How to solve examples with fractions. Complex expressions with fractions. Procedure. Multiplication of mixed fractions

Students are introduced to fractions in 5th grade. Previously, people who knew how to perform actions with fractions were considered very smart. The first fraction was 1/2, that is, half, then 1/3 appeared, and so on. For several centuries, the examples were considered too complex. Now detailed rules have been developed for converting fractions, addition, multiplication and other actions. It is enough to understand the material a little, and the solution will be given easily.

An ordinary fraction, which is called a simple fraction, is written as a division of two numbers: m and n.

M is the dividend, that is, the numerator of the fraction, and the divisor n is called the denominator.

Select proper fractions (m< n) а также неправильные (m >n).

A proper fraction is less than one (for example, 5/6 - this means that 5 parts are taken from one; 2/8 - 2 parts are taken from one). An improper fraction is equal to or greater than 1 (8/7 - the unit will be 7/7 and one more part is taken as a plus).

So, a unit is when the numerator and denominator matched (3/3, 12/12, 100/100 and others).

Actions with ordinary fractions Grade 6

With simple fractions, you can do the following:

• Expand fraction. If you multiply the upper and lower parts of the fraction by any identical number (but not by zero), then the value of the fraction will not change (3/5 = 6/10 (just multiplied by 2).
• Reducing fractions is similar to expanding, but here they are divided by a number.
• Compare. If two fractions have the same numerator, then the fraction with the smaller denominator will be larger. If the denominators are the same, then the fraction with the largest numerator will be larger.
• Perform addition and subtraction. With the same denominators, this is easy to do (we sum the upper parts, and the lower part does not change). For different ones, you will have to find a common denominator and additional factors.
• Multiply and divide fractions.

Examples of operations with fractions are considered below.

Reduced fractions Grade 6

To reduce means to divide the top and bottom of a fraction by some equal number.

The figure shows simple examples of reduction. In the first option, you can immediately guess that the numerator and denominator are divisible by 2.

On a note! If the number is even, then it is divisible by 2 in any way. Even numbers are 2, 4, 6 ... 32 8 (ends in even), etc.

In the second case, when dividing 6 by 18, it is immediately clear that the numbers are divisible by 2. Dividing, we get 3/9. This fraction is also divisible by 3. Then the answer is 1/3. If you multiply both divisors: 2 by 3, then 6 will come out. It turns out that the fraction was divided by six. This gradual division is called successive reduction of a fraction by common divisors.

Someone will immediately divide by 6, someone will need division by parts. The main thing is that at the end there is a fraction that cannot be reduced in any way.

Note that if the number consists of digits, the addition of which will result in a number divisible by 3, then the original can also be reduced by 3. Example: the number 341. Add the numbers: 3 + 4 + 1 = 8 (8 is not divisible by 3, so the number 341 cannot be reduced by 3 without a remainder). Another example: 264. Add: 2 + 6 + 4 = 12 (divided by 3). We get: 264: 3 = 88. This will simplify the reduction of large numbers.

In addition to the method of successive reduction of a fraction by common divisors, there are other ways.

NOD is the most big divisor for number. Having found the GCD for the denominator and numerator, you can immediately reduce the fraction by the desired number. The search is carried out by gradually dividing each number. Next, they look at which divisors match, if there are several of them (as in the picture below), then you need to multiply.

Mixed fractions grade 6

All improper fractions can be converted into mixed fractions by isolating the whole part in them. The integer is written on the left.

Often you have to make a mixed number from an improper fraction. The conversion process in the example below: 22/4 = 22 divided by 4, we get 5 integers (5 * 4 = 20). 22 - 20 = 2. We get 5 integers and 2/4 (the denominator does not change). Since the fraction can be reduced, we divide the upper and lower parts by 2.

It is easy to turn a mixed number into an improper fraction (this is necessary when dividing and multiplying fractions). To do this: multiply the whole number by the lower part of the fraction and add the numerator to this. Ready. The denominator does not change.

Calculations with fractions Grade 6

Mixed numbers can be added. If the denominators are the same, then this is easy to do: add up the integer parts and numerators, the denominator remains in place.

