Adding positive decimals examples. Addition and subtraction of decimal fractions. Adding a natural number with a decimal fraction

In this tutorial, we will look at each of these operations separately.

Lesson content

Adding Decimals

As we know, the decimal fraction has a whole and a fractional part. When adding decimal fractions, whole and fractional parts are added separately.

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

Let's first write these two fractions in a column, while the whole parts must necessarily be under the whole, and the fractional part under the fractional. At school this requirement is called Comma under comma.

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

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

Now we add the whole parts: 3 + 5 \u003d 8. We write the eight in the whole part of our answer:

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

The answer was 8.5. So expressions 3.2 + 5.3 is 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.

Decimal places

Decimal fractions, like ordinary numbers, have their places. These are tenths, hundredths, thousandths. In this case, the digits begin after the decimal point.

The first digit after the decimal point is responsible for the tenth place, the second digit after the decimal point for the hundredth place, the third digit after the decimal point for the thousandth place.

The decimal places hold some useful information. In particular, they report how many tenths, hundredths and thousandths are in decimal fraction.

For example, consider the decimal 0.345

The position where the triplet is located is called in tenths

The position where the four is located is called hundredths

The position where the five is located is called thousandths

Let's take a look at this figure. We see that in the tenth place there is a three. This suggests that the decimal 0.345 contains three tenths.

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

It can be seen that at first we received 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. Decimal fractions are added in digits: tenths are added with tenths, hundredths with hundredths, thousandths with thousandths.

Therefore, when adding decimal fractions, you must observe the rule Comma under comma... The comma below the 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, add the fractional parts 5 + 4 \u003d 9. Write the nine in the fractional part of our answer:

Now we add the whole parts 1 + 3 \u003d 4. We write the four in the whole part of our answer:

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

The answer was 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 down this expression in a column, observing the "comma under the comma" rule

First of all, add the fractional part, namely the hundredths 1 + 2 \u003d 3. We write the three in the hundredth part of our answer:

Now add the tenths 5 + 2 \u003d 7. We write the seven in the tenth part of our answer:

Now add the whole parts 3 + 1 \u003d 4. We write a four in the whole part of our answer:

Separate the whole part from the fractional part with a comma, observing the “comma under the comma” rule:

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

3,51 + 1,22 = 4,73

As with normal numbers, adding decimal fractions can occur. 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 5 + 7 \u003d 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 we transfer the unit to the next digit:

Now we add the tenths 6 + 2 \u003d 8 plus the one that came from the previous operation, we get 9. We write the number 9 in the tenth part of our answer:

Now add the whole parts 2 + 3 \u003d 5. We write the number 5 in the whole part of our answer:

The answer was 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

We write this expression in a column

We add the fractional parts 5 + 8 \u003d 13. The number 13 will not fit into the fractional part of our answer, so first we write down the number 3, and we transfer the unit to the next digit, or rather we transfer it to the whole part:

Now we add the whole parts 9 + 2 \u003d 11 plus the one that came from the previous operation, we get 12. We write the number 12 in the integer part of our answer:

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

The answer was 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 numbers, 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 down this expression in a column, let's make the number of digits after the decimal point in both fractions the same. There are three digits in the decimal fraction 12.725 after the decimal point, and in the fraction 1.7 there is 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 down this expression in a column and start calculating:

Add the thousandths 5 + 0 \u003d 5. We write down the number 5 in the thousandth part of our answer:

Add the hundredths 2 + 0 \u003d 2. We write down the number 2 in the hundredth part of our answer:

Add tenths 7 + 7 \u003d 14. The number 14 will not fit in a tenth of our answer. Therefore, first we write down the number 4, and transfer the unit to the next digit:

Now we add the whole parts 12 + 1 \u003d 13 plus the one that got from the previous operation, we get 14. We write the number 14 in the whole part of our answer:

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

The answer was 14.425. So the value of the expression 12.725 + 1.700 is equal to 14.425

12,725+ 1,700 = 14,425

Subtracting decimal fractions

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

Example 1. Find the value of expression 2.5 - 2.2

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

Evaluate the fractional part 5−2 \u003d 3. We write the number 3 in the tenth part of our answer:

Evaluate the integer part 2−2 \u003d 0. We write zero in the integer part of our answer:

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

The answer was 0.3. So the value of the expression 2.5 - 2.2 is 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. There are three digits after the decimal point in the fraction 7.353, and in the fraction 3.1 there is only one. This means that in the fraction 3.1 at the end, you need to add two zeros 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:

The answer was 4.253. So the value of the expression 7.353 - 3.1 is equal to 4.253

7,353 — 3,1 = 4,253

As with ordinary numbers, sometimes you have to occupy one from the adjacent digit if subtraction becomes impossible.

