Can fractions contain decimal places

Math: Actions with Fractions. Actions with decimals and fractions. Decimal. Decimal actions

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

Lesson content

Add decimal places

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

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

First we write these two fractions in one column, while the whole parts must be below the whole and the fraction below the fractions. This requirement is called in school Comma below comma.

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

We start adding the fractions: 2 + 3 \ u003d 5. We write the five in the fraction 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 fraction with a comma. For this we follow the rule again Comma below comma:

The answer was 8.5. So the expressions 3.2 + 5.3 are 8.5

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

Decimal places

Like ordinary numbers, decimal fractions have their place. 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, and the third digit after the decimal point for the thousandth place.

Digits in decimal fractions store some useful information ... Specifically, they tell you how many tenths, hundredths, and thousandths are in decimal fractions.

For example, consider the decimal place 0.345

The position at which the triplet is located is called in tenths

The position where the four are is called hundredth

The position where the five are is called thousandths

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

If we add the fractions together, we get the original decimal place, 0.345

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

The same principles and rules are followed when adding decimal fractions 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 need to follow the rule Comma below comma... The comma below the comma indicates the same order in which tenths to tenths, hundredths to hundredths, thousandths to thousandths are added.

Example 1. Find the value of the expression 1.5 + 3.4

First, add the fractions 5 + 4 \ u003d 9.Write the nine in the fraction 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 let's separate the whole part from the fraction with a comma. To do this, we again observe the rule "comma below the comma":

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 this expression in a column and observe the rule "comma under comma"

First, add the fraction, 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 all the parts 3 + 1 \ u003d 4. We write a four throughout the part of our answer:

Separate the entire part from the fraction with a comma, observing the rule "comma below the comma":

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 is carried over 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 does not fit into the hundredth part of our answer. Therefore we write the number 2 in the hundredth part and transfer the unit to the next digit:

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

Now add all the parts 2 + 3 \ u003d 5. We write the number 5 in the entire 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 fractions 5 + 8 \ u003d 13. The number 13 does not fit in the fraction of our answer, so first we write down the number 3 and transfer the unit to the next digit, or rather to the whole part:

Now we add the whole parts 9 + 2 \ u003d 11 plus those 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 fraction with a comma:

The answer was 3/12. 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 must be the same in both fractions. If there are not enough numbers, these fractional digits are filled with zeros.

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

Before writing that 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 after the decimal point has three digits and the fraction 1.7 has only one. So in faction 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 the thousandths 5 + 0 \ u003d 5. We write the number 5 in the thousandth part of our answer:

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

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

Now we add the whole parts 12 + 1 \ u003d 13 plus those obtained 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 fraction 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

Subtract decimal fractions

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

Example 1. Find the value of the expression 2.5 - 2.2

We write this expression in a column and observe the rule "comma below the comma":

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

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

Separate the whole part from the fraction 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 decimal places. There are three digits after the decimal point in Group 7.353 and in Group 3.1 there is only one. This means that in faction 3.1 you have to add two zeros at the end to make the number of digits the same in both factions. Then we get 3,100.

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

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 normal numbers, sometimes you have to borrow one from the neighboring digit when subtraction becomes impossible.

Example 3. Find the value of Expression 3.46 - 2.39

Subtract the hundredths from 6-9. Do not subtract the number 9 from the number 6, so you need to take one from the adjacent bit. After taking one from the adjacent bit, the number 6 becomes the number 16. Now you can calculate the hundredths from 16-9 \ u003d 7. We write the seven in the hundredth part of our answer:

Now we're subtracting tenths. Since we occupied a unit in tenth place, the number there was reduced by one unit. In other words, in the tenth place there 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 fraction 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 Expression 3 - 1.2

In this example, one decimal place is subtracted from an integer. We write this expression in a column so that the integer part of the decimal fraction 1.23 is below the number 3

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

Now we subtract the tenths: 0-2. You can't subtract the number 2 from zero. Hence, you need to take one from the adjacent bit. Taking one from the neighboring 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 let's subtract whole parts. The integer used to contain the number 3, but we borrowed a unit of it. As a result, it became 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 fraction 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 decimals, multiply them like regular numbers, ignoring the commas.

After you get the answer, you need to separate the whole part from the fraction with a comma. To do this, you have to count the number of digits after the decimal point in both fractions, then count the same number of digits to the right in the answer and put a comma.

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

Let's multiply these decimal fractions as usual and ignore the commas. In order to ignore the commas, you can imagine for a while that they are not there at all:

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

Return to number 375 and move from right to left. We have 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:

Obtained 34695. In this number you must separate the whole part from the fraction with a comma. To do this, you need to count the number of digits after the decimal point in the fractions 12.85 and 2.7. In Group 12.85 there are two digits after the decimal point, in Group 2.7 there is one digit - a total of three digits.

