How to multiply decimals. Actions with decimals Make three examples of multiplying decimals

Like regular numbers.

2. We count the number of decimal places for the 1st decimal fraction and for the 2nd. We add up their number.

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

Rules for multiplying decimals.

1. Multiply without paying attention to the comma.

2. In the product, we separate as many digits after the decimal point as there are after the commas in both factors together.

Multiplying a decimal fraction by a natural number, you must:

1. Multiply numbers, ignoring the comma;

2. As a result, we put a comma so that there are as many digits to the right of it as in a decimal fraction.

Multiplication of decimal fractions by a column.

Let's look at an example:

We write decimal fractions in a column and multiply them as natural numbers, ignoring the commas. Those. We consider 3.11 as 311, and 0.01 as 1.

The result is 311. Next, we count the number of decimal places (digits) for both fractions. There are 2 digits in the 1st decimal and 2 in the 2nd. The total number of digits after the decimal points:

2 + 2 = 4

We count from right to left four characters of the result. In the final result, there are fewer digits than you need to separate with a comma. In this case, it is necessary to add the missing number of zeros on the left.

In our case, the 1st digit is missing, so we add 1 zero on the left.

Note:

Multiplying any decimal fraction by 10, 100, 1000, and so on, the comma in the decimal fraction is moved to the right by as many places as there are zeros after the one.

For example:

70,1 . 10 = 701

0,023 . 100 = 2,3

5,6 . 1 000 = 5 600

Note:

To multiply a decimal by 0.1; 0.01; 0.001; and so on, you need to move the comma to the left in this fraction by as many characters as there are zeros in front of the unit.

We count zero integers!

For example:

12 . 0,1 = 1,2

0,05 . 0,1 = 0,005

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

Let's move on to studying the next action with decimal fractions, now we will comprehensively consider multiplying decimals. First, let's discuss the general principles of multiplying decimal fractions. After that, let's move on to multiplying a decimal fraction by a decimal fraction, show how the multiplication of decimal fractions by a column is performed, consider the solutions of examples. Next, we will analyze the multiplication of decimal fractions by natural numbers, in particular by 10, 100, etc. In conclusion, let's talk about multiplying decimal fractions by ordinary fractions and mixed numbers.

Let's say right away that in this article we will only talk about multiplying positive decimal fractions (see positive and negative numbers). The remaining cases are analyzed in the articles multiplication of rational numbers and multiplication of real numbers.

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General principles for multiplying decimals

Let's discuss the general principles that should be followed when performing multiplication with decimal fractions.

Since finite decimals and infinite periodic fractions are the decimal form of ordinary fractions, the multiplication of such decimal fractions is essentially the multiplication of ordinary fractions. In other words, multiplication of final decimals, multiplication of final and periodic decimal fractions, as well as multiplying periodic decimals comes down to multiplying ordinary fractions after converting decimal fractions to ordinary.

Consider examples of the application of the voiced principle of multiplying decimal fractions.

Example.

Perform the multiplication of decimals 1.5 and 0.75.

Solution.

Let us replace the multiplied decimal fractions with the corresponding ordinary fractions. Since 1.5=15/10 and 0.75=75/100, then . You can reduce the fraction, and then select the whole part from the improper fraction, and it is more convenient to write the resulting ordinary fraction 1 125/1 000 as a decimal fraction 1.125.

Answer:

1.5 0.75=1.125.

It should be noted that it is convenient to multiply the final decimal fractions in a column; we will talk about this method of multiplying decimal fractions in.

Consider an example of multiplying periodic decimal fractions.

Example.

Compute the product of the periodic decimals 0,(3) and 2,(36) .

Solution.

Let's convert periodic decimal fractions to ordinary fractions:

Then . You can convert the resulting ordinary fraction to a decimal fraction:

Answer:

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

If there are infinite non-periodic fractions among the multiplied decimal fractions, then all multiplied fractions, including finite and periodic ones, should be rounded up to a certain digit (see rounding numbers), and then perform the multiplication of the final decimal fractions obtained after rounding.

Example.

Multiply the decimals 5.382… and 0.2.

Solution.

First, we round off an infinite non-periodic decimal fraction, rounding can be done to hundredths, we have 5.382 ... ≈5.38. The final decimal fraction 0.2 does not need to be rounded to hundredths. Thus, 5.382… 0.2≈5.38 0.2. It remains to calculate the product of final decimal fractions: 5.38 0.2 \u003d 538 / 100 2 / 10 \u003d 1,076/1,000 \u003d 1.076.

Answer:

5.382… 0.2≈1.076.

Multiplication of decimal fractions by a column

Multiplication of trailing decimals can be done by a column, similar to column multiplication of natural numbers.

Let's formulate multiplication rule for decimal fractions. To multiply decimal fractions by a column, you need:

• ignoring commas, perform multiplication according to all the rules of multiplication by a column of natural numbers;
• in the resulting number, separate as many digits on the right with a decimal point as there are decimal places in both factors together, and if there are not enough digits in the product, then the required number of zeros must be added on the left.

