To understand how to multiply decimal fractions, let's look at specific examples.

Decimal multiplication rule

1) We multiply, ignoring the comma.

2) As a result, we separate as many digits after the comma as there are after the commas in both factors together.

Examples.

Find the product of decimal fractions:

To multiply decimal fractions, we multiply, ignoring the commas. That is, we are not multiplying 6.8 and 3.4, but 68 and 34. As a result, we separate as many digits after the comma as there are after the commas in both factors together. The first multiplier after the decimal point has one digit, the second - also one. In total, we separate two digits after the decimal point. Thus, we got the final answer: 6.8 ∙ 3.4 = 23.12.

Multiply decimals without taking into account the comma. That is, in fact, instead of multiplying 36.85 by 1.14, we multiply 3685 by 14. We get 51590. Now, in this result, we need to separate as many digits with a comma as there are in both factors together. The first number after the decimal point has two digits, the second - one. In total, we separate three digits with a comma. Since there is a zero at the end of the entry after the decimal point, we do not write it in response: 36.85 ∙ 1.4 = 51.59.

To multiply these decimal fractions, we multiply the numbers, ignoring the commas. That is, we multiply the natural numbers 2315 and 7. We get 16205. In this number, you need to separate four digits after the decimal point - as many as there are in both factors together (two in each). The final answer: 23.15 ∙ 0.07 = 1.6205.

Multiplication of a decimal fraction by a natural number is performed in the same way. We multiply the numbers, not paying attention to the comma, that is, we multiply 75 by 16. In the result, after the comma, there should be as many digits as there are in both factors together - one. Thus, 75 ∙ 1.6 = 120.0 = 120.

We start multiplying decimal fractions by multiplying natural numbers, since we do not pay attention to commas. After that, we separate as many digits after the decimal point as there are in both factors together. In the first number after the decimal point, there are two digits, in the second - also two. In total, as a result, there should be four digits after the decimal point: 4.72 ∙ 5.04 = 23.7888.

In the course of secondary and high school students passed the topic "Fractions". However, this concept is much broader than it is given in the learning process. Today, the concept of a fraction is encountered quite often, and not everyone can carry out calculations of any expression, for example, multiplication of fractions.

What is a fraction?

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

Initially, students are introduced to the concept of share. For example, if you divide a watermelon into 8 parts, then each will get one-eighth of the watermelon. This one part out of eight is called a fraction.

A fraction equal to ½ of any value is called half; ⅓ - third; ¼ - a quarter. Records of the form 5/8, 4/5, 2/4 are called ordinary fractions. A common fraction is divided into a numerator and a denominator. Between them is a fractional line, or fractional line. A slash can be drawn as either a horizontal or an oblique line. In this case, it denotes the division sign.

The denominator represents by how many equal shares the value, the object is divided; and the numerator is how many equal shares are taken. The numerator is written above the fractional line, the denominator below it.

It is most convenient to show ordinary fractions on the coordinate ray. If you divide a unit segment into 4 equal shares, designate each share with a Latin letter, then as a result you can get an excellent visual aid. So, point A shows a fraction equal to 1/4 of the entire unit segment, and point B marks 2/8 of this segment.

Varieties of fractions

Fractions can be ordinary, decimal, and mixed numbers. In addition, fractions can be divided into correct and incorrect. This classification is more suitable for ordinary fractions.

A correct fraction is understood as a number whose numerator is less than the denominator. Accordingly, an improper fraction is a number whose numerator is greater than the denominator. The second type is usually written as mixed number... Such an expression consists of an integer and a fractional part. For example, 1½. 1 - whole part, ½ - fractional. However, if you need to carry out some manipulations with the expression (division or multiplication of fractions, their reduction or transformation), the mixed number is converted into an improper fraction.

A correct fractional expression is always less than one, and an incorrect one is always greater than or equal to 1.

As for that, this expression means a record in which any number is represented, the denominator of a fractional expression of which can be expressed through one with several zeros. If the fraction is correct, then the integer part in decimal notation will be zero.

To write down a decimal fraction, you must first write the whole part, separate it from the fractional part with a comma, and then write down the fractional expression. It must be remembered that after the comma, the numerator must contain the same number of digital characters as there are zeros in the denominator.

