Multiplication and division into columns: examples

Mathematics is akin to puzzles. Especially it concerns division and multiplication in a column. In school, these actions are studied from simple to complex. Therefore, it is certainly necessary to well understand the algorithm for performing these operations on simple examples. That then there were no difficulties with division of decimal fractions in a column. After all, this is the most difficult version of such tasks.

Tips for those who want to know math well

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

The second obligatory condition for the successful study of mathematics is to go over to examples by division into a column only after the addition, subtraction and multiplication have been mastered.

The child will be difficult to divide if he has not learned the multiplication table. By the way, it is better to learn from the Pythagorean table. There is nothing superfluous, and the multiplication is assimilated in this case is simpler.

How are natural numbers multiplied in a column?

If there is a difficulty in solving the examples in the column for division and multiplication, then starting to fix the problem is relying on multiplication. Since division is the inverse operation of multiplication:

1. Before you multiply two numbers, you need to carefully look at them. Select the one with more digits (longer), write it first. Place the second below it. And the digits of the corresponding digit should be under the same rank. That is, the rightmost digit of the first number must be above the rightmost second.
2. Multiply the rightmost digit of the bottom number by each upper digit, starting from the right. Write down the answer below the line so that its last digit is under the one you multiplied.
3. Do the same with another lower figure. But the result of multiplication must be shifted one digit to the left. At the same time, his last figure will be under the one to which they multiplied.

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

The algorithm of multiplication in a column of decimals

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

The difference begins when the answer is recorded. At this point, you need to count all the numbers that are after the commas in both fractions. That's how much they need to be counted from the end of the answer and there put a comma.

It is convenient to illustrate this algorithm by the example: 0.25 x 0.33:

• To write these fractions it is necessary so that number 33 was under 25.
• Now the right-hand triple must be multiplied by 25. It turns out 75. To write it is supposed to be so that the five is under the triplet, to which the multiplication was performed.
• Then multiply 25 by the first 3. Again there will be 75, but it will be written so that 5 turned up under the 7 previous number.
• After adding these two numbers, we get 825. In decimal fractions, 4 digits are separated by commas. Therefore, in the answer you need to separate the comma, too, with 4 digits. But there are only three of them. To do this, before 8 it is necessary to write 0, put a comma, before it one more 0.
• The answer in the example is the number 0.0825.

How to start teaching the division?

Before you decide on examples of division into a column, you are supposed to remember the names of the numbers that stand in the example of division. The first of them (the one that divides) is the dividend. The second (divided into it) is a divisor. The answer is private.

After this, on a simple domestic example, let us explain the essence of this mathematical operation. For example, if you take 10 chocolates, then you can divide them equally between your mother and father. But what if you need to give them to your parents and brother?

After that, you can get acquainted with the rules of division and master them on specific examples. First simple, and then move on to an increasingly complex.

Algorithm of dividing numbers into columns

First, we will represent the order of actions for natural numbers that are divisible by a single-valued number. They will be the basis for many-valued dividers or decimals. Only then is it necessary to make small changes, but more about this later:

• Before dividing into a column, you need to find out where the dividend and divisor are.
• Write a dividend. To the right of it is a divider.
• Draw the left and the bottom near the last corner.
• Identify an incomplete dividend, that is, a number that will be minimal for division. Usually it consists of one number, a maximum of two.
• Find the number that will be first written in the answer. It must be the number of times the divisor is placed in a divisible.
• Write the result from multiplying this number by a divisor.
• Write it under incomplete divisibility. Perform the subtraction.
• Take down the first digit to the remainder after the part that is already divided.
• Again, choose a number for the answer.
• Repeat the multiplication and subtraction. If the remainder is zero and the dividend is ended, then the example is done. Otherwise, repeat the action: demolish the figure, pick up the number, multiply, subtract.

How to solve the division in a column, if the divisor is more than one digit?

The algorithm itself is identical to the one described above. The difference is the number of digits in incomplete divisibility. There should now be at least two of them, but if they are less than a divisor, then it works with the first three digits.