Consider examples of multiplying decimal fractions by a column.

Example.

Multiply the decimals 63.37 and 0.12.

Solution.

Let's carry out the multiplication of decimal fractions by a column. First, we multiply the numbers, ignoring the commas:

It remains to put a comma in the resulting product. She needs to separate 4 digits on the right, since there are four decimal places in the factors (two in the fraction 3.37 and two in the fraction 0.12). There are enough numbers there, so you don’t have to add zeros on the left. Let's finish the record:

As a result, we have 3.37 0.12 = 7.6044.

Answer:

3.37 0.12=7.6044.

Example.

Calculate the product of decimals 3.2601 and 0.0254 .

Solution.

Having performed multiplication by a column without taking into account commas, we get the following picture:

Now in the product you need to separate 8 digits on the right with a comma, since the total number of decimal places of the multiplied fractions is eight. But there are only 7 digits in the product, therefore, you need to assign as many zeros on the left so that 8 digits can be separated by a comma. In our case, we need to assign two zeros:

This completes the multiplication of decimal fractions by a column.

Answer:

3.2601 0.0254=0.08280654 .

Multiplying decimals by 0.1, 0.01, etc.

Quite often you have to multiply decimals by 0.1, 0.01, and so on. Therefore, it is advisable to formulate a rule for multiplying a decimal fraction by these numbers, which follows from the principles of multiplication of decimal fractions discussed above.

So, multiplying a given decimal by 0.1, 0.01, 0.001, and so on gives a fraction, which is obtained from the original one, if in its entry the comma is moved to the left by 1, 2, 3 and so on digits, respectively, and if there are not enough digits to move the comma, then you need to add the required number of zeros to the left.

For example, to multiply the decimal fraction 54.34 by 0.1, you need to move the decimal point to the left by 1 digit in the fraction 54.34, and you get the fraction 5.434, that is, 54.34 0.1 \u003d 5.434. Let's take another example. Multiply the decimal fraction 9.3 by 0.0001. To do this, we need to move the comma 4 digits to the left in the multiplied decimal fraction 9.3, but the record of the fraction 9.3 does not contain such a number of characters. Therefore, we need to assign as many zeros in the record of the fraction 9.3 on the left so that we can easily transfer the comma to 4 digits, we have 9.3 0.0001 \u003d 0.00093.

Note that the announced rule for multiplying a decimal fraction by 0.1, 0.01, ... is also valid for infinite decimal fractions. For example, 0,(18) 0.01=0.00(18) or 93.938… 0.1=9.3938… .

Multiplying a decimal by a natural number

At its core multiplying decimals by natural numbers is no different from multiplying a decimal by a decimal.

It is most convenient to multiply a finite decimal fraction by a natural number by a column, while you should follow the rules for multiplying by a column of decimal fractions discussed in one of the previous paragraphs.

Example.

Calculate the product 15 2.27 .

Solution.

Let's carry out the multiplication of a natural number by a decimal fraction in a column:

Answer:

15 2.27=34.05.

When multiplying a periodic decimal fraction by a natural number, the periodic fraction should be replaced with an ordinary fraction.

Example.

Multiply the decimal fraction 0,(42) by the natural number 22.

Solution.

First, let's convert the periodic decimal to a common fraction:

Now let's do the multiplication: . This decimal result is 9,(3) .

Answer:

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

And when multiplying an infinite non-periodic decimal fraction by a natural number, you must first round off.

Example.

Do the multiplication 4 2.145….

Solution.

Rounding up to hundredths the original infinite decimal fraction, we will come to the multiplication of a natural number and a final decimal fraction. We have 4 2.145…≈4 2.15=8.60.

Answer:

4 2.145…≈8.60.

Multiplying a decimal by 10, 100, ...

Quite often you have to multiply decimal fractions by 10, 100, ... Therefore, it is advisable to dwell on these cases in detail.

Let's voice rule for multiplying a decimal by 10, 100, 1,000, etc. When multiplying a decimal fraction by 10, 100, ... in its entry, you need to move the comma to the right by 1, 2, 3, ... digits, respectively, and discard extra zeros on the left; if there are not enough digits in the record of the multiplied fraction to transfer the comma, then you need to add the required number of zeros to the right.

Example.

Multiply the decimal 0.0783 by 100.

Solution.

Let's transfer the fraction 0.0783 two digits to the right into the record, and we get 007.83. Dropping two zeros on the left, we get the decimal fraction 7.38. Thus, 0.0783 100=7.83.

Answer:

0.0783 100=7.83.

Example.

Multiply the decimal fraction 0.02 by 10,000.

Solution.