Example 3. Find the value of expression 3.46 - 2.39

Subtract the hundredths of 6-9. From the number 6, do not subtract the number 9. Therefore, you need to take one from the adjacent bit. Having taken one from the adjacent bit, the number 6 turns into the number 16. Now you can calculate the hundredths of 16-9 \u003d 7. We write the seven in the hundredth part of our answer:

Now let's subtract tenths. Since we occupied one unit in the tenth place, the figure that was located there decreased by one unit. In other words, in the tenth place is now not the number 4, but the number 3. Let's calculate the tenths of 3−3 \u003d 0. We write zero in the tenth part of our answer:

Now we subtract the whole parts 3−2 \u003d 1. We write one in the integer part of our answer:

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

The answer was 1.07. So the value of expression 3.46−2.39 is 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. We 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 we subtract the tenths: 0−2. You cannot subtract the number 2 from zero. Therefore, you need to take one from the adjacent bit. Taking one from the adjacent bit, 0 becomes 10. Now we can calculate the tenths of 10−2 \u003d 8. We write the eight in the tenth part of our answer:

Now we subtract whole parts. Previously, the integer contained the number 3, but we borrowed one unit from it. As a result, it turned into number 2. Therefore, subtract 1.2 from 2. 2−1 \u003d 1. We write one in the integer part of our answer:

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

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

Decimal multiplication

Decimal multiplication is easy and even fun. To multiply decimal fractions, you multiply them like ordinary numbers, ignoring the commas.

Having received the answer, 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 both fractions, then in the answer, count the same number of digits from the right and put a comma.

Example 1. Find the value of the expression 2.5 × 1.5

Let's multiply these decimal fractions as usual numbers, ignoring the commas. In order not to pay attention to the commas, you can imagine for a while that they are absent at all:

Received 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 2.5 and 1.5. In the first fraction after the decimal point there is one digit, in the second fraction there is also one. There are two digits in total.

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

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

2.5 x 1.5 \u003d 3.75

Example 2. Find the value of the expression 12.85 × 2.7

Let's multiply these decimal fractions, ignoring the commas:

Received 34695. 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 fractions 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 start moving from right to left. We need to count three digits to the right and put a comma:

The answer was 34.695. So the value of the expression 12.85 × 2.7 is 34.695

12.85 × 2.7 \u003d 34.695

Decimal multiplication by an ordinary number

Sometimes situations arise when you need to multiply a decimal fraction by an ordinary number.

To multiply a decimal fraction and an ordinary number, you need to multiply them, ignoring the comma in the decimal fraction. Having received the answer, 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 the decimal fraction, then in the answer, count the same number of digits on 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 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 2.54. There are two digits after the decimal point in the fraction 2.54.

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

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

2.54 x 2 \u003d 5.08

Decimal multiplication by 10, 100, 1000

Multiplying decimal fractions by 10, 100, or 1000 is done in the same way as multiplying decimal fractions by regular numbers. You need to perform multiplication, not paying attention to the comma in the decimal fraction, then in the answer separate the whole part from the fractional part, counting as many digits to the right as there were digits after the decimal point in the decimal fraction.

For example, multiply 2.88 by 10

Multiply the decimal 2.88 by 10, ignoring the decimal point:

Received 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 there are two digits after the decimal point in the fraction 2.88.

