Basic rules for adding decimals. Adding decimals

Adding and subtracting decimals is similar to adding and subtracting natural numbers, but with certain conditions.

Rule. is made by the digits of the integer and fractional parts as natural numbers.

When written adding and subtracting decimals the comma separating the integer part from the fractional part must be in the terms and the sum or the minuend, subtrahend and difference in one column (a comma under a comma from the condition to the end of the calculation).

Adding and subtracting decimals to the line:

243,625 + 24,026 = 200 + 40 + 3 + 0,6 + 0,02 + 0,005 + 20 + 4 + 0,02 + 0,006 = 200 + (40 + 20) + (3 + 4)+ 0,6 + (0,02 + 0,02) + (0,005 + 0,006) = 200 + 60 + 7 + 0,6 + 0,04 + 0,011 = 200 + 60 + 7 + 0,6 + (0,04 + 0,01) + 0,001 = 200 + 60 + 7 + 0,6 + 0,05 + 0,001 = 267,651

843,217 — 700,628 = (800 — 700) + 40 + 3 + (0,2 — 0,6) + (0,01 — 0,02) + (0,007 — 0,008) = 100 + 40 + 2 + (1,2 — 0,6) + (0,01 — 0,02) + (0,007 — 0,008) = 100 + 40 + 2 + 0,5 + (0,11 — 0,02) + (0,007 — 0,008) = 100 + 40 + 2 + 0,5 + 0,09 + (0,007 — 0,008) = 100 + 40 + 2 + 0,5 + 0,08 + (0,017 — 0,008) = 100 + 40 + 2 + 0,5 + 0,08 + 0,009 = 142,589

Adding and subtracting decimals in a column:

Adding decimal fractions requires an upper extra line to write numbers when the sum of the digit goes through a ten. Subtracting decimals requires the top extra line to mark the digit in which the 1 is being borrowed.

If there are not enough digits of the fractional part to the right of the term or reduced, then as many zeros can be added to the right in the fractional part (increase the bit depth of the fractional part) as there are digits in another term or reduced.

Decimal multiplication is performed in the same way as the multiplication of natural numbers, according to the same rules, but in the product a comma is placed according to the sum of the digits of the factors in the fractional part, counting from right to left (the sum of the digits of the factors is the number of digits after the decimal point for the factors taken together).

Example:

At multiplying decimals in a column, the first significant digit on the right is signed under the first significant digit on the right, as in natural numbers:

Recording multiplying decimals in a column:

Recording decimal division in a column:

The underlined characters are comma wrapping characters because the divisor must be an integer.

Rule. At division of fractions the divisor of a decimal fraction increases by as many digits as there are digits in its fractional part. So that the fraction does not change, the dividend increases by the same number of digits (in the dividend and divisor, the comma is transferred to the same number of characters). A comma is placed in the quotient at the stage of division when the whole part of the fraction is divided.

For decimal fractions, as well as for natural numbers, the rule is preserved: You can't divide a decimal by zero!

We study other actions that can be performed with decimal fractions. In this article, we will learn how to correctly calculate the difference between decimal fractions. We will separately analyze the rules for finite and infinite fractions (both periodic and non-periodic), and also see how to count the difference of fractions as a column. In the second part, we will explain how to subtract a decimal from a natural number, a common fraction, a mixed number.

We note in advance that in this article only cases are considered when a smaller fraction is subtracted from a larger one, i.e. the result of this action is positive; other cases refer to finding the difference between rational and real numbers and must be explained separately.

The process of calculating both finite and infinite periodic decimal fractions can be reduced to finding the difference between ordinary fractions. Earlier we talked about how decimal fractions can be written as ordinary fractions. Based on this rule, we will analyze several examples of finding the difference.

Example 1

Find the difference 3.7 - 0.31.

Solution

We rewrite decimal fractions in the form of ordinary ones: 3, 7 \u003d 37 10 and 0, 31 \u003d 31 100.

What to do next, we have already studied. We got the answer, which we translate back into a decimal: 339 100 = 3 , 39 .

It is convenient to make calculations related to decimal fractions in a column. How to use this method? Let's show by solving the problem.

Example 2

Calculate the difference between the periodic fraction 0 , (4) and the periodic decimal fraction 0 , 41 (6) .

Solution

Let's translate the records of periodic fractions into ordinary ones and calculate.

0 , 4 (4) = 0 , 4 + 0 , 004 + . . . = 0 , 4 1 - 0 , 1 = 0 , 4 0 , 9 = 4 9 . 0 , 41 (6) = 0 , 41 + (0 , 006 + 0 , 0006 + . . .) = 41 100 + 0 , 006 0 , 9 = = 41 100 + 6 900 = 41 100 + 1 150 = 123 300 + 2 300 = 125 300 = 5 12

Total: 0 , (4) - 0 , 41 (6) = 4 9 - 5 12 = 16 36 - 15 36 = 1 36

If necessary, we can express the answer as a decimal fraction:

Answer: 0 , (4) − 0 . 41 (6) = 0 . 02 (7) .

