Column division(you can also see the name division corner) is a standard procedure inarithmetic, designed to divide simple or complex multi-digit numbers by breakingdivision into a number of simpler steps. As in all division problems, a single number, calleddivisible, is divided into another, calleddivider, producing a result calledprivate.

The column can be used to divide both natural numbers without a remainder, and the division of natural numbers with the rest.

Rules for recording when dividing by a column.

Let's start by studying the rules for writing the dividend, divisor, all intermediate calculations and results whendivision of natural numbers by a column. Let's say right away that in writing to perform division by a columnit is most convenient on paper with a checkered line - so there is less chance of straying from the desired row and column.

First, the dividend and the divisor are written in one line from left to right, after which between the writtennumbers represent the symbol of the form.

For example, if the dividend is the number 6105, and the divisor is 55, then their correct notation when dividing inthe column will look like this:

Look at the following diagram illustrating the places to write the dividend, divisor, quotient,remainder and intermediate calculations when dividing by a column:

It can be seen from the above diagram that the desired quotient (or incomplete quotient when dividing with a remainder) will bewritten below the divisor under the horizontal bar. And intermediate calculations will be carried out belowdivisible, and you need to take care of the availability of space on the page in advance. In doing so, one should be guidedrule: the greater the difference in the number of characters in the records of the dividend and divisor, the morespace is required.

Division by a column of a natural number by a single-digit natural number, column division algorithm.

How to divide into a column is best explained with an example.Calculate:

512:8=?

First, write down the dividend and the divisor in a column. It will look like this:

Their quotient (result) will be written under the divisor. Our number is 8.

1. We define an incomplete quotient. First, we look at the first digit from the left in the dividend entry.If the number defined by this figure is greater than the divisor, then in the next paragraph we have to workwith this number. If this number is less than the divisor, then we need to add to the consideration the followingon the left, the digit in the record of the dividend, and work further with the number determined by the two considerednumbers. For convenience, we select in our record the number with which we will work.

2. Take 5. The number 5 is less than 8, so you need to take one more digit from the dividend. 51 is greater than 8. So.this is an incomplete quotient. We put a point in the quotient (under the corner of the divider).

After 51 there is only one number 2. So we add one more point to the result.

3. Now, remembering multiplication table by 8, we find the product closest to 51 → 6 x 8 = 48→ write the number 6 in the quotient:

We write 48 under 51 (if we multiply 6 from the quotient by 8 from the divisor, we get 48).

Attention! When written under an incomplete quotient, the rightmost digit of the incomplete quotient must be aboverightmost digit works.

4. Between 51 and 48 on the left, put "-" (minus). Subtract according to the rules of subtraction in column 48 and below the linewrite down the result.

However, if the result of the subtraction is zero, then it need not be written down (unless the subtraction inthis paragraph is not the very last action that completely completes the division process column).

The remainder turned out to be 3. Let's compare the remainder with the divisor. 3 is less than 8.

Attention!If the remainder is greater than the divisor, then we made a mistake in the calculation and there is a productcloser than the one we took.

5. Now under the horizontal line to the right of the numbers located there (or to the right of the place where we do notbegan to write down zero) we write down the figure located in the same column in the record of the dividend. If inthere are no digits in this column, then the division by a column ends here.

The number 32 is greater than 8. And again, using the multiplication table for 8, we find the nearest product → 8 x 4 = 32:

The remainder is zero. This means that the numbers are divided completely (without a remainder). If after the lastsubtracting zero, and there are no more digits left, then this is the remainder. We add it to the private inbrackets (e.g. 64(2)).

Division by a column of multivalued natural numbers.

Division by a natural multi-digit number is done in a similar way. At the same time, in the firstThe “intermediate” dividend includes so many high-order digits that it turns out to be more than the divisor.

For example, 1976 divided by 26.

• The number 1 in the most significant digit is less than 26, so consider a number made up of two digits senior ranks - 19.
• The number 19 is also less than 26, so consider the number made up of the digits of the three most significant digits - 197.
• The number 197 is greater than 26, divide 197 tens by 26: 197: 26 = 7 (15 tens left).
• We translate 15 tens into units, add 6 units from the category of units, we get 156.
• Divide 156 by 26 to get 6.