Example... Present the fraction 7 21/1000 in decimal notation.

Algorithm for converting an improper fraction to a mixed number and vice versa

It is incorrect to write down the wrong fraction in the answer to the problem, so it must be converted to a mixed number:

• divide the numerator by the existing denominator;
• in a specific example, the incomplete quotient is the whole;
• and the remainder is the numerator of the fractional part, and the denominator remains unchanged.

Example... Convert improper fraction to mixed number: 47/5.

Solution... 47: 5. The incomplete quotient equals 9, the remainder = 2. Hence, 47/5 = 9 2/5.

Sometimes you want to represent a mixed number as an improper fraction. Then you need to use the following algorithm:

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

Example... Provide a mixed number as an improper fraction: 9 8/10.

Solution... 9 x 10 + 8 = 90 + 8 = 98 - numerator.

Answer: 98 / 10.

Multiplication of ordinary fractions

Various algebraic operations can be performed on ordinary fractions. To multiply two numbers, you need to multiply the numerator with the numerator, and the denominator with the denominator. Moreover, the multiplication of fractions with different denominators does not differ from the product fractional numbers with the same denominators.

It happens that after finding the result, you need to cancel the fraction. It is imperative to simplify the resulting expression as much as possible. Of course, one cannot say that an incorrect fraction in an answer is a mistake, but it is also difficult to call it a correct answer.

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

As you can see from the example, after finding the work, you get an abbreviated fractional notation. Both the numerator and the denominator in this case are divided by 4, and the answer is 5/9.

Multiplication of decimal fractions

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

• two decimal fractions must be written under each other so that the rightmost digits are one under the other;
• you need to multiply the written numbers, despite the commas, that is, as natural;
• count the number of digits after the comma in each of the numbers;
• in the result obtained after multiplication, you need to count as many digital symbols from the right as is contained in the sum in both factors after the decimal point, and put a separating sign;
• if there are fewer numbers in the product, then in front of them you need to write as many zeros to cover this amount, put a comma and assign the whole part equal to zero.

Example... Calculate the product of two decimal fractions, 2.25 and 3.6.

Solution.

Multiplication of mixed fractions

To calculate the product of two mixed fractions, you need to use the rule for multiplying fractions:

• Convert mixed numbers to improper fractions;
• find the product of the numerators;
• find the product of the denominators;
• write down the resulting result;
• Simplify the expression as much as possible.

Example... Find the product of 4½ and 6 2/5.

Multiplying a number by a fraction (fractions by a number)

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

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

• write the number under the fraction so that the rightmost digits are one above the other;
• find a work despite the comma;
• in the resulting result, separate the integer part from the fractional part using a comma, counting the number of digits from the right that is after the decimal point in the fraction.

To multiply common fraction by a number, you should find the product of the numerator and the natural factor. If the answer contains a cancellation fraction, it should be converted.

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

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

Answer: 7 1 / 2.

As you can see from the previous example, it was necessary to shorten the resulting result and convert the incorrect fractional expression to a mixed number.

Also, the multiplication of fractions also applies to finding the product of a number in mixed form and a natural factor. To multiply these two numbers, you should multiply the integer part of the mixed factor by a number, multiply the numerator by the same value, and leave the denominator unchanged. If required, you need to simplify the resulting result as much as possible.

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

Solution... 9 5/6 x 9 = 9 x 9 + (5 x 9) / 6 = 81 + 45/6 = 81 + 7 3/6 = 88 1/2.

Answer: 88 1 / 2.

Multiplication by factors of 10, 100, 1000 or 0.1; 0.01; 0.001

The following rule follows from the previous paragraph. To multiply a decimal fraction by 10, 100, 1000, 10000, etc., you need to move the comma to the right by as many digits as there are zeros in the multiplier after one.

Example 1... Find the product of 0.065 and 1000.

Solution... 0.065 x 1000 = 0065 = 65.

Answer: 65.

Example 2... Find the product of 3.9 and 1000.

