by the same pizza slices, but this time they will be divided into equal parts (reduced to the same denominator):

The first drawing depicts a fraction (eight out of twelve pieces), and the second drawing depicts a fraction (three out of twelve pieces). Cutting off three pieces from eight pieces, we get five pieces out of twelve. Fraction and describes these five pieces.

Example 2. Find the value of an expression

These fractions have different denominators, so you first need to bring them to the same (common) denominator.

Let's find the LCM of the denominators of these fractions.

The denominators of the fractions are 10, 3, and 5. The least common multiple of these numbers is 30

LCM (10, 3, 5) = 30

Now we find additional factors for each fraction. To do this, we divide the LCM by the denominator of each fraction.

Let's find an additional factor for the first fraction. LCM is the number 30, and the denominator of the first fraction is 10. Divide 30 by 10, we get the first additional factor 3. We write it over the first fraction:

Now we find an additional factor for the second fraction. Divide the LCM by the denominator of the second fraction. LCM is the number 30, and the denominator of the second fraction is the number 3. Divide 30 by 3, we get the second additional factor 10. We write it over the second fraction:

Now we find an additional factor for the third fraction. Divide the LCM by the denominator of the third fraction. LCM is the number 30, and the denominator of the third fraction is 5. Divide 30 by 5, we get the third additional factor 6. We write it over the third fraction:

Everything is now ready for subtraction. It remains to multiply the fractions by their additional factors:

We came to the conclusion that fractions with different denominators turned into fractions with the same (common) denominators. We already know how to subtract such fractions. Let's finish this example.

The continuation of the example will not fit on one line, so we transfer the continuation to the next line. Don't forget about the equal sign (=) on a new line:

In the answer, we got the correct fraction, and everything seems to suit us, but it is too cumbersome and ugly. We should have made it easier. What can be done? You can shorten this fraction.

To reduce a fraction, you need to divide its numerator and denominator by (GCD) numbers 20 and 30.

So, we find the GCD of numbers 20 and 30:

Now we return to our example and divide the numerator and denominator of the fraction by the found GCD, that is, by 10

Received an answer

Multiplying a fraction by a number

To multiply a fraction by a number, you need to multiply the numerator of this fraction by this number, and leave the denominator unchanged.

Example 1... Multiply the fraction by 1.

Multiply the numerator of the fraction by 1

Recording can be understood as taking half 1 time. For example, if you take pizzas 1 time, you get pizzas

From the laws of multiplication, we know that if the multiplier and the factor are reversed, then the product will not change. If the expression is written as, then the product will still be equal. Again, the rule for multiplying an integer and a fraction works:

This record can be understood as taking half of one. For example, if there is 1 whole pizza and we take half of it, then we will have pizza:

Example 2... Find the value of an expression

Multiply the numerator of your fraction by 4

The answer is an incorrect fraction. Let's select the whole part in it:

Expression can be understood as taking two quarters 4 times. For example, if you take pizzas 4 times, you get two whole pizzas.

And if we swap the multiplier and the multiplier in places, we get the expression. It will also be equal to 2. This expression can be understood as taking two pizzas from four whole pizzas:

A number that is multiplied by a fraction and the denominator of a fraction is allowed if they have a common factor greater than one.

For example, an expression can be evaluated in two ways.

The first way... Multiply 4 by the numerator of the fraction, and leave the denominator of the fraction unchanged:

Second way... The multiplied four and four in the denominator of the fraction can be canceled. You can cancel these fours by 4, since the greatest common divisor for two fours is the four itself:

The same result was obtained 3. After the reduction of the fours, new numbers are formed in their place: two ones. But multiplying one with three, and then dividing by one, does not change anything. Therefore, the solution can be written shorter:

The reduction can be performed even when we decided to use the first method, but at the stage of multiplying the number 4 and the numerator 3 we decided to use the reduction:

But, for example, the expression can be calculated only in the first way - multiply 7 by the denominator of the fraction, and leave the denominator unchanged:

This is due to the fact that the number 7 and the denominator of the fraction do not have a common divisor, greater than one, and, accordingly, do not cancel.

Some students mistakenly abbreviate the multiplication number and the numerator of the fraction. This cannot be done. For example, the following is not correct:

Fraction reduction implies that and the numerator and denominator will be divided by the same number. In a situation with an expression, division is performed only in the numerator, since writing it down is the same as writing it down. We see that division is performed only in the numerator, and no division occurs in the denominator.

Multiplication of fractions

To multiply fractions, you need to multiply their numerators and denominators. If the answer turns out to be an incorrect fraction, you need to select the whole part in it.

Example 1. Find the value of the expression.

