Rules for moving decimals when dividing decimals. Division by decimal fraction – Knowledge Hypermarket
Adding decimals is the same as adding whole numbers. Let's see this with examples.
1) 0.132 + 2.354. Let's label the terms one below the other.
Here, adding 2 thousandths to 4 thousandths resulted in 6 thousandths;
from adding 3 hundredths with 5 hundredths the result is 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 labeling terms one after another.
3) 1,2357 + 0,469 + 2,08 + 3,90701.
Here, when adding thousandths, the result is 21 thousandths; we wrote 1 under the thousandths, and added 2 to the hundredths, so in the hundredths place we got the following terms: 2 + 3 + 6 + 8 + 0; in total they give 19 hundredths, we signed 9 under hundredths, and 1 counted as tenths, etc.
Thus, when adding decimal fractions, the following order must be observed: sign the fractions one below the other so that in all terms the same digits are located under each other and all commas are in the same vertical column; to the right of the decimal places of some terms, such a number of zeros are assigned, at least mentally, so that all terms after the decimal point have same number numbers Then they perform addition by digits, starting from the right side, and in the resulting sum they put a comma in the same vertical column in which it is located in these terms.
§ 108. Subtraction of decimal fractions.
Subtracting decimals works 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 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, you 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 turn to the nearest digit on the left, i.e., hundred thousandths, but in place of hundred thousandths there is also zero, so we take 1 ten thousandth from 3 ten thousandths and We break it up into hundred thousandths, we get 10 hundred thousandths, of which we leave 9 hundred thousandths in the hundred thousandths category, and we break 1 hundred thousandth into millionths, we get 10 millionths. Thus, in the last three digits we have: millionths 10, hundred thousandths 9, ten thousandths 2. For greater clarity and convenience (so as not to forget), these numbers are written above the corresponding fractional digits of the minuend. Now you can start subtracting. From 10 millionths we subtract 2 millionths, we get 8 millionths; from 9 hundred thousandths we subtract 1 hundred thousandth, we get 8 hundred thousandths, etc.
Thus, when subtracting decimal fractions, the following order is observed: sign the subtrahend under the minuend so that the same digits are located under each other and all commas are in the same vertical column; on the right they add, at least mentally, so many zeros in the minuend or subtrahend so that they have the same number of digits, then they subtract by digits, starting from the right side, and in the resulting difference they put a comma in the same vertical column in which it is located in minuend and subtract.
§ 109. Multiplication of decimal fractions.
Let's look at some 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 increases several times, the product increases by the same amount. This means that the number that is obtained from multiplying the integer factors, i.e. 28 by 23, is 10 times greater than the true product, and in order to obtain the true product, the found product must be reduced by 10 times. Therefore, here you will have to multiply by 10 once and divide by 10 once, but multiplying and dividing by 10 is done by moving the decimal point to the right and left by one place. Therefore, you need to do this: in the factor, move the comma to the right one place, this will make it 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 place 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 the 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 as decimal fractions. But here, in the process of multiplication, we will not pay attention to commas, i.e. we will temporarily increase the multiplicand by 100 times, and the multiplier by 1,000 times, which will increase the product by 100,000 times. Thus, multiplying 1,227 by 21, we get:
1 227 21 = 25 767.
Considering that the resulting product is 100,000 times larger than the true product, we must now reduce it by 100,000 times 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 whole numbers and in the product to separate as many decimal places with a comma on the right side as there were in the multiplicand and in the multiplier together.
The last example resulted in a product with five decimal places. If such great precision is not required, then the decimal fraction is rounded. When rounding, you should use the same rule as 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 for 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 53 by 15, but our second factor was 10 times smaller, which means 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 2,491. But since we increased the multiplicand and the multiplier by a total of 100 times, the resulting product is 100 times larger than it should be; so we must reduce this product by 100 times:
5,3 4,7 = 24,91.
3) Multiply 0.53 by 7.4.
First, we find in the table the product 53 by 74; it will be 3,922. But since we increased the multiplicand by 100 times, and the multiplier by 10 times, the product increased by 1,000 times; so we now have to reduce it by 1,000 times:
0,53 7,4 = 3,922.
