Math is akin to puzzles. This is especially true for division and multiplication in a column. At school, these actions are studied from simple to complex. Therefore, it is imperative to rely on a good grasp of the algorithm for performing these operations with simple examples. So that later there would be no difficulties with dividing decimal fractions into 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 consistent study. Knowledge gaps are unacceptable here. This principle should be adopted by every student in the first grade. Therefore, if you skip several lessons in a row, you will have to master the material yourself. Otherwise, later problems will arise 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 the division by column examples only after addition, subtraction, and multiplication are 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 from the Pythagorean table. There is nothing superfluous, and multiplication is assimilated in this case easier.
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 it is necessary to begin to eliminate the problem with multiplication. Since division is the inverse of multiplication:
- Before you multiply two numbers, you need to carefully look at them. Choose the one in which more digits (longer), write it first. Place a second 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 second.
- Multiply the rightmost digit of the bottom number by each digit of the top, starting from the right. Write the answer under the bar so that its last digit is under the one by which we multiplied.
- Repeat the same with the other digit of the lower number. But the result of multiplication must be shifted by one digit to the left. At the same time, its last figure will be under the one by which it was multiplied.
Continue this multiplication in a column until the numbers in the second factor are over. Now they need to be folded. This will be the desired answer.
Column Decimal Algorithm
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 response is recorded. At this point, you need to count all the numbers that appear after the commas in both fractions. That is how many of them need to be counted 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:
- These fractions should be written so that the number 33 is under 25.
- Now the right three must be multiplied by 25. It will turn out 75. It is supposed to be written so that the five is under the three 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, 4 digits must also be separated by 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 one more 0.
- The answer in the example is the number 0.0825.
Where to start fission training?
Before deciding on the division examples in a column, it is necessary to remember the names of the numbers that stand in the division example. The first of them (the one that is divided) is the dividend. The second (divided into it) is the divisor. The answer is private.
After that, we explain the essence of this mathematical operation using a simple everyday example. For example, if you take 10 candies, then it is easy to divide them equally between mom and dad. But what if you need to give them to your parents and brother?
After that, you can familiarize yourself with the rules of division and master them with specific examples. First simple, and then move on to more and more complex.
Column division algorithm
First, imagine the procedure for natural numbers divisible by a single number. They will be the basis for multi-valued divisors or decimal fractions. Only then it is supposed to make small changes, but more on that later:
- Before doing the division in the column, you need to find out where the dividend and the divisor are.
- Record the dividend. To his right is the divisor.
- Draw a corner on the left and bottom near the last.
- Determine the incomplete dividend, that is, the number that will be the minimum for division. Usually it consists of one digit, a maximum of two.
- Choose the number that will be first written in the answer. It should be how many times the divisor is placed in the dividend.
- Record the result of multiplying this number by the divisor.
- Write it under an incomplete dividend. Perform a subtraction.
- Take to the remainder the first digit after the part that is already divided.
- Pick up the number to answer again.
- Repeat multiplication and subtraction. If the remainder is zero and the dividend is over, then an example is done. Otherwise, repeat the steps: tear down the number, pick up the number, multiply, subtract.
How to solve the division in the column, if the divider has more than one digit?
The algorithm itself completely coincides with what was described above. The difference is the number of digits in the incomplete divisible. There should now 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 one more nuance in this division. The fact is that the remainder and the digit carried to it are sometimes not divided by a divisor. Then it is supposed to ascribe one more figure in order. But at the same time, it is necessary to put zero in response. If you are dividing three-digit numbers in a column, you may need to carry more than two digits. Then the rule is introduced: the zeros in the answer should be one less than the number of digits removed.
Consider this division can be an example - 12082: 863.
- The number 1208 is incomplete divisible in it. The number 863 is placed in it only once. Therefore, in response, it is supposed to put 1, and under 1208 write 863.
- After subtraction, the remainder is 345.
- To it you need to carry the number 2.
- Among 3452 four times fits 863.
- The four must be written in response. Moreover, when multiplied by 4, this is exactly the number obtained.
- The residue after subtraction is zero. That is, the division is completed.
The answer in the example is the number 14.
What if the dividend ends in zero?
Or a few zeros? In this case, a zero remainder is obtained, and in the dividend there are still zeros. Do not despair, everything is easier than it might seem. It is enough to simply attribute to the answer all the zeros that have remained undivided.
For example, you need to divide 400 by 5. Incomplete dividend 40. Five is placed in it 8 times. So, the answer is supposed to write 8. When subtracting the remainder is not left. That is, the division is completed, but the dividend remains zero. It will have to be attributed to the answer. Thus, dividing 400 by 5 results in 80.
What if decimal fraction is needed?
Again, this number is similar to the natural, if not for the comma, separating the integer part from the fractional. This suggests that dividing decimal fractions into a column is similar to that described above.
The only difference is the semicolon. It is supposed to be answered immediately as soon as the first digit from the fractional part is demolished. In another way, it can be said this way: the division of the whole part has ended - put a comma and continue the solution further.
When solving examples for dividing in a column with decimal fractions, you need to remember that in the part after the decimal point, you can attribute any number of zeros. Sometimes this is necessary in order to finish the numbers to the end.
Division of two decimal fractions
It may seem complicated. But only at the beginning. After all, how to perform the division into a column of fractions by a natural number is already clear. So, we need to reduce this example to an already familiar form.
It is 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 selected based on how many zeros are in the decimal part of the divisor. That is, as a result it turns out that the fraction will have to be divided 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 fractional division example reduces to the simplest option: operations with natural numbers.
As an example: 28.4 divided by 3.2:
- First you need to multiply them by 10, because in the second number, there is only one digit after the decimal point. Multiplication will give 284 and 32.
- They are supposed to be divided. And at once the whole number is 284 by 32.
- The first number selected for the answer is 8. From multiplying it, 256. The remainder is 28.
- The division of the whole part is over, and in response it is supposed to put a comma.
- Take to the remainder 0.
- Take 8 again.
- Residue: 24. Ascribe 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 and get just 160. The remainder is 0.
The division is completed. 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, division by column is not needed here. Simply transfer the comma in the desired direction by 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 1,000, then the comma is transferred to the left by 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 the 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 shifted 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 given in the dividend may be insufficient. Then on the left (in the integer part) or on the right (after the decimal point) you can attribute the missing zeros.
Division of Periodic Fractions
In this case, you cannot get the exact answer when dividing by column. How to solve an example if a fraction with a period is encountered? Here it is supposed to pass to ordinary fractions. And then perform their division according to the rules previously studied.
For example, you need to divide 0, (3) by 0.6. The first fraction is periodic. It is converted to a fraction of 3/9, which after reduction will give 1/3. The second fraction is the final decimal. It is even simpler to write it down: 6/10, which is 3/5. The rule for dividing ordinary fractions requires replacing division with multiplication and the divider with the inverse number. That is, the example reduces to multiplying 1/3 by 5/3. The answer is 5/9.
If the example has different fractions ...
Then several solutions are possible. Firstly, you can try to convert the ordinary fraction to decimal. Then divide already two decimal by the above algorithm.
Secondly, each final decimal fraction can be written as an ordinary fraction. Only it is not always convenient. Most often, such fractions are huge. And the answers are cumbersome. Therefore, the first approach is considered more preferable.