Solution... 3.9 x 1000 = 3.900 x 1000 = 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 comma to the left in the resulting product by as many digits as there are zeros up to one. If necessary, sufficient zeros are written in front of the natural number.

Example 1... Find the product of 56 and 0.01.

Solution... 56 x 0.01 = 0056 = 0.56.

Answer: 0,56.

Example 2... Find the product of 4 and 0.001.

Solution... 4 x 0.001 = 0004 = 0.004.

Answer: 0,004.

So, finding the product of different fractions should not cause any difficulties, except perhaps calculating the result; in this case, you simply cannot do without a calculator.

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

Within the framework of this material, we will only touch on the rules for multiplying positive fractions. Cases with negative ones are dealt with separately in articles on the multiplication of rational and real numbers.

Let's formulate general principles, which must be adhered to when solving problems on the multiplication of decimal fractions.

To begin with, remember that decimal fractions are nothing more than special form writing of ordinary fractions, therefore, the process of their multiplication can be reduced to the same for ordinary fractions. This rule works for both finite and infinite fractions: after converting them into ordinary fractions, it is easy to perform multiplication with them according to the rules we have already learned.

Let's see how such tasks are solved.

Example 1

Calculate the product of 1, 5 and 0.75.

Solution: first, let's replace the decimal fractions with ordinary ones. We know that 0.75 is 75/100 and 1.5 is 15 10. We can cancel the fraction and select the whole part. We will write the received result 125 1000 as 1, 125.

Answer: 1 , 125 .

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

Example 2

Multiply one periodic fraction 0, (3) by the other 2, (36).

To begin with, we bring the original fractions to ordinary ones. We will get:

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

Therefore, 0, (3) 2, (36) = 1 3 26 11 = 26 33.

The resulting ordinary fraction can be reduced to decimal form by dividing the numerator by the denominator in a column:

Answer: 0, (3) 2, (36) = 0, (78).

If we have infinite non-periodic fractions in the problem statement, then we need to pre-round them (see the article on rounding numbers if you forgot how to do this). After that, you can perform the multiplication action with already rounded decimal fractions. Let's give an example.

Example 3

Calculate the product of 5, 382 ... and 0, 2.

Solution

We have an infinite fraction in our problem, which must first be rounded to the nearest hundredths. It turns out that 5, 382 ... ≈ 5, 38. The second factor does not make sense to round to hundredths. Now you can calculate the desired product and write down the answer: 5, 38 · 0, 2 = 538 100 · 2 10 = 1 076 1000 = 1, 076.

Answer: 5, 382 ... · 0.2 ≈ 1.076.

The column counting method can be used not only for natural numbers. If we have decimal fractions, we can multiply them in exactly the same way. Let's derive the rule:

Definition 1

Multiplication of decimal fractions with a column is performed in 2 steps:

1. We carry out multiplication with a column, not paying attention to commas.

2. We put a decimal point in the final number, separating it as many digits on the right side as both factors contain decimal places together. If, as a result, there are not enough numbers for this, add zeros to the left.

Let's look at examples of such calculations in practice.

Example 4

Multiply decimals 63, 37 and 0, 12 by a column.

Solution

The first step is to multiply the numbers, ignoring the decimal points.

Now we need to put the comma in the right place. It will separate the four digits from the right side, since the sum of the decimal places in both factors is 4. You don't have to add zeros, because enough signs:

Answer: 3.37 0.12 = 7.5044.

Example 5

Calculate how much 3.2601 is multiplied by 0.0254.

Solution

We count without regard to commas. We get the following number:

We will put a comma separating 8 digits from the right side, because the original fractions together have 8 decimal places. But in our result there are only seven digits, and we cannot do without additional zeros:

Answer: 3.601 0 .0254 = 0. 08280654.

How to multiply a decimal by 0.001, 0.01, 01, etc

Decimal fractions are often multiplied by such numbers, so it is important to be able to do it quickly and accurately. Let's write down a special rule that we will use in this multiplication:

Definition 2

If we multiply the decimal fraction by 0, 1, 0, 01, etc., we end up with a number similar to the original fraction, with the comma shifted to the left by the required number of digits. If there are not enough numbers for transfer, you need to add zeros to the left.