When adding numbers with different denominators, the process is more complicated. First, we bring the numbers to one smallest denominator (NOD).

In the example below, for the numbers 9 and 6, the denominator will be 18. After that, additional factors are needed. To find them, you should divide 18 by 9, so an additional number is found - 2. We multiply it by the numerator 4, we get the fraction 8/18). The same is done with the second fraction. We already add the converted fractions (whole numbers and numerators separately, we do not change the denominator). In the example, the answer had to be converted to a proper fraction (initially, the numerator turned out to be greater than the denominator).

Please note that with the difference of fractions, the algorithm of actions is the same.

When multiplying fractions, it is important to place both under the same line. If the number is mixed, then we turn it into a simple fraction. Next, multiply the top and bottom parts and write down the answer. If it is clear that fractions can be reduced, then we reduce immediately.

In this example, we didn’t have to cut anything, we just wrote down the answer and highlighted the whole part.

In this example, I had to reduce the numbers under one line. Though it is possible to reduce also the ready answer.

When dividing, the algorithm is almost the same. First, we turn the mixed fraction into an improper one, then we write the numbers under one line, replacing the division with multiplication. Do not forget to swap the upper and lower parts of the second fraction (this is the rule for dividing fractions).

If necessary, we reduce the numbers (in the example below, they reduced it by five and two). We transform the improper fraction by highlighting the integer part.

Basic tasks for fractions Grade 6

The video shows a few more tasks. For clarity, graphic images of solutions are used to help visualize fractions.

Examples of fraction multiplication Grade 6 with explanations

Multiplying fractions are written under one line. After that, they are reduced by dividing by the same numbers (for example, 15 in the denominator and 5 in the numerator can be divided by five).

Comparison of fractions Grade 6

To compare fractions, you need to remember two simple rules.

Rule 1. If the denominators are different

Rule 2. When the denominators are the same

For example, let's compare the fractions 7/12 and 2/3.

1. We look at the denominators, they do not match. So you need to find a common one.
2. For fractions, the common denominator is 12.
3. We divide 12 first by the lower part of the first fraction: 12: 12 = 1 (this is an additional factor for the 1st fraction).
4. Now we divide 12 by 3, we get 4 - add. multiplier of the 2nd fraction.
5. We multiply the resulting numbers by numerators to convert fractions: 1 x 7 \u003d 7 (first fraction: 7/12); 4 x 2 = 8 (second fraction: 8/12).
6. Now we can compare: 7/12 and 8/12. Turned out: 7/12< 8/12.

To represent fractions better, you can use drawings for clarity, where an object is divided into parts (for example, a cake). If you want to compare 4/7 and 2/3, then in the first case, the cake is divided into 7 parts and 4 of them are chosen. In the second, they divide into 3 parts and take 2. With the naked eye, it will be clear that 2/3 will be more than 4/7.

Examples with fractions grade 6 for training

As an exercise, you can perform the following tasks.

• Compare fractions

• do the multiplication

Tip: if it is difficult to find the lowest common denominator of fractions (especially if their values ​​are small), then you can multiply the denominator of the first and second fractions. Example: 2/8 and 5/9. Finding their denominator is simple: multiply 8 by 9, you get 72.

Solving equations with fractions Grade 6

In solving equations, you need to remember the actions with fractions: multiplication, division, subtraction and addition. If one of the factors is unknown, then the product (total) is divided by the known factor, that is, the fractions are multiplied (the second is turned over).

If the dividend is unknown, then the denominator is multiplied by the divisor, and to find the divisor, you need to divide the dividend by the quotient.

Let's imagine simple examples of solving equations:

Here it is only required to produce the difference of fractions, without leading to a common denominator.

The answer is an improper fraction. It can be converted to 1 whole and 3/5.

In the second method, the numerator and denominator were multiplied by 4 to shorten the bottom rather than flip the denominator.

One of the most important sciences, the application of which can be seen in disciplines such as chemistry, physics and even biology, is mathematics. The study of this science allows you to develop some mental qualities, improve the ability to concentrate. One of the topics that deserve special attention in the course "Mathematics" is the addition and subtraction of fractions. Many students find it difficult to study. Perhaps our article will help to better understand this topic.