We return to number 34695 and move from right to left. We have 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 with an ordinary number

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

To multiply a decimal fraction and a regular number, you need to multiply them and ignore the comma in the decimal fraction. After receiving the answer, the whole part must be separated from the fraction by a comma. To do this, you need to count the number of digits after the decimal point in the decimal fraction, then count the same number of digits to the right in the answer and put a comma.

For example, multiply 2.54 by 2

We multiply the decimal fraction 2.54 by the usual number 2 and ignore the comma:

We have the number 508. In this number you have to separate the whole part from the fraction with a comma. To do this, you need to count the number of digits after the decimal point in the fraction 2.54. After the decimal point in the fraction 2.54 there are two digits.

We return to number 508 and move from right to left. We have 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 the same way as multiplying decimal fractions by regular numbers. You need to do a multiplication without paying attention to the comma in the decimal fraction, and then in the answer separate the whole part from the fraction and count 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 place 2.88 by 10 ignoring the decimal point:

2880 received. In this number you have to separate the integer part from the fraction 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 the fraction 2.88 has two decimal places.

We return to number 2880 and move from right to left. We have 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 shifting the comma in the decimal fraction to the right by as many places as the factor contains zeros.

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

2.88 x 10 \ u003d 28.8

Let's try to multiply 2.88 by 100. We immediately consider the factor 100. We are interested in how many zeros there are in it. We see that there are two zeros. Now move the comma two digits to the right in the fraction 2.88, 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. Now move the comma three places to the right in the fraction 2.88. The third digit isn't there, so we're adding another zero. The result is 2880.

2.88 × 1000 \ u003d 2880

Decimal multiplication by 0.1, 0.01 and 0.001

Decimal places are multiplied by 0.1, 0.01, and 0.001 in the same way as decimal places are multiplied by decimal places. It is necessary to multiply fractions like ordinary numbers and insert a comma in the answer, counting as many digits to the right as there are in both fractions after the decimal point.

For example, multiply 3.25 by 0.1

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

Obtained 325. In this number you need to separate the whole part from the fraction with a comma. To do this, you need to count the number of digits after the decimal point in the 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 number 325 and move from right to left. We have to count three numbers from the right and put a comma. After counting three digits, we find that the digits are over. In this case you need to add a zero and put a comma:

The answer was .325. So the value of the expression 3.25 × 0.1 is equal to 0.325

3.25 × 0.1 \ u003d 0.325

There's 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 shifting the comma in the decimal fraction to the left by as many places as there are zeros in the multiplier.

For example, let's solve the previous example with 3.25 × 0.1 this way. Without calculations, we immediately give the factor 0.1. We are interested in how many zeros there are. We see that there is a zero. Now move the comma one digit to the left in the fraction 3.25. If you move the comma one digit to the left, you will see that there are no more digits in front of the three. In this case, add a zero and add a comma. The result is 0.325

3.25 × 0.1 \ u003d 0.325

Let's try to multiply 3.25 by 0.01. Immediately take a look at the 0.01 multiplier. We are interested in how many zeros there are. We see that there are two zeros. Now move the comma two digits to the left in fraction 3.25. You will 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. We see that there are three zeros. Now move the comma three digits to the left in fraction 3.25. 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. Typical mistake most people make.

When multiplying by 10, 100, 1000, the decimal point is shifted to the right by the same number of places as the factor contains zeros.

And when multiplied by 0.1, 0.01 and 0.001, the decimal point is carried to the left with the same number of digits as zeros in the multiplier.

If it is difficult to remember at first, you can use the first method, where the multiplication is done like ordinary numbers. In the answer, you need to separate the integer part from the fractional part and count as many digits from the right as the digits after the decimal point in both fractions.

Divide 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 whose numerator is the dividend and the denominator is the divisor.

For example, to divide an apple by two, you need to write 1 (an apple) in the numerator and 2 (two friends) in the denominator. As a result, we get a fraction. So every friend gets an apple. In other words, half an apple. Fraction is the answer to the problem "How to share an apple for two"

It turns out that if you divide 1 by 2, you can further solve this problem. After all, a breakbar in any fraction means division, and therefore that division is acceptable in a fraction. But how? We're used to the dividend always being higher than the divisor. And here, on the contrary, the dividend is less than the divisor.

Everything becomes clear when we remember that fraction means division, division, division. This means that a unit can be divided into any number of parts and not just into two parts.

If you divide a smaller number by a larger one, you get a decimal fraction with the integer part being 0 (zero). The fraction can be anything.

