Math is like a puzzle. This is especially true for division and multiplication in a column. 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.
Advice for those who want to be good at math
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 long division examples only after addition, subtraction and multiplication have been mastered.
Childit will be difficult 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 start solving the problem with multiplication. Because division is the inverse of multiplication:
- Before you multiply 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 should be above the rightmost digit of the second.
- 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 you multiplied by.
- 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 previouscase.
The difference starts when the answer is recorded. 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:
- Write down these fractions so that the number 33 is under 25.
- Now the right triple should be multiplied by 25. It turns out 75. It is supposed to be written so that the five is under the triple by which the multiplication was performed.
- Then multiply 25 by the first 3. Again it will be 75, but it will be written so that 5 is under 7 of the previous number.
- After adding these two numbers, we get 825. In decimal fractions, 4 digits are separated by commas. Therefore, in the answer, you must also separate 4 digits with a comma. But there are only three of them. To do this, you will have to write 0 before 8, put a comma, before it another 0.
- The answer in the example will be the number 0, 0825.
How to start learning to divide?
Before solving long division examples, you should remember the names of the numbers used in the division example. The first of them (the one that is divisible) is the divisible. The second (divided into it) is a divisor. The answer is a quotient.
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 rulesdivisions and master them with specific examples. First simple ones, and then move on to more and more complex ones.
Algorithm for dividing numbers into a column
First, let's imagine the procedure for natural numbers divisible by a single digit. They will also be the basis for multi-digit divisors or decimal fractions. Only then are small changes supposed to be made, but more on that later:
- Before doing long division, you need to figure out where the dividend and divisor are.
- Write the dividend. To the right of it is the divisor.
- Draw 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 the first written 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 the divisor.
- Write it under the incomplete divisor. Subtract.
- Remove the first digit after the part that is already 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 divisor has more than one digit?
The algorithm itself completely coincides with what was described above. The difference will be the number of digits in the incomplete dividend. Themnow there should be at least two, but if they turn out to be less than the divisor, then it is supposed to work with the first three digits.
There is one more 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 a rule is introduced: there should be one less number of zeros in the answer than the number of digits taken down.
You can consider such a division using the example - 12082: 863.
- 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 under 1208 write 863.
- After subtracting, the remainder is 345.
- You need to demolish the number 2 to it.
- The number 3452 fits four times 863.
- The 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 over.
The answer in the example will be the number 14.
What if the dividend ends in zero?
Or some 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 add 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. Whenthere is no remainder to subtract. That is, the division is over, but zero remains in the dividend. It will have to be added to the answer. So 400 divided by 5 is 80.
What if you need to divide a decimal?
Again, this number looks like a natural number, except for the comma separating the integer part from the fractional part. This suggests that the long division of decimals 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 is over - put a comma and continue the solution further.
When solving examples for division 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.
It's easy to do. 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 the fraction by a natural number.
And thiswill 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 the second number has only one digit after the decimal point. Multiplying will give 284 and 32.
- They are supposed to be separated. And at once the whole number 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 has ended, and a comma is supposed to be put in the answer.
- Dash to balance 0.
- Take 8 again.
- Remainder: 24. Add another 0 to it.
- Now you need to take 7.
- The result of multiplication is 224, the remainder is 16.
- Demolish another 0. Take 5 each and get exactly 160. The remainder is 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?
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 commashould move two digits to the left. 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 parts).
It is worth noting that the number of digits given in the dividend may not be sufficient. Then the missing zeros can be added to the left (in the integer part) or to the right (after the decimal point).
Recurring fraction division
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. That is, the example boils down to multiplying 1/3 by 5/3. The answer will be 5/9.
If the example has different fractions…
Then there are several possible solutions. First, an ordinary fraction can betry to convert to 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.