To multiply 0.02 by 10,000 we need to move the comma 4 digits to the right. Obviously, in the record of the fraction 0.02 there are not enough digits to transfer the comma to 4 digits, so we will add a few zeros to the right so that the comma can be transferred. In our example, it is enough to add three zeros, we have 0.02000. After moving the comma, we get the entry 00200.0 . Dropping the zeros on the left, we have the number 200.0, which is equal to the natural number 200, it is the result of multiplying the decimal fraction 0.02 by 10,000.

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, they estimated how much the calculations are simplified compared to the usual “two-story” fractions.

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

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

The significant part of a number is everything between the first and last non-zero digit, including the trailers. We are only talking 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 their corresponding significant parts:

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

Please note: 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 to ordinary ones (see the lesson “ Decimal Fractions”).

This point is so important, and errors 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 a significant part, will proceed, in fact, to the topic of the lesson.

Decimal multiplication

The multiplication operation consists of three consecutive steps:

1. For each fraction, write down the significant part. You will get 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 a column. We get the significant part of the desired fraction;
3. Find out where and by how many digits the decimal point is shifted in the original fractions to obtain the corresponding significant part. Perform reverse shifts on 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 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. 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 multiplier, the decimal point is shifted 2 digits to the right (0.28 → 28), and in the second - by another 1 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 as many as 4. Total: 5 to the right. We perform a shift by 5 to the left: 45050 → .45050 = 0.4505. Zero was removed at the end, and added to the front so as not to leave a “bare” decimal point.

The following expression: 0.0108 1600.5.

1. We write 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 fraction 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).

Pay attention to 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 digits to the left. As a result, we stepped 2 − 1 = 1 digit to the left.

Decimal division

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.

So let's look at a generic algorithm that is a little longer, but much more reliable:

1. Convert all decimals to common fractions. 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 numerical fractions");
3. If possible, return the result as a decimal. This step is also fast, because often the denominator already has a power of ten.

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. First, let's convert obi fractions to decimals:

We do the same with the second expression. The numerator of the first fraction is again decomposed into factors:

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 perform this reduction.

The last example is interesting because the numerator of the second fraction is a prime number. There is simply nothing to factorize here, so we consider it “blank through”:

Sometimes division results in an integer (I'm talking about the last example). In this case, the third step is not performed at all.

In addition, when dividing, “ugly” fractions often appear that cannot be converted to decimals. This is where division differs from multiplication, where the results are always expressed in decimal form. Of course, in this case, the last step is again not performed.

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

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

To understand how to multiply decimals, 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 decimals:

To multiply decimals, we multiply without paying attention to commas. That is, we do not multiply 6.8 and 3.4, but 68 and 34. As a result, we separate as many digits after the decimal point as there are after the commas in both factors together. In the first factor after the decimal point there is one digit, in the second there is also one. In total, we separate two digits after the decimal point. Thus, we got the final answer: 6.8∙3.4=23.12.

Multiplying 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 has two digits after the decimal point, the second has 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 decimals, we multiply the numbers without paying attention to the commas. That is, we multiply the natural numbers 2315 and 7. We get 16205. In this number, four digits must be separated after the decimal point - as many as there are in both factors together (two in each). Final answer: 23.15∙0.07=1.6205.

Multiplying a decimal fraction by a natural number is done in the same way. We multiply the numbers without paying attention to the comma, that is, we multiply 75 by 16. In the result obtained, after the comma there should be as many signs as there are in both factors together - one. Thus, 75∙1.6=120.0=120.

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

In this tutorial, we'll look at each of these operations one by one.

Adding decimals

As we know, a decimal has an integer part and a fractional part. When adding decimals, the integer and fractional parts are added separately.

For example, let's add the decimals 3.2 and 5.3. It is more convenient to add decimal fractions in a column.

First, we write these two fractions in a column, while the integer parts must be under the integer parts, and the fractional ones under the fractional parts. In school, this requirement is called "comma under comma".

Let's write the fractions in a column so that the comma is under the comma:

We begin to add the fractional parts: 2 + 3 \u003d 5. We write down the five in the fractional part of our answer:

Now we add up the integer parts: 3 + 5 = 8. We write the eight in the integer part of our answer:

Now we separate the integer part from the fractional part with a comma. To do this, we again follow the rule "comma under comma":

Got the answer 8.5. So the expression 3.2 + 5.3 is equal to 8.5

In fact, not everything is as simple as it seems at first glance. Here, too, there are pitfalls, which we will now talk about.

Places in decimals

Decimals, like ordinary numbers, have their own digits. These are tenth places, hundredth places, thousandth places. In this case, the digits begin after the decimal point.

The first digit after the decimal point is responsible for the tenths place, the second digit after the decimal point for the hundredths place, the third digit after the decimal point for the thousandths place.

Decimal digits store some useful information. In particular, they report how many tenths, hundredths, and thousandths are in a decimal.

For example, consider the decimal 0.345

The position where the triple is located is called tenth place

The position where the four is located is called hundredths place

The position where the five is located is called thousandths

Let's look at this figure. We see that in the category of tenths there is a three. This suggests that there are three tenths in the decimal fraction 0.345.