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

You cannot just be completely divided into two parts. When you ask a question "How many twos are in one", then the answer is 0. Therefore we write 0 in the quotient and put a comma:

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

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

We have 10. We divide 10 by 2, we get 5. We write the five in the fraction 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. The proportion is therefore 0.5

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

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 will get 0.5 cm

Example 2. Find the value of the expression 4: 5

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

Multiply 0 by 5, we get 0. Write zero under the four. We immediately subtract that zero from the dividend:

Now let's start by dividing the four into five. To do this, add zero to the right of 4 and divide 40 by 5, we get 8. Write the eight in the quotient.

End 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 there 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 by dividing the five into 125 parts. To do this, we write to the right of these five zeros:

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

Multiply 0 by 125, we get 0. Write that zero below 50. Immediately subtract 0 from 50

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

Divide 500 by 125. How many numbers 125 does the number 500 contain? The number 500 contains four numbers 125. We 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 the numbers without a remainder

So we put a comma in the quotient after the one to indicate that the division of the whole parts is finished, and we continue with the fraction:

Add zero to the remainder 4

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

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

9: 5 = 1,8

Example 2... Divide 84 by 5 with no remainder

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

Get 16 privately 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. We write the eight in the quotient after the decimal point:

and finish the example by checking if there is anything left:

Divide a decimal place by a regular number

As is well known, the decimal fraction consists of an integer and a fraction. The first thing you need to do when dividing a decimal fraction by an ordinary number is:

  • divide the entire part of the decimal fraction by this number.
  • after the whole part has been divided, you need to immediately put a comma in the quotient and continue the calculation as with normal 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 the two in the quotient and immediately put a comma:

Now let's 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 a zero yet because the answer is not complete. Then we continue to calculate as in ordinary division. Take 8 down and divide 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: \ u200b \ u200b2 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 let's 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're not writing a zero yet. If you divide the last 3 of the dividend and divide by 3 you 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 place by a decimal place

To divide a decimal fraction by a decimal fraction, in the dividend and in the divisor, the comma must be shifted to the right by the same number of places that are after the decimal point in the divisor, and then divided by an ordinary number.

For example, divide 5.95 by 1.7

Let's write this phrase in a corner

Now move the comma in the dividend and in the divisor by the same number of places to the right as after the comma in the divisor. There is a number after the decimal point. So we have to move the comma in the dividend and in the divisor one digit to the right. We transmit:

After moving the comma one digit to the right, the decimal fraction 5.95 became 59.5. And the decimal fraction 1.7 after moving the comma to the right by one digit became the usual number 17. And we already know how to divide the decimal fraction by the usual number. Further calculations are not difficult:

The comma is broken to the right to make it easier to divide. This is permissible due to the fact that the quotient does not change when the dividend and the divisor are multiplied or divided by the same number. What does that mean?

This is one of the interesting functions of the division. It is called the property of the quotient. Consider the expression 9: 3 \ u003d 3. In this expression, if the dividend and divisor are multiplied or divided by the same number, then the quotient 3 does 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 thing happens when we put the decimal point in the dividend and in the divisor. In the previous example where we divided 5.91 by 1.7, we moved the decimal point in dividend and divisor one digit to the right. After moving the decimal point, the fraction 5.91 was converted into a fraction of 59.1 and the fraction 1.7 into the usual number 17.

In fact, there was a multiplication by 10 within that process. Here's what it looked like:

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 divisor are multiplied by. In other words, the number of places after the decimal point in the divisor determines how many places in the dividend and in the divisor the decimal point is shifted to the right.

Divide a decimal place by 10, 100, 1000

Dividing a decimal place 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 easier. 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 this way. 2.1: 10. We consider the divisor. We are interested in how many zeros there are. We see that there is a zero. So in dividend 2.1 you have to move the decimal point one digit to the left. We move the comma one digit to the left and see that there are no numbers left. In this case, add another zero before the number. The result is 0.21

Let's try dividing 2.1 by 100. There are two zeros in 100. So in dividend 2.1 you have to move the decimal point to the left by two digits:

2,1: 100 = 0,021

Let's try dividing 2.1 by 1000. 1000 contains three zeros. In the dividend 2.1 you have to move the decimal point three digits to the left:

2,1: 1000 = 0,0021

Division of a decimal place by 0.1, 0.01 and 0.001

Dividing the decimal place 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 shifted to the right by as many places as it is after the comma in the divisor.

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

After moving the comma one digit to the right, the decimal fraction 6.3 becomes the usual number 63, and the decimal fraction 0.1 after moving the comma to the right becomes a digit. And dividing 63 by 1 is very easy:

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

But there is also a second way. It is easier. The essence of this method is that the decimal point in the dividend is shifted to the right by as many places as there are zeros in the divisor.

