In this article, we will consider such an action as multiplication decimal fractions. Let's start with the wording of general principles, then we show how to multiply one decimal fraction to another and consider the multiplication method by the column. All definitions will be illustrated by examples. Then we will analyze how to multiply the decimal fractions to ordinary, as well as on mixed and integers (including 100, 10, etc.)

As part of this material we will touch only the rules of multiplying positive fractions. Cases with negative disassemble separately in articles on multiplication of rational and valid numbers.

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Formulate general principleswhich must be adhere to when solving problems for multiplying decimal fractions.

Recall to begin with that decimal fractions are nothing but a special form of the recording of ordinary fractions, therefore, the process of their multiplication can be reduced to similar for the fractions of ordinary. This rule also works for the ultimate, and for endless fractions: after their transfer to ordinary with them it is easy to perform multiplication of the rules already studied by us.

Let's see how such tasks are solved.

Example 1.

Calculate the work 1, 5 and 0, 75.

Solution: To begin with, replace decimal fractions to ordinary. We know that 0, 75 is 75/100, and 1, 5 is 15 10. We can reduce the fraction and produce the whole part. The resulting result of 125 1000 we will write as 1, 125.

Answer: 1 , 125 .

We can use a column counting method as for natural numbers.

Example 2.

Multiply one periodic fraction 0, (3) to another 2, (36).

To begin with, we present the original fractions to ordinary. We will have:

0 , (3) = 0 , 3 + 0 , 03 + 0 , 003 + 0 , 003 + . . . = 0 , 3 1 - 0 , 1 = 0 , 3 9 = 3 9 = 1 3 2 , (36) = 2 + 0 , 36 + 0 , 0036 + . . . = 2 + 0 , 36 1 - 0 , 01 = 2 + 36 99 = 2 + 4 11 = 2 4 11 = 26 11

Consequently, 0, (3) · 2, (36) \u003d 1 3 · 26 11 \u003d 26 33.

The usual fraction obtained in the end can be brought to a decimal form by dividing the numerator to the denominator in the column:

Answer: 0, (3) · 2, (36) \u003d 0, (78).

If we have infinite non-periodic fractions in the condition of the problem, then you need to perform their preliminary rounding (see the article on rounding numbers if you forgot how it is done). After that, it is possible to perform multiplication with already rounded decimal fractions. Let us give an example.

Example 3.

Calculate the work 5, 382 ... and 0, 2.

Decision

In our task, there is an infinite fraction that you need to first round up to the hundredths. It turns out that 5, 382 ... ≈ 5, 38. The second factor is rounded to the hundredths of meaning. Now you can count the desired work and write out the answer: 5, 38 · 0, 2 \u003d 538 100 · 2 10 \u003d 1 076 1000 \u003d 1, 076.

Answer: 5, 382 ... · 0, 2 ≈ 1, 076.

The counting method of the column can be applied not only for natural numbers. If we have decimal fractions, we can multiply them in the same way. We bring the rule:

Definition 1.

The multiplication of decimal fractions by the column is performed in 2 steps:

1. We perform multiplication by a column, not paying for commas.

2. We put in the final number of the decimal comma, separating it so much figures on the right side, how much both factors contain decimal signs together. If the result is not enough for this numbers, add the left of the zeros.

We will analyze examples of such calculations in practice.

Example 4.

Multiply decimal fractions 63, 37 and 0, 12 column.

Decision

First of all, you will perform multiplication of numbers by ignoring decimal commas.

Now we need to put a comma on right place. It will separate four numbers on the right side, since the sum of decimal signs in both multipliers is 4. Drop the zeros do not have to do, because Signs enough:

Answer: 3, 37 · 0, 12 \u003d 7, 6044.

Example 5.

Calculate how much it will be 3, 2601 multiply by 0, 0254.

Decision

We consider without registering commas. We get the following:

We will put a comma separating 8 digits on the right side, because the initial fractions together have 8 signs after the comma. But in our result, only seven digits, and we can not do without extra zeros:

Answer: 3, 2601 · 0, 0254 \u003d 0, 08280654.

How to multiply decimal fraction 0.001, 0.01, 01 ,, etc

Multiplying decimal fractions on such numbers has often, so it is important to be able to do it quickly and accurately. We write a special rule that we will use with such multiplication:

Definition 2.

If we multiply the decimal fraction on 0, 1, 0, 01, etc., as a result, it turns out the number similar to the original fraction, the comma of which is transferred to the left the right amount signs. When lacking digits for transfer, you need to add zeros to the left.