§ 111. Division of decimal fractions.
We will look at dividing decimal fractions in this order:
1. Dividing a decimal fraction by a whole number,
1. Divide a decimal fraction by a whole number.
1) Divide 2.46 by 2.
We divided by 2 first whole, then tenths and finally hundredths.
2) Divide 32.46 by 3.
32,46: 3 = 10,82.
We divided 3 tens by 3, then began to divide 2 units by 3; since the number of units of the dividend (2) is less than the divisor (3), we had to put 0 in the quotient; further, to the remainder we took 4 tenths and divided 24 tenths by 3; received 8 tenths in the quotient and finally divided 6 hundredths.
3) Divide 1.2345 by 5.
1,2345: 5 = 0,2469.
Here in the quotient the first place is zero integers, 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 the quotient, we discovered a remainder equal to 2 hundredths, we split this remainder into thousandths, got 20 thousandths and completed the division.
Rule. Dividing a decimal fraction by an integer is performed in the same way as dividing integers, and the resulting remainders are converted into decimal fractions, smaller and smaller; Division continues until the remainder is zero.
2. Divide a decimal by a decimal.
1) Divide 2.46 by 0.2.
We already know how to divide a decimal fraction by a whole number. Let's think, is it possible to reduce this new case of division to the previous one? At one time, we considered the remarkable property of a quotient, which consists in the fact that it remains unchanged when the dividend and divisor simultaneously increase or decrease by the same number of times. We could easily divide the numbers given to us if the divisor were an integer. To do this, it is enough to increase it by 10 times, and to obtain the correct quotient, it is necessary to increase the dividend by the same amount, i.e., 10 times. Then the division of these numbers will be replaced by the division of the following numbers:
Moreover, there will no longer be any need to make any amendments to the particulars.
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 0 in the quotient and divide 125 tenths by 16, we get 7 tenths in the quotient and the remainder 13. We split 13 tenths into hundredths by assigning zero and divide 130 hundredths by 16, etc. Please note to the following:
a) when there are no integers in a particular, then zero integers are written in their place;
b) when, after adding 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 removing the last digit of the dividend, the division does not end, then, adding zeros to the remainder, the division continues;
d) if the dividend is an integer, then when dividing it by a decimal fraction, it is increased by adding zeros to it.
Thus, to divide a number by a decimal fraction, you need to discard the comma in the divisor, and then increase the dividend by as many times as the divisor increased when discarding the comma in it, and then perform the division according to the rule for dividing a decimal fraction by a whole number.
§ 112. Approximate quotients.
In the previous paragraph, we looked at the division of decimal fractions, and in all the examples we solved the division was completed, i.e., an exact quotient was obtained. However, in most cases, an exact quotient cannot be obtained, no matter how far we continue 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 in the remainder we begin to have numbers that have already been encountered before. In the quotient, numbers will also be repeated: it is obvious that after the number 7 the number 5 will appear, then 2, etc. endlessly. In such cases, the division is interrupted and limited to the first few digits of the quotient. This quotient is called close ones. We will show with examples how to perform division.
Let it be necessary to divide 25 by 3. Obviously, an exact quotient, expressed as an integer or a 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 1. To obtain the exact quotient, you need to add the fraction that is obtained by dividing the remainder equal to 1 by 3 to the found approximate quotient, i.e., to 8; this 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 represents correct fraction, i.e. fraction, less than one, then, discarding it, we will allow error, which less than one. The quotient 8 will be approximate quotient up to unity with a disadvantage. If instead of 8 we take 9 in the quotient, then we will also allow an error that is less than one, since we will not add the whole unit, but 2/3. Such a private will approximate quotient to within one with excess.
Let's now take another example. Let’s say we need to divide 27 by 8. Since here we won’t 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 equal to 3/8, it is less than one, which means that the approximate quotient (3) was found accurate to one with a disadvantage. Let's continue the division: split the remainder 3 into tenths, we get 30 tenths; divide them by 8.