Let's solve the previous example this way. 6.3: 0.1. We look at the divider. We are interested in how many zeros it contains. We see that there is a zero. This means that on the dividend of 6.3 you have to move the decimal point one digit to the right. Move the comma right one digit and get 63

Let's try dividing 6.3 by 0.01. The divisor 0.01 has two zeros. This means that in the dividend of 6.3 you have to move the decimal point two digits to the right. In the dividend, however, there is only one number after the decimal point. In this case, another zero must be added at the end. As a result we get 630

Let's try dividing 6.3 by 0.001. The divisor 0.001 has three zeros. This means that in the dividend of 6.3 you have to move the decimal point three digits to the right:

6,3: 0,001 = 6300

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In this article we will find out what a decimal fraction is, what its characteristics and properties are. Walk! 🙂

The decimal fraction is a special case of common fractions (where the denominator is a multiple of 10).

definition

Fractions are called decimal places, the denominators of which are numbers made up of one and a number of zeros. That is, these are fractions with the denominator 10, 100, 1000, etc. Otherwise, a decimal fraction can be described as a fraction with a denominator of 10 or one of the powers of ten.

Examples of political groups:

, ,

The decimal fraction is written differently than the usual one. Operations with these fractions are also different from operations with ordinary ones. The rules for actions on them are largely close to the rules for actions on integers. This particularly determines their relevance for solving practical problems.

Fractional representation in decimal notation

The decimal notation does not contain a denominator. It shows the number of the meter. IM overall view The decimal fraction is recorded according to the following scheme:

where X is the entire part of the fraction, Y is its fraction, "," is the decimal point.

Correct representation of an ordinary fraction as a decimal requires that it be correct, that is, with a highlighted integer part (if possible) and a numerator that is less than the denominator. Then in decimal notation, the entire part is written on the decimal point (X), and the numerator of the common fraction is written after the decimal point (Y).

If the numerator contains a number with fewer digits than the number of zeros in the denominator, the missing number of digits in part Y is filled in decimal notation with zeros in front of the numerator digits.

Example:

If the portion is less than 1, i.e. has no integer part, then 0 is written in decimal form for X.

Any number of zeros can be entered in the fraction (Y) after the last significant digit (not equal to zero). This does not affect the value of the fraction. Conversely, all zeros at the end of the decimal fraction can be omitted.

Read decimal fractions

Part X is generally read like this: "X whole".

Part Y is read according to the number in the denominator. Read for the denominator 10: "Y tenths", for the denominator 100: "Y hundredths", for the denominator 1000: "Y thousandths" and so on ... 😉

Another approach to reading is believed to be more correct based on counting the number of digits of the fraction. To do this, you need to understand that fraction digits are in a mirror image with respect to the digits of the integer part of the fraction.

The names for correct reading are given in the table:

Based on this, the reading should be based on the equivalent of the category name to the last digit of the fraction.

  • 3.5 reads like "three point five tenths"
  • 0.016 reads zero point 16 thousandths

Convert any fraction to a decimal number

If the denominator of an ordinary fraction is 10 or a power of ten, the fraction is translated as described above. In other situations, additional transformations are required.

There are two ways to transfer.

First translation method

The numerator and denominator must be multiplied by an integer to get 10 or one of the powers of ten in the denominator. And then the fraction is represented in decimal notation.

This method is applicable to fractions whose denominator can only be expanded to 2 and 5. So in the previous example ... If the expansion contains other prime factors (e.g.), you have to resort to the 2nd method.

Second translation method

The second method is to divide the numerator by the denominator in a column or on a calculator. The entire part, if any, is not involved in the conversion.

The long division rule that results in a decimal number is described below (see Decimal division).

Convert a decimal place to a fraction

To do this, its fraction (to the right of the decimal point) should be written in the form of the numerator, and the result of reading the fraction in the form of the corresponding number in the denominator. If possible, you need to reduce the resulting proportion.

Final and infinite decimal numbers

The finale is called a decimal fraction, the fraction of which consists of a finite number of digits.

All of the above examples contain exactly the final decimal fractions. However, not every common fraction can be represented as the last decimal place. If the 1st translation method is not applicable to a particular fraction and the 2nd method shows that the division cannot be completed, then only an infinite decimal fraction can be obtained.

It is impossible to write down an infinite fraction in its full form. Such fractions can be represented in incomplete form:

  1. as a result of reducing to the desired number of decimal places;
  2. as a periodic parliamentary group.

Periodic is a fraction that allows you to choose an endlessly repeating sequence of numbers after the decimal point.

The rest of the fractions are said to be non-periodic. For non-periodic fractions, only the 1st representation (rounding) is permitted.