So, for multiplication 45, 34 to 0, 1 must be transferred in the original decimal fraction with a comma one sign. We will result in 4, 534.

Example 6.

Multiply 9, 4 to 0, 0001.

Decision

We will have to endure the comma for four signs by the number of zeros in the second multiplier, but the numbers in the first will not be enough for this. We attribute the necessary zeros and we obtain that 9, 4 · 0, 0001 \u003d 0, 00094.

Answer: 0 , 00094 .

For infinite decimal fractions, we use the same rule. So, for example, 0, (18) · 0, 01 \u003d 0, 00 (18) or 94, 938 ... · 0, 1 \u003d 9, 4938 .... and etc.

The process of such multiplication is no different effect of multiplying two decimal fractions. It is convenient to use the multiplication method in the column, if the ultimate decimal fraction is worth in the task condition. At the same time, we must take into account all those rules about which we told in the previous paragraph.

Example 7.

Calculate how much it will be 15 · 2, 27.

Decision

Multiply column source numbers and separable two seaste.

Answer: 15 · 2, 27 \u003d 34, 05.

If we are multiplying a periodic decimal fraction on a natural number, you must first change the decimal fraction on the ordinary one.

Example 8.

Calculate the product 0, (42) and 22.

Let us give a periodic fraction to the form of ordinary.

0 , (42) = 0 , 42 + 0 , 0042 + 0 , 000042 + . . . = 0 , 42 1 - 0 , 01 = 0 , 42 0 , 99 = 42 99 = 14 33

0, 42 · 22 \u003d 14 33 · 22 \u003d 14 · 22 3 \u003d 28 3 \u003d 9 1 3

The final result can be written in the form of a periodic decimal fraction as 9, (3).

Answer: 0, (42) · 22 \u003d 9, (3).

Infinite fractions before counting must be pre-rounded.

Example 9.

Calculate how much 4 · 2, 145 ....

Decision

Rounded to the hundredths of the original infinite decimal fraction. After that, we will come to the multiplication of a natural number and the ultimate decimal fraction:

4 · 2, 145 ... ≈ 4 · 2, 15 \u003d 8, 60.

Answer: 4 · 2, 145 ... ≈ 8, 60.

How to multiply decimal fraction per 1000, 100, 10, etc.

Multiplying decimal fraction 10, 100, etc. It is often found in tasks, so we will analyze this case separately. The main rule of multiplication sounds like this:

Definition 3.

To multiply the decimal fraction per 1000, 100, 10, etc., you need to transfer it to the comma on 3, 2, 1 numbers depending on the multiplier and discard the left of the extra zeros. If the digits for the transfer of the comma are not enough, we add so much zeros, how much we need.

Let's show on the example of how to do it.

Example 10.

Perform multiplication 100 and 0, 0783.

Decision

To do this, we need to move in a decimal fraction with a comma on 2 digits to the right side. We obtain in the end 007, 83 zeros, standing on the left, can be discarded and record the result as 7, 38.

Answer: 0, 0783 · 100 \u003d 7, 83.

Example 11.

Multiply 0, 02 by 10 thousand.

Solution: We will carry the comma for four digits to the right. In the original decimal fraction, we will not be enough for this signs, so you have to add zeros. In this case, it will be enough three 0. As a result, it turned out 0, 02000, we move the comma and get 00200, 0. Ignoring zeros on the left, we can write out the answer as 200.

Answer: 0, 02 · 10 000 \u003d 200.

The rule given by us will work as well as in the case of endless decimal fractions, but here you should be very attentive to the period of the final fraction, as it is easy to make an error.

Example 12.

Calculate the work 5, 32 (672) per 1,000.

Solution: First of all, we will write a periodic fraction like 5, 32672672672 ... so the probability will be mistaken less. After that, we can carry the comma for the desired number of signs (for three). As a result, it turns out 5326, 726726 ... We conclude the period in the brackets and write the answer as 5 326, (726).

Answer: 5, 32 (672) · 1 000 \u003d 5 326, (726).

If in the conditions of the problem there are endless non-periodic fractions, which must be multiplied by ten, a hundred, thousand, etc., do not forget to round them before multiplying.

To multiply this type, you need to submit a decimal fraction in the form of an ordinary and continue to act on the already familiar rules.

Example 13.

Multiply 0, 4 to 3 5 6

Decision

In the beginning we will transfer the decimal fraction in the ordinary. We have: 0, 4 \u003d 4 10 \u003d 2 5.