So, to multiply 45, 34 by 0, 1, you need to move the comma in the original decimal fraction by one digit. We end up with 4,534.

Example 6

Multiply 9.4 by 0.0001.

Solution

We will have to move the comma by four decimal places according to the number of zeros in the second factor, but the numbers in the first one will not be enough for this. We assign the necessary zeros and get that 9.4 · 0, 0001 = 0, 00094.

Answer: 0 , 00094 .

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

The process of such multiplication is no different then the action of multiplying two decimal fractions. It is convenient to use the column multiplication method if there is a finite decimal fraction in the problem statement. In this case, it is necessary to take into account all those rules that we talked about in the previous paragraph.

Example 7

Calculate how much is 15 2, 27.

Solution

Multiply the original numbers with a column and separate the two decimal places.

Answer: 15 2, 27 = 34, 05.

If we carry out the multiplication of a periodic decimal fraction by a natural number, we must first change the decimal fraction to an ordinary one.

Example 8

Calculate the product of 0, (42) and 22.

Let us bring the periodic fraction to the form of an ordinary one.

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

0, 42 22 = 14 33 22 = 14 22 3 = 28 3 = 9 1 3

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

Answer: 0, (42) 22 = 9, (3).

Infinite fractions must be rounded off before counting.

Example 9

Calculate how much will be 4 · 2, 145….

Solution

Let's round up the original infinite decimal fraction to hundredths. After that, we come to the multiplication of a natural number and a final decimal fraction:

4 · 2, 145 ... ≈ 4 · 2, 15 = 8, 60.

Answer: 4 · 2, 145 ... ≈ 8, 60.

How to multiply a decimal by 1000, 100, 10, etc.

Decimal multiplication by 10, 100, etc. is often encountered in problems, so we will analyze this case separately. The basic rule of multiplication is as follows:

Definition 3

To multiply a decimal fraction by 1000, 100, 10, etc., you need to move its comma by 3, 2, 1 digits depending on the multiplier and discard the extra zeros on the left. If there are not enough digits to carry the comma, add as many zeros to the right as we need.

Let's show with an example how exactly to do this.

Example 10

Multiply 100 and 0.0783.

Solution

To do this, we need to move the decimal point by 2 digits to the right side. We will end up with 007, 83 The zeros on the left can be discarded and the result is written as 7, 38.

Answer: 0.0783 100 = 7.83.

Example 11

Multiply 0.02 by 10 thousand.

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

Answer: 0.02 10,000 = 200.

The rule we have given will work the same in the case of infinite decimal fractions, but here you should be very careful about the period of the final fraction, since it is easy to make a mistake in it.

Example 12

Calculate the product 5, 32 (672) times 1,000.

Solution: first of all, we will write the periodic fraction as 5, 32672672672 ..., so the probability of making a mistake will be less. After that, we can transfer the comma to the required number of characters (three). As a result, we get 5326, 726726 ... Let's put the period in brackets and write the answer as 5 326, (726).

Answer: 5, 32 (672) 1000 = 5 326, (726).

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

To perform this type of multiplication, you need to represent the decimal fraction in the form of an ordinary fraction and then proceed according to the already familiar rules.

Example 13

Multiply 0.4 by 3 5 6

Solution

First, let's convert the decimal fraction to a common one. We have: 0, 4 = 4 10 = 2 5.

We got a mixed number answer. 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, you need to round it to a certain figure and only then multiply.

Example 14

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

Solution

We can represent the second factor as 2 3 = 0, 6666…. Next, let's round off both factors to the thousandth place. After that, we will need to calculate the product of two final decimal fractions 3, 568 and 0, 667. Let's count in a column and get the answer:

The final result must be rounded to thousandths, since it was up to this digit that we rounded the original numbers. We get that 2.379856 ≈ 2.380.

Answer: 3, 5678. ... ... 2 3 ≈ 2, 380

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In the last lesson, we learned how to add and subtract decimal fractions (see the lesson "Adding and subtracting decimal fractions"). At the same time, we appreciated how much easier the calculations are compared to the usual "two-level" fractions.

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

First, let's introduce a new definition. We will meet with him quite often, and not only in this lesson.