An example of a periodic fraction: 0.8888888 ... there is a repeating number 8 that obviously repeats indefinitely as there is no reason to assume otherwise. This figure is called break time.

Periodic fractions can be clean or mixed. A decimal fraction is a clean fraction where the period starts immediately after the decimal point. Have mixed fraction There is one or more digits before the decimal point.

54.33333 ... - periodic pure decimal fraction

2.5621212121 ... - mixed periodic fraction

Examples of writing infinite decimal fractions:

The second example shows how to correctly format the period in the notation of a periodic fraction.

Converting periodic decimal places to fractions

To translate a pure periodic fraction into an ordinary fraction, its period is written in the numerator and in the denominator a number consisting of nines in a quantity equal to the number of digits in the period is written.

The mixed periodic decimal fraction is translated as follows:

  1. they must form a number consisting of the number after the decimal point before the point and the first point.
  2. subtract the number after the decimal point before the point from the resulting number. The sum is the numerator of the common fraction;
  3. in the denominator you need to enter a number consisting of a number of nines equal to the number of period digits, followed by zeros, the number of which is equal to the number of digits of the number after the decimal point up to the first period.

Comparison of decimal places

Decimal fractions are first compared according to their whole parts. The larger the fraction, the more of its integral part it has.

If the total parts are equal, the digits of the corresponding digits of the fraction starting with the first (from the tenth) are compared. The same principle applies here: the larger the fraction with a larger tenth place; If the tenth digits are the same, the hundredth digits are compared, and so on.

So far

, because with the same whole parts and the same tenths in the fraction, the 2nd fraction has a larger number of hundredths.

Adding and subtracting decimal fractions

Decimal fractions are added and subtracted like whole numbers, whereby the corresponding digits are written one below the other. To do this, you must have decimal places below each other. Then the units (tens, etc.) of the entire part and the tenths (hundredths, etc.) of the fraction will be the same. The missing digits of the fraction are filled with zeros. Directly the addition and subtraction process is the same as for whole numbers.

Decimal multiplication

To multiply decimal fractions, you need to write them one below the other, align them with the last digit and not pay attention to the position of the decimal places. Then you need to multiply the numbers in the same way as when multiplying whole numbers. After receiving the result, recalculate the number of decimal places in both fractions and separate the total number of decimal places with a comma in the resulting number. If there are not enough digits, they are replaced with zeros.

Multiplying and dividing decimal fractions by 10 n

These actions are simple and are limited to transferring the decimal point. P. when multiplying, the decimal point is shifted to the right by the number of digits that corresponds to the number of zeros in 10 n (the fraction is increased), where n is any integer power. That is, a certain number of digits is transferred from the fraction to the whole. When dividing, the comma is transferred to the left (the number decreases) and part of the digits are transferred from the integer part to the fractional part. If there are not enough digits, the missing digits are filled with zeros.

Division of a decimal place and an integer by an integer and a decimal number

The division in a column of a decimal fraction by an integer corresponds to the division of two integers. In addition, only the position of the decimal point needs to be taken into account: if you tear off a digit of a digit followed by a comma, you must put a comma after the current digit of the generated answer. Next, you have to keep dividing until you get zero. If the dividend does not have enough signs to fully divide, zeros should be used as such.

Similarly, if all digits of the dividend have been removed and full division has not yet been completed, 2 integers are divided into one column. In this case, after canceling the last digit of the dividend, a decimal point is inserted into the answer formed, and zeros are used as dismantled digits. Those. In fact, the dividend is represented as a decimal fraction with a fraction of zero.

To divide a decimal fraction (or whole number) by a decimal number, you need to multiply the dividend and divisor by 10 n, where the number of zeros is equal to the number of digits after the decimal point in the divisor. This way you remove the decimal point in the fraction you want to divide by. Furthermore, the dividing process is the same as described above.

Graphic representation of decimal fractions

Decimal fractions are shown graphically with a coordinate line. For this purpose, the unit segments are also divided into 10 equal parts, just as centimeters and millimeters are placed on a ruler at the same time. This ensures that decimal fractions are displayed accurately and can be compared objectively.

In order for the subdivisions in unit segments to be the same, you should carefully consider the length of the unit segment itself. It should be such that it is possible to provide the convenience of additional partitioning.