We got 3 in the quotient in place of tenths and 6 tenths in the remainder. If we limit ourselves to the number 3.3 and discard the remainder 6, then we will allow an error of 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; this division would yield 6/80, which is less than one tenth. (Check!) Thus, if in the quotient we limit ourselves to tenths, then we can say that we have found the quotient accurate to one tenth(with a disadvantage).
Let's continue division to find another decimal place. To do this, we split 6 tenths into hundredths and get 60 hundredths; divide them by 8.
In the quotient in third place it turned out to be 7 and the remainder 4 hundredths; if we discard them, we will allow an error of less than one hundredth, because 4 hundredths divided by 8 is less than one hundredth. In such cases they say that the quotient has been found accurate to one hundredth(with a disadvantage).
In the example we are now looking at, we 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 look at this example:
40: 7 = 5,71428571...
The dots placed at the end of the number indicate that the division is not completed, i.e. the equality is approximate. Usually the approximate equality is written as follows:
40: 7 = 5,71428571.
We took the quotient with eight decimal places. But if such great accuracy is not required, you can limit yourself to only whole part quotient, i.e. the number 5 (more precisely 6); for greater accuracy, one could take into account tenths and take the quotient equal to 5.7; if for some reason this accuracy is insufficient, then you can stop at hundredths and take 5.71, etc. Let’s write out the individual quotients and name them.
The first approximate quotient accurate to one 6.
Second » » » to one tenth 5.7.
Third » » » to one hundredth 5.71.
Fourth » » » to one thousandth 5.714.
Thus, in order to find an approximate quotient accurate to some, for example, 3rd decimal place (i.e., up to one thousandth), stop division as soon as this sign is found. In this case, you need to remember the rule set out in § 40.
§ 113. The simplest problems involving percentages.
After learning about decimals, we'll do some more percent problems.
These problems are similar to those we solved in the fractions department; 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 move from an ordinary fraction to a decimal 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 fraction with a denominator of 100:
Let us now consider several problems.
1. Finding the percentage of a given number.
Task 1. Only 1,600 people live in one village. Number of children school age makes up 25% of the total number of residents. How many school-age children are there 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.
To clearly understand 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 there 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 provide depositors with a 2% return annually. How much income will a depositor receive in a year if he puts in the cash register: a) 200 rubles? b) 500 rubles? c) 750 rubles? d) 1000 rub.?
In all four cases, to solve the problem you will need 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 (rub.),
b) 500 0.02 = 10 (rub.),
c) 750 0.02 = 15 (rub.),
d) 1,000 0.02 = 20 (rub.).
Each of these cases can be verified by the following considerations. Savings banks give investors 2% income, i.e. 0.02 of the amount deposited in savings. If the amount was 100 rubles, then 0.02 of it would be 2 rubles. This means that every hundred brings the investor 2 rubles. income. Therefore, in each of the cases considered, it is enough to figure out how many hundreds there are in a given number, and multiply 2 rubles by this number of hundreds. In example a) there are 2 hundreds, which means
2 2 = 4 (rub.).
In example d) there are 10 hundreds, which means
2 10 = 20 (rub.).
2. Finding a number by its percentage.
Task 1. The school graduated 54 students in the spring, representing 6% of its total enrollment. How many students were there in the school last school year?
Let us first clarify the meaning of this task. 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 at 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 entire number. Thus, we have before us an ordinary task of finding a number from its fraction (§90, paragraph 6). Problems of this type are solved by division:
This means that there were only 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 head, solve a problem of the first type (finding the percentage of a given number): take the found number (900) as given and find the percentage of it indicated in the solved problem , namely:
900 0,06 = 54.
Task 2. A family spends 780 rubles on food during the month, which is 65% monthly earnings father. 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 what you are looking for is all the earnings:
780: 0,65 = 1 200.
Therefore, the required income is 1200 rubles.
3. Finding the percentage of numbers.
Task 1. IN school library only 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 problem we need to find the percentage ratio of the numbers 1,200 and 6,000.
Let's 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, let’s 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.
I. To divide a decimal fraction by a natural number, you need to divide the fraction by this number, as natural numbers are divided, and put a comma in the quotient when the division of the whole part is completed.
