Dividing decimals by an integer examples. Drawing up a system of equations

Adding decimals is done in the same way as adding whole numbers. Let's see this with examples.

1) 0.132 + 2.354. Let's sign the terms one under the other.

Here, from the addition of 2 thousandths with 4 thousandths, 6 thousandths were obtained;
from the addition of 3 hundredths with 5 hundredths, it turned out 8 hundredths;
from adding 1 tenth with 3 tenths -4 tenths and
from adding 0 integers with 2 integers - 2 integers.

2) 5,065 + 7,83.

There are no thousandths in the second term, so it is important not to make mistakes when signing the terms under each other.

3) 1,2357 + 0,469 + 2,08 + 3,90701.

Here, when adding thousandths, we get 21 thousandths; we wrote 1 under the thousandths, and 2 added to the hundredths, so in the hundredth place we got the following terms: 2 + 3 + 6 + 8 + 0; in sum, they give 19 hundredths, we signed 9 under hundredths, and 1 was counted as tenths, etc.

Thus, when adding decimal fractions, one must observe next order: sign fractions one under the other so that in all terms the same digits are under each other and all commas are in the same vertical column; to the right of the decimal places of some terms, they attribute, at least mentally, such a number of zeros so that all terms after the decimal point have the same number digits. Then, addition is performed by digits, starting from the right side, and in the resulting sum they put a comma in the same vertical column as it is in these terms.

§ 108. Subtraction of decimal fractions.

Subtracting decimals is done in the same way as subtracting whole numbers. Let's show this with examples.

1) 9.87 - 7.32. Let's sign the subtrahend under the minuend so that the units of the same digit are under each other:

2) 16.29 - 4.75. Let's sign the subtrahend under the minuend, as in the first example:

To subtract tenths, one had to take one whole unit from 6 and split it into tenths.

3) 14.0213-5.350712. Let's sign the subtrahend under the minuend:

The subtraction was performed as follows: since we cannot subtract 2 millionths from 0, we should refer to the nearest digit to the left, i.e., to hundred-thousandths, but there is also zero in place of hundred-thousandths, so we take 1 ten-thousandth from 3 ten-thousandths and we split it into hundred-thousandths, we get 10 hundred-thousandths, of which 9 hundred-thousandths are left in the category of hundred-thousandths, and 1 hundred-thousandth is crushed into millionths, we get 10 millionths. Thus, in last three digits we got: millionths 10, hundred-thousandths 9, ten-thousandths 2. For greater clarity and convenience (not to forget), these numbers are written on top of the corresponding fractional digits of the reduced. Now we can start subtracting. We subtract 2 millionths from 10 millionths, we get 8 millionths; subtract 1 hundred-thousandth from 9 hundred-thousandths, we get 8 hundred-thousandths, etc.

Thus, when subtracting decimal fractions, the following order is observed: the subtrahend is signed under the reduced so that the same digits are one under the other and all the commas are in the same vertical column; on the right, they attribute, at least mentally, in the reduced or subtracted so many zeros so that they have the same number of digits, then subtract by digits, starting from the right side, and in the resulting difference put a comma in the same vertical column in which it is located in reduced and subtracted.

§ 109. Multiplication of decimal fractions.

Consider a few examples of multiplying decimal fractions.

To find the product of these numbers, we can reason as follows: if the factor is increased by 10 times, then both factors will be integers and we can then multiply them according to the rules for multiplying integers. But we know that when one of the factors is increased several times, the product increases by the same amount. This means that the number that results from multiplying integer factors, i.e. 28 by 23, is 10 times greater than the true product, and in order to get the true product, you need to reduce the found product by 10 times. Therefore, here you have to perform a multiplication by 10 once and a division by 10 once, but multiplication and division by 10 is performed by moving the comma to the right and left by one sign. Therefore, you need to do this: in the multiplier, move the comma to the right by one sign, from this it will be equal to 23, then you need to multiply the resulting integers:

This product is 10 times larger than the true one. Therefore, it must be reduced by 10 times, for which we move the comma one character to the left. Thus, we get

28 2,3 = 64,4.

For verification purposes, you can write a decimal fraction with a denominator and perform an action according to the rule for multiplying ordinary fractions, i.e.

2) 12,27 0,021.

The difference between this example and the previous one is that here both factors are represented by decimal fractions. But here, in the process of multiplication, we will not pay attention to commas, that is, we will temporarily increase the multiplier by 100 times, and the multiplier by 1,000 times, which will increase the product by 100,000 times. Thus, multiplying 1227 by 21, we get:

1 227 21 = 25 767.

Considering that the resulting product is 100,000 times the true product, we must now reduce it by a factor of 100,000 by properly placing a comma in it, then we get:

32,27 0,021 = 0,25767.