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

The answer was 28.80. If we drop the last zero, we get 28.8. So the value of the expression 2.88 × 10 is 28.8

2.88 x 10 \u003d 28.8

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

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

2.88 x 10 \u003d 28.8

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

2.88 × 100 \u003d 288

Let's try to multiply 2.88 by 1000. Immediately look at the multiplier 1000. We are interested in how many zeros there are. We see that there are three zeros in it. Now in the fraction 2.88, move the comma 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 × 1000 \u003d 2880

Decimal multiplication by 0.1 0.01 and 0.001

Multiplying decimal fractions by 0.1, 0.01, and 0.001 works in the same way as multiplying a decimal fraction by a decimal fraction. It is necessary to multiply fractions like ordinary numbers, and put a comma in the answer, counting as many digits to 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:

Received 325. 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 fractions 3.25 and 0.1. There are two digits after the decimal point in the fraction 3.25, in the fraction 0.1 there is one digit. There are three numbers in total.

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

The answer was 0.325. So the value of the expression 3.25 × 0.1 is equal to 0.325

3.25 × 0.1 \u003d 0.325

There is also a second way to multiply decimal fractions 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 is moved to the left by as many digits as there are zeros in the multiplier.

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

3.25 × 0.1 \u003d 0.325

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

3.25 × 0.01 \u003d 0.0325

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

3.25 × 0.001 \u003d 0.00325

Multiplying decimal fractions by 0.1, 0.001 and 0.001 should not be confused with multiplying by 10, 100, 1000. This is a typical mistake most people make.

When multiplying by 10, 100, 1000, the comma is shifted to the right by the same number of digits as there are zeros in the factor.

And when multiplied by 0.1, 0.01 and 0.001, the comma is transferred to the left by the same number of digits as 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, counting as many digits from the right as the 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 you divide a smaller number by a larger one, you get a fraction, in the numerator of which is the dividend and in the denominator - the divisor.

For example, to divide one apple by two, you need to write 1 (one apple) in the numerator and 2 (two friends) in the denominator. As a result, we get a fraction. So each friend will get an apple. In other words, half an apple. Fraction is the answer to the problem "How to split one apple for 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, and therefore this division is 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 fraction means division, division, division. This means that a 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, you get a decimal fraction, in which the integer part will be 0 (zero). The fractional part can be any.

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

One cannot simply be divided into two completely. If you ask a question "How many twos are in one" , then the answer will be 0. Therefore, in the quotient we write 0 and put a comma:

Now, as usual, we multiply the quotient by the divisor in order 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 resulting one:

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

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

The answer was 0.5. So the fraction is 0.5

Half an apple can also be written using a decimal fraction of 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 you 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 the expression 4: 5

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

Multiply 0 by 5, we get 0. Write zero under the four. We 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, add zero and divide 40 by 5, we get 8. We write the eight in the quotient.

Finishing the example by multiplying 8 by 5 to get 40:

The answer was 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 the five? Not at all. We write 0 in the quotient and put a comma:

Multiply 0 by 5, we get 0. Write 0 under the five. Immediately subtract 0 from the five

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

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

Multiply 0 by 125, we get 0. Write this zero under 50. Immediately subtract 0 from 50

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

Divide 500 by 125. How many numbers 125 are in the number 500. There are four numbers 125 in the number 500. Write the four in the quotient:

Finish the example by multiplying 4 by 125 to get 500

The answer was 0.04. So the value of the expression 5: 125 is 0.04

Division of numbers without remainder

So, we put a comma in the quotient after the one, thereby indicating that the division of the whole parts is over and we proceed to the fractional part:

Add zero to remainder 4

Now we divide 40 by 5, we get 8. Write the eight in the quotient:

40-40 \u003d 0. Got 0 in the remainder. This means that the division is completely completed. Dividing 9 by 5 gives the decimal 1.8:

9: 5 = 1,8

Example 2... Divide 84 by 5 without a remainder

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

Received in private 16 and 4 more in the remainder. Now divide this remainder by 5. Put a comma in the quotient, and add 0 to the remainder 4

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

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

Division of a decimal by a regular number

The decimal fraction, as we know, consists of an integer and a fractional part. When dividing a decimal fraction by an ordinary number, you first need to:

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

For example, divide 4.8 by 2

Let's write this example in a corner:

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

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

4−4 \u003d 0. The remainder is zero. We do not write down zero yet, since the solution is not complete. Then we continue to calculate as in ordinary division. Take down 8 and divide it by 2