Examples.
Perform division: 1) 96,25: 5; 2) 4,78: 4; 3) 183,06: 45.
Solution.
Example 1) 96,25: 5.
We divide with a “corner” in the same way as natural numbers are divided. After we take down the number 2 (the number of tenths is the first digit after the decimal point in the dividend 96, 2 5), in the quotient we put a comma and continue the division.
Answer: 19,25.
Example 2) 4,78: 4.
We divide as natural numbers are divided. In the quotient we will put a comma as soon as we remove it 7
— the first digit after the decimal point in the dividend 4, 7
8. We continue the division further. When subtracting 38-36 we get 2, but the division is not completed. How do we proceed? We know that zeros can be added to the end of a decimal fraction - this will not change the value of the fraction. We assign zero and divide 20 by 4. We get 5 - the division is over.
Answer: 1,195.
Example 3) 183,06: 45.
Divide as 18306 by 45. In the quotient we put a comma as soon as we remove the number 0
— the first digit after the decimal point in the dividend 183, 0
6. Just as in example 2), we had to assign zero to the number 36 - the difference between the numbers 306 and 270.
Answer: 4,068.
Conclusion: when dividing a decimal fraction by a natural number in private we put a comma immediately after we take down the figure in the tenths place of the dividend. Please note: all highlighted numbers in red in these three examples belong to the category tenths of the dividend.
II. To divide a decimal fraction by 10, 100, 1000, etc., you need to move the decimal point to the left by 1, 2, 3, etc. digits.
Examples.
Perform division: 1) 41,56: 10; 2) 123,45: 100; 3) 0,47: 100; 4) 8,5: 1000; 5) 631,2: 10000.
Solution.
Moving the decimal point to the left depends on how many zeros after the one are in the divisor. So, when dividing a decimal fraction by 10
we will carry over in the dividend comma to the left one digit; when divided by 100
- move the comma left two digits; when divided by 1000
convert to this decimal fraction comma three digits to the left.
At school these actions are studied from simple to complex. Therefore, it is imperative to thoroughly understand the algorithm for performing these operations on 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. Every student should learn this principle already in the first grade. Therefore, if you miss several lessons in a row, you will have to master the material on your own. Otherwise, later problems will arise not only with mathematics, but also with other subjects related to it.
Second required condition successful study mathematics - move on to examples of long division 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 teach it using the Pythagorean table. There is nothing superfluous, and multiplication is easier to learn in this case.
How are natural numbers multiplied in a column?
If difficulty arises in solving examples in a column for division and multiplication, then you should begin to solve the problem with multiplication. Since division is the inverse operation of multiplication:
- Before multiplying two numbers, you need to look at them carefully. Choose the one with more digits (longer) and write it down first. Place the second one under it. Moreover, the numbers of the corresponding category must 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 below the line so that its last digit is under the one you multiplied by.
- Repeat the same with another digit of the lower number. But the result of 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 factor run out. Now they need to be folded. This will be the answer you are looking for.
Algorithm for multiplying decimals
First, you need to imagine that the given fractions are not decimals, but natural ones. That is, remove the commas from them and then proceed as described in the previous case.
The difference begins when the answer is written down. At this moment, it is necessary to count all the numbers that appear after the decimal points in both fractions. This is exactly 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 using an example: 0.25 x 0.33:
Where to start learning division?
Before solving long division examples, you need to remember the names of the numbers that appear in the long division example. The first of them (the one that is divided) is divisible. The second (divided by) is the divisor. The answer is private.
After this, using a simple everyday example, we will explain the essence of this mathematical operation. For example, if you take 10 sweets, then it’s easy to divide them equally between mom and dad. But what if you need to give them to your parents and brother?
After this, you can get acquainted with the rules of division and master them in specific examples. First simple ones, and then move on to more and more complex ones.
Algorithm for dividing numbers into a column
First, let us present the procedure for natural numbers divisible by single digit number. They will also be the basis for multi-digit divisors or decimal fractions. Only then should you enter minor changes, but more on that later:
- Before doing long division, you need to figure out where the dividend and divisor are.