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

The digits included in the significant part of the number are called significant digits. They can be repeated and even be equal to zero.

For example, consider several decimal fractions and write out the corresponding significant parts:

1. 91.25 → 9125 (significant digits: 9; 1; 2; 5);
2. 0.008241 → 8241 (significant digits: 8; 2; 4; 1);
3. 15.0075 → 150075 (significant digits: 1; 5; 0; 0; 7; 5);
4. 0.0304 → 304 (significant digits: 3; 0; 4);
5. 3000 → 3 (there is only one significant digit: 3).

Please note: the zeros inside the significant part of the number do not go anywhere. We have already encountered something similar when we learned to convert decimal fractions into ordinary ones (see the lesson "Decimal fractions").

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

Decimal multiplication

The multiplication operation consists of three consecutive steps:

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

Let me remind you once again that zeros on the sides of the significant part are never counted. 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. Let's write out the significant parts for the numbers from this expression: 28 and 125;
2. Their product: 28 · 125 = 3500;
3. In the first factor, the decimal point is shifted by 2 digits to the right (0.28 → 28), and in the second - by 1 more digit. In total, a shift to the left by three digits is needed: 3500 → 3.500 = 3.5.

Now let's deal with the expression 6.3 · 1.08.

1. Let's write out the significant parts: 63 and 108;
2. Their product: 63 · 108 = 6804;
3. Again, two shifts to the right: by 2 and 1 digits, respectively. In total - again 3 digits to the right, so the reverse shift will be 3 digits to the left: 6804 → 6.804. This time there are no zeros at the end.

We got to the third expression: 132.5 · 0.0034.

1. Significant parts: 1325 and 34;
2. Their product: 1325 · 34 = 45,050;
3. In the first fraction, the decimal point goes to the right by 1 digit, and in the second - by whole 4. Total: 5 to the right. Shift 5 to the left: 45,050 →, 45050 = 0.4505. Zero was removed at the end, and added in front, so as not to leave a "bare" decimal point.

The following expression is 0.0108 1600.5.

1. We write the significant parts: 108 and 16 005;
2. We multiply them: 108 16 005 = 1 728 540;
3. We count the numbers after the decimal point: in the first number there are 4, in the second - 1. In total - again 5. We have: 1 728 540 → 17.28540 = 17.2854. At the end, the "extra" zero was removed.

Finally, the last expression: 5.25 · 10,000.

1. Significant parts: 525 and 1;
2. We multiply them: 525 · 1 = 525;
3. The first fraction is shifted 2 digits to the right, and the second is shifted 4 digits to the left (10,000 → 1.0000 = 1). Total 4 - 2 = 2 digits to the left. We perform a reverse shift by 2 digits to the right: 525, → 52,500 (we had to add zeros).

Notice the last example: since the decimal point moves in different directions, the total shift is through the difference. This is a very important point! Here's another example:

Consider the numbers 1.5 and 12,500. We have: 1.5 → 15 (shift by 1 to the right); 12,500 → 125 (shift 2 to the left). We "step" 1 digit to the right, and then 2 to the left. As a result, we stepped 2 - 1 = 1 bit to the left.

Division of decimal fractions

Division is perhaps the most difficult operation. Of course, here you can act by analogy with multiplication: divide the significant parts, and then "move" the decimal point. But in this case, there are many subtleties that negate the potential savings.

Therefore, let's consider a universal algorithm that is slightly longer, but much more reliable:

1. Convert all decimal fractions to common ones. With a little practice, this step will take you a matter of seconds;
2. Divide the resulting fractions in the classical way. In other words, multiply the first fraction by the "inverted" second (see the lesson "Multiplication and division of numeric fractions");
3. If possible, present the result as a decimal again. This step is also fast, because often the denominator is already a power of ten.

Task. Find the meaning 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 count the first expression. First, let's convert the obi fractions to decimal:

Let's do the same with the second expression. The numerator of the first fraction is again factorized:

There is an important point in the third and fourth examples: after getting rid of the decimal notation, cancellable fractions appear. However, we will not be implementing this reduction.