So 1976: 26 = 76.

If at some division step the “intermediate” dividend turned out to be less than the divisor, then in the quotient0 is written, and the number from this digit is transferred to the next, lower digit.

Division with a decimal fraction in a quotient.

Decimal fractions online. Convert decimals to common fractions and common fractions to decimals.

If a natural number is not evenly divisible by a single-digit natural number, you can continuebitwise division and get a quotient decimal.

For example, 64 divided by 5.

• Divide 6 tens by 5 to get 1 tens and 1 tens remainder.
• We translate the remaining ten into units, add 4 from the category of units, we get 14.
• 14 units divided by 5, we get 2 units and 4 units in the remainder.
• We translate 4 units into tenths, we get 40 tenths.
• Divide 40 tenths by 5 to get 8 tenths.

So 64:5 = 12.8

Thus, if when dividing a natural number by a natural one-digit or many-digit numberthe remainder is obtained, then you can put in a private comma, convert the remainder to the units of the next,smaller digit and continue dividing.

Instruction

Before teaching how to divide two-digit numbers, it is necessary to explain to the child that a number is the sum of tens and ones. This will save him from a future rather common mistake that many children make. They begin to divide the first and second digits of the dividend and divisor into each other.

First, work from numbers to single digits. This technique is best practiced using the knowledge of the multiplication table. The more such practice, the better. The skills of such division should be brought to automaticity, then it will be easier for the child to move on to the more complex topic of the two-digit divisor, which, like the dividend, is the sum of tens and ones.

The most common way of dividing two-digit numbers is the selection method, which involves dividing successively by numbers from 2 to 9 so that the final product equals the dividend. Example: Divide 87 by 29. Reason as follows:

29 times 2 equals 54 - not enough;
29 x 3 = 87 is correct.

Pay the student's attention to the second digits (units) of the dividend and divisor, which are convenient to navigate when using the multiplication table. For example, in the example above, the second digit of the divisor is 9. Think about how much you need to multiply the number 9 so that the number of units of the product is 7? The answer in this case is only one - by 3. This greatly simplifies the task of two-digit division. Test your guess by multiplying the whole number 29.

If the task is performed in writing, then it is advisable to use the method of dividing into a column. This approach is similar to the previous one, except that the student does not need to keep the numbers in his head and do mental calculations. It is better to arm yourself with a pencil or a draft sheet for written work.

Sources:

• multiplication of two-digit numbers by two-digit tables

The topic of dividing numbers is one of the most important in the 5th grade math program. Without mastering this knowledge, further study of mathematics is impossible. Divide numbers come into life every day. And don't always rely on a calculator. To separate two numbers, you need to remember a certain sequence of actions.

You will need

• Checkered sheet of paper
• pen or pencil

Instruction

Write the dividend and on one line. Separate them with a vertical bar two lines high. Draw a horizontal line under the divisor and dividend perpendicular to the previous line. To the right, under this line, the quotient will be written. Below and to the left of the dividend, under the horizontal line, write zero.

Move one leftmost, but not yet transferred, digit of the dividend down under the last horizontal line. Mark the transferred digit of the dividend with a dot.

Compare the number under the last horizontal bar with the divisor. If the number is less than the divisor, then continue with step 4, otherwise go to step 5.

The division of natural numbers, especially multi-valued ones, is conveniently carried out by a special method, which is called division by a column (in a column). You can also see the name corner division. Immediately, we note that the column can be carried out both division of natural numbers without a remainder, and division of natural numbers with a remainder.

In this article, we will understand how division by a column is performed. Here we will talk about the writing rules, and about all intermediate calculations. First, let us dwell on the division of a multi-valued natural number by a single-digit number by a column. After that, we will focus on cases where both the dividend and the divisor are multi-valued natural numbers. The whole theory of this article is provided with characteristic examples of division by a column of natural numbers with detailed explanations of the solution and illustrations.