How to subtract fractions whose denominators are the same

Fractions are the same numbers with which you can perform various actions. Their difference from integers lies in the presence of a denominator. That is why when performing actions with fractions, you need to study some of their features and rules. The simplest case is the subtraction of ordinary fractions, the denominators of which are represented as the same number. It will not be difficult to perform this action if you know a simple rule:

• In order to subtract the second from one fraction, it is necessary to subtract the numerator of the fraction to be subtracted from the numerator of the reduced fraction. We write this number into the numerator of the difference, and leave the denominator the same: k / m - b / m = (k-b) / m.

Examples of subtracting fractions whose denominators are the same

7/19 - 3/19 = (7 - 3)/19 = 4/19.

From the numerator of the reduced fraction "7" subtract the numerator of the subtracted fraction "3", we get "4". We write this number in the numerator of the answer, and put in the denominator the same number that was in the denominators of the first and second fractions - "19".

The picture below shows a few more such examples.

Consider a more complex example where fractions with the same denominators are subtracted:

29/47 - 3/47 - 8/47 - 2/47 - 7/47 = (29 - 3 - 8 - 2 - 7)/47 = 9/47.

From the numerator of the reduced fraction "29" by subtracting in turn the numerators of all subsequent fractions - "3", "8", "2", "7". As a result, we get the result "9", which we write in the numerator of the answer, and in the denominator we write the number that is in the denominators of all these fractions - "47".

Adding fractions with the same denominator

Addition and subtraction of ordinary fractions is carried out according to the same principle.

• To add fractions with the same denominators, you need to add the numerators. The resulting number is the numerator of the sum, and the denominator remains the same: k/m + b/m = (k + b)/m.

Let's see how it looks like in an example:

1/4 + 2/4 = 3/4.

To the numerator of the first term of the fraction - "1" - we add the numerator of the second term of the fraction - "2". The result - "3" - is written in the numerator of the amount, and the denominator is left the same as that was present in the fractions - "4".

Fractions with different denominators and their subtraction

We have already considered the action with fractions that have the same denominator. As you can see, knowing simple rules, solving such examples is quite easy. But what if you need to perform an action with fractions that have different denominators? Many high school students are confused by such examples. But even here, if you know the principle of the solution, the examples will no longer be difficult for you. There is also a rule here, without which the solution of such fractions is simply impossible.

To subtract fractions with different denominators, they must be reduced to the same smallest denominator.



We will talk in more detail about how to do this.

Fraction property

In order to reduce several fractions to the same denominator, you need to use the main property of the fraction in the solution: after dividing or multiplying the numerator and denominator by the same number, you get a fraction equal to the given one.

So, for example, the fraction 2/3 can have denominators such as "6", "9", "12", etc., that is, it can look like any number that is a multiple of "3". After we multiply the numerator and denominator by "2", we get a fraction of 4/6. After we multiply the numerator and denominator of the original fraction by "3", we get 6/9, and if we perform a similar action with the number "4", we get 8/12. In one equation, this can be written as:

2/3 = 4/6 = 6/9 = 8/12…

How to bring multiple fractions to the same denominator

Consider how to reduce several fractions to the same denominator. For example, take the fractions shown in the picture below. First you need to determine what number can become the denominator for all of them. To make it easier, let's decompose the available denominators into factors.

The denominator of the fraction 1/2 and the fraction 2/3 cannot be factored. The denominator of 7/9 has two factors 7/9 = 7/(3 x 3), the denominator of the fraction 5/6 = 5/(2 x 3). Now you need to determine which factors will be the smallest for all these four fractions. Since the first fraction has the number “2” in the denominator, it means that it must be present in all denominators, in the fraction 7/9 there are two triples, which means that they must also be present in the denominator. Given the above, we determine that the denominator consists of three factors: 3, 2, 3 and is equal to 3 x 2 x 3 = 18.



Consider the first fraction - 1/2. Its denominator contains "2", but there is not a single "3", but there should be two. To do this, we multiply the denominator by two triples, but, according to the property of the fraction, we must multiply the numerator by two triples:
1/2 = (1 x 3 x 3)/(2 x 3 x 3) = 9/18.