We got an answer. It is desirable to shorten this fraction. The fraction can be reduced by 2. Then the final decision will take the following form:

The expression can be understood as taking pizza from half of the pizza. Let's say we have half a pizza:

How to get two-thirds of this half? First, you need to divide this half into three equal parts:

And take two from these three pieces:

We'll make pizza. Remember what a pizza looks like when divided into three parts:

One slice from this pizza and the two slices we took will have the same dimensions:

In other words, we are talking about the same pizza size. Therefore, the value of the expression is

Example 2... Find the value of an expression

We multiply the numerator of the first fraction by the numerator of the second fraction, and the denominator of the first fraction by the denominator of the second fraction:

The answer is an incorrect fraction. Let's select the whole part in it:

Example 3. Find the value of an expression

We multiply the numerator of the first fraction by the numerator of the second fraction, and the denominator of the first fraction by the denominator of the second fraction:

The answer is a correct fraction, but it will be good if you reduce it. To reduce this fraction, you need to divide the numerator and denominator of this fraction by the greatest common divisor (GCD) of 105 and 450.

So, let's find the GCD of numbers 105 and 450:

Now we divide the numerator and denominator of our answer to the GCD, which we have now found, that is, by 15

Fraction representation of an integer

Any integer can be represented as a fraction. For example, the number 5 can be represented as. From this, the five will not change its value, since the expression means "the number five divided by one", and this, as you know, is equal to five:

Reverse numbers

Now we will get acquainted with a very interesting topic in mathematics. It is called "back numbers".

Definition. The inverse of the numbera is a number that, when multiplied bya gives one.

Let's substitute in this definition instead of a variable a number 5 and try to read the definition:

The inverse of the number 5 is a number that, when multiplied by 5 gives one.

Can you find a number that, when multiplied by 5, gives one? It turns out you can. Let's represent the five as a fraction:

Then multiply this fraction by itself, just change the places of the numerator and denominator. In other words, we multiply the fraction by itself, only inverted:

What will be the result of this? If we continue to solve this example, we get one:

This means that the inverse of 5 is a number, because when 5 is multiplied by, one is obtained.

The reciprocal can also be found for any other integer.

You can also find the reciprocal for any other fraction. To do this, just turn it over.

Dividing a fraction by a number

Let's say we have half a pizza:

Let's divide it equally in two. How much pizza will each get?

It can be seen that after splitting half of the pizza, there are two equal slices, each of which makes up a pizza. So everyone gets a pizza.

Actions with fractions.

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

So, what are fractions, types of fractions, transformations - we remembered. Let's get down to the main issue.

What can you do with fractions? Yes, everything that is with ordinary numbers. Add, subtract, multiply, divide.

All these actions with decimal fractions are no different from operations with integers. Actually, that's why they are good, decimal. The only thing is that you need to put the comma correctly.

Mixed numbers, as I said, are of little use for most actions. They still need to be converted into fractions.

But the actions with ordinary fractions will be more cunning. And much more important! Let me remind you: all actions with fractional expressions with letters, sines, unknowns and so on and so on are no different from actions with ordinary fractions! Fractional operations are the foundation for all algebra. It is for this reason that we will analyze all this arithmetic in great detail here.

Addition and subtraction of fractions.

Everyone can add (subtract) fractions with the same denominators (I really hope!). Well, let me remind you completely forgetful: when adding (subtracting) the denominator does not change. The numerators are added (subtracted) to give the numerator of the result. Type:

In short, in general terms:

And if the denominators are different? Then, using the basic property of the fraction (here it came in handy again!), We make the denominators the same! For example:

Here we had to make 4/10 from the fraction 2/5. For the sole purpose of making the denominators the same. Note, just in case, that 2/5 and 4/10 are the same fraction! Only 2/5 is uncomfortable for us, and 4/10 is nothing at all.

By the way, this is the essence of solving any problems in mathematics. When we are from uncomfortable expressions do the same, but already convenient for solution.

Another example:

The situation is similar. Here we make 48 out of 16. By simple multiplication by 3. It's all clear. But here we came across something like:

How to be ?! It is difficult to make nine out of seven! But we are smart, we know the rules! We transform every fraction so that the denominators become the same. This is called "converting to a common denominator":

How! How did I know about 63? Very simple! 63 is a number that is evenly divisible by 7 and 9 at the same time. Such a number can always be obtained by multiplying the denominators. If we multiplied some number by 7, for example, then the result will certainly be divisible by 7!