We received an answer in the form of a mixed number. You can write it as a periodic fraction 1, 5 (3).

Answer: 1 , 5 (3) .

If an infinite non-periodic fraction is involved in the calculation, it is necessary to round it up to some numbers and then multiply.

Example 14.

Calculate the work 3, 5678. . . · 2 3.

Decision

We can imagine the second factor as 2 3 \u003d 0, 6666 .... Next, rounded up to the thousandth discharge of both factor. After that, we need to calculate the product of the two finite decimal fractions 3, 568 and 0, 667. Calculate the column and get the answer:

The final result should be rounded up to thousands of stakes, since it is before that discharge we have rounded the initial numbers. We obtain that 2, 379856 ≈ 2, 380.

Answer: 3, 5678. . . · 2 3 ≈ 2, 380

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In the course of the middle and older school students passed the topic "Frui". However, this concept is much wider than given in the learning process. Today, the concept of fraction is found quite often, and not everyone can calculate any expression, for example, multiplication of fractions.

What is a fraction?

So historically it happened that fractional numbers appeared due to the need to measure. As practice shows, there are often examples for determining the length of the segment, the volume of the rectangular rectangle.

Initially, students get acquainted with such a concept as a share. For example, if splitting the watermelon on 8 parts, each will get each eighth watermelon. This one is one of the eight and is called a fraction.

A fraction of ½ from any value is called half; ⅓ - Third; ¼ - Quarter. Records of the form 5/8, 4/5, 2/4 are called ordinary fractions. The ordinary fraction is divided into a numerator and denominator. Between them is the feature of a fraction, or a fractional trait. The fractional feature can be drawn in the form of both horizontal and inclined line. In this case, it denotes a fission sign.

The denominator represents how much the same shares are separated by the value; And the numerator is how many identical fractions taken. The numerator is written above a fractional feature, denominator - under it.

It is most convenient to show ordinary fractions on the coordinate beam. If a single segment is divided into 4 equal shares, designate every fraction of a Latin letter, then as a result, you can get a great visual allowance. So, point A shows a share equal to 1/4 from the entire unit segment, and the point B notes 2/8 from this segment.

Varieties of fractions

Fruit is ordinary, decimal, as well as mixed numbers. In addition, the fraction can be divided into the correct and incorrect. This classification is more suitable for ordinary fractions.

Under proper shot understand the number whose numerator less denominator. Accordingly, the wrong fraction is the number whose numerator is greater than the denominator. The second form is usually written in the form of a mixed number. Such an expression consists of a whole and fractional part. For example, 1½. one - whole part, ½ - fractional. However, if you need to carry out some manipulations with the expression (division or multiplication of fractions, their abbreviation or transformation), the mixed number is translated into the wrong fraction.

Proper fractional expression is always less than a unit, and incorrect - more either equal to 1.

As for this expression, they understand the record, in which any number is represented by the denominator of the fractional expression of which can be expressed by a unit with several zeros. If the fraction is correct, then the whole part in the decimal record will be zero.

To record a decimal fraction, you must first write a whole part, separating it from fractional with a comma and then write a fractional expression. It must be remembered that after the semicolons the numerator must contain as many digital characters as zeros in the denominator.

Example. Present fraction 7 21/1000 in a decimal record.

Algorithm for the transfer of incorrect fraction in a mixed number and vice versa

To record the task in response, the wrong fraction incorrectly, so it must be translated into a mixed number:

• split the numerator on the existing denominator;
• in specific example incomplete private - whole;
• and the residue is the numerator of the fractional part, and the denominator remains unchanged.

Example. Translate the wrong fraction into a mixed number: 47/5.

Decision. 47: 5. Incomplete private equals 9, the residue \u003d 2. So, 47/5 \u003d 9 2/5.

Sometimes it is necessary to present a mixed number as incorrect fraction. Then you need to use the following algorithm:

• the whole part is multiplied by the denominator of the fractional expression;
• the resulting product is added to the numerator;
• the result is written in a numerator, the denominator remains unchanged.

Example. Present a mixed form as an incorrect fraction: 9 8/10.

Decision. 9 x 10 + 8 \u003d 90 + 8 \u003d 98 - numerator.

Answer: 98 / 10.

Multiplication of fractions ordinary

Over ordinary fractions, various algebraic operations can be performed. To multiply two numbers, you need to multiply the numerator with a numerator, and the denominator with the denominator. Moreover, the multiplication of fractions with different denominators is different from the work. fractional numbers with the same denominators.