We will analyze further how to find the difference if we have infinite non-periodic fractions in the conditions. This case can also be reduced to finding the difference between finite decimals, for which you need to round the infinite fractions to a certain digit (usually the smallest possible).

Example 3

Find the difference 2.77369... - 0.52.

Solution

The second fraction in the condition is finite, and the first is infinite non-periodic. We can round it up to four decimal places: 2.77369 ... ≈ 2.7737. After that, you can subtract: 2, 77369 ... - 0, 52 ≈ 2, 7737 - 0, 52.

Answer: 2, 2537.

Column subtraction is a quick and visual way to find out the difference between final decimals. The counting process is very similar to that for natural numbers.

if in the specified decimal fractions the number of decimal places differs, we equalize it. To do this, add zeros to the desired fraction;
write the fraction to be subtracted under the reduced one, placing the values of the digits strictly under each other, and the comma under the comma;
we will perform the column count in the same way as we do for natural numbers, while ignoring the comma;
in the answer, we separate the required number of numbers with a comma so that it is located in the same place.

Let's look at a specific example of using this method in practice.

Example 4

Find the difference 4452.294 - 10.30501.

Solution

First, let's do the first step - equalize the number of decimal places. Let's add two zeros to the first fraction and get a fraction of the form 4 452 , 29400 , the value of which is identical to the original one.

Let's write the resulting numbers under each other in the right order to get a column:

We count as usual, ignoring commas:

In the resulting answer, put a comma in the right place:

The calculations are over.

Our result: 4452.294 − 10.30501 = 4441.98899.

Finding the difference between a final decimal fraction and a natural number is easiest in the way described above - a column. To do this, the number from which we subtract must be written as a decimal fraction, in the fractional part of which there are zeros.

Example 5

Calculate 15 - 7, 32.

Let's write the reduced number 15 as a fraction 15, 00, since the fraction we need to subtract has two decimal places. Next, we perform the counting in a column, as usual:

So 15 − 7.32 = 7.68.

If we need to subtract an infinite periodic fraction from a natural number, then we again reduce this problem to a similar calculation. We replace the periodic decimal fraction with an ordinary one.

Example 6

Compute the difference 1 - 0 , (6) .

Solution

The periodic decimal fraction specified in the condition corresponds to the usual 2 3 .

We consider: 1 − 0 , (6) = 1 − 2 3 = 1 3 .

The received answer can be translated into a periodic fraction 0 , (3) .

If the fraction given in the condition is non-periodic, we proceed in the same way, having previously rounded it to the desired digit.

Example 7

Subtract 4, 274... from 5.

Solution

We will round the indicated infinite fraction to hundredths and get 4, 274 ... ≈ 4, 27.

After that, we calculate 5 − 4 , 274 ... ≈ 5 − 4 , 27 .

Let's convert 5 to 5, 00 and write down the column:

As a result, 5 − 4.274 ... ≈ 0.73.

If we are faced with the inverse task - to subtract a natural number from a decimal fraction, then we subtract from the integer part of the fraction, and do not touch the fractional part at all. We do this with both finite and infinite fractions.

Example 8

Find the difference 37, 505 - 17.

Solution

We separate the integer part 37 from the fraction and subtract the required number from it. We get 37 , 505 − 17 = 20 , 505 .

This problem also needs to be reduced to the subtraction of ordinary fractions - both in the case of mixed numbers and decimal fractions.

Example 9

Calculate the difference 0 . 25 - 4 5 .

Solution

Let's represent 0, 25 as an ordinary fraction - 0, 25 \u003d 25 100 \u003d 1 4.

Now we need to find the difference between 1 4 and 4 5 .

We consider: 4 5 - 0, 25 \u003d 4 5 - 1 4 \u003d 16 20 - 5 20 \u003d 11 20.

Let's write the answer as a decimal notation: 0, 55.

If the condition contains a mixed number, from which it is necessary to subtract a finite or periodic decimal fraction, then we proceed similarly.

Example 10

Condition: Subtract 0 , (18) from 8 4 11 .

Let's rewrite the periodic fraction in the form of an ordinary fraction. 0 , (18) = 0 , 18 + 0 , 0018 + 0 , 000018 + . . . = 0, 18 1 - 0, 01 = 0, 18 0, 99 = 18 99 = 2 11

It turns out that 8 4 11 - 0 , (18) = 8 4 11 - 2 11 = 8 2 11 .

In decimal form, the answer can be written as 8 , (18) .

We proceed in the same way when we subtract a mixed number or a common fraction from a finite or periodic fraction.

Example 11

Calculate 9 40 - 0.03 .

Solution

We replace the fraction 0.03 with an ordinary 3100.