If you need to add (subtract) several fractions, there is no need to do it in pairs, in steps. You just need to find a denominator common to all fractions, and bring each fraction to this very denominator. For example:

And what is the common denominator? You can, of course, multiply 2, 4, 8, and 16. We get 1024. Nightmare. It’s easier to figure out that the number 16 is perfectly divisible by 2, and 4, and 8. Therefore, from these numbers it is easy to get 16. This number will be the common denominator. 1/2 will turn into 8/16, 3/4 into 12/16, and so on.

By the way, if we take 1024 as the common denominator, everything will work out too, in the end everything will shrink. Only not everyone will get to this end, because of the calculations ...

Complete the example yourself. Not a logarithm ... It should be 29/16.

So, adding (subtracting) fractions is clear, I hope? Of course, it is easier to work in a shortened version, with additional factors. But this pleasure is available to those who honestly worked in the lower grades ... And have not forgotten anything.

And now we will do the same actions, but not with fractions, but with fractional expressions... There will be a new rake here, yes ...

So, we need to add two fractional expressions:

We need to make the denominators the same. And only with the help multiplication! So the basic property of a fraction dictates. Therefore, I cannot add one to the first fraction in the denominator to the x. (but it would be nice!). But if you multiply the denominators, you see, everything will grow together! So we write down, a line of the fraction, we leave an empty space on top, then we add it, and below we write the product of the denominators, so as not to forget:

And, of course, we don’t multiply anything on the right side, we don’t open the parentheses! And now, looking at the common denominator of the right side, we figure out: in order to get the denominator x (x + 1) in the first fraction, the numerator and denominator of this fraction must be multiplied by (x + 1). And in the second fraction - by x. Here's what happens:

Note! Brackets appeared here! This is the rake that many are stepping on. Not brackets, of course, but their absence. The parentheses appear because we are multiplying the whole numerator and the whole denominator! And not their separate pieces ...

In the numerator of the right side, we write the sum of the numerators, everything is like in numeric fractions, then we open the brackets in the numerator of the right side, i.e. we multiply everything and give similar ones. You don't need to open parentheses in denominators, you don't need to multiply something! In general, a work is always more pleasant in the denominators (any)! We get:

So we got the answer. The process seems long and difficult, but it depends on the practice. Solve examples, get used to it, everything will become simple. Those who have mastered fractions in due time do all these operations with one hand, on the machine!

And one more note. Many famously deal with fractions, but hang on examples with whole numbers. Like: 2 + 1/2 + 3/4 =? Where to fasten the deuce? You don't need to fasten it anywhere, you need to make a fraction out of two. It is not easy, but very simple! 2 = 2/1. Like this. Any integer can be written as a fraction. The numerator is the number itself, the denominator is one. 7 is 7/1, 3 is 3/1, and so on. It's the same with letters. (a + b) = (a + b) / 1, x = x / 1, etc. And then we work with these fractions according to all the rules.

Well, in addition - subtraction of fractions, knowledge has been refreshed. We repeated the conversion of fractions from one type to another. You can and check. Shall we solve a little?)

Calculate:

Answers (in disarray):

71/20; 3/5; 17/12; -5/4; 11/6

Multiplication / division of fractions - in the next lesson. There are also tasks for all actions with fractions.

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. Instant validation testing. Learning - with interest!)

you can get acquainted with functions and derivatives.

Class: 5

Lesson presentation

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Attention! Slide previews are for informational purposes only and may not represent all the presentation options. If you are interested in this work, please download the full version.

Lesson objectives:

Educational:

systematize knowledge about ordinary fractions;
repeat the rules for adding and subtracting fractions with the same denominators;
repeat the rules of addition and subtraction of fractions with different denominators.

Developing:

develop attention, speech, memory, logical thinking, independence.

Educational:

to cultivate the desire to achieve the set goal; self-confidence, ability to work in a team.

Know: rules for adding and subtracting fractions with the same and different denominators.

Lesson type: a lesson in generalization and systematization of knowledge.

Equipment: screen, multimedia, presentation "Addition and subtraction of ordinary fractions" (Appendix 1), model of an ordinary fraction (Figure 1); a form with a test, a table of answers (Figure 2), emoticons for reflection (Figure 3), a drawn Christmas tree (Figure 4).

P / p No.	Lesson stage	Time	Stage objectives
1.	Organizing time.	3 min.	Set up students for a lesson.
2.	Knowledge update. Repetition of the passed material.	10 min.	To repeat correct, incorrect fractions, reduction of fractions, reduction of fractions to a new denominator, highlighting the whole part.
3.	Applying the rules for adding and subtracting common fractions with the same denominators.	10 min.	Repeat additions and subtractions of common fractions with the same denominators.
4.	Physical education.	3 min.	Relieve fatigue of the child, provide active rest and increase the mental performance of students.
5.	Applying the rules for adding and subtracting ordinary fractions with different denominators.	13 minutes	Repeat additions and subtractions of common fractions with different denominators.
6.	Homework.	2 minutes.	Homework briefing.
7.	Lesson summary.	4 minutes	Summing up. Grading. Reflection.