Let's check:

Thus, in order to multiply two decimal fractions, it is enough, without paying attention to commas, to multiply them as integers and in the product to separate with a comma on the right side as many decimal places as there were in the multiplicand and in the factor together.

In the last example, the result is a product with five decimal places. If such greater precision is not required, then rounding is done. decimal fraction. When rounding, you should use the same rule that was indicated for integers.

§ 110. Multiplication using tables.

Multiplying decimals can sometimes be done using tables. For this purpose, you can, for example, use those multiplication tables two-digit numbers, the description of which was given earlier.

1) Multiply 53 by 1.5.

We will multiply 53 by 15. In the table, this product is equal to 795. We found the product of 53 by 15, but our second factor was 10 times less, which means that the product must be reduced by 10 times, i.e.

53 1,5 = 79,5.

2) Multiply 5.3 by 4.7.

First, we find in the table the product of 53 by 47, it will be 2491. But since we increased the multiplicand and the multiplier by a total of 100 times, then the resulting product is 100 times larger than it should be; so we have to reduce this product by a factor of 100:

5,3 4,7 = 24,91.

3) Multiply 0.53 by 7.4.

First we find in the table the product of 53 by 74; it will be 3,922. But since we have increased the multiplier by 100 times, and the multiplier by 10 times, the product has increased by 1,000 times; so we now have to reduce it by a factor of 1,000:

0,53 7,4 = 3,922.

§ 111. Division of decimals.

We will look at decimal division in this order:

1. Decimal division by integer,

1. Division of a decimal fraction by an integer.

1) Divide 2.46 by 2.

We divided by 2 first integers, then tenths and finally hundredths.

2) Divide 32.46 by 3.

32,46: 3 = 10,82.

We divided 3 tens by 3, then we began to divide 2 units by 3; since the number of units of the dividend (2) less divisor(3), then I had to put 0 in the quotient; further, to the remainder we demolished 4 tenths and divided 24 tenths by 3; received in private 8 tenths and finally divided 6 hundredths.

3) Divide 1.2345 by 5.

1,2345: 5 = 0,2469.

Here, in the quotient in the first place, zero integers turned out, since one integer is not divisible by 5.

4) Divide 13.58 by 4.

The peculiarity of this example is that when we received 9 hundredths in private, then a remainder equal to 2 hundredths was found, we split this remainder into thousandths, got 20 thousandths and brought the division to the end.

Rule. The division of a decimal fraction by an integer is carried out in the same way as the division of integers, and the resulting remainders are converted into decimal fractions, more and more small; division continues until the remainder is zero.

2. Division of a decimal fraction by a decimal fraction.

1) Divide 2.46 by 0.2.

We already know how to divide a decimal fraction by an integer. Let's think about whether this new case of division can also be reduced to the previous one? At one time, we considered the remarkable property of the quotient, which consists in the fact that it remains unchanged while increasing or decreasing the dividend and divisor by the same number of times. We would easily perform the division of the numbers offered to us if the divisor were an integer. To do this, it is enough to increase it 10 times, and to obtain the correct quotient, it is necessary to increase the dividend by the same number of times, that is, 10 times. Then the division of these numbers will be replaced by the division of such numbers:

and there is no need to make any amendments in private.

Let's do this division:

So 2.46: 0.2 = 12.3.

2) Divide 1.25 by 1.6.

We increase the divisor (1.6) by 10 times; so that the quotient does not change, we increase the dividend by 10 times; 12 integers are not divisible by 16, so we write in quotient 0 and divide 125 tenths by 16, we get 7 tenths in quotient and the remainder is 13. We split 13 tenths into hundredths by assigning zero and divide 130 hundredths by 16, etc. Pay attention to the following:

a) when integers are not obtained in the quotient, then zero integers are written in their place;

b) when, after taking the digit of the dividend to the remainder, a number is obtained that is not divisible by the divisor, then zero is written in the quotient;

c) when, after the removal of the last digit of the dividend, the division does not end, then, by assigning zeros to the remainders, the division continues;

d) if the dividend is an integer, then when dividing it by a decimal fraction, its increase is carried out by assigning zeros to it.

Thus, in order to divide a number by a decimal fraction, you need to discard a comma in the divisor, and then increase the dividend as many times as the divisor increased when the comma was dropped in it, and then perform the division according to the rule of dividing the decimal fraction by an integer.

§ 112. Approximate quotient.

In the previous paragraph, we considered the division of decimal fractions, and in all the examples we solved, the division was brought to the end, i.e., an exact quotient was obtained. However, in most cases the exact quotient cannot be obtained, no matter how far we extend the division. Here is one such case: Divide 53 by 101.