We get that: 9 40 - 0, 03 = 9 40 - 3 100 = 90 400 - 12 400 = 78 400 = 39 200

The answer can be left as is or converted to decimal 0 , 195 .

If we need to perform a subtraction involving infinite non-periodic fractions, then we will need to reduce them to finite ones. We do the same with mixed numbers. To do this, we write an ordinary fraction or a mixed number as a decimal fraction and round the fraction to be subtracted to a certain digit. Let's illustrate our idea with an example:

Example 12

Subtract 4 , 38475603 ... . out of 10 2 7 .

Solution

Convert the mixed number to an improper fraction.

The result is 10 2 7 - 4 , 38475603 . . . = 10 , (285714) - 4 , 38475603 . . . .

Now let's round the subtracted numbers to the seventh decimal place: 10, (285714) = 10, 285714285714 … ≈ 10, 2857143 and 4, 38475603 … ≈ 4, 3847560

Then 10 , (285714) − 4 , 38475603 … ≈ 10 , 2857143 − 4 , 3847560 .

The only thing left to do is subtract one final decimal from the other. Let's do the column count:

Answer: 10 2 7 - 4, 38475603. . . ≈ 5.9009583

If you notice a mistake in the text, please highlight it and press Ctrl+Enter

The school mathematics course is large enough, so as soon as students get used to adding ordinary fractions and mixed numbers, they need to learn new rules for adding decimal fractions. In order not to relearn, you need to understand the topic once and never make mistakes again.

Types of fractions

There are two major subtypes of fractions:

Ordinary fractions. This includes numbers that are written through a fractional line. These numbers always have a numerator and a denominator.
Decimals. For decimal fractions, the numerator is written into the line, and the denominator can be determined by the position of the comma. The number of decimal places is equal to the power to which you need to raise the number 10 to get the denominator.

There are mixed numbers among both ordinary and decimal fractions. In this case, there cannot be an improper decimal fraction. The notation system is such that the integer part of the decimal fraction is highlighted automatically.

So the denominator of the number 0.17 is the number 100, since the fraction has 2 decimal places. The decimal fraction is called because the denominator is always the power of 10, this is implied by the very system of writing such numbers.

Rules for adding ordinary fractions

To add common fractions, you need to make sure that both numbers have the same denominator.

If ordinary fractions have different denominators, then you cannot add them!

The first step is to bring fractions with different denominators under the same denominator. The next step is to add the numerators. The denominators remain the same. The common denominator of two or more numbers is the LCM of the denominators.

Adding decimals

With decimal fractions, the issue is more complicated. As already mentioned, the denominator is not visible here. It is denoted by a comma. To add two decimals, you need to make sure that both numbers have the same number of decimal places.

For this, a fraction with the largest number of signs is selected, all signs are recalculated. After that, the required number of zeros is assigned to the number with fewer characters on the right. After that, the fractions are added like ordinary numbers, and the comma is moved to the same position.

To add two decimal fractions in a column, write one number under the other so that the comma is under the comma. After such an addition, the sign will not move to another place, and you will not make a mistake.

Consider a small example of adding decimals:

0.12 + 0.1258 - the largest number of decimal places 4. So, to solve the example, you need to write it like this:

0.1200 + 0.1258 - in order not to confuse the position of the comma in the result, you can use a trick and take out the common factor

0.1200+0.1258=0.0001*(1200+1258)=0.0001*2458=0.2458 - You don't have to use this trick. When calculating in a column, there should be no error. But this trick will help you correctly add decimal fractions to a string.

What have we learned?

We talked about the differences in adding decimals and common fractions. They told how to correctly add decimal fractions in a column and in a row. They also gave an example where they considered a little trick to simplify the calculation.

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Adding decimals produced according to the rules of addition in a column.

Decimal fractions are added in a column, like natural numbers, without paying attention to commas.

In the final result, a comma is placed under the commas, as in the original fractions.

Note! If the initial decimal fractions have a different number of decimal places (digits), then the required number of zeros must be added to the fraction in which the number of decimal places is less in order to equalize the number of decimal places in the fractions.

Consider an example. Determine the sum of decimals:

0,678 + 13,7 =

Equalize the number of decimal places in decimal fractions. Add 2 zeros to the right of the decimal 13,7 :

0,678 + 13,700 =

Write down the answer:

0,678 + 13,7 = 14,378

Basic rules for adding decimals:

Equalize the number of decimal places.
Write the decimal fractions under each other so that the commas are under each other.
Perform the addition of decimal fractions, ignoring the commas, according to the rules of addition in a column of natural numbers.
Put a comma under the commas in your answer.

In written addition and subtraction of decimal fractions, the comma, which separates the integer part from the fractional part, must be located at the terms and the sum in one column (a comma under the comma from the condition to the end of the calculation).

For example.Adding decimals to a string:

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

Lesson content

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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