It happens that after finding the result you need to reduce the fraction. In mandatory, you need to simplify the resulting expression. Of course, it is impossible to say that the wrong fraction in the answer is an error, but also to call it the right answer is also difficult.

Example. Find a product of two ordinary fractions: ½ and 20/18.

As can be seen from the example, after finding the work it turned out a reduced fractional entry. And the numerator, and the denominator in this case is divided into 4, and the result is the answer 5/9.

Multiplication of fractions decimal

The product of decimal fractions is quite different from the work of ordinary on its principle. So, the multiplication of fractions is as follows:

• two decimal fractions should be written in each other so that the extreme right numbers are one to another;
• it is necessary to multiply the recorded numbers, despite the commas, that is, as natural;
• calculate the number of numbers after the semicolon in each of the numbers;
• in the resulting step after multiplying the result, it is necessary to count so much digital characters as it is contained in the amount in both factors after the comma, and put the separating sign;
• if the numbers in the work turned out less, then before them need to write so much zeros to cover this amount, put the comma and attribute a whole part equal to zero.

Example. Calculate the work of two decimal fractions: 2.25 and 3.6.

Decision.

Multiplying mixed fractions

To calculate the work of two mixed fractions, You need to use the fraction multiplication rule:

• translate numbers mixed into incorrect fractions;
• find a product of numerals;
• find a product of denominators;
• record the resulting result;
• maximum simplify expression.

Example. Find a product 41 and 6 2/5.

Multiplication of the number of fraction (fractions by number)

In addition to finding the work of two fractions, mixed numbers, there are tasks where you need to multiply by fraction.

So, to find a product of a decimal fraction and a natural number, you need:

• record a number under the fraction so that the extreme right numbers turned out to be one above the other;
• find a work, despite the comma;
• in the resulting result, it is possible to separate the whole part of the fraction with the help of a semicolon, counting on the right then the number of characters that is after the comma in the fraction.

To multiply the ordinary fraction to the number, you should find a product of a numerator and a natural multiplier. If the response is reduced fraction, it should be converted.

Example. Calculate the work of 5/8 and 12.

Decision. 5 / 8 * 12 = (5*12) / 8 = 60 / 8 = 30 / 4 = 15 / 2 = 7 1 / 2.

Answer: 7 1 / 2.

As can be seen from the previous example, it was necessary to reduce the resulting result and convert an incorrect fractional expression into a mixed number.

Also multiplication of fractions concerns and finding a product of a mixed form and a natural multiplier. To multiply these two numbers, you follow the integer part of the mixed multiplier to multiply by the number, multiply the numerator to the same value, and the denominator is left unchanged. If required, you need to easily simplify the result.

Example. Find a product 9 5/6 and 9.

Decision. 9 5/6 x 9 \u003d 9 x 9 + (5 x 9) / 6 \u003d 81 + 45/6 \u003d 81 + 7 3/6 \u003d 88 1/2.

Answer: 88 1 / 2.

Multiplication of multipliers 10, 100, 1000 or 0.1; 0.01; 0.001.

From the previous item follows the following rule. For multiplication of the fraction decimal, 10, 100, 1000, 10,000, etc., you need to move the comma to the right to so many characters of numbers, how many zeros in the multiplier after a unit.

Example 1.. Find a product 0.065 and 1000.

Decision. 0.065 x 1000 \u003d 0065 \u003d 65.

Answer: 65.

Example 2.. Find a product 3.9 and 1000.

Decision. 3.9 x 1000 \u003d 3,900 x 1000 \u003d 3900.

Answer: 3900.

If you need to multiply a natural number and 0.1; 0.01; 0.001; 0.0001, etc., you should move the left comma in the resulting product to so many characters of numbers, how many zeros is up to one. If necessary, the zeros are recorded in sufficient quantity in a natural number.

Example 1.. Find a product 56 and 0.01.

Decision. 56 x 0.01 \u003d 0056 \u003d 0.56.

Answer: 0,56.

Example 2.. Find a product 4 and 0.001.

Decision. 4 x 0.001 \u003d 0004 \u003d 0.004.

Answer: 0,004.

So, finding the work of various fractions should not cause difficulties, except for counting the result; In this case, without a calculator, it's just not to do.

In the last lesson, we learned to fold and subtract decimal fractions (see the lesson "Addition and subtraction of decimal fractions"). At the same time, they appreciated how simplifying the calculations compared to the usual "two-storey" fractions.