If we add the fractions, and then we get the original decimal fraction 0.345

It can be seen that at first we got the answer, but converted it to a decimal fraction and got 0.345.

When adding decimal fractions, the same principles and rules are followed as when adding ordinary numbers. The addition of decimal fractions occurs by digits: tenths are added to tenths, hundredths to hundredths, thousandths to thousandths.

Therefore, when adding decimal fractions, it is required to follow the rule "comma under comma". A comma under a comma provides the same order in which tenths are added to tenths, hundredths to hundredths, thousandths to thousandths.

Example 1 Find the value of the expression 1.5 + 3.4

First of all, we add the fractional parts 5 + 4 = 9. We write the nine in the fractional part of our answer:

Now we add up the integer parts 1 + 3 = 4. We write down the four in the integer part of our answer:

Now we separate the integer part from the fractional part with a comma. To do this, we again observe the rule "comma under a comma":

Got the answer 4.9. So the value of the expression 1.5 + 3.4 is 4.9

Example 2 Find the value of the expression: 3.51 + 1.22

We write this expression in a column, observing the rule "comma under a comma"

First of all, add the fractional part, namely the hundredths 1+2=3. We write the triple in the hundredth part of our answer:

Now add tenths of 5+2=7. We write down the seven in the tenth part of our answer:

Now add the whole parts 3+1=4. We write down the four in the whole part of our answer:

We separate the integer part from the fractional part with a comma, observing the “comma under the comma” rule:

Got the answer 4.73. So the value of the expression 3.51 + 1.22 is 4.73

3,51 + 1,22 = 4,73

As with ordinary numbers, when adding decimal fractions, . In this case, one digit is written in the answer, and the rest are transferred to the next digit.

Example 3 Find the value of the expression 2.65 + 3.27

We write this expression in a column:

Add hundredths of 5+7=12. The number 12 will not fit in the hundredth part of our answer. Therefore, in the hundredth part, we write the number 2, and transfer the unit to the next bit:

Now we add the tenths of 6+2=8 plus the unit that we got from the previous operation, we get 9. We write the number 9 in the tenth of our answer:

Now add the whole parts 2+3=5. We write the number 5 in the integer part of our answer:

Got the answer 5.92. So the value of the expression 2.65 + 3.27 is 5.92

2,65 + 3,27 = 5,92

Example 4 Find the value of the expression 9.5 + 2.8

Write this expression in a column

We add the fractional parts 5 + 8 = 13. The number 13 will not fit in the fractional part of our answer, so we first write down the number 3, and transfer the unit to the next digit, or rather transfer it to the integer part:

Now we add the integer parts 9+2=11 plus the unit that we got from the previous operation, we get 12. We write the number 12 in the integer part of our answer:

Separate the integer part from the fractional part with a comma:

Got the answer 12.3. So the value of the expression 9.5 + 2.8 is 12.3

9,5 + 2,8 = 12,3

When adding decimal fractions, the number of digits after the decimal point in both fractions must be the same. If there are not enough digits, then these places in the fractional part are filled with zeros.

Example 5. Find the value of the expression: 12.725 + 1.7

Before writing this expression in a column, let's make the number of digits after the decimal point in both fractions the same. The decimal fraction 12.725 has three digits after the decimal point, while the fraction 1.7 has only one. So in the fraction 1.7 at the end you need to add two zeros. Then we get the fraction 1,700. Now you can write this expression in a column and start calculating:

Add thousandths of 5+0=5. We write the number 5 in the thousandth part of our answer:

Add hundredths of 2+0=2. We write the number 2 in the hundredth part of our answer:

Add tenths of 7+7=14. The number 14 will not fit in a tenth of our answer. Therefore, we first write down the number 4, and transfer the unit to the next bit:

Now we add the integer parts 12+1=13 plus the unit that we got from the previous operation, we get 14. We write the number 14 in the integer part of our answer:

Separate the integer part from the fractional part with a comma:

Got the answer 14,425. So the value of the expression 12.725+1.700 is 14.425

12,725+ 1,700 = 14,425

Subtraction of decimals

When subtracting decimal fractions, you must follow the same rules as when adding: “a comma under a comma” and “an equal number of digits after a decimal point”.

Example 1 Find the value of the expression 2.5 − 2.2

We write this expression in a column, observing the “comma under comma” rule:

We calculate the fractional part 5−2=3. We write the number 3 in the tenth part of our answer:

Calculate the integer part 2−2=0. We write zero in the integer part of our answer:

Separate the integer part from the fractional part with a comma:

We got the answer 0.3. So the value of the expression 2.5 − 2.2 is equal to 0.3

2,5 − 2,2 = 0,3

Example 2 Find the value of the expression 7.353 - 3.1

This expression has a different number of digits after the decimal point. In the fraction 7.353 there are three digits after the decimal point, and in the fraction 3.1 there is only one. This means that in the fraction 3.1, two zeros must be added at the end to make the number of digits in both fractions the same. Then we get 3,100.