We have already received five digits in the quotient, but the division has not yet ended and there is no hope that it will ever end, since the numbers that we have met before begin to appear in the remainder. Numbers will also be repeated in the quotient: obviously, after the number 7, the number 5 will appear, then 2, and so on without end. In such cases, division is interrupted and limited to the first few digits of the quotient. This private is called approximate. How to perform division in this case, we will show with examples.

Let it be required to divide 25 by 3. It is obvious that the exact quotient, expressed as an integer or decimal fraction, cannot be obtained from such a division. Therefore, we will look for an approximate quotient:

25: 3 = 8 and remainder 1

The approximate quotient is 8; it is, of course, less than the exact quotient, because there is a remainder of 1. To get the exact quotient, you need to add to the found approximate quotient, that is, to 8, the fraction that results from dividing the remainder, equal to 1, by 3; it will be a fraction 1/3. This means that the exact quotient will be expressed as a mixed number 8 1 / 3 . Since 1/3 is proper fraction, i.e. fraction, less than one, then, discarding it, we assume error, which less than one . Private 8 will approximate quotient up to one with a drawback. If we take 9 instead of 8, then we also allow an error that is less than one, since we will add not a whole unit, but 2 / 3. Such a private will approximate quotient up to one with an excess.

Let's take another example now. Let it be required to divide 27 by 8. Since here we will not get an exact quotient expressed as an integer, we will look for an approximate quotient:

27: 8 = 3 and remainder 3.

Here the error is 3 / 8 , it is less than one, which means that the approximate quotient (3) is found up to one with a drawback. We continue the division: we split the remainder of 3 into tenths, we get 30 tenths; Let's divide them by 8.

We got in private on the spot tenths 3 and in the remainder b tenths. If we confine ourselves to the number 3.3 in particular, and discard the remainder 6, then we will allow an error less than one tenth. Why? Because the exact quotient would be obtained when we added to 3.3 the result of dividing 6 tenths by 8; from this division would be 6/80, which is less than one tenth. (Check!) Thus, if we limit ourselves to tenths in the quotient, then we can say that we have found the quotient accurate to one tenth(with disadvantage).

Let's continue the division to find one more decimal place. To do this, we split 6 tenths into hundredths and get 60 hundredths; Let's divide them by 8.

In private in third place it turned out 7 and in the remainder 4 hundredths; if we discard them, then we allow an error of less than one hundredth, because 4 hundredths divided by 8 is less than one hundredth. In such cases, the quotient is said to be found. accurate to one hundredth(with disadvantage).

In the example that we are now considering, you can get the exact quotient, expressed as a decimal fraction. To do this, it is enough to split the last remainder, 4 hundredths, into thousandths and divide by 8.

However, in the vast majority of cases, it is impossible to obtain an exact quotient and one has to limit oneself to its approximate values. We will now consider such an example:

40: 7 = 5,71428571...

The dots at the end of the number indicate that the division is not completed, that is, the equality is approximate. Usually approximate equality is written like this:

40: 7 = 5,71428571.

We took the quotient with eight decimal places. But if such great precision is not required, one can confine oneself to the whole part of the quotient, i.e., the number 5 (more precisely, 6); for greater accuracy, tenths could be taken into account and the quotient taken equal to 5.7; if for some reason this accuracy is insufficient, then we can stop at hundredths and take 5.71, etc. Let's write out the individual quotients and name them.

The first approximate quotient up to one 6.

The second » » » to one tenth 5.7.

Third » » » up to one hundredth 5.71.

Fourth » » » up to one thousandth of 5.714.

Thus, in order to find an approximate quotient with an accuracy of some, for example, the 3rd decimal place (i.e., up to one thousandth), division is stopped as soon as this sign is found. In this case, one must remember the rule set forth in § 40.

§ 113. The simplest problems for interest.

After studying decimal fractions, we will solve a few more percentage problems.

These problems are similar to those we solved in the department of ordinary fractions; but now we will write hundredths in the form of decimal fractions, that is, without an explicitly designated denominator.

First of all, you need to be able to easily switch from an ordinary fraction to a decimal fraction with a denominator of 100. To do this, you need to divide the numerator by the denominator:

The table below shows how a number with a % (percentage) symbol is replaced by a decimal with a denominator of 100:

Let's now consider a few problems.

1. Finding percentages of a given number.

Task 1. Only 1,600 people live in one village. The number of school-age children is 25% of total number residents. How many school-age children are in this village?

In this problem, you need to find 25%, or 0.25, of 1,600. The problem is solved by multiplying:

1,600 0.25 = 400 (children).

Therefore, 25% of 1,600 is 400.

For a clear understanding of this task, it is useful to recall that for every hundred of the population there are 25 school-age children. Therefore, to find the number of all school-age children, you can first find out how many hundreds are in the number 1,600 (16), and then multiply 25 by the number of hundreds (25 x 16 = 400). This way you can check the validity of the solution.