During the classes

1). Organizing time.

- "Addition and subtraction of ordinary fractions".

It is proposed to formulate the goals and objectives of the lesson, during the discussion they are formulated (the teacher can write them down on the board).

2). Knowledge update. Repetition of the passed material. (Slide number 1).

a) Today we will start the lesson with an auction. The only lot is "ordinary fraction" (picture 1)... Let's remember what we know about ordinary fractions:

Numerator;

Denominator;

Fractional line - division;

On b parts we divide, we take a such parts;

Correct;

Wrong;

Select the whole part;

Shrink;

Lead to a new denominator;

Examples.

Whoever said the last about an ordinary fraction gets the model of an ordinary fraction.

b) We will consolidate our knowledge when performing the test(answer form, task number 1, slide number 2).

TEST

1. Find the correct fraction:

A); B); V) .

2. Find the improper fraction:

A); B); V) .

3. Reduce the fraction:

A); B); V) .

4. Bring the fraction to the denominator 28:

A); B); V) .

5. Select the whole part:

A); B); V) .

The answers are entered into the table.

1 2 3 4 5

Summarize:

5 "+" mark 5,
4 "+" mark 4,
3 "+" mark 3.

3). Application of the rules of addition and subtraction of ordinary fractions with the same denominators.

What common fractions can we add?

Fractions with the same and different denominators (slide number 3).

Let's repeat the addition of fractions with the same denominators.

To add two fractions with the same denominator, add their numerators and leave the denominator unchanged.

To subtract fractions with the same denominator, you need to subtract the numerator of the subtracted from the numerator of the reduced, and leave the denominator unchanged.

Let's consolidate our knowledge in practice.

Students are invited to calculate orally examples and write down answers in the answer form for assignment number 2.

Exchange notebooks, carry out a mutual check.

Summarize:

9-8 "+" mark 5,
7-6 "+" mark 4,
5 "+" mark 3.

4). Physical education.

5). Applying the rules for adding and subtracting ordinary fractions with different denominators.

We added fractions with the same denominators. What needs to be done to add common fractions with different denominators?(slide number 4).

To add and subtract fractions with different denominators, you need to bring the fractions to a common denominator by finding additional factors. Add and subtract common fractions with the same denominators.

It is quite important even in Everyday life... Subtraction can often come in handy when calculating change in a store. For example, you have one thousand (1000) rubles with you, and your purchases are 870. You, having not paid yet, ask: "How much change will I have left?" So, 1000-870 will be 130. And such calculations are many different and without mastering this topic, it will be difficult in real life. Subtraction is an arithmetic operation, during which the second number is subtracted from the first number, and the result will be the third.

The addition formula is expressed as follows: a - b = c

a- Vasya had apples initially.

b- the number of apples given to Petya.

c- Vasya has apples after the transfer.

Let's substitute in the formula:

Subtracting numbers

Subtraction of numbers is easy for any first grader to learn. For example, from 6 you need to subtract 5.6-5 = 1, 6 more numbers 5 per one, which means that the answer will be one. You can add 1 + 5 = 6 to check. If you are not familiar with addition, you can read ours.

Big number divided into parts, take the number 1234, and in it: 4-units, 3-tens, 2-hundreds, 1-thousand. If you subtract units, then everything is easy and simple. But let's say an example: 14-7. In the number 14: 1 is ten, and 4 is one. 1 dozen - 10 units. Then we get 10 + 4-7, let's do it like this: 10-7 + 4, 10 - 7 = 3, and 3 + 4 = 7. The answer was found correctly!

Consider example 23-16. The first number is 2 tens and 3 units, and the second is 1 dozen and 6 units. Let's represent the number 23 as 10 + 10 + 3, and 16 as 10 + 6, then let's represent 23-16 as 10 + 10 + 3-10-6. Then 10-10 = 0, there will remain 10 + 3-6, 10-6 = 4, then 4 + 3 = 7. The answer has been found!

The same is done with hundreds and thousands.

Column subtraction

Answer: 3411.

Subtraction of fractions

Let's imagine a watermelon. The watermelon is one whole, and if we cut it in half, we get something less than one, right? Half of the unit. How to write this down?

½, so we denote half of one whole watermelon, and if we divide the watermelon into 4 equal parts, then each of them will be denoted by ¼. Etc…

subtraction of fractions like this?