There were 5 ribbon colors in the sewing workshop. The bureaucracy was eight feet higher than the blue, but three feet less than the green. The white band was 1.5 meters higher than the black, but 1.9 meters less than the green. How many yards of tape was there in the workshop if the white tape was 7.3 meters?

    decision
  • 1) 7.3 + 1.9 \ u003d 9.2 (m) of the green ribbon were in the workshop;
  • 2) 7.3 - 1.5 \ u003d 5.8 (m) black tape;
  • 3) 9.2 - 3.8 \ u003d 5.4 (m) bureaucracy;
  • 4) 5.4 - 2.4 \ u003d 3 (m) blue ribbon;
  • 5) 7.3 + 9.2 + 5.8 + 5.4 + 3 \ u003d 30.7 (m).
  • Answer: There was a total of 30.7 meters of tape in the workshop.

exercise 2

The length of the rectangular section is 19.4 meters and the width 2.8 meters less. Calculate the size of the lot.

    decision
  • 1) 19.4 - 2.8 \ u003d 16.6 (m) section width;
  • 2) 16.6 * 2 + 19.4 * 2 \ u003d 33.2 + 38.8 \ u003d 72 (m).
  • Answer: The perimeter of the site is 72 meters.

Problem 3

The jump length of a kangaroo can be 13.5 meters. The world record for one person is 8.95 meters. How far does the kangaroo jump?

    decision
  • 1) 13.5 - 8.95 \ u003d 4.55 (m).
  • 2) Answer: The kangaroo jumps 4.55 meters further.

Problem 4

The lowest temperature on the planet was measured in the summer of July 21, 1983 at Vostok station in Antarctica and was -89.2 ° C, and the hottest in the city of Al-Aziziya was +57 on September 13, 1922, 8 ° C. Calculate the difference between the temperatures.

    decision
  • 1) 89.2 + 57.8 \ u003d 147 ° C.
  • Answer: The temperature difference is 147 ° C.


Problem 5

The load capacity of the Gazel delivery truck is 1.5 tons and the BelAZ dump truck 24 times more. Calculate the load capacity of the BelAZ dump truck.

    decision
  • 1) 1.5 24 \ u003d 36 (tons).
  • Answer: The load capacity of BelAZ is 36 tons.

Problem 6

The maximum speed of the earth in its orbit is 30.27 km / s and the speed of Mercury is 17.73 km higher. How fast is Mercury moving in its orbit?

    decision
  • 1) 30.27 + 17.73 \ u003d 48 (km / s).
  • Answer: The speed of movement of mercury in orbit is 48 km / s.

Problem 7

Deep Mariana Trench is 11,023 km and the height of the highest mountain in the world - Chomolungma is 8,848 km above sea level. Calculate the difference between these two points.

    decision
  • 1) 11.023 + 8.848 \ u003d 19.871 (km).
  • Answer: 19, 871 km.

Problem 8

For Kolya, as for any other healthy person, the normal body temperature is 36.6 ° C, and for his four-legged friend Sharik 2.2 ° C. What is the normal temperature for the ball?

    decision
  • 1) 36.6 + 2.2 \ u003d 38.8 ° C.
  • Answer: For the ball, normal body temperature is 38.8 ° C.

Problem 9

The painter painted 18.6 m² of the fence in one day, his assistant painted 4.4 m² less. How many square meters of fence do the painter and his assistant paint in one working week, if it corresponds to five days?

    decision
  • 1) 18.6 - 4.4 \ u003d 14.2 (m²) The painter assistant paints in 1 day;
  • 2) 14.2 + 18.6 \ u003d 32.8 (m²) are painted together in one day;
  • 3) 32.8 · 5 \ u003d 164 (m²).
  • Answer: During the working week, the painter and his assistant paint together 164 m² of the fence.

Problem 10

Two boats drove at the same time from the two pillars towards each other. The speed of one boat is 42.2 km / h, the second 6 km / h more. What is the distance between the boats in 2.5 hours if the distance between the marinas is 140.5 km?

    decision
  • 1) 42.2 + 6 \ u003d 48.2 (km / h) speed of the second boat;
  • 2) 42.2 * 2.5 \ u003d 105.5 (km) pass the first boat in 2.5 hours;
  • 3) 48.2 * 2.5 \ u003d 120.5 (km) pass the second boat in 2.5 hours;
  • 4) 140.5 - 105.5 \ u003d 35 (km) distance from the first boat to the opposite pier;
  • 5) 140.5 - 120.5 \ u003d 20 (km) distance from the second boat to the opposite pier;
  • 6) 35 + 20 \ u003d 55 (km);
  • 7) 140 - 55 \ u003d 85 (km).
  • Answer: There are 85 km between the boats.

Exercise 11

Every day the cyclist covers 30.2 km. If it took the same amount of time, a motorcyclist would cover a distance 2.5 times longer than a cyclist. How much distance can a motorcyclist cover in 4 days?

    decision
  • 1) 30.2 * 2.5 \ u003d 75.5 (km) are overcome by a motorcyclist in 1 day;
  • 2) 75.5 * 4 \ u003d 302 (km).
  • Answer: A motorcyclist can cover 302 km in 4 days.