Now you can write this expression in a column and calculate it:

Got the answer 4,253. So the value of the expression 7.353 − 3.1 is 4.253

7,353 — 3,1 = 4,253

As with ordinary numbers, sometimes you will have to borrow one from the adjacent bit if subtraction becomes impossible.

Example 3 Find the value of the expression 3.46 − 2.39

Subtract hundredths of 6−9. From the number 6 do not subtract the number 9. Therefore, you need to take a unit from the adjacent digit. Having borrowed one from the neighboring digit, the number 6 turns into the number 16. Now we can calculate the hundredths of 16−9=7. We write down the seven in the hundredth part of our answer:

Now subtract tenths. Since we took one unit in the category of tenths, the figure that was located there decreased by one unit. In other words, the tenth place is now not the number 4, but the number 3. Let's calculate the tenths of 3−3=0. We write zero in the tenth part of our answer:

Now subtract the integer parts 3−2=1. We write the unit in the integer part of our answer:

Separate the integer part from the fractional part with a comma:

Got the answer 1.07. So the value of the expression 3.46−2.39 is equal to 1.07

3,46−2,39=1,07

Example 4. Find the value of the expression 3−1.2

This example subtracts a decimal from an integer. Let's write this expression in a column so that the integer part of the decimal fraction 1.23 is under the number 3

Now let's make the number of digits after the decimal point the same. To do this, after the number 3, put a comma and add one zero:

Now subtract tenths: 0−2. Do not subtract the number 2 from zero. Therefore, you need to take a unit from the adjacent digit. By borrowing one from the adjacent digit, 0 turns into the number 10. Now you can calculate the tenths of 10−2=8. We write down the eight in the tenth part of our answer:

Now subtract the whole parts. Previously, the number 3 was located in the integer, but we borrowed one unit from it. As a result, it turned into the number 2. Therefore, we subtract 1 from 2. 2−1=1. We write the unit in the integer part of our answer:

Separate the integer part from the fractional part with a comma:

Got the answer 1.8. So the value of the expression 3−1.2 is 1.8

Decimal multiplication

Multiplying decimals is easy and even fun. To multiply decimals, you need to multiply them like regular numbers, ignoring the commas.

Having received the answer, it is necessary to separate the integer part from the fractional part with a comma. To do this, you need to count the number of digits after the decimal point in both fractions, then count the same number of digits on the right in the answer and put a comma.

Example 1 Find the value of the expression 2.5 × 1.5

We multiply these decimal fractions as ordinary numbers, ignoring the commas. To ignore the commas, you can temporarily imagine that they are absent altogether:

We got 375. In this number, it is necessary to separate the whole part from the fractional part with a comma. To do this, you need to count the number of digits after the decimal point in fractions of 2.5 and 1.5. In the first fraction there is one digit after the decimal point, in the second fraction there is also one. A total of two numbers.

We return to the number 375 and begin to move from right to left. We need to count two digits from the right and put a comma:

Got the answer 3.75. So the value of the expression 2.5 × 1.5 is 3.75

2.5 x 1.5 = 3.75

Example 2 Find the value of the expression 12.85 × 2.7

Let's multiply these decimals, ignoring the commas:

We got 34695. In this number, you need to separate the integer part from the fractional part with a comma. To do this, you need to calculate the number of digits after the decimal point in fractions of 12.85 and 2.7. In the fraction 12.85 there are two digits after the decimal point, in the fraction 2.7 there is one digit - a total of three digits.

We return to the number 34695 and begin to move from right to left. We need to count three digits from the right and put a comma:

Got the answer 34,695. So the value of the expression 12.85 × 2.7 is 34.695

12.85 x 2.7 = 34.695

Multiplying a decimal by a regular number

Sometimes there are situations when you need to multiply a decimal fraction by a regular number.

To multiply a decimal and an ordinary number, you need to multiply them, regardless of the comma in the decimal. Having received the answer, it is necessary to separate the integer part from the fractional part with a comma. To do this, you need to count the number of digits after the decimal point in the decimal fraction, then in the answer, count the same number of digits to the right and put a comma.

For example, multiply 2.54 by 2

We multiply the decimal fraction 2.54 by the usual number 2, ignoring the comma:

We got the number 508. In this number, you need to separate the integer part from the fractional part with a comma. To do this, you need to count the number of digits after the decimal point in the fraction 2.54. The fraction 2.54 has two digits after the decimal point.

We return to the number 508 and begin to move from right to left. We need to count two digits from the right and put a comma:

Got the answer 5.08. So the value of the expression 2.54 × 2 is 5.08

2.54 x 2 = 5.08

Multiplying decimals by 10, 100, 1000

Multiplying decimals by 10, 100, or 1000 is done in the same way as multiplying decimals by regular numbers. It is necessary to perform the multiplication, ignoring the comma in the decimal fraction, then in the answer, separate the integer part from the fractional part, counting the same number of digits on the right as there were digits after the decimal point in the decimal fraction.