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Rules for recording when dividing by a column

Let's start by studying the rules for writing the dividend, divisor, all intermediate calculations and results when dividing natural numbers by a column. Let's say right away that it is most convenient to divide in a column in writing on paper with a checkered line - so there is less chance of going astray from the desired row and column.

First, the dividend and the divisor are written in one line from left to right, after which a symbol of the form is displayed between the written numbers. For example, if the dividend is the number 6 105, and the divisor is 5 5, then their correct notation when divided into a column will be:

Look at the following diagram, which illustrates the places for writing the dividend, divisor, quotient, remainder, and intermediate calculations when dividing by a column.

It can be seen from the above diagram that the desired quotient (or incomplete quotient when dividing with a remainder) will be written below the divisor under the horizontal line. And intermediate calculations will be carried out below the dividend, and you need to take care of the availability of space on the page in advance. In this case, one should be guided by the rule: the greater the difference in the number of characters in the entries of the dividend and divisor, the more space is required. For example, when dividing a natural number 614,808 by 51,234 by a column (614,808 is a six-digit number, 51,234 is a five-digit number, the difference in the number of characters in the records is 6−5=1), intermediate calculations will require less space than when dividing numbers 8 058 and 4 (here the difference in the number of characters is 4−1=3 ). To confirm our words, we present the completed records of division by a column of these natural numbers:

Now you can go directly to the process of dividing natural numbers by a column.

Division by a column of a natural number by a single-digit natural number, division by a column algorithm

It is clear that dividing one single-digit natural number by another is quite simple, and there is no reason to divide these numbers into a column. However, it will be useful to practice the initial skills of division by a column on these simple examples.

Example.

Let us need to divide by a column 8 by 2.

Solution.

Of course, we can perform division using the multiplication table, and immediately write down the answer 8:2=4.

But we are interested in how to divide these numbers by a column.

First, we write the dividend 8 and the divisor 2 as required by the method:

Now we start to figure out how many times the divisor is in the dividend. To do this, we successively multiply the divisor by the numbers 0, 1, 2, 3, ... until the result is a number equal to the dividend (or a number greater than the dividend, if there is a division with a remainder). If we get a number equal to the dividend, then we immediately write it under the dividend, and in place of the private we write the number by which we multiplied the divisor. If we get a number greater than the divisible, then under the divisor we write the number calculated at the penultimate step, and in place of the incomplete quotient we write the number by which the divisor was multiplied at the penultimate step.

Let's go: 2 0=0 ; 2 1=2 ; 2 2=4 ; 2 3=6 ; 2 4=8 . We got a number equal to the dividend, so we write it under the dividend, and in place of the private we write the number 4. The record will then look like this:

The final stage of dividing single-digit natural numbers by a column remains. Under the number written under the dividend, you need to draw a horizontal line, and subtract numbers above this line in the same way as it is done when subtracting natural numbers with a column. The number obtained after subtraction will be the remainder of the division. If it is equal to zero, then the original numbers are divided without a remainder.

In our example, we get

Now we have a finished record of division by a column of the number 8 by 2. We see that the quotient 8:2 is 4 (and the remainder is 0 ).

Answer:

8:2=4 .

Now consider how the division by a column of single-digit natural numbers with a remainder is carried out.

Example.

Divide by a column 7 by 3.

Solution.

At the initial stage, the entry looks like this:

We begin to find out how many times the dividend contains a divisor. We will multiply 3 by 0, 1, 2, 3, etc. until we get a number equal to or greater than the dividend 7. We get 3 0=0<7 ; 3·1=3<7 ; 3·2=6<7 ; 3·3=9>7 (if necessary, refer to the article comparison of natural numbers). Under the dividend we write the number 6 (it was obtained at the penultimate step), and in the place of the incomplete quotient we write the number 2 (multiplication was carried out on it at the penultimate step).

It remains to carry out the subtraction, and the division by a column of single-digit natural numbers 7 and 3 will be completed.

So the partial quotient is 2 , and the remainder is 1 .

Answer:

7:3=2 (rest. 1) .

Now we can move on to dividing multi-valued natural numbers by single-digit natural numbers by a column.