Unfortunately, with multiplication and division of decimal fractions of this effect does not occur. In some cases, the decimal record of the number even complicates these operations.

To begin with, we introduce a new definition. We will meet with him quite often, and not only in this lesson.

The meaningful part of the number is everything that is between the first and last nonzero digit, including the ends. We are talking only about numbers, the decimal point is not taken into account.

Figures included in the meaningful part of the number are called meaningful numbers. They can be repeated and even to be zero.

For example, consider several decimal fractions and discourage the most significant parts:

1. 91.25 → 9125 (meaningful numbers: 9; 1; 2; 5);
2. 0.008241 → 8241 (meaning numbers: 8; 2; 4; 1);
3. 15,0075 → 150075 (meaningful numbers: 1; 5; 0; 0; 7; 5);
4. 0.0304 → 304 (meaningful numbers: 3; 0; 4);
5. 3000 → 3 (meaningful number one: 3).

Please note: zeros, standing inside the meaningful part of the number, do not go anywhere. We have already come across something similar when they learned to translate decimal fractions to ordinary (see the lesson "Decimal fractions").

This moment is so important, and the mistakes here are allowed so often that in the near future I will publish a test on this topic. Be sure to practice! And we, armed with the concept of a meaningful part, proceed, in fact, to the topic of the lesson.

Multiplying decimal fractions

The multiplication operation consists of three consecutive steps:

1. For each fraction, write a meaningful part. It will turn out two ordinary integers - without any denominators and decimal points;
2. Multiply these numbers any in a convenient way. Directly, if the numbers are small, or a column. We obtain a significant part of the desired fraction;
3. Find out where and how many digits is shifted by a decimal point in the original fractions to obtain the appropriate significant part. Run reverse shifts for a significant part obtained in the previous step.

Once again I remind you that zeros, standing on the sides of the meaningful part, are never taken into account. Ignoring this rule leads to errors.

1. 0.28 · 12.5;
2. 6.3 · 1.08;
3. 132,5 · 0.0034;
4. 0.0108 · 1600.5;
5. 5.25 · 10 000.

We work with the first expression: 0.28 · 12.5.

1. We repel the most significant parts for numbers from this expression: 28 and 125;
2. Their work: 28 · 125 \u003d 3500;
3. In the first multiplier, the decimal point is shifted to 2 digits to the right (0.28 → 28), and in the second - another 1 digit. It is a shift to the left of three digits: 3500 → 3,500 \u003d 3.5.

Now we will deal with the expression 6.3 · 1.08.

1. We repel the meaning parts: 63 and 108;
2. Their work: 63 · 108 \u003d 6804;
3. Again two shifts to the right: 2 and 1 digit, respectively. Total - again 3 digits to the right, so the reverse shift will be 3 digits to the left: 6804 → 6,804. This time zeros at the end is not.

Reached to the third expression: 132,5 · 0.0034.

1. Significant parts: 1325 and 34;
2. Their work: 1325 · 34 \u003d 45 050;
3. In the first fraction, the decimal point goes to the right to 1 digit, and in the second - by as much as 4. Total: 5 to the right. We perform a shift to 5 left: 45 050 →, 45050 \u003d 0.4505. At the end removed zero, and in front - adds not to leave a "naked" decimal point.

The following expression: 0.0108 · 1600.5.

1. We write significant parts: 108 and 16 005;
2. Multiply them: 108 · 16 005 \u003d 1 728 540;
3. We consider the numbers after the decimal point: in the first number there are 4, in the second - 1. total - again 5. We have: 1 728 540 → 17,28540 \u003d 17,2854. At the end removed the "extra" zero.

Finally, the last expression: 5.25 · 10 000.

1. Significant parts: 525 and 1;
2. Multiply them: 525 · 1 \u003d 525;
3. In the first fraction, a shift is made on 2 digits to the right, and in the second - on 4 digits to the left (10 000 → 1.0000 \u003d 1). Total 4 - 2 \u003d 2 digits left. We carry out the return shift to 2 digits to the right: 525, → 52 500 (I had to add zeros).

Pay attention to the last example: since the decimal point moves to different areasThe total shift is through the difference. This is very important moment! Here is another example:

Consider the numbers 1.5 and 12 500. We have: 1.5 → 15 (shift to 1 to the right); 12 500 → 125 (shift 2 to the left). We "walk" by 1 category to the right, and then - 2 to the left. As a result, we stepped at 2 - 1 \u003d 1 category left.