Task 2. Savings banks give depositors 2% of income annually. How much income per year will be received by a depositor who has deposited: a) 200 rubles? b) 500 rubles? c) 750 rubles? d) 1000 rubles?

In all four cases, to solve the problem, it will be necessary to calculate 0.02 of the indicated amounts, i.e., each of these numbers will have to be multiplied by 0.02. Let's do it:

a) 200 0.02 = 4 (rubles),

b) 500 0.02 = 10 (rubles),

c) 750 0.02 = 15 (rubles),

d) 1,000 0.02 = 20 (rubles).

Each of these cases can be verified by the following considerations. Savings banks give depositors 2% of income, that is, 0.02 of the amount put into savings. If the amount were 100 rubles, then 0.02 of it would be 2 rubles. This means that every hundred brings the depositor 2 rubles. income. Therefore, in each of the cases considered, it is enough to figure out how many hundreds are in a given number, and multiply 2 rubles by this number of hundreds. In example a) hundreds of 2, so

2 2 \u003d 4 (rubles).

In example d) hundreds are 10, which means

2 10 \u003d 20 (rubles).

2. Finding a number by its percentage.

Task 1. In the spring, the school graduated 54 students, which is 6% of the total number of students. How many students were in the school in the past academic year?

Let us first clarify the meaning of this problem. The school graduated 54 students, which is 6% of the total number of students, or, in other words, 6 hundredths (0.06) of all students in the school. This means that we know the part of the students expressed by the number (54) and the fraction (0.06), and from this fraction we must find the whole number. Thus, before us is an ordinary problem of finding a number by its fraction (§ 90 p. 6). Problems of this type are solved by division:

This means that there were 900 students in the school.

It is useful to check such problems by solving the inverse problem, i.e. after solving the problem, you should, at least in your mind, solve the problem of the first type (finding the percentage of a given number): take the found number (900) as given and find the percentage indicated in the solved problem from it , namely:

900 0,06 = 54.

Task 2. The family spends 780 rubles on food during the month, which is 65% of the father's monthly income. Determine his monthly income.

This task has the same meaning as the previous one. It gives part of the monthly earnings, expressed in rubles (780 rubles), and indicates that this part is 65%, or 0.65, of the total earnings. And the desired is the entire earnings:

780: 0,65 = 1 200.

Therefore, the desired earnings is 1200 rubles.

3. Finding the percentage of numbers.

Task 1. The school library has a total of 6,000 books. Among them are 1,200 books on mathematics. What percentage of math books make up the total number of books in the library?

We have already considered (§97) problems of this kind and came to the conclusion that to calculate the percentage of two numbers, you need to find the ratio of these numbers and multiply it by 100.

In our task, we need to find the percentage of the numbers 1,200 and 6,000.

We first find their ratio, and then multiply it by 100:

Thus, the percentage of the numbers 1,200 and 6,000 is 20. In other words, math books make up 20% of the total number of all books.

To check, we solve the inverse problem: find 20% of 6,000:

6 000 0,2 = 1 200.

Task 2. The plant should receive 200 tons of coal. 80 tons have already been delivered. What percentage of coal has been delivered to the plant?

This problem asks what percentage one number (80) is of another (200). The ratio of these numbers will be 80/200. Let's multiply it by 100:

This means that 40% of the coal has been delivered.

In this article, we will analyze such an important action with decimal fractions as division. First we formulate general principles, then we will analyze how to correctly divide decimal fractions by a column both into other fractions and into natural numbers. Next, we will analyze the division of ordinary fractions into decimals and vice versa, and at the end we will see how to correctly divide fractions ending in 0, 1, 0, 01, 100, 10, etc.

Here we take only cases with positive fractions. If there is a minus before the fraction, then to act with it, you need to study the material on the division of rational and real numbers.

Yandex.RTB R-A-339285-1

All decimal fractions, both finite and periodic, are just a special form of writing ordinary fractions. Therefore, the same principles apply to them as to their corresponding ordinary fractions. Thus, we reduce the whole process of dividing decimal fractions to replacing them with ordinary ones, followed by calculation by methods already known to us. Let's take a specific example.

Example 1

Divide 1.2 by 0.48.

Solution

We write decimal fractions in the form of ordinary fractions. We will be able to:

1 , 2 = 12 10 = 6 5

0 , 48 = 48 100 = 12 25 .

Thus, we need to divide 6 5 by 12 25 . We believe:

1, 2: 0, 48 = 6 2: 12 25 = 6 5 25 12 = 6 25 5 12 = 5 2

From the resulting improper fraction you can select the whole part and get mixed number 2 1 2, or you can represent it as a decimal fraction so that it matches the original numbers: 5 2 \u003d 2, 5. How to do this, we have already written earlier.