Now we will analyze column division algorithm. At each stage, we will present the results obtained by dividing the many-valued natural number 140 288 by the single-valued natural number 4 . This example was not chosen by chance, since when solving it, we will encounter all possible nuances, we will be able to analyze them in detail.

First, we look at the first digit from the left in the dividend entry. If the number defined by this figure is greater than the divisor, then in the next paragraph we have to work with this number. If this number is less than the divisor, then we need to add the next digit to the left in the dividend record, and work further with the number determined by the two digits in question. For convenience, we select in our record the number with which we will work.

The first digit from the left in the dividend 140,288 is the number 1. The number 1 is less than the divisor 4, so we also look at the next digit on the left in the dividend record. At the same time, we see the number 14, with which we have to work further. We select this number in the notation of the dividend.

The following points from the second to the fourth are repeated cyclically until the division of natural numbers by a column is completed.

Now we need to determine how many times the divisor is contained in the number we are working with (for convenience, let's denote this number as x ). To do this, we successively multiply the divisor by 0, 1, 2, 3, ... until we get the number x or a number greater than x. When the number x is obtained, then we write it under the selected number according to the notation rules used when subtracting by a column of natural numbers. The number by which the multiplication was carried out is written in place of the quotient during the first pass of the algorithm (during subsequent passes of 2-4 points of the algorithm, this number is written to the right of the numbers already there). When a number is obtained that is greater than the number x, then under the selected number we write the number obtained at the penultimate step, and in place of the quotient (or to the right of the numbers already there) we write the number by which the multiplication was carried out at the penultimate step. (We carried out similar actions in the two examples discussed above).

We multiply the divisor of 4 by the numbers 0 , 1 , 2 , ... until we get a number that is equal to 14 or greater than 14 . We have 4 0=0<14 , 4·1=4<14 , 4·2=8<14 , 4·3=12<14 , 4·4=16>fourteen . Since at the last step we got the number 16, which is greater than 14, then under the selected number we write the number 12, which turned out at the penultimate step, and in place of the quotient we write the number 3, since in the penultimate paragraph the multiplication was carried out exactly on it.


At this stage, from the selected number, subtract the number below it in a column. Below the horizontal line is the result of the subtraction. However, if the result of the subtraction is zero, then it does not need to be written down (unless the subtraction at this point is the very last action that completely completes the division by a column). Here, for your control, it will not be superfluous to compare the result of subtraction with the divisor and make sure that it is less than the divisor. Otherwise, a mistake has been made somewhere.

We need to subtract the number 12 from the number 14 in a column (for the correct notation, you must not forget to put a minus sign to the left of the subtracted numbers). After the completion of this action, the number 2 appeared under the horizontal line. Now we check our calculations by comparing the resulting number with a divisor. Since the number 2 is less than the divisor 4, you can safely move on to the next item.


Now, under the horizontal line to the right of the numbers located there (or to the right of the place where we did not write zero), we write down the number located in the same column in the record of the dividend. If there are no numbers in the record of the dividend in this column, then the division by a column ends here. After that, we select the number formed under the horizontal line, take it as a working number, and repeat with it from 2 to 4 points of the algorithm.

Under the horizontal line to the right of the number 2 already there, we write the number 0, since it is the number 0 that is in the record of the dividend 140 288 in this column. Thus, the number 20 is formed under the horizontal line.


We select this number 20, take it as a working number, and repeat the actions of the second, third and fourth points of the algorithm with it.

We multiply the divisor of 4 by 0 , 1 , 2 , ... until we get the number 20 or a number that is greater than 20 . We have 4 0=0<20 , 4·1=4<20 , 4·2=8<20 , 4·3=12<20 , 4·4=16<20 , 4·5=20 . Так как мы получили число, равное числу 20 , то записываем его под отмеченным числом, а на месте частного, справа от уже имеющегося там числа 3 записываем число 5 (на него производилось умножение).


We carry out subtraction by a column. Since we subtract equal natural numbers, then, due to the property of subtracting equal natural numbers, we get zero as a result. We do not write down zero (since this is not yet the final stage of dividing by a column), but we remember the place where we could write it down (for convenience, we will mark this place with a black rectangle).