8: 2 \u003d 4. We write the four in the quotient and immediately multiply it by the divisor:

The answer was 2.4. The expression value 4.8: 2 is 2.4

Example 2. Find the value of the expression 8.43: 3

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

Now we multiply the quotient by the divisor 2 × 3 \u003d 6. Write the six under the eight and find the remainder:

Divide 24 by 3, we get 8. We write the eight in the quotient. Immediately multiply it by the divisor to find the remainder of the division:

24-24 \u003d 0. The remainder is zero. We do not write down zero yet. Dividing the last 3 from the dividend and dividing by 3, we get 1. Immediately multiply 1 by 3 to complete this example:

The answer was 2.81. So the value of the expression 8.43: 3 is 2.81

Division of a decimal by a decimal

To divide a decimal fraction by a decimal fraction, it is necessary in the dividend and in the divisor to 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 an ordinary number.

For example, divide 5.95 by 1.7

Let's write this expression in a corner

Now, in the dividend and in the divisor, move the comma to the right by the same number of digits as there are after the comma in the divisor. There is one digit after the decimal point. So we have to move the comma to the right by one digit in the dividend and in the divisor. We transfer:

After moving the comma to the right one digit, the decimal fraction 5.95 turned into a fraction 59.5. And the decimal fraction 1.7 after transferring the comma 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 wrapped 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 property of the quotient. Consider the expression 9: 3 \u003d 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) \u003d 18: 6 \u003d 3

As you can see from the example, the quotient has not changed.

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

In fact, inside this process, there was a multiplication by 10. This is how it looked:

5.91 x 10 \u003d 59.1

Therefore, the number of digits after the decimal point in the divisor depends on what the dividend and the 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.

Dividing a decimal 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 is lighter. The essence of this method is that the comma in the dividend is shifted 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 divisor. We are interested in how many zeros there are in it. We see that there is one zero. So in the dividend 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 numbers left. In this case, 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 100. So in the dividend 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 1000. This means that in the dividend 2,1 it is necessary to move the comma to the left by three digits:

2,1: 1000 = 0,0021

Division of a decimal by 0.1, 0.01, and 0.001

Dividing a decimal fraction by 0.1, 0.01, and 0.001 is done in the same way as. In the dividend and in the divisor, the comma must be moved to the right by as many digits as there are after the comma in the divisor.

For example, divide 6.3 by 0.1. First of all, move the commas in the dividend and in the divisor to the right by the same number of digits as there are after the comma in the divisor. There is one digit after the decimal point. So we transfer commas in the dividend and in the divisor to the right by one digit.

After moving the comma 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 comma to the right one digit turns into one. And dividing 63 by 1 is very simple:

So the value of the expression 6.3: 0.1 is 63

But there is also a second way. It is lighter. The essence of this method is that the comma in the dividend is shifted 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. We look at the divider. We are interested in how many zeros there are in it. We see that there is one zero. This means that in the dividend 6,3, you need to move the comma to the right by one digit. Move the comma to the right one digit and get 63

Let's try to divide 6.3 by 0.01. The divisor 0.01 has two zeros. This means that in the dividend 6,3 it is necessary to move the comma to the right by two digits. But there is only one digit after the comma in the dividend. In this case, one more zero must be added at the end. As a result, we get 630

Let's try to divide 6.3 by 0.001. The divisor 0.001 has three zeros. This means that in the dividend of 6.3, you need to move the comma to the right by three digits:

6,3: 0,001 = 6300

Self-help assignments

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Decimal addition can be done in two ways:

Present decimal fractions as fractions and add them.
Perform column addition of decimal fractions.

Addition by conversion to fractions

When adding decimal fractions by converting them into ordinary fractions, the following rule should be followed:

You need to compare the number of decimal places for decimal fractions.
If the number of decimal places is the same, then we convert the decimal fractions into ordinary ones and add them.
If the number of decimal places is different, then first you need to equalize their number by assigning the required number of zeros to the decimal fraction with fewer digits on the right.

Example 1... Add numbers 3.1 and 4.7.

Decision... Since the number of decimal places is the same, we simply convert the decimal fractions into ordinary ones and add them. The decimal fraction 3.1 corresponds to an ordinary fraction, and the decimal fraction 4.7 corresponds to an ordinary fraction, which means:

Example 2... Add the numbers 3.45 and 7.368.