Division of decimal fractions

The division is perhaps the most difficult operation. Of course, here you can act by analogy with multiplication: to divide the meaning parts, and then "move" the decimal point. But in this case there are many subtleties, which are reduced to no potential savings.

So let's consider a universal algorithm that a little longer, but much more reliable:

1. Translate all decimal fractions to ordinary. If you practice a little, you will have a few seconds for this step;
2. Split the resulting fraratically by the classical way. In other words, multiply the first fraction on the "inverted" second (see the lesson "Multiplication and division of numerical fractions");
3. If possible, the result is again submitted in the form of a decimal fraction. This step is also performed quickly, since it is often a dozen degrees in the denominator.

A task. Find the value of the expression:

1. 3,51: 3,9;
2. 1,47: 2,1;
3. 6,4: 25,6:
4. 0,0425: 2,5;
5. 0,25: 0,002.

We consider the first expression. To begin with, we will transfer Obproba to decimal:

Similarly, adopted with the second expression. The numerator of the first fraction will decompose again on multipliers:

In the third and fourth examples there is an important point: after getting rid of decimal records there are short-lived fractions. However, we will not fulfill this reduction.

The last example is interesting in that the second fraction has a simple number. There is simply nothing to decompose on multipliers, so we consider "alprint":

Sometimes as a result of division, an integer is obtained (this is what I am about the last example). In this case, the third step is not fulfilled at all.

In addition, the division often occurs "ugly" fractions, which cannot be translated into decimal. This division differs from multiplication, where the results are always representable in decimal form. Of course, in this case, the last step is not performed again.

Pay attention to the 3rd and 4th examples. In them, we intentionally do not reduce the usual fractions derived from decimal. Otherwise, this will complicate the inverse task - the representation of the final response is again in decimal form.

Remember: The main property of the fraction (like any other rule in mathematics) in itself does not mean that it must be applied everywhere and always, with each convenient case.

The decimal fraction is used when you need to perform actions with the neuroys. This may seem irrational. But this type of numbers significantly facilitates mathematical operations that need to be performed with them. This understanding comes with time when their recording becomes usual, and reading does not cause difficulties, and the rules of decimal fractions are mastered. Moreover, all actions repeat already known, which are learned with natural numbers. Only you need to remember some features.

Definition of decimal fractions

The decimal fraction is a special representation of a neurotic number with a denominator, which is divided into 10, and the answer is obtained in the form of a unit and, possibly, zeros. In other words, if in the denominator 10, 100, 1000, and so on, it is more convenient to rewrite a semicolon. Then before it will be located a whole part, and then - fractional. Moreover, the record of the second half of the number will depend on the denominator. The number of numbers that are in the fractional part must be equal to the discharge of the denominator.

You can illustrate the above these numbers:

9/10=0,9; 178/10000=0,0178; 3,05; 56 003,7006.

The reasons for which the application of decimal frains took

Mathematics required decimal fractions on several grounds:

Simplify recording. Such a fraction is located along one line without a dash between the denominator and the numerator, and the visibility does not suffer.

Easy in comparison. It is enough to simply relate the numbers in the same positions, while with ordinary fractions would have to bring them to a common denominator.

Simplify computing.

Calculators are not designed to introduce ordinary fractions, they use a decimal record of numbers for all operations.

How to read such numbers?

The answer is simple: just like an ordinary mixed number with a denominator, multiple 10. The exceptions are only a fraction without a whole value, then when reading you need to say "zero integers."

For example, 45/1000 need to pronounce as forty five thousands, at the same time, 0.045 will sound like zero forty five thousandth.

Mixed number with an integer equal to 7 and scream 17/100, which is recorded as 7.17, in both cases it will be read as seven whole seventeen hundredths.

The role of discharges in the records of fractions

It is true to note the discharge - this is what mathematics requires. Decimal fractions and their value can change significantly if you write the number in the wrong place. However, it was fair before.

To read the discharges of a whole part of the decade, you need to simply use the rules known for natural numbers. And on the right side, they are mirrored and read differently. If "dozens" sounded in the whole part, then after the comma, it will be "tenths".

It can be clearly seen in this table.

How to record a mixed number of decimal fraction?

If the denominator costs a number 10 or 100, and others, then the question of how to translate into decimal fraction is simple. For this, it is quite different to rewrite all its components. Such items will help:

a little in the side of writing a numerator of a fraction, at that moment a decimal comma is located on the right, after the last digit;

move the comma to the left, here the most important thing is to check the numbers correctly - you need to move it on so many positions as the nole in the denominator;

if they are missing, it should be zeros on empty positions;

zeros, which were at the end of the numerator, are no longer needed, and they can be shocked;

before the comma to attribute a whole part, if it was not, then zero will also be here.