It's simple. Subtract the ¼ th from 2/4. When subtracting, it is important that the denominator (4) of one fraction coincides with the denominator of the second. (1) and (2) are called numerators.

So, subtract. We made sure that the denominators are the same. Then subtract the numerators (2-1) / 4, so we get 1/4.

Subtraction limits

Subtracting limits isn't hard. Here is a fairly simple formula, which says that if the limit of the difference of functions tends to the number a, then this is equivalent to the difference of these functions, the limit of each of which tends to the number a.

Subtraction of mixed numbers

A mixed number is an integer with a fractional part. That is, if the numerator is less than the denominator, then the fraction is less than one, and if the numerator is greater than the denominator, then the fraction is greater than one. A mixed number is a fraction that is greater than one and has an integer part highlighted, for example:

To subtract mixed numbers, you need:

Bring fractions to a common denominator.

Enter the whole part into the numerator

Calculate

Subtraction lesson

Subtraction is an arithmetic operation, in the process of which the difference of 2 numbers is sought and the answers are the third. The addition formula is expressed as follows: a - b = c.

Examples and tasks can be found below.

At subtracting fractions it should be remembered that:

Given the fraction 7/4, we get that 7 is more than 4, which means 7/4 is more than 1. How to select the whole part? (4 + 3) / 4, then we get the sum of fractions 4/4 + 3/4, 4: 4 + 3/4 = 1 + 3/4. Result: one whole, three quarters.

Subtraction grade 1

The first grade is the beginning of the path, the beginning of learning and learning the basics, including subtraction. Learning should be done in a playful way. Always in the first grade, calculations begin with simple examples on apples, sweets, pears. This method is not used in vain, but because children are much more interested in playing with them. And this is not the only reason. Children saw apples, sweets and the like very often in their lives and dealt with the transfer and quantity, so it will not be difficult to teach how to add such things.

You can think of a whole cloud of subtraction problems for first graders, for example:

Objective 1. In the morning, walking through the forest, the hedgehog found 4 mushrooms, and in the evening, when he came home, the hedgehog ate 2 mushrooms for dinner. How many mushrooms are left?

Objective 2. Masha went to the store for bread. Mom gave mache 10 rubles, and the bread costs 7 rubles. How much money should Masha bring home?

Objective 3. In the morning, there were 7 kilograms of cheese on the counter in the store. Before lunch, the visitors bought 5 kilograms. How many kilos are left?

Task 4. Roma took out into the yard the candy that his dad had given him. Roma had 9 sweets, and he gave his friend Nikita 4. How many sweets did Roma have left?

First graders mostly solve problems in which the answer is a number from 1 to 10.

Subtraction grade 2

The second class is already higher than the first, and, accordingly, examples for the solution too. So let's get started:

Numerical assignments:

Single-digit numbers:

10 - 5 =
7 - 2 =
8 - 6 =
9 - 1 =
9 - 3 - 4 =
8 - 2 - 3 =
9 - 9 - 0 =
4 - 1 - 3 =

Double figures:

10 - 10 =
17 - 12 =
19 - 7 =
15 - 8 =
13 - 7 =
64 - 37 =
55 - 53 =
43 - 12 =
34 - 25 =
51 - 17 - 18 =
47 - 12 - 19 =
31 - 19 - 2 =
99 - 55 - 33 =

Text tasks

Subtraction 3-4 grade

The essence of subtraction in grade 3-4 is subtraction in a column of large numbers.

Consider example 4312-901. To begin with, let's write the numbers under each other, so that from the number 901, the unit is under 2, 0 under 1, 9 under 3.

Then we subtract from right to left, that is, from the number 2 the number 1. We get one:

Subtracting nine from the three, you need to borrow 1 dozen. That is, subtract 1 dozen from 4. 10 + 3-9 = 4.

And since 4 took 1, then 4-1 = 3

Answer: 3411.

Subtraction grade 5

The fifth grade is time for working on complex fractions with different denominators. Let's repeat the rules: 1. Numerators are subtracted, not denominators.

So, subtract. We made sure that the denominators are the same. Then subtract the numerators (2-1) / 4, so we get 1/4. When adding fractions, only the numerators are subtracted!

2. Make sure the denominators are equal to perform the subtraction.

If you come across the difference of fractions, for example, 1/2 and 1/3, then you will have to multiply not one fraction, but both, in order to bring to a common denominator. The easiest way to do this: multiply the first fraction by the denominator of the second, and the second fraction by the denominator of the first, we get: 3/6 and 2/6. Add (3-2) / 6 to get 1/6.