Under the horizontal line to the right of the memorized place, we write down the number 2, since it is she who is in the record of the dividend 140 288 in this column. Thus, under the horizontal line we have the number 2 .


We take the number 2 as a working number, mark it, and once again we will have to perform the steps from 2-4 points of the algorithm.

We multiply the divisor by 0 , 1 , 2 and so on, and compare the resulting numbers with the marked number 2 . We have 4 0=0<2 , 4·1=4>2. Therefore, under the marked number, we write the number 0 (it was obtained at the penultimate step), and in place of the quotient to the right of the number already there, we write the number 0 (we multiplied by 0 at the penultimate step).


We perform subtraction by a column, we get the number 2 under the horizontal line. We check ourselves by comparing the resulting number with the divisor 4 . Since 2<4 , то можно спокойно двигаться дальше.


Under the horizontal line to the right of the number 2, we add the number 8 (since it is in this column in the record of the dividend 140 288). Thus, under the horizontal line is the number 28.


We accept this number as a worker, mark it, and repeat steps 2-4 of paragraphs.

There shouldn't be any problems here if you've been careful up to now. Having done all the necessary actions, the following result is obtained.

It remains for the last time to carry out the actions from points 2, 3, 4 (we leave it to you), after which we get a complete picture of dividing natural numbers 140 288 and 4 in a column:

Please note that the number 0 is written at the very bottom of the line. If this were not the last step of division by a column (that is, if there were numbers in the columns on the right in the record of the dividend), then we would not write this zero.

Thus, looking at the completed record of dividing the multi-digit natural number 140 288 by the single-valued natural number 4, we see that the number 35 072 is private (and the remainder of the division is zero, it is in the very bottom line).

Of course, when dividing natural numbers by a column, you will not describe all your actions in such detail. Your solutions will look something like the following examples.

Example.

Perform long division if the dividend is 7136 and the divisor is a single natural number 9.

Solution.

At the first step of the algorithm for dividing natural numbers by a column, we get a record of the form

After performing the actions from the second, third and fourth points of the algorithm, the record of division by a column will take the form

Repeating the cycle, we will have

One more pass will give us a complete picture of division by a column of natural numbers 7 136 and 9

Thus, the partial quotient is 792 , and the remainder of the division is 8 .

Answer:

7 136:9=792 (rest 8) .

And this example demonstrates how long division should look like.

Example.

Divide the natural number 7 042 035 by the single digit natural number 7 .

Solution.

It is most convenient to perform division by a column.

Answer:

7 042 035:7=1 006 005 .

Division by a column of multivalued natural numbers

We hasten to please you: if you have well mastered the algorithm for dividing by a column from the previous paragraph of this article, then you already almost know how to perform division by a column of multivalued natural numbers. This is true, since steps 2 to 4 of the algorithm remain unchanged, and only minor changes appear in the first step.

At the first stage of dividing into a column of multi-valued natural numbers, you need to look not at the first digit on the left in the dividend entry, but at as many of them as there are digits in the divisor entry. If the number defined by these numbers is greater than the divisor, then in the next paragraph we have to work with this number. If this number is less than the divisor, then we need to add to the consideration the next digit on the left in the record of the dividend. After that, the actions indicated in paragraphs 2, 3 and 4 of the algorithm are performed until the final result is obtained.

It remains only to see the application of the algorithm for dividing by a column of multi-valued natural numbers in practice when solving examples.

Example.

Let's perform division by a column of multivalued natural numbers 5562 and 206.

Solution.

Since 3 characters are involved in the record of the divisor 206, we look at the first 3 digits on the left in the record of the dividend 5 562. These numbers correspond to the number 556. Since 556 is greater than the divisor 206, we take the number 556 as a working one, select it, and proceed to the next stage of the algorithm.