For example, multiply 2.88 by 10

Let's multiply the decimal fraction 2.88 by 10, ignoring the comma in the decimal fraction:

We got 2880. In this number, you need to separate the whole part from the fractional part with a comma. To do this, you need to count the number of digits after the decimal point in the fraction 2.88. We see that in the fraction 2.88 there are two digits after the decimal point.

We return to the number 2880 and begin to move from right to left. We need to count two digits from the right and put a comma:

Got the answer 28.80. We discard the last zero - we get 28.8. So the value of the expression 2.88 × 10 is 28.8

2.88 x 10 = 28.8

There is a second way to multiply decimal fractions by 10, 100, 1000. This method is much simpler and more convenient. It consists in the fact that the comma in the decimal fraction moves to the right by as many digits as there are zeros in the multiplier.

For example, let's solve the previous example 2.88×10 in this way. Without giving any calculations, we immediately look at the factor 10. We are interested in how many zeros are in it. We see that it has one zero. Now in the fraction 2.88 we move the decimal point to the right by one digit, we get 28.8.

2.88 x 10 = 28.8

Let's try to multiply 2.88 by 100. We immediately look at the factor 100. We are interested in how many zeros are in it. We see that it has two zeros. Now in the fraction 2.88 we move the decimal point to the right by two digits, we get 288

2.88 x 100 = 288

Let's try to multiply 2.88 by 1000. We immediately look at the factor 1000. We are interested in how many zeros are in it. We see that it has three zeros. Now in the fraction 2.88 we move the decimal point to the right by three digits. The third digit is not there, so we add another zero. As a result, we get 2880.

2.88 x 1000 = 2880

Multiplying decimals by 0.1 0.01 and 0.001

Multiplying decimals by 0.1, 0.01, and 0.001 works in the same way as multiplying a decimal by a decimal. It is necessary to multiply fractions like ordinary numbers, and put a comma in the answer, counting as many digits on the right as there are digits after the decimal point in both fractions.

For example, multiply 3.25 by 0.1

We multiply these fractions like ordinary numbers, ignoring the commas:

We got 325. In this number, you need to separate the whole part from the fractional part with a comma. To do this, you need to calculate the number of digits after the decimal point in fractions of 3.25 and 0.1. In the fraction 3.25 there are two digits after the decimal point, in the fraction 0.1 there is one digit. A total of three numbers.

We return to the number 325 and begin to move from right to left. We need to count three digits on the right and put a comma. After counting three digits, we find that the numbers are over. In this case, you need to add one zero and put a comma:

We got the answer 0.325. So the value of the expression 3.25 × 0.1 is 0.325

3.25 x 0.1 = 0.325

There is a second way to multiply decimals by 0.1, 0.01 and 0.001. This method is much easier and more convenient. It consists in the fact that the comma in the decimal fraction moves to the left by as many digits as there are zeros in the multiplier.

For example, let's solve the previous example 3.25 × 0.1 in this way. Without giving any calculations, we immediately look at the factor 0.1. We are interested in how many zeros are in it. We see that it has one zero. Now in the fraction 3.25 we move the decimal point to the left by one digit. Moving the comma one digit to the left, we see that there are no more digits before the three. In this case, add one zero and put a comma. As a result, we get 0.325

3.25 x 0.1 = 0.325

Let's try multiplying 3.25 by 0.01. Immediately look at the multiplier of 0.01. We are interested in how many zeros are in it. We see that it has two zeros. Now in the fraction 3.25 we move the comma to the left by two digits, we get 0.0325

3.25 x 0.01 = 0.0325

Let's try multiplying 3.25 by 0.001. Immediately look at the multiplier of 0.001. We are interested in how many zeros are in it. We see that it has three zeros. Now in the fraction 3.25 we move the decimal point to the left by three digits, we get 0.00325

3.25 × 0.001 = 0.00325

Do not confuse multiplying decimals by 0.1, 0.001 and 0.001 with multiplying by 10, 100, 1000. A common mistake most people make.

When multiplying by 10, 100, 1000, the comma is moved to the right by as many digits as there are zeros in the multiplier.

And when multiplying by 0.1, 0.01 and 0.001, the comma is moved to the left by as many digits as there are zeros in the multiplier.

If at first it is difficult to remember, you can use the first method, in which the multiplication is performed as with ordinary numbers. In the answer, you will need to separate the integer part from the fractional part by counting as many digits on the right as there are digits after the decimal point in both fractions.

Dividing a smaller number by a larger one. Advanced level.

In one of the previous lessons, we said that when dividing a smaller number by a larger one, a fraction is obtained, in the numerator of which is the dividend, and in the denominator is the divisor.