Decision... Since the number of decimal places is different, we first equalize their number by assigning the digit 0 to the fraction 3.45 on the right. A decimal fraction 3.450 corresponds to an ordinary fraction, and a decimal fraction 7.368 is an ordinary fraction, which means:

Column addition of decimals

Decimal fractions can be added in columns.

When adding decimal fractions in a column, the following rule should be followed:

Write the decimal fractions in a column so that the numbers of the same digits are one under the other. The commas of the decimal fractions must also appear underneath each other.
If the number of decimal places for fractions is different, for convenience, you can equalize their number by assigning the required number of zeros to the decimal fraction with fewer decimal places on the right.
Ignoring the commas, perform the addition as you would add a column of natural numbers.
In the resulting sum, put a comma so that it is under the commas of the terms.

Example 1... Add 3.1 and 4.7.

Decision... We carry out the addition in the same way as the addition of a column of natural numbers is performed, not paying attention to the commas:

Example 2... Add 3.45 and 7.368.

Decision... We perform the addition in the same way as the addition of a column of natural numbers is performed. For convenience, you can equalize the number of decimal places in the added fractions:

Adding a decimal to a natural number

The rule for adding decimal fractions with natural numbers:

To add a decimal fraction and a natural number, you need to add this natural number to the whole part of the decimal fraction, and leave the fractional part unchanged.

Example... Calculate the sum of 14.3 and 29.

Decision... For the convenience of addition, any natural number can be represented as a decimal fraction. To do this, you need to put a comma after the ones place and add the required number of zeros after the decimal point. The addition is performed according to the rule for adding decimal fractions in a column:

Adding a Decimal to a Fraction

The rule for adding decimal fractions with a common fraction.

Like addition, subtraction of decimal fractions depends on how the numbers are written correctly.

Decimal subtraction rule

1) COMMA COMMA!

This part of the rule is the most important. When subtracting decimal fractions, they should be written so that the commas of the reduced and subtracted are strictly one below the other.

2) Equalize the number of digits after the decimal point. To do this, including where the number of digits after the decimal point is less, we add zeros after the decimal point.

3) Subtract the numbers without paying attention to the comma.

4) Remove the comma under the commas.

Examples for subtracting decimal fractions.

To find the difference between decimal fractions 9.7 and 3.5, we write them down so that the commas in both numbers are strictly one below the other. Then we subtract, ignoring the comma. In the resulting result, we demolish the comma, that is, we write under the commas of the reduced and subtracted:

2) 23,45 — 1,5

To subtract another from one decimal fraction, you need to write them down so that the commas are located exactly one below the other. Since 23.45 after the decimal point has two digits, and 1.5 has only one, we add zero to 1.5. After that, we carry out subtractions, not paying attention to the comma. As a result, remove the comma under the commas:

23,45 — 1,5=21,95.

We start subtracting decimal fractions by writing them so that the commas are located exactly one under one. In the first number after the decimal point, there is one digit, in the second - three, so we write zeros in place of the missing two digits in the first number. Then we subtract the numbers, ignoring the comma. In the resulting result, remove the comma under the commas:

63,5-8,921=54,579.

4) 2,8703 — 0,507

To subtract these decimal fractions, write them down so that the comma of the second number is located exactly below the comma of the first. The first number after the decimal point has four digits, the second - three, so we supplement the second number after the decimal point with a zero at the end. After that, we subtract these numbers as usual natural numbers, excluding the comma. In the resulting result, write a comma under the commas:

2,8703 — 0,507 = 2,3663.

5) 35,46 — 7,372

We start subtracting decimal fractions by writing numbers in such a way that the commas are one below the other. We supplement the first number with a zero after the decimal point, so that both fractions after the decimal point have three digits. Then we subtract, ignoring the comma. In the answer, remove the comma under the commas:

35,46 — 7,372 = 28,088.

In order to subtract a decimal fraction from a natural number, put a comma in its entry at the end and assign the required number of zeros after the decimal point. Why subtract without taking into account the comma. In response, we remove the comma exactly under the commas:

45 — 7,303 = 37,698.