Similarly, we perform actions with the remaining fractions.

• 2/3 - one three and one two are missing in the denominator:
  2/3 = (2 x 3 x 2)/(3 x 3 x 2) = 12/18.
• 7/9 or 7/(3 x 3) - the denominator is missing two:
  7/9 = (7 x 2)/(9 x 2) = 14/18.
• 5/6 or 5/(2 x 3) - the denominator is missing a triple:
  5/6 = (5 x 3)/(6 x 3) = 15/18.

All together it looks like this:



How to subtract and add fractions with different denominators

As mentioned above, in order to add or subtract fractions with different denominators, they must be reduced to the same denominator, and then use the rules for subtracting fractions with the same denominator, which have already been described.

Consider this with an example: 4/18 - 3/15.

Finding multiples of 18 and 15:

• The number 18 consists of 3 x 2 x 3.
• The number 15 consists of 5 x 3.
• The common multiple will consist of the following factors 5 x 3 x 3 x 2 = 90.

After the denominator is found, it is necessary to calculate a factor that will be different for each fraction, that is, the number by which it will be necessary to multiply not only the denominator, but also the numerator. To do this, we divide the number that we found (common multiple) by the denominator of the fraction for which additional factors need to be determined.

• 90 divided by 15. The resulting number "6" will be a multiplier for 3/15.
• 90 divided by 18. The resulting number "5" will be a multiplier for 4/18.

The next step in our solution is to bring each fraction to the denominator "90".

We have already discussed how this is done. Let's see how this is written in an example:

(4 x 5) / (18 x 5) - (3 x 6) / (15 x 6) = 20/90 - 18/90 = 2/90 = 1/45.

If fractions with small numbers, then you can determine the common denominator, as in the example shown in the picture below.



Similarly produced and having different denominators.

Subtraction and having integer parts

Subtraction of fractions and their addition, we have already analyzed in detail. But how to subtract if the fraction has an integer part? Again, let's use a few rules:

• Convert all fractions that have an integer part to improper ones. In simple words, remove the whole part. To do this, the number of the integer part is multiplied by the denominator of the fraction, the resulting product is added to the numerator. The number that will be obtained after these actions is the numerator of an improper fraction. The denominator remains unchanged.
• If fractions have different denominators, they should be reduced to the same.
• Perform addition or subtraction with the same denominators.
• When receiving an improper fraction, select the whole part.



There is another way by which you can add and subtract fractions with integer parts. For this, actions are performed separately with integer parts, and separately with fractions, and the results are recorded together.



The above example consists of fractions that have the same denominator. In the case when the denominators are different, they must be reduced to the same, and then follow the steps as shown in the example.

Subtracting fractions from a whole number

Another of the varieties of actions with fractions is the case when the fraction must be subtracted from At first glance, such an example seems difficult to solve. However, everything is quite simple here. To solve it, it is necessary to convert an integer into a fraction, and with such a denominator, which is in the fraction to be subtracted. Next, we perform a subtraction similar to subtraction with the same denominators. For example, it looks like this:

7 - 4/9 = (7 x 9)/9 - 4/9 = 53/9 - 4/9 = 49/9.

The subtraction of fractions given in this article (Grade 6) is the basis for solving more complex examples, which are considered in subsequent classes. Knowledge of this topic is used subsequently to solve functions, derivatives, and so on. Therefore, it is very important to understand and understand the actions with fractions discussed above.

Almost every fifth grader after the first acquaintance with ordinary fractions is in a little shock. Not only do you still need to understand the essence of fractions, but you still have to perform arithmetic operations with them. After that, little students will systematically interrogate their teacher, find out when these fractions will run out.

To avoid such situations, it is enough just to explain this difficult topic to children as simply as possible, but rather in game form.

The essence of the fraction

Before you learn what a fraction is, the child must get acquainted with the concept share . Here the associative method is best suited.

Imagine a whole cake that has been divided into several equal parts, let's say four. Then each piece of the cake can be called a share. If you take one of the four pieces of cake, then it will be one-fourth of a share.