3. The reduction of a fraction is made by dividing the numerator and denominator by the same number.

The fraction 2/4 can be reduced to ½. Why? What is a fraction? 1/2 = 1: 2, and dividing 2 by 4 is the same as dividing 1 by 2. Therefore, the fraction 2/4 = 1/2.

4. If the fraction is greater than one, then you can select the whole part.

Subtraction presentation

The link to the presentation is below. The presentation addresses basic sixth grade subtraction issues: Download presentation

Presentation addition and subtraction

Examples for addition and subtraction

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This article begins the study of actions with algebraic fractions: we will consider in detail such actions as addition and subtraction of algebraic fractions. Let us analyze the scheme of addition and subtraction of algebraic fractions with both the same denominators and different ones. Let's learn how to add an algebraic fraction with a polynomial and how to subtract them. Let us explain each step of the search for a solution to problems with specific examples.

Yandex.RTB R-A-339285-1

Addition and Subtraction Actions with the Same Denominators

The scheme for adding ordinary fractions is also applicable to algebraic ones. We know that when adding or subtracting ordinary fractions with the same denominators, you must add or subtract their numerators, and the denominator remains the original.

For example: 3 7 + 2 7 = 3 + 2 7 = 5 7 and 5 11 - 4 11 = 5 - 4 11 = 1 11.

Accordingly, the rule for addition and subtraction of algebraic fractions with the same denominators is written in a similar way:

Definition 1

To add or subtract algebraic fractions with the same denominators, you need to add or subtract the numerators of the original fractions, respectively, and write the denominator unchanged.

This rule makes it possible to conclude that the result of addition or subtraction of algebraic fractions is a new algebraic fraction (in a particular case: a polynomial, monomial or number).

Let us indicate an example of the application of the formulated rule.

Example 1

Algebraic fractions are given: x 2 + 2 x y - 5 x 2 y - 2 and 3 - x y x 2 y - 2. It is necessary to add them together.

Solution

The original fractions contain the same denominators. According to the rule, let's add the numerators of the given fractions, and leave the denominator unchanged.

Adding the polynomials that are the numerators of the original fractions, we get: x 2 + 2 x y - 5 + 3 - x y = x 2 + (2 x y - x y) - 5 + 3 = x 2 + x y - 2.

Then the required sum will be written as: x 2 + x · y - 2 x 2 · y - 2.

In practice, as in many cases, the solution is given by a chain of equalities, clearly showing all the stages of the solution:

x 2 + 2 x y - 5 x 2 y - 2 + 3 - x yx 2 y - 2 = x 2 + 2 x y - 5 + 3 - x yx 2 y - 2 = x 2 + x y - 2 x 2 y - 2

Answer: x 2 + 2 x y - 5 x 2 y - 2 + 3 - x y x 2 y - 2 = x 2 + x y - 2 x 2 y - 2.

The result of addition or subtraction can be a cancellable fraction, in this case it is optimal to reduce it.

Example 2

It is necessary to subtract from the algebraic fraction x x 2 - 4 · y 2 the fraction 2 · y x 2 - 4 · y 2.

Solution

The denominators of the original fractions are equal. Let's perform actions with the numerators, namely: subtract the numerator of the second from the numerator of the first fraction, and then write down the result, leaving the denominator unchanged:

x x 2 - 4 y 2 - 2 y x 2 - 4 y 2 = x - 2 y x 2 - 4 y 2

We see that the resulting fraction is a cancellable one. Let's carry out its reduction by transforming the denominator using the formula for the difference of squares:

x - 2 y x 2 - 4 y 2 = x - 2 y (x - 2 y) (x + 2 y) = 1 x + 2 y

Answer: x x 2 - 4 y 2 - 2 y x 2 - 4 y 2 = 1 x + 2 y.

By the same principle, three or more algebraic fractions are added or subtracted with the same denominators. For example:

1 x 5 + 2 x 3 - 1 + 3 x - x 4 x 5 + 2 x 3 - 1 - x 2 x 5 + 2 x 3 - 1 - 2 x 3 x 5 + 2 x 3 - 1 = 1 + 3 x - x 4 - x 2 - 2 x 3 x 5 + 2 x 3 - 1

Addition and Subtraction Actions for Different Denominators

Let us again turn to the scheme of actions with ordinary fractions: in order to add or subtract ordinary fractions with different denominators, it is necessary to bring them to a common denominator, and then add the resulting fractions with the same denominators.

For example, 2 5 + 1 3 = 6 15 + 5 15 = 11 15 or 1 2 - 3 7 = 7 14 - 6 14 = 1 14.