Answer: 1 , 2: 0 , 48 = 2 , 5 .

Example 2

Calculate how many will be 0 , (504) 0 , 56 .

Solution

First, we need to convert a periodic decimal fraction to an ordinary one.

0 , (504) = 0 , 504 1 - 0 , 001 = 0 , 504 0 , 999 = 504 999 = 56 111

After that, we will also translate the final decimal fraction into another form: 0, 56 = 56 100. Now we have two numbers with which it will be easy for us to carry out the necessary calculations:

0 , (504) : 1 , 11 = 56 111: 56 100 = 56 111 100 56 = 100 111

We have a result that we can also convert to decimal. To do this, divide the numerator by the denominator using the column method:

Answer: 0 , (504) : 0 , 56 = 0 , (900) .

If, in the division example, we met non-periodic decimal fractions, then we will act a little differently. We cannot bring them to the usual ordinary fractions, so when dividing, we have to first round them up to a certain digit. This action must be performed both with the dividend and with the divisor: we will also round the existing finite or periodic fraction in the interests of accuracy.

Example 3

Find how much will be 0, 779 ... / 1, 5602.

Solution

First of all, we round both fractions to hundredths. This is how we move from infinite non-recurring fractions to finite decimals:

0 , 779 … ≈ 0 , 78

1 , 5602 ≈ 1 , 56

We can continue the calculations and get an approximate result: 0, 779 ...: 1, 5602 ≈ 0, 78: 1, 56 = 78 100: 156 100 = 78 100 100 156 = 78 156 = 1 2 = 0, 5.

The accuracy of the result will depend on the degree of rounding.

Answer: 0 , 779 … : 1 , 5602 ≈ 0 , 5 .

How to divide a natural number by a decimal and vice versa

The approach to division in this case is almost the same: we replace finite and periodic fractions with ordinary ones, and round off infinite non-periodic ones. Let's start with the example of division with a natural number and a decimal fraction.

Example 4

Divide 2.5 by 45.

Solution

Let's bring 2, 5 to the form of an ordinary fraction: 255 10 \u003d 51 2. Next, we just need to divide it into natural number. We already know how to do this:

25, 5: 45 = 51 2: 45 = 51 2 1 45 = 17 30

If we translate the result into decimal notation, then we get 0 , 5 (6) .

Answer: 25 , 5: 45 = 0 , 5 (6) .

The method of division by a column is good not only for natural numbers. By analogy, we can use it for fractions as well. Below we will indicate the sequence of actions that need to be carried out for this.

Definition 1

To divide a column of decimal fractions by natural numbers, you must:

1. Add a few zeros to the decimal fraction on the right (for division, we can add any number of them that we need).

2. Divide a decimal fraction by a natural number using an algorithm. When the division of the integer part of the fraction comes to an end, we put a comma in the resulting quotient and count further.

The result of such a division can be either a finite or an infinite periodic decimal fraction. It depends on the remainder: if it is zero, then the result will be finite, and if the remainders begin to repeat, then the answer will be a periodic fraction.

Let's take a few tasks as an example and try to complete these steps with specific numbers.

Example 5

Calculate how much will be 65 , 14 4 .

Solution

We use the column method. To do this, add two zeros to the fraction and get the decimal fraction 65, 1400, which will be equal to the original. Now we write a column for dividing by 4:

The resulting number will be the result of dividing the integer part we need. We put a comma, separating it, and continue:

We have reached the zero remainder, therefore, the division process is completed.

Answer: 65 , 14: 4 = 16 , 285 .

Example 6

Divide 164.5 by 27.

Solution

We divide the fractional part first and get:

We separate the resulting figure with a comma and continue to divide:

We see that the remainders began to repeat periodically, and the numbers nine, two and five began to alternate in the quotient. We will stop there and write the answer as a periodic fraction 6, 0 (925) .

Answer: 164 , 5: 27 = 6 , 0 (925) .

Such a division can be reduced to the process of finding a private decimal fraction and a natural number already described above. To do this, we need to multiply the dividend and the divisor by 10, 100, etc. so that the divisor turns into a natural number. Then we perform the above sequence of actions. This approach is possible due to the properties of division and multiplication. In literal form, we wrote them like this:

a: b = (a 10) : (b 10) , a: b = (a 100) : (b 100) and so on.

Let's formulate the rule:

Definition 2

To divide one final decimal fraction by another, you must:

1. Move the comma in the dividend and divisor to the right by the number of characters that is necessary to turn the divisor into a natural number. If there are not enough signs in the dividend, we add zeros to it on the right side.

2. After that, we divide the fraction by a column by the resulting natural number.

Let's take a look at a specific problem.

Example 7

Divide 7, 287 by 2, 1.