The shares are different, because the whole can be divided into a completely different number of parts. The more shares in general, the smaller they are, and vice versa.

So that the shares could be designated, they came up with such a mathematical concept as common fraction. The fraction will allow us to write down as many shares as needed.

The components of a fraction are the numerator and denominator, which are separated by a fractional bar or a slash. Many children do not understand their meaning, and therefore the essence of the fraction is not clear to them. The fractional bar indicates division, there is nothing complicated here.

It is customary to write the denominator below, under the fractional line or to the right of the overlay line. It shows the number of parts of the whole. The numerator, it is written above the fractional line or to the left of the oblique line, determines how many shares were taken. For example, the fraction 4/7. AT this case 7 is the denominator, shows that there are only 7 shares, and the numerator 4 indicates that four of the seven shares were taken.

The main shares and their record in fractions:

In addition to the ordinary, there is also a decimal fraction.

Actions with fractions Grade 5

In the fifth grade, they learn to perform all arithmetic operations with fractions.

All actions with fractions are performed according to the rules, and it’s not worth hoping that without learning the rule everything will turn out by itself. Therefore, do not neglect the oral part homework mathematics.

We have already understood that the decimal and ordinary fractions are different, therefore, arithmetic operations will be performed differently. Actions with ordinary fractions depend on those numbers that are in the denominator, and in decimal, after the decimal point on the right.

For fractions that have the same denominators, the addition and subtraction algorithm is very simple. Actions are performed only with numerators.

For fractions with different denominators, find Least Common Denominator (LCD). This is the number that will be divided without a remainder by all denominators, and will be the smallest of such numbers, if there are several of them.

To add or subtract decimals, you need to write them in a column, comma under comma, and equalize the number of decimal places if necessary.

To multiply ordinary fractions, simply find the product of the numerators and denominators. A very simple rule.

The division is performed according to the following algorithm:

1. Dividend to write without change
2. Division turn into multiplication
3. Flip the divisor (write the reciprocal of the divisor)
4. Perform multiplication

Addition of fractions, explanation

Let's take a closer look at how to add common and decimal fractions.

As you can see in the image above, the fractions one third and two thirds have a common denominator three. So it is required to add only the numerators one and two, and leave the denominator unchanged. The result is three thirds. Such an answer, when the numerator and denominator of the fraction are equal, can be written as 1, since 3:3 = 1.

It is required to find the sum of fractions two thirds and two ninths. In this case, the denominators are different, 3 and 9. To perform the addition, you need to find a common one. There is a very simple way. We choose the largest denominator, this is 9. We check whether it is divisible by 3. Since 9:3 = 3 without a remainder, therefore 9 is suitable as a common denominator.

The next step is to find additional factors for each numerator. To do this, we divide the common denominator 9 in turn by the denominator of each fraction, the resulting numbers will be added. plural For the first fraction: 9:3 \u003d 3, we add 3 to the numerator of the first fraction. For the second fraction: 9:9 \u003d 1, one can not be added, since when multiplied by it, the same number will be obtained.

Now we multiply the numerators by their complementary factors and add the results. The resulting amount is a fraction of eight ninths.

Adding decimals follows the same rules as adding natural numbers. In a column, the discharge is written below the discharge. The only difference is that in decimal fractions, you need to correctly put a comma in the result. To do this, the fractions are written comma under the comma, and in the sum it is only required to carry the comma down.

Let's find the sum of fractions 38, 251 and 1, 56. To make it more convenient to perform the actions, we leveled the number of decimal places on the right by adding 0.

Adding fractions, ignoring the comma. And in the resulting amount, simply drop the comma down. Answer: 39, 811.

Subtraction of fractions, explanation

To find the difference between two-thirds and one-third fractions, you need to calculate the difference between the numerators 2-1 = 1, and leave the denominator unchanged. In the answer we get a difference of one third.

Find the difference between five sixths and seven tenths. We find a common denominator. We use the selection method, out of 6 and 10, the largest is 10. We check: 10: 6 is not divisible without a remainder. We add another 10, it turns out 20:6, it also cannot be divided without a remainder. Again we increase by 10, we got 30:6 = 5. The common denominator is 30. The NOZ can also be found from the multiplication table.