The last example is interesting because the numerator of the second fraction contains a prime number. There is simply nothing to factor out here, so we think ahead:

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

In addition, division often produces "ugly" fractions that cannot be converted to decimal. This is how division differs from multiplication, where the results are always represented in decimal form. Of course, in this case, the last step is again not performed.

Note also the 3rd and 4th examples. In them, we deliberately do not abbreviate ordinary fractions derived from decimals. Otherwise, it will complicate the inverse problem - representing the final answer in decimal form again.

Remember: the basic property of a fraction (like any other rule in mathematics) in itself does not mean that it should be applied everywhere and always, at every opportunity.

You already know that a * 10 = a + a + a + a + a + a + a + a + a + a. For example, 0.2 * 10 = 0.2 + 0.2 + 0.2 + 0.2 + 0.2 + 0.2 + 0.2 + 0.2 + 0.2 + 0.2. It is easy to guess that this sum is equal to 2, i.e. 0.2 * 10 = 2.

Similarly, you can make sure that:

5,2 * 10 = 52 ;

0,27 * 10 = 2,7 ;

1,253 * 10 = 12,53 ;

64,95 * 10 = 649,5 .

You probably guessed that when multiplying a decimal fraction by 10, you need to move the comma to the right by one digit in this fraction.

How do you multiply a decimal by 100?

We have: a * 100 = a * 10 * 10. Then:

2,375 * 100 = 2,375 * 10 * 10 = 23,75 * 10 = 237,5 .

Arguing similarly, we obtain that:

3,2 * 100 = 320 ;

28,431 * 100 = 2843,1 ;

0,57964 * 100 = 57,964 .

Multiply the fraction 7.1212 by 1000.

We have: 7.1212 * 1000 = 7.1212 * 100 * 10 = 712.12 * 10 = 7121.2.

These examples illustrate the following rule.

To multiply a decimal fraction by 10, 100, 1,000, etc., you need to move the comma to the right in this fraction by 1, 2, 3, etc., respectively. numbers.

So, if the comma is moved to the right by 1, 2, 3, etc. digits, then the fraction will increase respectively by 10, 100, 1,000, etc. once.

Hence, if the comma is moved to the left by 1, 2, 3, etc. digits, then the fraction will decrease by 10, 100, 1,000, etc., respectively. once .

Let us show that the decimal form of writing fractions gives the opportunity to multiply them, guided by the rule of multiplying natural numbers.

Let's find, for example, the product 3.4 * 1.23. Let's increase the first factor by 10 times, and the second by 100 times. This means that we have enlarged the work 1000 times.

Therefore, the product of natural numbers 34 and 123 is 1,000 times larger than the desired product.

We have: 34 * 123 = 4182. Then, to get the answer, the number 4 182 must be reduced by 1000 times. We write down: 4 182 = 4 182.0. Moving the comma in the number 4 182.0 three digits to the left, we get the number 4.182, which is 1,000 times less than the number 4 182. Therefore 3.4 * 1.23 = 4.182.

The same result can be obtained using the following rule.

To multiply two decimal fractions, you need:

1) multiply them as natural numbers, ignoring the commas;

2) in the resulting product, separate as many digits with a comma on the right as they are after the commas in both factors together.

In cases where the product contains fewer digits than it is required to separate with a comma, on the left, before this, the product is added the required number of zeros, and then the comma is moved to the left by the required number of digits.

For example, 2 * 3 = 6, then 0.2 * 3 = 0.006; 25 * 33 = 825, then 0.025 * 0.33 = 0.00825.

In cases where one of the factors is 0.1; 0.01; 0.001, etc., it is convenient to use the following rule.

To multiply a decimal by 0.1; 0.01; 0.001, etc., it is necessary in this fraction to move the comma to the left, respectively, by 1, 2, 3, etc. numbers.

For example, 1.58 * 0.1 = 0.158; 324.7 * 0.01 = 3.247.

The properties of multiplication of natural numbers are also fulfilled for fractional numbers:

ab = ba is the displacement property of multiplication,

(ab) c = a (b c) is the combination property of multiplication,

a (b + c) = ab + ac - the distributive property of multiplication with respect to addition.