Exercise 12

The store sold 18.3 kg of cookies and 2.4 kg less candy in one day. How many candies and cookies were sold together in the store that day?

    decision
  • 1) 18.3 - 2.4 \ u003d 15.9 (kg) candies were sold in the store;
  • 2) 15.9 + 18.3 \ u003d 34.2 (kg).
  • Answer: A total of 34.2 kg of sweets and biscuits were sold.


Fractions written in the form 0.8; 0.13; 2,856; 5.2; 0.04 means decimal. In fact, decimal fractions are a simplified notation for common fractions. It is convenient to use this notation for all fractions with denominators of 10, 100, 1000, and so on.

Consider examples (0.5 is read as zero five);

(0.15 reads, zero fifteen hundredths);

(5.3 reads, five point three).

Note that in decimal notation, the comma separates the integer part of the number from the fraction, the integer part of the regular fraction 0. The fraction of the decimal fraction contains as many digits as the denominator of zeros contains the corresponding ordinary fraction.

Consider an example , , .

In some cases it is necessary to consider a natural number as a decimal fraction where the fraction is zero. It is customary to write down that 5 \ u003d 5.0; 245 \ u003d 245.0 and so on. Note that in the decimal notation of a natural number, the unit of the least significant digit is ten times smaller than the unit of the adjacent most significant digit. Writing decimal fractions has the same property. So immediately after the decimal point there is a tenth place, then a hundredth place, then a thousandth place, and so on. Below are the names of the digits of the number 31.85431, the first two columns are the integer part, the remaining columns are the fractional part.

That fraction is read as thirty-one points eighty-five thousand four hundred thirty-one hundred thousandths.

Adding and subtracting decimal fractions

The first way is to convert decimals to fractions and add them together.

as you can see from the example, this method is very inconvenient and it is better to use the second method, which is more correct, without converting decimals to ordinary ones. To add two decimal fractions you will need:

  • equalize the number of digits after the decimal point in the terms;
  • write the terms one below the other so that each category of the second term is below the corresponding category of the first term.
  • add the resulting numbers as you add natural numbers.
  • put a comma in the resulting sum under the commas in the terms.

Let's look at some examples:

  • equalize in the reduced and subtracted number of digits after the decimal point;
  • write the subtracted under the decreasing so that each digit of the subtracted is below the corresponding digit of the decreasing.
  • subtract when subtracting natural numbers;
  • put a comma below the commas in reduced and subtracted in the resulting difference.

Let's look at some examples:

In the examples discussed above, it can be seen that the addition and subtraction of decimal fractions was done bit by bit, that is, in the same way that we did similar actions with natural numbers. This is the main advantage of decimal notation for fractions.

Decimal multiplication

To multiply a decimal fraction by 10, 100, 1000, etc., you need to move the decimal point in that fraction by 1, 2, 3, etc. to the right. Therefore, if the comma is shifted to the right by 1, 2, 3, etc., the fraction will increase by 10, 100, 1000, etc. To multiply two decimal fractions, you will need:

  • multiply them as natural numbers and ignore the commas;
  • Separate as many digits in the resulting product with a comma to the right as there are after the commas in both factors combined.

There are cases when the product contains fewer digits than are required to be separated by a comma.Add the required number of zeros to the left of this product, then move the comma to the left after the correct amount of digits.

Consider examples: 2 * 4 \ u003d 8, then 0.2 * 0.4 \ u003d 0.08; 23 * 35 \ u003d 805, then 0.023 * 0.35 \ u003d 0.00805.

There are cases where one of the factors is 0.1; 0.01; 0.001 and so on, it is more convenient to use the following rule.

  • Multiply the decimal place by 0.1; 0.01; 0.001 etc. In this decimal fraction, the decimal point must be shifted by 1, 2, 3 etc. to the left.

Let's consider examples: 2.65 * 0.1 \ u003d 0.265; 457.6 * 0.01 \ u003d 4.576.

Multiplication Properties Natural numbers are also performed on decimal fractions.

  • ab \ u003d ba - the displacement property of the multiplication;
  • (ab) c \ u003d a (bc) - the combined property of multiplication;
  • a (b + c) \ u003d ab + ac - the distribution property of multiplication relative to addition.

Division of decimal fractions

It is known whether we divide the natural number a for a natural number b means finding such a natural number cwhat if multiplied by b indicates the number a... This rule stays true if at least one of the numbers is a, b, c is a decimal fraction.

Consider an example, you want to divide 43.52 by 17 vertices without paying attention to the decimal point. In this case, the comma in the quotient should come immediately before the first digit after the decimal point was used in the dividend.