We find additional factors. 30:6 = 5 - for the first fraction. 30:10 = 3 - for the second. We multiply the numerators and their additional multiplier. We get 25/30 reduced and 21/30 subtracted. Next, we subtract the numerators, and leave the denominator unchanged.

The result is a difference of 4/30. The fraction is abbreviated. Divide it by 2. The answer is 2/15.

Division of decimal fractions Grade 5

There are two options for this topic:

Multiplication of decimal fractions Grade 5

Remember how you multiply natural numbers, in exactly the same way you find the product of decimal fractions. First, let's figure out how to multiply a decimal fraction by a natural number. For this:

When multiplying a decimal by a decimal, we act in the same way.

Mixed fractions Grade 5

Five-graders like to call such fractions not mixed, but<<смешные>> probably easier to remember. Mixed fractions are called so because they are obtained by combining a whole natural number and an ordinary fraction.

A mixed fraction consists of an integer part and a fractional part.

When reading such fractions, the whole part is first called, then the fractional part: one whole two thirds, two whole one fifth, three whole two fifths, four point three fourths.

How are they obtained, these mixed fractions? Everything is pretty simple. When we get an improper fraction in the answer (a fraction whose numerator is greater than the denominator), we must always convert it to a mixed one. Just divide the numerator by the denominator. This action is called extracting the integer part:

Converting a mixed fraction back to an improper one is also easy:

Examples with decimals Grade 5 with explanation

Many questions in children are caused by examples of several actions. Let's look at a couple of such examples.

(0.4 8.25 - 2.025) : 0.5 =

The first step is to find the product of the numbers 8.25 and 0.4. We carry out multiplication according to the rule. In the answer, we count from right to left three characters and put a comma.

The second action is in the same place in brackets, this is the difference. Subtract 2.025 from 3.300. We write the action in a column, a comma under a comma.

The third action is division. The resulting difference in the second action is divided by 0.5. The comma is carried over by one character. Result 2.55.

Answer: 2.55.

(0, 93 + 0, 07) : (0, 93 — 0, 805) =

The first action is the sum in brackets. We put it in a column, remember that the comma is under the comma. We get the answer 1.00.

The second action is the difference from the second parenthesis. Since the minuend has fewer decimal places than the subtrahend, we add the missing one. The result of the subtraction is 0.125.

The third step is to divide the sum by the difference. The comma is carried over to three digits. The result was a division of 1000 by 125.

Answer: 8.

Examples with ordinary fractions with different denominators Grade 5 with explanation

In the first example, we find the sum of fractions 5/8 and 3/7. The common denominator will be the number 56. We find additional multipliers, divide 56:8 \u003d 7 and 56:7 \u003d 8. We add them to the first and second fractions, respectively. We multiply the numerators and their factors, we get the sum of fractions 35/56 and 24/56. We got the sum 59/56. The fraction is incorrect, we translate it into a mixed number. The rest of the examples are solved in a similar way.

Examples with fractions grade 5 for training

For convenience, convert mixed fractions to improper and follow the steps.

How to teach a child to easily solve fractions with Lego

With the help of such a constructor, you can not only develop the child’s imagination well, but also explain clearly in a playful way what a fraction and a fraction are.

The picture below shows that one part with eight circles is a whole. So, taking a puzzle with four circles, you get half, or 1/2. The picture clearly shows how to solve examples with Lego, if you count the circles on the details.

You can build turrets from a certain number of parts and label each of them, as in the picture below. For example, take a turret of seven parts. Each part of the green constructor will be 1/7. If you add two more to one such part, you get 3/7. Visual explanation of the example 1/7+2/7 = 3/7.

To get A's in math, don't forget to learn the rules and practice them.

To express a part as a fraction of the whole, you need to divide the part by the whole.

Task 1. There are 30 students in the class, four are missing. What proportion of students are missing?

Decision:

Answer: there are no students in the class.

Finding a fraction from a number

To solve problems in which it is required to find a part of a whole, the following rule is true:

If a part of the whole is expressed as a fraction, then to find this part, you can divide the whole by the denominator of the fraction and multiply the result by its numerator.