Similarly, we will formulate the rule for addition and subtraction of algebraic fractions with different denominators:

Definition 2

To carry out the addition or subtraction of algebraic fractions with different denominators, you must:

bring the original fractions to a common denominator;
perform addition or subtraction of the resulting fractions with the same denominators.

Obviously, the key here will be the skill of bringing algebraic fractions to a common denominator. Let's take a closer look.

Common denominator of algebraic fractions

To bring algebraic fractions to a common denominator, it is necessary to carry out an identical transformation of the given fractions, as a result of which the denominators of the original fractions become the same. Here it is optimal to act according to the following algorithm for reducing algebraic fractions to a common denominator:

first, we determine the common denominator of algebraic fractions;
then we find additional factors for each of the fractions by dividing the common denominator by the denominators of the original fractions;
by the last action, the numerators and denominators of the given algebraic fractions are multiplied by the corresponding additional factors.

Example 3

Algebraic fractions are given: a + 2 2 a 3 - 4 a 2, a + 3 3 a 2 - 6 a and a + 1 4 a 5 - 16 a 3. It is necessary to bring them to a common denominator.

Solution

We act according to the above algorithm. Let's determine the common denominator of the original fractions. For this purpose, we factor out the denominators of the given fractions: 2 a 3 - 4 a 2 = 2 a 2 (a - 2), 3 a 2 - 6 a = 3 a (a - 2) and 4 a 5 - 16 a 3 = 4 a 3 (a - 2) (a + 2)... From here we can write down the common denominator: 12 a 3 (a - 2) (a + 2).

Now we have to find additional factors. Let us divide, according to the algorithm, the found common denominator into the denominators of the original fractions:

for the first fraction: 12 a 3 (a - 2) (a + 2): (2 a 2 (a - 2)) = 6 a (a + 2);
for the second fraction: 12 a 3 (a - 2) (a + 2): (3 a (a - 2)) = 4 a 2 (a + 2);
for the third fraction: 12 a 3 (a - 2) (a + 2): (4 a 3 (a - 2) (a + 2)) = 3 .

The next step is to multiply the numerators and denominators of the given fractions by the additional factors found:

a + 2 2 a 3 - 4 a 2 = (a + 2) 6 a (a + 2) (2 a 3 - 4 a 2) 6 a (a + 2) = 6 a (a + 2) 2 12 a 3 (a - 2) (a + 2) a + 3 3 a 2 - 6 a = (a + 3) 4 a 2 ( a + 2) 3 a 2 - 6 a 4 a 2 (a + 2) = 4 a 2 (a + 3) (a + 2) 12 a 3 (a - 2) (A + 2) a + 1 4 a 5 - 16 a 3 = (a + 1) 3 (4 a 5 - 16 a 3) 3 = 3 (a + 1) 12 a 3 (a - 2) (a + 2)

Answer: a + 2 2 a 3 - 4 a 2 = 6 a (a + 2) 2 12 a 3 (a - 2) (a + 2); a + 3 3 a 2 - 6 a = 4 a 2 (a + 3) (a + 2) 12 a 3 (a - 2) (a + 2); a + 1 4 a 5 - 16 a 3 = 3 (a + 1) 12 a 3 (a - 2) (a + 2).

So, we brought the original fractions to a common denominator. If necessary, you can further transform the result into the form of algebraic fractions by multiplying polynomials and monomials in the numerators and denominators.

Let us also clarify the following point: it is optimal to leave the found common denominator in the form of a product in case it is necessary to cancel the finite fraction.

We examined in detail the scheme for reducing the original algebraic fractions to a common denominator, now we can proceed to the analysis of examples for addition and subtraction of fractions with different denominators.

Example 4

Algebraic fractions are given: 1 - 2 x x 2 + x and 2 x + 5 x 2 + 3 x + 2. It is necessary to carry out the action of their addition.

Solution

The original fractions have different denominators, so the first step is to bring them to a common denominator. Factor the denominators: x 2 + x = x (x + 1), and x 2 + 3 x + 2 = (x + 1) (x + 2), since square trinomial roots x 2 + 3 x + 2 these are numbers: - 1 and - 2. Determine the common denominator: x (x + 1) (x + 2), then the additional factors will be: x + 2 and - x for the first and second fractions, respectively.