For example, to divide one apple into two, you need to write 1 (one apple) in the numerator, and write 2 (two friends) in the denominator. The result is a fraction. So each friend will get an apple. In other words, half an apple. A fraction is the answer to a problem how to split one apple between two

It turns out that you can solve this problem further if you divide 1 by 2. After all, a fractional bar in any fraction means division, which means that this division is also allowed in a fraction. But how? We are used to the fact that the dividend is always greater than the divisor. And here, on the contrary, the dividend is less than the divisor.

Everything will become clear if we remember that a fraction means crushing, dividing, dividing. This means that the unit can be split into as many parts as you like, and not just into two parts.

When dividing a smaller number by a larger one, a decimal fraction is obtained, in which the integer part will be 0 (zero). The fractional part can be anything.

So, let's divide 1 by 2. Let's solve this example with a corner:

One cannot be divided into two just like that. If you ask a question "how many twos are in one" , then the answer will be 0. Therefore, in private we write 0 and put a comma:

Now, as usual, we multiply the quotient by the divisor to pull out the remainder:

The moment has come when the unit can be split into two parts. To do this, add another zero to the right of the received one:

We got 10. We divide 10 by 2, we get 5. We write down the five in the fractional part of our answer:

Now we take out the last remainder to complete the calculation. Multiply 5 by 2, we get 10

We got the answer 0.5. So the fraction is 0.5

Half an apple can also be written using the decimal fraction 0.5. If we add these two halves (0.5 and 0.5), we again get the original one whole apple:

This point can also be understood if we imagine how 1 cm is divided into two parts. If you divide 1 centimeter into 2 parts, you get 0.5 cm

Example 2 Find the value of expression 4:5

How many fives are in four? Not at all. We write in private 0 and put a comma:

We multiply 0 by 5, we get 0. We write zero under the four. Immediately subtract this zero from the dividend:

Now let's start splitting (dividing) the four into 5 parts. To do this, to the right of 4, we add zero and divide 40 by 5, we get 8. We write the eight in private.

We complete the example by multiplying 8 by 5, and get 40:

We got the answer 0.8. So the value of the expression 4: 5 is 0.8

Example 3 Find the value of expression 5: 125

How many numbers 125 are in five? Not at all. We write 0 in private and put a comma:

We multiply 0 by 5, we get 0. We write 0 under the five. Immediately subtract from the five 0

Now let's start splitting (dividing) the five into 125 parts. To do this, to the right of this five, we write zero:

Divide 50 by 125. How many numbers 125 are in 50? Not at all. So in the quotient we again write 0

We multiply 0 by 125, we get 0. We write this zero under 50. Immediately subtract 0 from 50

Now we divide the number 50 into 125 parts. To do this, to the right of 50, we write another zero:

Divide 500 by 125. How many numbers are 125 in the number 500. In the number 500 there are four numbers 125. We write the four in private:

We complete the example by multiplying 4 by 125, and get 500

We got the answer 0.04. So the value of the expression 5: 125 is 0.04

Division of numbers without a remainder

So, let's put a comma in the quotient after the unit, thereby indicating that the division of integer parts is over and we proceed to the fractional part:

Add zero to the remainder 4

Now we divide 40 by 5, we get 8. We write the eight in private:

40−40=0. Received 0 in the remainder. So the division is completely completed. Dividing 9 by 5 results in a decimal of 1.8:

9: 5 = 1,8

Example 2. Divide 84 by 5 without a remainder

First we divide 84 by 5 as usual with a remainder:

Received in private 16 and 4 more in the balance. Now we divide this remainder by 5. We put a comma in the private, and add 0 to the remainder 4

Now we divide 40 by 5, we get 8. We write the eight in the quotient after the decimal point:

and complete the example by checking if there is still a remainder:

Dividing a decimal by a regular number

A decimal fraction, as we know, consists of an integer and a fractional part. When dividing a decimal fraction by a regular number, first of all you need:

• divide the integer part of the decimal fraction by this number;
• after the integer part is divided, you need to immediately put a comma in the private part and continue the calculation, as in ordinary division.

For example, let's divide 4.8 by 2

Let's write this example as a corner:

Now let's divide the whole part by 2. Four divided by two is two. We write the deuce in private and immediately put a comma:

Now we multiply the quotient by the divisor and see if there is a remainder from the division:

4−4=0. The remainder is zero. We do not write zero yet, since the solution is not completed. Then we continue to calculate, as in ordinary division. Take down 8 and divide it by 2

8: 2 = 4. We write the four in the quotient and immediately multiply it by the divisor:

Got the answer 2.4. Expression value 4.8: 2 equals 2.4

Example 2 Find the value of the expression 8.43:3

We divide 8 by 3, we get 2. Immediately put a comma after the two:

Now we multiply the quotient by the divisor 2 × 3 = 6. We write the six under the eight and find the remainder:

We divide 24 by 3, we get 8. We write the eight in private. We immediately multiply it by the divisor to find the remainder of the division:

24−24=0. The remainder is zero. Zero is not recorded yet. Take the last three of the dividend and divide by 3, we get 1. Immediately multiply 1 by 3 to complete this example:

Got the answer 2.81. So the value of the expression 8.43: 3 is equal to 2.81

Dividing a decimal by a decimal

To divide a decimal fraction into a decimal fraction, in the dividend and in the divisor, move the comma to the right by the same number of digits as there are after the decimal point in the divisor, and then divide by a regular number.