Solution: To make the divisor a natural number, we need to move the comma one character to the right. So we moved on to dividing the decimal fraction 72, 87 by 21. Let's write down the obtained numbers in a column and calculate

Answer: 7 , 287: 2 , 1 = 3 , 47

Example 8

Calculate 16 , 3 0 , 021 .

Solution

We will have to move the comma to three digits. There are not enough numbers in the divisor for this, so you need to use extra zeros. We think the end result will be:

We see the periodic repetition of residues 4 , 19 , 1 , 10 , 16 , 13 . The quotient repeats 1 , 9 , 0 , 4 , 7 and 5 . Then our result is the periodic decimal 776 , (190476) .

Answer: 16 , 3: 0 , 021 = 776 , (190476) ​​​​​​

The method described by us allows you to do the opposite, that is, divide a natural number by a final decimal fraction. Let's see how it's done.

Example 9

Calculate how many will be 3 5 , 4 .

Solution

Obviously, we will have to move the comma to the right by one character. After that we can start dividing 30 , 0 by 54 . Let's write the data in a column and calculate the result:

Repeating the remainder gives us the number 0 , (5) , which is a periodic decimal.

Answer: 3: 5 , 4 = 0 , (5) .

How to divide decimals by 1000, 100, 10, etc.

According to the already studied rules for dividing ordinary fractions, dividing a fraction into tens, hundreds, thousands is similar to multiplying it by 1/1000, 1/100, 1/10, etc. It turns out that in order to perform the division, in this case, it is enough just to move the comma to right amount digits. If there are not enough values ​​in the number to transfer, you need to add the required number of zeros.

Example 10

So, 56, 21: 10 = 5, 621, and 0, 32: 100,000 = 0, 0000032.

In the case of infinite decimals, we do the same.

Example 11

For example, 3 , (56) : 1000 = 0 , 003 (56) and 593 , 374 ... : 100 = 5 , 93374 ... .

How to divide decimals by 0.001, 0.01, 0.1, etc.

Using the same rule, we can also divide fractions by the specified values. This action will be similar to multiplying by 1000 , 100 , 10 respectively. To do this, we move the comma to one, two or three digits, depending on the conditions of the problem, and add zeros if there are not enough digits in the number.

Example 12

For example, 5, 739: 0, 1 = 57, 39 and 0, 21: 0, 00001 = 21,000.

This rule also applies to infinite decimals. We only advise you to be careful with the period of the fraction that is obtained in the answer.

So, 7 , 5 (716) : 0 , 01 = 757 , (167) , because after we moved the comma in the decimal notation 7 , 5716716716 ... two digits to the right, we got 757 , 167167 ... .

If we have non-periodic fractions in the example, then everything is simpler: 394 , 38283 ... : 0 , 001 = 394382 , 83 ... .

How to divide a mixed number or a common fraction by a decimal and vice versa

We also reduce this action to operations with ordinary fractions. To do this, you need to replace decimal numbers corresponding ordinary fractions, and write the mixed number as an improper fraction.

If we divide a non-periodic fraction by an ordinary or mixed number, we need to do the opposite, replacing common fraction or a mixed number with their corresponding decimal fraction.

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Find the first digit of the quotient (the result of division). To do this, divide the first digit of the dividend by the divisor. Write the result under the divisor.

• In our example, the first digit of the dividend is 3. Divide 3 by 12. Since 3 is less than 12, then the result of the division will be 0. Write 0 under the divisor - this is the first digit of the quotient.

Consider examples of dividing decimals in this light.

Example.

Divide decimal 1.2 by decimal 0.48.

Solution.

Answer:

1,2:0,48=2,5 .

Example.

Divide the periodic decimal 0.(504) by the decimal 0.56 .

Solution.

Let's translate the periodic decimal fraction into an ordinary:. We also translate the final decimal fraction 0.56 into an ordinary one, we have 0.56 \u003d 56/100. Now we can move from dividing the original decimals to dividing ordinary fractions and finish the calculations: .

Let's translate the resulting ordinary fraction into a decimal fraction by dividing the numerator by the denominator in a column:

Answer:

0,(504):0,56=0,(900) .

The principle of dividing infinite non-periodic decimal fractions differs from the principle of dividing finite and periodic decimal fractions, since non-repeating decimal fractions cannot be converted to ordinary fractions. The division of infinite non-periodic decimal fractions is reduced to the division of finite decimal fractions, for which it is carried out rounding numbers up to a certain level. Moreover, if one of the numbers with which the division is carried out is a final or periodic decimal fraction, then it is also rounded to the same digit as the non-periodic decimal fraction.

Example.

Divide the infinite non-recurring decimal 0.779... by the final decimal 1.5602.

Solution.