Attention. It is impossible to burn zeros, which were surrounded by other numbers.

About how to be in a situation where the number in the denominator is not only from the unit and zeros, as the fraction to translate into decimal, you can read just below. it important informationWith which it is necessary to familiarize yourself.

How to translate into decimal if the denominator is an arbitrary number?

Here are two options:

When the denominator can be represented as a number that is ten to any degree.

If such an operation can not be done.

How to check it? You need to decompose the denominator for multipliers. If only 2 and 5 are present in the work, then everything is fine, and the fraction is easily converted into a finite decimal. Otherwise, if 3, 7 and other simple numbers appear, the result will be infinite. This decimal fraction for the convenience of use in mathematical transactions is made round. This will be a little lower.

He studies how such decimal fractions are obtained, grade 5. Examples here will be very useful.

Let there be numbers in the denominar: 40, 24 and 75. The decomposition of simple multipliers will be such:

• 40 \u003d 2 · 2 · 2 · 5;
• 24 \u003d 2 · 2 · 2 · 3;
• 75 \u003d 5 · 5 · 3.

In these examples, only the first fraction can be represented as a finite.

Algorithm for the transfer of ordinary fraction in the final decimal

Check the decomposition of the denominator to simple multipliers and make sure that it will consist of 2 and 5.

Add to these numbers so much 2 and 5 so that they become equal to the amount. They will give the value of an additional multiplier.

Making multiplication of the denominator and the numerator to this number. As a result, it turns out an ordinary fraction, under the feature that costs 10 to some extent.

If these actions are performed in the task with a mixed number, then it must be represented as an incorrect fraction. And then act according to the described scenario.

Representation of an ordinary fraction in the form of a rounded decimal

This method of how to translate into decimal, someone will seem even easier to someone. Because there is no large number actions. You only need to split the value of the numerator to the denominator.

To any number with a decimal part of the right of the semicolons, you can attribute an infinite number of zeros. This property should be used.

First, write a whole part and put a comma after it. If the fraction is correct, then write zero.

Then it is supposed to divide the numerator to the denominator. So that the number of numbers they have the same. That is, attribute to the right in the numerator the necessary number of nonols.

Perform division into a column until the required number of digits are scored. For example, if you need to round down to hundredths, then in response there should be 3. In general, the numbers should be one more than you need to get in the end.

Record an intermediate response after the comma and rounded according to the rules. If the last digit is from 0 to 4, then it needs to be simply discarded. And when it is equal to 5-9, then standing in front of it need to be increased by one, throwing the latter.

Returns from decimal fractions to ordinary

In mathematics, there are tasks when decimal fractions are more convenient to present in the form of ordinary, in which there is a numerator with a denominator. You can sigh with relief: this operation is always possible.

For this procedure, you need to do the following:

record a whole part, if it is zero, then nothing needs to write;

conduct a fractional line;

on it, write numbers from the right part, if the first go zeros, then they need to shovel;

under the line, write a unit with such a number of nonols, how many figures are after the comma in the initial fraction.

It's all you need to do to translate the decimal fraction in ordinary.

What can be done with decimal fractions?

In mathematics, these will have certain actions with decimal fractions that were previously carried out for other numbers.

They are:

comparison;

addition and subtraction;

multiplication and division.

The first action, comparison, similar to how it was done for natural numbers. To determine what more, you need to compare the discharges of the whole part. If they are equal, they are moving to fractional and also according to discharges compare them. The number where the big figure will be in the older discharge, and will be the answer.

Addition and subtraction of decimal fractions

This is perhaps the most simple actions. Because they are executed according to the rules for natural numbers.



So, to perform the addition of decimal fractions, they need to be recorded with each other, placing commas in the column. With such an entry to the left of the commas, the whole parts are found, and on the right - fractional. And now you need to fold the numbers are boiled, as is done with natural numbers, driving down the comma. Begin additionally needed from the smallest discharge of the fractional part of the number. If there are not enough numbers in the right half, then add zeros.

When subtracting, apply the same. And here is a rule that describes the opportunity to take a unit from the older discharge. If in the reduced fraction after the semicolons less numbers than that of the subtracted, then the zeros are simply attributed to it.

A little more difficult is the case with the tasks, where you need to perform multiplication and division of decimal fractions.

How to multiply decimal fraction in different examples?