There are times when the dividend is less than the divisor and the entire part of the quotient is zero. Let's consider an example:

Let's look at another interesting example.

The division process is stopped because the digits of the dividend have ended, but they have not received a zero in the remainder. It is known that if any number of zeros are assigned to it on the right, the decimal fraction does not change. Then it becomes clear that the dividend numbers can't end.

To divide the decimal fraction by 10, 100, 1000, etc., you need to move the decimal point in that fraction by 1, 2, 3, etc. Consider an example: 5.14: 10 \ u003d 0.514; 2: 100 \ u003d 0.02; 37.51: 1000 \ u003d 0.03751.

If the dividend and divisor are increased by 10, 100, 1000, etc. at the same time, the quotient does not change.

Consider an example: 39.44: 1.6 \ u003d 24.65, increase the dividend and divisor 10 times. 394.4: 16 \ u003d 24.65 It should be noted that dividing a decimal fraction by a natural number is easier in the second example.

To divide a decimal fraction by a decimal number, you need:

  • move the commas in the dividend and in the divisor to the right by as many places as there are after the comma in the divider.
  • perform division by a natural number.

Consider an example: 23.6: 0.02, notice that the divisor has two decimal places. Therefore, we multiply both numbers by 100. We get 2360: 2 \ u003d 1180, divide the result by 100 and get the answer 11.80 or 23.6: 0, 02 \ u003d 11.8.

Comparison of decimal places

There are two ways to compare decimal fractions. Method one: it is necessary to compare two decimal fractions 4.321 and 4.32, balance the number of decimal places and start digitally comparing tenths to tenths, hundredths to hundredths, and so on. As a result, we get 4,321 \ u003e 4,320.

The second way to compare decimal fractions is through multiplication. Multiply the above example by 1000 and compare 4321 \ u003e 4320. Which method is more convenient, everyone chooses for themselves.

When adding decimal fractions, you need to write them one below the other so that the same digits are below each other and the comma below the decimal point, and add the fractions when adding natural numbers. For example, add fractions 12.7 and 3.442. The first fraction contains one decimal place and the second three. To do an addition, we transform the first fraction so that there are three digits after the decimal point :, then

The subtraction of decimal fractions is done in the same way. Find the difference between the numbers 13.1 and 0.37:

If you multiply decimal fractions, it is sufficient to multiply the specified numbers, ignore the commas (like natural numbers) and then separate as many digits with a comma on the right as they are after the comma in both factors in total.

For example, multiply 2.7 by 1.3. We have. Separate two digits on the right with a comma (the sum of the digits in the multipliers after the decimal point is two). As a result, we get 2.7 1.3 \ u003d 3.51.

If the product contains fewer digits than need to be separated by a comma, the missing zeros are prefixed, for example:

Consider multiplying a decimal fraction by 10, 100, 1000, etc. For example, suppose you need to multiply the fraction 12,733 by 10. If we separate three digits to the right with a comma, we get no. Means,

12,733 10 \ u003d 127.33. Multiplying a decimal fraction by 10 is reduced to moving a comma one digit to the right.

To multiply a decimal fraction by 10, 100, 1000, the comma in this fraction must generally be shifted 1, 2, 3 places to the right, adding a certain number of zeros if necessary (fraction right). For example,

Dividing a decimal fraction by a natural number is done in the same way as dividing a natural number by a natural number, and the comma in the quotient is placed after the division of the integer part is completed. Let's divide 22.1 by 13:

If the integer part of the dividend is less than the divisor, the answer is zero integers, for example:

Now divide a decimal fraction by a decimal number. Let's divide 2.576 by 1.12. To do this, in both the dividend and the divisor, we transfer the comma to the right by as many digits as after the comma in the divisor (in this example by two). In other words, multiply the dividend and divisor by 100 - this doesn't change the quotient. Then you have to divide the fraction 257.6 by the natural number 112, that is, the problem is reduced to the case already considered:

To divide the decimal fraction by, the comma in this fraction must be shifted to the left of the digits (in this case, the required number of zeros is assigned to the left, if necessary). For example, .

Division is not always possible with natural numbers, so it is not always possible for decimal fractions. For example, divide 2.8 by 0.09:

The result is what is known as an infinite decimal fraction. In such cases they switch to ordinary fractions. For example:

It may turn out that some numbers are written in the form of ordinary fractions, others in the form of mixed numbers, and still others in the form of decimal fractions. When taking actions on such numbers, there are several ways you can act: either convert decimals to ordinary fractions and apply the rules for actions ordinary fractions, or convert common fractions and mixed numbers to decimals (if possible) and apply the rules for dealing with Decimal fractions.