First, you need to round the decimal fractions in order to go from dividing an infinite non-repeating decimal fraction to dividing finite decimal fractions. We can round to hundredths: 0.779…≈0.78 and 1.5602≈1.56. Thus, 0.779…:1.5602≈0.78:1.56= 78/100:156/100=78/100 100/156= 78/156=1/2=0,5 .

Answer:

0,779…:1,5602≈0,5 .

Dividing a natural number by a decimal fraction and vice versa

The essence of the approach to dividing a natural number by a decimal fraction and to dividing a decimal fraction by a natural number is no different from the essence of dividing decimal fractions. That is, finite and periodic fractions are replaced by ordinary fractions, and infinite non-periodic fractions are rounded.

To illustrate, consider the example of dividing a decimal fraction by a natural number.

Example.

Divide the decimal fraction 25.5 by the natural number 45.

Solution.

Replacing the decimal fraction 25.5 with an ordinary fraction 255/10=51/2, division is reduced to dividing an ordinary fraction by a natural number: . The resulting fraction in decimal notation is 0.5(6) .

Answer:

25,5:45=0,5(6) .

Division of a decimal fraction by a natural number by a column

Division of final decimal fractions by natural numbers is conveniently carried out by a column by analogy with division by a column of natural numbers. Here is the division rule.

To divide a decimal by a natural number by a column, necessary:

• add a few digits to the right in the divisible decimal fraction 0, (during the division, if necessary, you can add any number of zeros, but these zeros may not be needed);
• perform division by a column of a decimal fraction by a natural number according to all the rules for dividing by a column of natural numbers, but when the division of the integer part of the decimal fraction is completed, then in the private one you need to put a comma and continue the division.

Let's say right away that as a result of dividing a finite decimal fraction by a natural number, either a final decimal fraction or an infinite periodic decimal fraction can be obtained. Indeed, after the division of all decimal places of the divisible fraction other than 0, we can get either a remainder 0, and we will get a final decimal fraction, or the remainder will begin to repeat periodically, and we will get a periodic decimal fraction.

Let's deal with all the intricacies of dividing decimal fractions into natural numbers by a column when solving examples.

Example.

Divide the decimal 65.14 by 4 .

Solution.

Let's perform the division of a decimal fraction by a natural number by a column. Let's add a pair of zeros to the right in the record of the fraction 65.14, while we get the decimal fraction equal to it 65.1400 (see equal and unequal decimal fractions). Now you can start dividing the integer part of the decimal fraction 65.1400 by a natural number 4 by a column:

This completes the division of the integer part of the decimal fraction. Here in private you need to put a decimal point and continue the division:

We have come to a remainder of 0, at this stage the division by a column ends. As a result, we have 65.14:4=16.285.

Answer:

65,14:4=16,285 .

Example.

Divide 164.5 by 27.

Solution.

Let's divide a decimal fraction by a natural number by a column. After dividing the integer part, we get the following picture:

Now we put a comma in private and continue the division with a column:

Now it is clearly seen that the remnants of 25, 7 and 16 have begun to repeat, while the numbers 9, 2 and 5 are repeated in the quotient. So dividing the decimal 164.5 by 27 gives us the periodic decimal 6.0(925) .

Answer:

164,5:27=6,0(925) .

Division of decimal fractions by a column

The division of a decimal fraction by a decimal fraction can be reduced to dividing a decimal fraction by a natural number by a column. To do this, the dividend and the divisor must be multiplied by such a number 10, or 100, or 1000, etc., so that the divisor becomes a natural number, and then divide by a natural number by a column. We can do this due to the properties of division and multiplication, since a:b=(a 10):(b 10) , a:b=(a 100):(b 100) and so on.

In other words, to divide a ending decimal by a ending decimal, need:

• in the dividend and divisor, move the comma to the right by as many characters as there are after the decimal point in the divisor, if at the same time there are not enough characters in the dividend to move the comma, then you need to add required amount zeros on the right;
• after that, carry out the division by a column of a decimal fraction by a natural number.

Consider, when solving an example, the application of this rule for dividing by a decimal fraction.

Example.

Do column division 7.287 by 2.1.

Solution.

Let's move the comma in these decimal fractions one digit to the right, this will allow us to go from dividing the decimal fraction 7.287 by the decimal fraction 2.1 to dividing the decimal fraction 72.87 by the natural number 21. Let's divide by a column:

Answer:

7,287:2,1=3,47 .

Example.

Divide decimal 16.3 by decimal 0.021.

Solution.

Move the comma in the dividend and divisor to the right by 3 digits. Obviously, there are not enough digits in the divisor to carry the comma, so let's add the required number of zeros to the right. Now let's divide the column of the fraction 16300.0 by the natural number 21:

From this moment, the remainders 4, 19, 1, 10, 16 and 13 begin to repeat, which means that the numbers 1, 9, 0, 4, 7 and 6 in the quotient will also repeat. As a result, we get a periodic decimal fraction 776,(190476) .