Thus: 1 - 2 xx 2 + x = 1 - 2 xx (x + 1) = (1 - 2 x) (x + 2) x (x + 1) (x + 2) = x + 2 - 2 x 2 - 4 xx (x + 1) x + 2 = 2 - 2 x 2 - 3 xx (x + 1) (x + 2) and 2 x + 5 x 2 + 3 x + 2 = 2 x + 5 (x + 1) (x + 2) = 2 x + 5 x (x + 1) (x + 2) x = 2 X 2 + 5 xx (x + 1) (x + 2)

Now let's add the fractions that we brought to a common denominator:

2 - 2 x 2 - 3 xx (x + 1) (x + 2) + 2 x 2 + 5 xx (x + 1) (x + 2) = = 2 - 2 x 2 - 3 x + 2 x 2 + 5 xx (x + 1) (x + 2) = 2 2 xx (x + 1) (x + 2)

The resulting fraction can be reduced by a common factor x + 1:

2 + 2 x x (x + 1) (x + 2) = 2 (x + 1) x (x + 1) (x + 2) = 2 x (x + 2)

And, finally, we write the result obtained in the form of an algebraic fraction, replacing the product in the denominator with a polynomial:

2 x (x + 2) = 2 x 2 + 2 x

Let us write down the course of the solution briefly in the form of a chain of equalities:

1 - 2 xx 2 + x + 2 x + 5 x 2 + 3 x + 2 = 1 - 2 xx (x + 1) + 2 x + 5 (x + 1) (x + 2 ) = = 1 - 2 x (x + 2) x x + 1 x + 2 + 2 x + 5 x (x + 1) (x + 2) x = 2 - 2 x 2 - 3 xx (x + 1) (x + 2) + 2 x 2 + 5 xx (x + 1) (x + 2) = = 2 - 2 x 2 - 3 x + 2 x 2 + 5 xx (x + 1) (x + 2) = 2 x + 1 x (x + 1) (x + 2) = 2 x (x + 2) = 2 x 2 + 2 x

Answer: 1 - 2 x x 2 + x + 2 x + 5 x 2 + 3 x + 2 = 2 x 2 + 2 x

Pay attention to the following detail: before adding or subtracting algebraic fractions, if possible, it is desirable to transform them in order to simplify.

Example 5

It is necessary to subtract fractions: 2 1 1 3 · x - 2 21 and 3 · x - 1 1 7 - 2 · x.

Solution

We transform the original algebraic fractions to simplify the further solution. Let's take out the numeric coefficients of the variables in the denominator outside the brackets:

2 1 1 3 x - 2 21 = 2 4 3 x - 2 21 = 2 4 3 x - 1 14 and 3 x - 1 1 7 - 2 x = 3 x - 1 - 2 x - 1 14

This transformation definitely gave us a benefit: we clearly see the presence of a common factor.

Let's get rid of the numerical coefficients in the denominators altogether. To do this, we use the main property of algebraic fractions: we multiply the numerator and denominator of the first fraction by 3 4, and the second by - 1 2, then we get:

2 4 3 x - 1 14 = 3 4 2 3 4 4 3 x - 1 14 = 3 2 x - 1 14 and 3 x - 1 - 2 x - 1 14 = - 1 2 3 x - 1 - 1 2 - 2 x - 1 14 = - 3 2 x + 1 2 x - 1 14.

Let's take an action that will allow us to get rid of fractional coefficients: multiply the resulting fractions by 14:

3 2 x - 1 14 = 14 3 2 14 x - 1 14 = 21 14 x - 1 and - 3 2 x + 1 2 x - 1 14 = 14 - 3 2 x + 1 2 x - 1 14 = - 21 x + 7 14 x - 1.

Finally, we perform the required action in the problem statement - subtraction:

2 1 1 3 x - 2 21 - 3 x - 1 1 7 - 2 x = 21 14 x - 1 - - 21 x + 7 14 x - 1 = 21 - - 21 x + 7 14 X - 1 = 21 x + 14 14 x - 1

Answer: 2 1 1 3 x - 2 21 - 3 x - 1 1 7 - 2 x = 21 x + 14 14 x - 1.

Addition and subtraction of an algebraic fraction and a polynomial

This action is also reduced to the addition or subtraction of algebraic fractions: it is necessary to represent the original polynomial as a fraction with denominator 1.

Example 6

It is necessary to add the polynomial x 2 - 3 with an algebraic fraction 3 x x + 2.

Solution

We write the polynomial as an algebraic fraction with denominator 1: x 2 - 3 1

Now we can perform addition according to the rule for adding fractions with different denominators:

x 2 - 3 + 3 xx + 2 = x 2 - 3 1 + 3 xx + 2 = x 2 - 3 (x + 2) 1 x + 2 + 3 xx + 2 = = x 3 + 2 X 2 - 3 x - 6 x + 2 + 3 xx + 2 = x 3 + 2 x 2 - 3 x - 6 + 3 xx + 2 = = x 3 + 2 x 2 - 6 x + 2

Answer: x 2 - 3 + 3 x x + 2 = x 3 + 2 x 2 - 6 x + 2.

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