There is one more nuance in this division. The fact is that the balance and demolished figure sometimes do not divide into a divisor. Then it is necessary to assign one more digit in order. But in this case, you must put a zero in response. If you divide three-digit numbers into a column, you may need to demolish more than two digits. Then the rule is entered: the zeroes in the answer must be one less than the number of demolished numbers.

Consider this division, for example - 12082: 863.

• Incomplete divisible in it is the number 1208. The number 863 is placed only once. Therefore, in the answer it is supposed to put 1, and under 1208 to write down 863.
• After subtraction, the remainder is 345.
• To him you need to take down the number 2.
• In the number of 3452 four times 863 fit.
• The four must be written down in response. And when multiplying by 4, this number is obtained.
• The remainder after subtraction is zero. That is, the division is over.

The answer in the example is number 14.

What if the dividend ends at zero?

Or some zeros? In this case, the zero remainder is obtained, but in the delimit there are still zeros. Despair is not worth it, it's easier than it might seem. It is enough simply to assign to the answer all the zeros that have not been separated.

For example, you need to divide 400 by 5. Incomplete dividend 40. It is 8 times placed five. So, in response, we are supposed to write 8. When subtracting the remainder does not remain. That is, the division is complete, but there was zero in the delimit. It will have to be attributed to the answer. Thus, when dividing 400 by 5, 80 is obtained.

What if you need to split a decimal fraction?

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

The only difference is the semicolon. It is supposed to be put in response immediately, as soon as the first digit is removed from the fractional part. In another way, it can be said so: the division of the whole part has ended - put a comma and continue the decision further.

During the decision of examples on division into a column with decimal fractions it is necessary to remember that in a part after a comma it is possible to attribute any number of zeros. Sometimes this is necessary in order to divide the numbers to the end.

The division of two decimals

It may seem complicated. But only in the beginning. After all, how to make division into a fraction column by a natural number is already clear. Hence, we need to reduce this example to the already familiar form.

Make it easy. It is necessary to multiply both fractions by 10, 100, 1 000 or 10 000, and maybe, in 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 need to split the fraction by a natural number.

And it will be in the worst case. After all, it may happen that the dividend from this operation will be an integer. Then the solution of the example with division into the fraction column will be reduced to the simplest version: operations with natural numbers.

As an example: 28.4 divide by 3.2:

• First, they must be multiplied by 10, because in the second number after the comma there is only one digit. Multiplication will yield 284 and 32.
• They are supposed to be divided. And at once all number 284 on 32.
• The first chosen number for the answer is 8. From its multiplication, 256 is obtained. The remainder is 28.
• The division of the whole part has ended, and in response it is supposed to put a comma.
• Demolish to the remainder 0.
• Again, take 8.
• Balance: 24. To him to assign one more 0.
• Now you need to take 7.
• The multiplication result is 224, the remainder is 16.
• Take down another 0. Take 5 and get just 160. Balance - 0.

The division is over. The result of Example 28.4: 3.2 is 8.875.

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

Just as with multiplication, division into a column is not needed here. It is enough simply to transfer the comma in the desired direction to a certain number of digits. And by this principle, you can solve examples with both integers and decimals.

So, if you need to divide by 10, 100 or 1,000, then the comma is carried to the left for as many digits as there are zeros in the divisor. That is, when the number is divided by 100, the comma should shift to the left by two digits. If the dividend is a natural number, then it is assumed that the comma is at its end.

This action gives the same result as if the number had to be multiplied by 0.1, 0.01 or 0.001. In these examples, the comma is also transferred to the left by the 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 in the data may be insufficient. Then you can assign the missing zeros to the left (in the whole part) or to the right (after the comma).

The division of periodic fractions

In this case, it will not be possible to obtain an exact answer when dividing into a column. How to solve an example if a fraction with a period met? Here it is assumed to proceed to ordinary fractions. And then carry out their division according to the previously learned rules.

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

If in an example different fractions ...

Then several solutions are possible. First, you can try to convert a common fraction into a decimal digit. Then divide the two decimal by the above algorithm.

Secondly, each final decimal fraction can be written in the form of an ordinary. Only this is not always convenient. Most often such fractions turn out to be huge. And the answers are cumbersome. Therefore, the first approach is considered preferable.