7) 17,256 — 4,756

We perform this example for subtracting decimal fractions in the same way. As a result, we got a number with zeros after the decimal point at the end. We do not write them in the answer: 17.256 - 4.756 \u003d 12.5.

In this article, we will focus on subtracting decimal fractions... Here we will look at the rules for subtracting finite decimal fractions, dwell on subtraction of decimal fractions in a column, and also consider how the subtraction of infinite periodic and non-periodic decimal fractions is carried out. Finally, let's talk about subtracting decimal fractions from natural numbers, fractions and mixed numbers, and subtracting natural numbers, fractions and mixed numbers from decimals.

We'll say right away that here we will only consider the subtraction of a smaller decimal fraction from a larger decimal fraction, we will analyze other cases in the articles subtraction of rational numbers and subtraction of real numbers.

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General principles of subtracting decimal fractions

At its core subtraction of trailing decimals and infinite periodic decimals represents the subtraction of the corresponding fractions. Indeed, the indicated decimal fractions are the decimal notation of ordinary fractions, as described in the article on the conversion of ordinary fractions to decimal fractions and vice versa.

Let's consider examples of subtraction of decimal fractions, starting from the stated principle.

Example.

Subtract from the decimal 3.7 decimal 0.31.

Decision.

Since 3.7 \u003d 37/10 and 0.31 \u003d 31/100, then. So the subtraction of decimal fractions was reduced to the subtraction of ordinary fractions with different denominators:. We represent the resulting fraction as a decimal fraction: 339/100 \u003d 3.39.

Answer:

3,7−0,31=3,39 .

Note that it is convenient to subtract final decimal fractions in a column, we will talk about this method in.

Now let's look at an example of subtracting periodic decimal fractions.

Example.

Subtract 0, (4) from the periodic decimal fraction 0.41 (6).

Decision.

Answer:

0,(4)−0,41(6)=0,02(7) .

It remains to voice principle of subtraction of infinite non-periodic fractions.

Subtracting infinite non-periodic fractions is reduced to subtracting trailing decimal fractions. To do this, subtracted infinite decimal fractions are rounded to a certain digit, usually to the lowest possible (see rounding numbers).

Example.

Subtract the final decimal 0.52 from the infinite non-periodic decimal 2.77369….

Decision.

Let's round the infinite non-periodic decimal fraction to 4 decimal places, we have 2.77369 ... ≈2.7737. Thus, 2,77369…−0,52≈2,7737−0,52 ... By calculating the difference between the final decimal fractions, we get 2.2537.

Answer:

2,77369…−0,52≈2,2537 .

Column subtraction of decimals

A very convenient way to subtract trailing decimals is column subtraction. Column subtraction of decimals is very similar to column subtraction of natural numbers.

To execute column subtraction of decimals, need to:

equalize the number of decimal places in the notation of decimal fractions (if it differs, of course) by adding a number of zeros to one of the fractions on the right;
write the subtracted under the reduced so that the digits of the corresponding digits are under each other, and the comma is under the comma;
perform subtraction in a column, ignoring the commas;
in the resulting difference, put a comma so that it is located under the commas of the reduced and subtracted.

Consider an example of column subtraction of decimal fractions.

Example.

Subtract the decimal 10.30501 from the decimal 4 452.294.

Decision.

Obviously, the number of decimal places in fractions is different. Let's equalize it by adding two zeros to the right in the fraction 4 452.294, and we get the decimal fraction equal to it 4 452.29400.

Now let's write the subtracted under the decrement, as the method of subtracting decimal fractions in a column suggests:

We carry out the subtraction, ignoring the commas:

It remains only to put a decimal point in the resulting difference:

At this stage, the record has taken on a finished form, and the subtraction of decimal fractions in a column is completed. The result is the following.

Answer:

4 452,294−10,30501=4 441,98899 .

Subtracting a decimal from a natural number and vice versa

Subtracting a final decimal from a natural number It is most convenient to do it in a column, writing down the reduced natural number as a decimal fraction with zeros in the fractional part. Let's deal with this when solving an example.

Example.

Subtract the decimal fraction 7.32 from the natural number 15.

Decision.