For example, divide 5.95 by 1.7

Let's write this expression as a corner

Now, in the dividend and in the divisor, we move the comma to the right by the same number of digits as there are after the decimal point in the divisor. The divisor has one digit after the decimal point. So we must move the comma to the right by one digit in the dividend and in the divisor. Transferring:

After moving the decimal point to the right by one digit, the decimal fraction 5.95 turned into a fraction 59.5. And the decimal fraction 1.7, after moving the decimal point to the right by one digit, turned into the usual number 17. And we already know how to divide the decimal fraction by the usual number. Further calculation is not difficult:

The comma is moved to the right to facilitate division. This is allowed due to the fact that when multiplying or dividing the dividend and the divisor by the same number, the quotient does not change. What does it mean?

This is one of the interesting features of division. It is called the private property. Consider expression 9: 3 = 3. If in this expression the dividend and the divisor are multiplied or divided by the same number, then the quotient 3 will not change.

Let's multiply the dividend and divisor by 2 and see what happens:

(9 × 2) : (3 × 2) = 18: 6 = 3

As can be seen from the example, the quotient has not changed.

The same thing happens when we carry a comma in the dividend and in the divisor. In the previous example, where we divided 5.91 by 1.7, we moved the comma one digit to the right in the dividend and divisor. After moving the comma, the fraction 5.91 was converted to the fraction 59.1 and the fraction 1.7 was converted to the usual number 17.

In fact, inside this process, multiplication by 10 took place. Here's what it looked like:

5.91 × 10 = 59.1

Therefore, the number of digits after the decimal point in the divisor depends on what the dividend and divisor will be multiplied by. In other words, the number of digits after the decimal point in the divisor will determine how many digits in the dividend and in the divisor the comma will be moved to the right.

Decimal division by 10, 100, 1000

Dividing a decimal by 10, 100, or 1000 is done in the same way as . For example, let's divide 2.1 by 10. Let's solve this example with a corner:

But there is also a second way. It's lighter. The essence of this method is that the comma in the dividend is moved to the left by as many digits as there are zeros in the divisor.

Let's solve the previous example in this way. 2.1: 10. We look at the divider. We are interested in how many zeros are in it. We see that there is one zero. So in the divisible 2.1, you need to move the comma to the left by one digit. We move the comma to the left by one digit and see that there are no more digits left. In this case, we add one more zero before the number. As a result, we get 0.21

Let's try to divide 2.1 by 100. There are two zeros in the number 100. So in the divisible 2.1, you need to move the comma to the left by two digits:

2,1: 100 = 0,021

Let's try to divide 2.1 by 1000. There are three zeros in the number 1000. So in the divisible 2.1, you need to move the comma to the left by three digits:

2,1: 1000 = 0,0021

Decimal division by 0.1, 0.01 and 0.001

Dividing a decimal by 0.1, 0.01, and 0.001 is done in the same way as . In the dividend and in the divisor, you need to move the comma to the right by as many digits as there are after the decimal point in the divisor.

For example, let's divide 6.3 by 0.1. First of all, we move the commas in the dividend and in the divisor to the right by the same number of digits as there are after the decimal point in the divisor. The divisor has one digit after the decimal point. So we move the commas in the dividend and in the divisor to the right by one digit.

After moving the decimal point to the right by one digit, the decimal fraction 6.3 turns into the usual number 63, and the decimal fraction 0.1, after moving the decimal point to the right by one digit, turns into one. And dividing 63 by 1 is very simple:

So the value of the expression 6.3: 0.1 is equal to 63

But there is also a second way. It's lighter. The essence of this method is that the comma in the dividend is transferred to the right by as many digits as there are zeros in the divisor.

Let's solve the previous example in this way. 6.3:0.1. Let's look at the divider. We are interested in how many zeros are in it. We see that there is one zero. So in the divisible 6.3, you need to move the comma to the right by one digit. We move the comma to the right by one digit and get 63

Let's try to divide 6.3 by 0.01. Divisor 0.01 has two zeros. So in the divisible 6.3, you need to move the comma to the right by two digits. But in the dividend there is only one digit after the decimal point. In this case, one more zero must be added at the end. As a result, we get 630

Let's try dividing 6.3 by 0.001. The divisor of 0.001 has three zeros. So in the divisible 6.3, you need to move the comma to the right by three digits:

6,3: 0,001 = 6300

Tasks for independent solution

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