Task 1. There were 600 rubles, this amount was spent. How much money have you spent?

Decision: to find from 600 rubles, you need to divide this amount into 4 parts, thereby we will find out how much money is one fourth:

600: 4 = 150 (p.)

Answer: spent 150 rubles.

Task 2. It was 1000 rubles, this amount was spent. How much money has been spent?

Decision: From the condition of the problem, we know that 1000 rubles consists of five equal parts. First we find how many rubles are one fifth of 1000, and then we find out how many rubles are two fifths:

1) 1000: 5 = 200 (p.) - one fifth.

2) 200 2 \u003d 400 (p.) - two fifths.

These two actions can be combined: 1000: 5 2 = 400 (p.).

Answer: 400 rubles were spent.

The second way to find a part of a whole:

To find a part of a whole, you can multiply the whole by a fraction expressing that part of the whole.

Task 3. According to the charter of the cooperative, for the validity of the reporting meeting, it must be attended by at least members of the organization. The cooperative has 120 members. With what composition can the reporting meeting be held?

Decision:

Answer: the reporting meeting can be held if there are 80 members of the organization.

Finding a number by its fraction

To solve problems in which it is required to find the whole by its part, the following rule is true:

If a part of the desired integer is expressed as a fraction, then to find this integer, you can divide this part by the numerator of the fraction and multiply the result by its denominator.

Task 1. We spent 50 rubles, this amounted to the original amount. Find the original amount of money.

Decision: from the description of the problem, we see that 50 rubles is 6 times less than the initial amount, i.e., the initial amount is 6 times more than 50 rubles. To find this amount, you need to multiply 50 by 6:

50 6 = 300 (r.)

Answer: the initial amount is 300 rubles.

Task 2. We spent 600 rubles, this amounted to the initial amount of money. Find the original amount.

Decision: we will assume that the desired number consists of three thirds. By condition, two-thirds of the number are equal to 600 rubles. First, we find one third of the initial amount, and then how many rubles are three-thirds (initial amount):

1) 600: 2 3 = 900 (p.)

Answer: the initial amount is 900 rubles.

The second way to find the whole by its part:

To find a whole by the value of its part, you can divide this value by a fraction that expresses this part.

Task 3. Line segment AB, equal to 42 cm, is the length of the segment CD. Find the length of a segment CD.

Decision:

Answer: segment length CD 70 cm

Task 4. Watermelons were brought to the store. Before lunch, the store sold, after lunch - brought watermelons, and it remains to sell 80 watermelons. How many watermelons were brought to the store in total?

Decision: first, we find out what part of the imported watermelons is the number 80. To do this, we take the total number of imported watermelons as a unit and subtract from it the number of watermelons that we managed to sell (sell):

And so, we learned that 80 watermelons are from the total number of watermelons brought. Now we will find out how many watermelons of the total amount is, and then how many watermelons are (the number of watermelons brought):

2) 80: 4 15 = 300 (watermelons)

Answer: in total, 300 watermelons were brought to the store.

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

Fraction multiplication formula:

For example:

Before proceeding with the multiplication of numerators and denominators, it is necessary to check for the possibility of fraction reduction. If you manage to reduce the fraction, then it will be easier for you to continue to make calculations.

Division of an ordinary fraction by a fraction.

Division of fractions involving a natural number.

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

Multiplication of mixed fractions.

Rules for multiplying fractions (mixed):

• convert mixed fractions to improper;
• multiply the numerators and denominators of fractions;
• we reduce the fraction;
• if we get an improper fraction, then we convert the improper fraction to a mixed one.

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

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

It is more convenient to use the second method of multiplying an ordinary fraction by a number.

Note! To multiply a fraction by a natural number, it is necessary to divide the denominator of the fraction by this number, and leave the numerator unchanged.

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

Multilevel fractions.

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

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

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

Note, for example:

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

Practical tips for multiplying and dividing fractions:

1. The most important thing in working with fractional expressions is accuracy and attentiveness. Do all calculations carefully and accurately, concentratedly and clearly. It is better to write down a few extra lines in a draft than to get confused in the calculations in your head.

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

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

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

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