We represent the natural number 15 as a decimal fraction, adding two digits 0 after the decimal point (since the decimal fraction to be subtracted has two digits in the fractional part), we have 15.00.

Now let's perform column subtraction of decimal fractions:

As a result, we get 15-7.32 \u003d 7.68.

Answer:

15−7,32=7,68 .

Subtracting an infinite periodic decimal from a natural number can be reduced to subtracting an ordinary fraction from a natural number. To do this, replace the periodic decimal fraction with the corresponding ordinary fraction.

Example.

Subtract the periodic decimal 0, (6) from the natural number 1.

Decision.

Periodic decimal fraction 0, (6) corresponds to the ordinary fraction 2/3. Thus, 1−0, (6) \u003d 1−2 / 3 \u003d 1/3. The resulting ordinary fraction can be written as a decimal fraction 0, (3).

Answer:

1−0,(6)=0,(3) .

Subtracting an infinite non-periodic decimal fraction from a natural number is reduced to subtracting the final decimal fraction. To do this, an infinite non-periodic decimal fraction must be rounded to a certain digit.

Example.

Subtract the infinite non-periodic decimal fraction 4,274… from the natural number 5.

Decision.

First, round off an infinite decimal fraction, we can round to hundredths, we have 4.274 ... ≈4.27. Then 5-4.274… ≈5-4.27.

Let's represent the natural number 5 as 5.00, and subtract the decimal fractions in a column:

Answer:

5−4,274…≈0,73 .

It remains to voice rule for subtracting a natural number from a decimal fraction: to subtract a natural number from a decimal fraction, subtract this natural number from the whole part of the decimal fraction to be reduced, and leave the fractional part unchanged. This rule applies to both finite decimal fractions and infinite ones. Let's consider the solution of an example.

Example.

Subtract the natural number 17 from the decimal fraction 37.505.

Decision.

The integer part of the decimal 37.505 is 37. Let us subtract the natural number 17 from it, we have 37−17 \u003d 20. Then 37.505−17 \u003d 20.505.

Answer:

37,505−17=20,505 .

Subtracting a decimal from a fraction or mixed number and vice versa

Subtract a finite decimal or infinite periodic decimal from a fraction can be reduced to the subtraction of ordinary fractions. To do this, it is enough to convert the subtracted decimal fraction into an ordinary fraction.

Example.

Subtract 0.25 from the decimal 4/5.

Decision.

Since 0.25 \u003d 25/100 \u003d 1/4, then the difference between the ordinary fraction 4/5 and the decimal fraction 0.25 is equal to the difference between the ordinary fractions 4/5 and 1/4. So, 4/5−0,25=4/5−1/4=16/20−5/20=11/20 ... In decimal notation, the resulting ordinary fraction is 0.55.

Answer:

4/5−0,25=11/20=0,55 .

Similarly subtracting a final decimal or a periodic decimal from a mixed number is reduced to subtracting an ordinary fraction from a mixed number.

Example.

Subtract the decimal 0, (18) from the mixed number.

Decision.

To begin with, let's convert the periodic decimal fraction 0, (18) into an ordinary fraction:. Thus, . The resulting mixed number in decimal notation is 8, (18).

Adding Decimals made according to the rules of addition to the column.

Decimal fractions are added in a column, like natural numbers, ignoring the commas.

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

Note! If the initial decimal fractions have a different number of decimal places, then the required number of zeros must be added to the fraction in which there are fewer decimal places in order to equalize the number of decimal places in the fractions.

If there are not enough digits of the fractional part to the right of the addend or the reduced one, then on the right in the fractional part, you can add as many zeros (increase the digit capacity of the fractional part) as there are digits in the other addend or the reduced one.

Let's look at an example. Determine the sum of decimal fractions:

0,678 + 13,7 =

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

0,678 + 13,700 =

We write down the answer:

0,678 + 13,7 = 14,378

Basic rules for adding decimal fractions:

Equalize the number of decimal places.
Write 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 for adding in a column of natural numbers.
Put a comma under the commas in response.

In written addition and subtraction of decimal fractions, the comma that separates the whole part from the fractional part must be located at the terms and the sum in one column (the comma under the comma from the condition record until the end of the calculation).

For example.Add decimal fractions to a string:

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.