Now we multiply the divisor 206 by the numbers 0 , 1 , 2 , 3 , ... until we get a number that is either equal to 556 or greater than 556 . We have (if the multiplication is difficult, then it is better to perform the multiplication of natural numbers in a column): 206 0=0<556 , 206·1=206<556 , 206·2=412<556 , 206·3=618>556 . Since we got a number that is greater than the number 556, then under the selected number we write the number 412 (it was obtained at the penultimate step), and in place of the quotient we write the number 2 (since it was multiplied at the penultimate step). The column division entry takes the following form:

Perform column subtraction. We get the difference 144, this number is less than the divisor, so you can safely continue to perform the required actions.

Under the horizontal line to the right of the number available there, we write the number 2, since it is in the record of the dividend 5 562 in this column:

Now we work with the number 1442, select it, and go through steps two through four again.

We multiply the divisor 206 by 0 , 1 , 2 , 3 , ... until we get the number 1442 or a number that is greater than 1442 . Let's go: 206 0=0<1 442 , 206·1=206<1 442 , 206·2=412<1 332 , 206·3=618<1 442 , 206·4=824<1 442 , 206·5=1 030<1 442 , 206·6=1 236<1 442 , 206·7=1 442 . Таким образом, под отмеченным числом записываем 1 442 , а на месте частного правее уже имеющегося там числа записываем 7 :

We subtract by a column, we get zero, but we don’t write it down right away, but only remember its position, because we don’t know if the division ends here, or we will have to repeat the steps of the algorithm again:

Now we see that under the horizontal line to the right of the memorized position, we cannot write down any number, since there are no numbers in the record of the dividend in this column. Therefore, this division by a column is over, and we complete the entry:

• Maths. Any textbooks for grades 1, 2, 3, 4 of educational institutions.
• Maths. Any textbooks for 5 classes of educational institutions.

At school, these actions are studied from simple to complex. Therefore, it is certainly necessary to master the algorithm for performing the above operations using simple examples. So that later there will be no difficulties with dividing decimal fractions into a column. After all, this is the most difficult version of such tasks.

This subject requires consistent study. Gaps in knowledge are unacceptable here. This principle should be learned by every student already in the first grade. Therefore, if you skip several lessons in a row, you will have to master the material yourself. Otherwise, later there will be problems not only with mathematics, but also with other subjects related to it.

The second prerequisite for a successful study of mathematics is to move on to examples of division in a column only after addition, subtraction and multiplication have been mastered.

It will be difficult for a child to divide if he has not learned the multiplication table. By the way, it is better to learn it from the Pythagorean table. There is nothing superfluous, and multiplication is easier to digest in this case.

How are natural numbers multiplied in a column?

If there is a difficulty in solving examples in a column for division and multiplication, then it is necessary to begin to fix the problem with multiplication. Because division is the inverse of multiplication:

1. Before multiplying two numbers, you need to look at them carefully. Choose the one with more digits (longer), write it down first. Place the second one under it. Moreover, the numbers of the corresponding category should be under the same category. That is, the rightmost digit of the first number must be above the rightmost digit of the second.
2. Multiply the rightmost digit of the bottom number by each digit of the top number, starting from the right. Write the answer under the line so that its last digit is under the one by which it was multiplied.
3. Repeat the same with the other digit of the bottom number. But the result of the multiplication must be shifted one digit to the left. In this case, its last digit will be under the one by which it was multiplied.

Continue this multiplication in a column until the numbers in the second multiplier run out. Now they need to be folded. This will be the desired answer.

Algorithm for multiplying into a column of decimal fractions

First, it is supposed to imagine that not decimal fractions are given, but natural ones. That is, remove commas from them and then proceed as described in the previous case.

The difference begins when the answer is written. At this point, it is necessary to count all the numbers that are after the decimal points in both fractions. That is how many of them you need to count from the end of the answer and put a comma there.

It is convenient to illustrate this algorithm with an example: 0.25 x 0.33:

How to start learning to divide?

Before solving examples for division in a column, it is supposed to remember the names of the numbers that are in the example for division. The first of them (the one that divides) is the divisible. The second (divided by it) is a divisor. The answer is private.

After that, using a simple everyday example, we will explain the essence of this mathematical operation. For example, if you take 10 sweets, then it is easy to divide them equally between mom and dad. But what if you need to distribute them to your parents and brother?