The rule for which the multiplication of decimal fractions on a natural number is carried out:

record them in a column, not paying attention to the comma;

multiply as if they were natural;

separate the semicolons as many numbers as they were in the fractional part of the initial number.

A special case is an example in which a natural number is equal to 10 to any extent. Then, to get the answer, you just need to move the comma to the right to so much positions as zeros in another multiplier. In other words, with multiplication by 10, the comma shifts on one digit, 100 there are already two of them, and so on. If the numbers in the fractional part are missing, then you need to write down on the empty positions of zeros.

The rule that enjoys when in the task you need to multiply the decimal fractions to the other of the same number:

record them in each other, not paying attention to the commas;

multiply as if they were natural;

separate the semicolons as many numbers as they were in fractional parts of both source fractions together.

An examples in which one of the multipliers is 0.1 or 0.01 and further are distinguished by a special case. They need to move the comma to the left to the number of numbers in the presented multipliers. That is, if multiplied by 0.1, the comma shifts on one position.

How to divide the decimal fraction in different tasks?

The division of decimal fractions on a natural number is carried out according to such a rule:

write them to divide them in a column, as if they were natural;

to share the familiar rule until the whole part ends;

put the comma in response;

continue dividing the fractional component to obtain in the remaining zero;

if necessary, you can attribute the desired number of zeros.

If the whole part is zero, it will not be in response either.

Separately there is a division in numbers equal to a dozen, hundred and so on. In such tasks you need to move the comma to the left by the number of zeros in the divider. It happens that the numbers are missing in the whole part, then zeros are used instead. It can be noted that this operation is similar to multiplication by 0.1 and similar to it.

To divide decimal fractions, you need to use this rule:

turn a divider into a natural number, and for this, to transfer the comma to the right to the end to the end;

perform the movement of the comma and in divide on the same number of numbers;

act over the previous scenario.

A division of 0.1 is distinguished; 0.01 and other similar numbers. In such examples, the comma shifts to the right to the number of numbers in the fractional part. If they are over, then you need to attribute the missing number of zeros. It is worth noting that this action repeats the division by 10 and similar to it.



Conclusion: It's all in practice

Nothing in their studies is easy and effortless. For reliable development of the new material, time and training is required. Mathematics is no exception.

To the topic about decimal fractions does not cause difficulties, you need to solve examples with them as much as possible. After all, there was a time when the addition of natural numbers put in a dead end. And now everything is fine.

Therefore, paraphrasing the famous phrase: to decide, decide and decide again. Then the tasks with such numbers will be performed easily and naturally as another puzzle.

By the way, the puzzles at first are solved difficult, and then you need to do the usual movements. Also in mathematical examples: passing one way several times, then you won't think about where to turn.

As ordinary numbers.

2. We consider the number of decimal places in the 1st decimal fraction and in the 2nd. Their number fold.

3. In the final result, we count on the right to left such a number of numbers as it turned out in paragraph above, and put the comma.

Rules multiplying decimal fractions.

1. Multiply, not paying attention to the comma.

2. In the work, we separate after the comma, such a number of numbers as they are after commas in both multipliers together.

Multiplying the decimal fraction on the natural number, it is necessary:

1. Multiply numbers, not paying attention to the comma;

2. As a result, we put the comma in such a way that it was so many numbers to the right, as in the decimal fraction.

Multiplying decimal fractions by a column.

Consider on the example:

We write down the decimal fractions in the column and multiply them as natural numbers, not paying attention to the commas. Those. 3.11 we consider as 311, and 0.01 as 1.

The result is 311. Next, we consider the number of signs (numbers) after the comma in both fractions. In the 1st decimal fraction 2 sign and 2 s 2. Total number Figures after commas:

2 + 2 = 4

We count on the right to left four signs from the result. In the final result of numbers less than to separate the comma. In this case, it is necessary to first add a lacking number of zeros.

In our case, it does not reach the 1st digit, so we add left 1 zero on the left.

Note:

Multiplying any decimal fraction at 10, 100, 1000, and so on, the comma in decimal fraction is transferred to the right to so many signs as zeros after a unit.

for example:

70,1 . 10 = 701

0,023 . 100 = 2,3

5,6 . 1 000 = 5 600

Note:

For multiplication of decimal fraction 0.1; 0.01; 0.001; And so on, you need to move the comma to the left for so many signs as zeros in front of the unit.

We consider zero whole!

For example:

12 . 0,1 = 1,2

0,05 . 0,1 = 0,005

1,256 . 0,01 = 0,012 56