Answer:

16,3:0,021=776,(190476) .

Note that the voiced rule allows you to divide a natural number by a final decimal fraction by a column.

Example.

Divide the natural number 3 by the decimal fraction 5.4.

Solution.

After moving the comma 1 digit to the right, we come to dividing the number 30.0 by 54. Let's divide by a column:
.

This rule can also be applied when dividing infinite decimal fractions by 10, 100, .... For example, 3,(56):1000=0.003(56) and 593.374…:100=5.93374… .

Dividing decimals by 0.1, 0.01, 0.001, etc.

Since 0.1 \u003d 1/10, 0.01 \u003d 1/100, etc., it follows from the rule of division by an ordinary fraction that dividing a decimal fraction by 0.1, 0.01, 0.001, etc. . it's like multiplying the given decimal by 10 , 100 , 1000 , etc. respectively.

In other words, to divide a decimal fraction by 0.1, 0.01, ... you need to move the comma to the right by 1, 2, 3, ... digits, and if there are not enough digits in the decimal fraction to move the comma, then you need to add the required number to the right zeros.

For example, 5.739:0.1=57.39 and 0.21:0.00001=21,000 .

The same rule can be applied when dividing infinite decimals by 0.1, 0.01, 0.001, etc. In this case, you should be very careful with the division of periodic fractions, so as not to be mistaken with the period of the fraction, which is obtained as a result of division. For example, 7.5(716):0.01=757,(167) , since after moving the comma in the decimal fraction record 7.5716716716 ... two digits to the right, we have the record 757.167167 ... . With infinite non-periodic decimals, everything is simpler: 394,38283…:0,001=394382,83… .

Dividing a fraction or mixed number by a decimal and vice versa

The division of an ordinary fraction or a mixed number by a finite or periodic decimal fraction, as well as the division of a finite or periodic decimal fraction by an ordinary fraction or a mixed number, is reduced to the division of ordinary fractions. To do this, decimal fractions are replaced by the corresponding ordinary fractions, and the mixed number is represented as an improper fraction.

When dividing an infinite non-periodic decimal fraction by an ordinary fraction or a mixed number and vice versa, one should proceed to the division of decimal fractions, replacing the ordinary fraction or mixed number with the corresponding decimal fraction.

Bibliography.

• Maths: studies. for 5 cells. general education institutions / N. Ya. Vilenkin, V. I. Zhokhov, A. S. Chesnokov, S. I. Shvartsburd. - 21st ed., erased. - M.: Mnemosyne, 2007. - 280 p.: ill. ISBN 5-346-00699-0.
• Maths. Grade 6: textbook. for general education institutions / [N. Ya. Vilenkin and others]. - 22nd ed., Rev. - M.: Mnemosyne, 2008. - 288 p.: ill. ISBN 978-5-346-00897-2.
• Algebra: textbook for 8 cells. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M. : Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
• Gusev V. A., Mordkovich A. G. Mathematics (a manual for applicants to technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.

In the last lesson, we learned how to add and subtract decimal fractions (see the lesson " Adding and subtracting decimal fractions"). At the same time, they estimated how much the calculations are simplified compared to the usual “two-story” fractions.

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

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

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

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

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

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

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

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

Decimal multiplication

The multiplication operation consists of three consecutive steps:

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

Let me remind you once again that zeros on the sides of the significant part are never taken into account. Ignoring this rule leads to errors.

1. 0.28 12.5;
2. 6.3 1.08;
3. 132.5 0.0034;
4. 0.0108 1600.5;
5. 5.25 10,000.

We work with the first expression: 0.28 12.5.

1. Let's write out the significant parts for the numbers from this expression: 28 and 125;
2. Their product: 28 125 = 3500;
3. In the first multiplier, the decimal point is shifted 2 digits to the right (0.28 → 28), and in the second - by another 1 digit. In total, a shift to the left by three digits is needed: 3500 → 3.500 = 3.5.

Now let's deal with the expression 6.3 1.08.

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

We got to the third expression: 132.5 0.0034.

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

The following expression: 0.0108 1600.5.

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

Finally, last expression: 5.25 10,000.

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

Note the last example: since the decimal point is moved to different directions, the total shift is found through the difference. This is very important point! Here's another example:

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

Decimal division

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

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

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

A task. Find the value of the expression:

1. 3,51: 3,9;
2. 1,47: 2,1;
3. 6,4: 25,6:
4. 0,0425: 2,5;
5. 0,25: 0,002.

We consider the first expression. First, let's convert obi fractions to decimals:

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

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

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

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

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

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

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