After that, you can get acquainted with the rules of division and master them with specific examples. Simple ones at first, and then moving on to more and more complex ones.

Algorithm for dividing numbers into a column

First, we present the procedure for natural numbers that are divisible by a single-digit number. They will also be the basis for multi-digit divisors or decimal fractions. Only then it is supposed to make small changes, but more on that later:

• Before doing division in a column, you need to find out where the dividend and divisor are.
• Write down the dividend. To the right of it is a divider.
• Draw a corner on the left and bottom near the last corner.
• Determine the incomplete dividend, that is, the number that will be the minimum for division. Usually it consists of one digit, maximum of two.
• Choose the number that will be written first in the answer. It must be the number of times the divisor fits in the dividend.
• Write down the result of multiplying this number by a divisor.
• Write it under an incomplete divisor. Perform subtraction.
• Carry to the remainder the first digit after the part that has already been divided.
• Pick up the answer again.
• Repeat multiplication and subtraction. If the remainder is zero and the dividend is over, then the example is done. Otherwise, repeat the steps: demolish the number, pick up the number, multiply, subtract.

How to solve long division if there is more than one digit in the divisor?

The algorithm itself completely coincides with what was described above. The difference will be the number of digits in the incomplete dividend. Now there should be at least two of them, but if they turn out to be less than the divisor, then it is supposed to work with the first three digits.

There is another nuance in this division. The fact is that the remainder and the figure carried to it are sometimes not divisible by a divisor. Then it is supposed to attribute one more figure in order. But at the same time, the answer must be zero. If three-digit numbers are divided into a column, then more than two digits may need to be demolished. Then the rule is introduced: zeros in the answer should be one less than the number of digits taken down.

You can consider such a division using the example - 12082: 863.

• The incomplete divisible in it is the number 1208. The number 863 is placed in it only once. Therefore, in response, it is supposed to put 1, and write 863 under 1208.
• After subtraction, the remainder is 345.
• To him you need to demolish the number 2.
• In the number 3452, 863 fits four times.
• Four must be written in response. Moreover, when multiplied by 4, this number is obtained.
• The remainder after subtraction is zero. That is, the division is completed.

The answer in the example is 14.

What if the dividend ends in zero?

Or a few zeros? In this case, a zero remainder is obtained, and there are still zeros in the dividend. Do not despair, everything is easier than it might seem. It is enough just to attribute to the answer all the zeros that remained undivided.

For example, you need to divide 400 by 5. The incomplete dividend is 40. Five is placed in it 8 times. This means that the answer is supposed to be written 8. When subtracting, there is no remainder. That is, the division is over, but zero remains in the dividend. It will have to be added to the answer. Thus, dividing 400 by 5 gives 80.

What if you need to divide a decimal?

Again, this number looks like a natural number, if not for the comma separating the integer part from the fractional part. This suggests that the division of decimal fractions into a column is similar to the one described above.

The only difference will be the semicolon. It is supposed to be answered immediately, as soon as the first digit from the fractional part is taken down. In another way, it can be said like this: the division of the integer part has ended - put a comma and continue the solution further.

When solving examples for dividing into a column with decimal fractions, you need to remember that any number of zeros can be assigned to the part after the decimal point. Sometimes this is necessary in order to complete the numbers to the end.

Division of two decimals

It may seem complicated. But only at the beginning. After all, how to perform division in a column of fractions by a natural number is already clear. So, we need to reduce this example to the already familiar form.

Make it easy. You need to multiply both fractions by 10, 100, 1,000, or 10,000, or maybe a million if the task requires it. The multiplier is supposed to be chosen based on how many zeros are in the decimal part of the divisor. That is, as a result, it turns out that you will have to divide a fraction by a natural number.

And it will be in the worst case. After all, it may turn out that the dividend from this operation becomes an integer. Then the solution of the example with division into a column of fractions will be reduced to the simplest option: operations with natural numbers.

As an example: 28.4 divided by 3.2:

• First, they must be multiplied by 10, since in the second number there is only one digit after the decimal point. Multiplying will give 284 and 32.
• They are supposed to be divided. And at once the whole number is 284 by 32.
• The first matched number for the answer is 8. Multiplying it gives 256. The remainder is 28.
• The division of the integer part is over, and a comma is supposed to be put in the answer.
• Demolish to remainder 0.
• Take 8 again.
• Remainder: 24. Add another 0 to it.
• Now you need to take 7.
• The result of the multiplication is 224, the remainder is 16.
• Demolish another 0. Take 5 and get exactly 160. The remainder is 0.

Division completed. The result of the 28.4:3.2 example is 8.875.

What if the divisor is 10, 100, 0.1, or 0.01?

As with multiplication, long division is not needed here. It is enough just to move the comma in the right direction for a certain number of digits. Moreover, according to this principle, you can solve examples with both integers and decimal fractions.

So, if you need to divide by 10, 100 or 1000, then the comma is moved to the left by as many digits as there are zeros in the divisor. That is, when a number is divisible by 100, the comma should move to the left by two digits. If the dividend is a natural number, then it is assumed that the comma is at the end of it.

This action produces the same result as if the number were to be multiplied by 0.1, 0.01, or 0.001. In these examples, the comma is also moved to the left by a number of digits equal to the length of the fractional part.

When dividing by 0.1 (etc.) or multiplying by 10 (etc.), the comma should move to the right by one digit (or two, three, depending on the number of zeros or the length of the fractional part).

It is worth noting that the number of digits given in the dividend may not be sufficient. Then the missing zeros can be assigned to the left (in the integer part) or to the right (after the decimal point).

Division of periodic fractions

In this case, you will not be able to get the exact answer when dividing into a column. How to solve an example if a fraction with a period is encountered? Here it is necessary to move on to ordinary fractions. And then perform their division according to the previously studied rules.

For example, you need to divide 0, (3) by 0.6. The first fraction is periodic. It is converted to the fraction 3/9, which after reduction will give 1/3. The second fraction is the final decimal. It is even easier to write down an ordinary one: 6/10, which is equal to 3/5. The rule for dividing ordinary fractions prescribes to replace division with multiplication and the divisor with the reciprocal of a number. That is, the example boils down to multiplying 1/3 by 5/3. The answer is 5/9.

If the example has different fractions...

Then there are several possible solutions. First, you can try to convert an ordinary fraction to a decimal. Then divide already two decimals according to the above algorithm.

Secondly, every final decimal fraction can be written as a common fraction. It's just not always convenient. Most often, such fractions turn out to be huge. Yes, and the answers are cumbersome. Therefore, the first approach is considered more preferable.

Of course, children learn the basics of mathematics in the classroom at school. But the teacher's explanations are not always clear to the kid. Or maybe the child got sick and missed the topic. In such cases, parents should remember their school years in order to help the child not miss important information, without which further education will be unrealistic.

Teaching a child with a column begins in the third grade. By this time, the student should already be able to use the multiplication table with ease. But if there are problems with this, it is worth immediately, because before you teach a child to divide by a column, there should not be any difficulties with multiplication.

How to teach column division?

Take for example the three-digit number 372 and divide it by 6. Choose any combination, but so that the division goes without a trace. At first, this can confuse a young mathematician.

We write down the numbers, separating them with a corner, and explain to the child that we will gradually divide this large number into six equal parts. Let's first try to divide the first digit of 3 by 6.

It is not divisible, which means we add the second one, that is, let's try to see if we can divide 37.

It is necessary to ask the child how many times the six will fit in the number 37. Anyone who knows mathematics without problems will immediately guess that the selection method can be used to select the desired multiplier. So, let's pick up, take, for example, 5 and multiply by 6 - it turns out 30, it seems that the result is not far from 37, but it's worth trying again. To do this, we multiply 6 by 6 - equal to 36. This suits us, and the first digit of the private has already been found - we write it under the divisor, behind the line.

We write the number 36 under 37 and when subtracting we get one. It is again not divisible by 6, which means that we demolish the remaining deuce to it. Now the number 12 is very easy to divide by 6. As a result, we get the second private number - two. Our division result will be 62.