- Write down the dividend. To the right of it is the divider.
- Draw a corner on the left and bottom near the last corner.
- Determine the incomplete dividend, that is, the number that will be minimal for division. Usually it consists of one digit, maximum of two.
- Choose the number that will be written first in the answer. It should be the number of times the divisor fits into the dividend.
- Write down the result of multiplying this number by the divisor.
- Write it under the incomplete dividend. Perform subtraction.
- Add to the remainder the first digit after the part that has already been divided.
- Choose the number for 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: remove the number, pick up the number, multiply, subtract.
How to solve long division if the 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. Now there should be at least two of them, but if they turn out to be less than the divisor, then you have to work with the first three digits.
There is one more nuance in this division. The fact is that the remainder and the number added to it are sometimes not divisible by the divisor. Then you have to add another number in order. But the answer must be zero. If division is carried out three-digit numbers in a column, you may need to remove more than two digits. Then a rule is introduced: there should be one less zero in the answer than the number of digits removed.
You can consider this division using the example - 12082: 863.
- The incomplete dividend in it turns out to be the number 1208. The number 863 is placed in it only once. Therefore, the answer is supposed to be 1, and under 1208 write 863.
- After subtraction, the remainder is 345.
- You need to add the number 2 to it.
- The number 3452 contains 863 four times.
- Four must be written down as an answer. Moreover, when multiplied by 4, this is exactly the number obtained.
- The remainder after subtraction is zero. That is, the division is completed.
The answer in the example would be the number 14.
What if the dividend ends in zero?
Or a few zeros? In this case, the remainder is zero, but the dividend still contains zeros. There is no need to despair, everything is simpler than it might seem. It is enough to simply add to the answer all the zeros that remain undivided.
For example, you need to divide 400 by 5. The incomplete dividend is 40. Five fits into it 8 times. This means that the answer should be written as 8. When subtracting, there is no remainder left. That is, the division is completed, but a zero remains in the dividend. It will have to be added to the answer. Thus, dividing 400 by 5 equals 80.
What to do if you need to divide a decimal fraction?
Again, this number looks like a natural number, if not for the comma separating the whole part from the fractional part. This suggests that the division of decimal fractions into a column is similar to that described above.
The only difference will be the semicolon. It is supposed to be put in the answer as soon as the first digit from the fractional part is removed. Another way to say this is this: if you have finished dividing the whole part, put a comma and continue the solution further.
When solving examples of long division with decimal fractions, you need to remember that any number of zeros can be added to the part after the decimal point. Sometimes this is necessary in order to complete the numbers.
Dividing two decimals
It may seem complicated. But only at the beginning. After all, how to divide a column of fractions by a natural number is already clear. This means that we need to reduce this example to an already familiar form.
It's easy to do. You need to multiply both fractions by 10, 100, 1,000 or 10,000, and maybe by a million if the problem requires it. The multiplier is supposed to be chosen based on how many zeros are in the decimal part of the divisor. That is, the result will be that you will have to divide the fraction by a natural number.
And this will be the worst case scenario. After all, it may happen that the dividend from this operation becomes an integer. Then the solution to the example with division into a column of fractions will be reduced to the very simple option: operations with natural numbers.
As an example: divide 28.4 by 3.2:
- They must first 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. Moreover, the whole number is 284 by 32.
- The first number chosen for the answer is 8. Multiplying it gives 256. The remainder is 28.
- The division of the whole part has ended, and a comma is required in the answer.
- Remove to remainder 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.
- Take down another 0. Take 5 each and you get exactly 160. The remainder is 0.
The division is complete. The result of example 28.4:3.2 is 8.875.
What if the divisor is 10, 100, 0.1, or 0.01?
Just like with multiplication, long division is not needed here. It is enough to simply move the comma in the desired direction for a certain number of digits. Moreover, using 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 decimal point is moved to the left by the same number of digits as there are zeros in the divisor. That is, when a number is divisible by 100, the decimal point must 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.
This action gives 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 the number of digits, equal to length fractional part.
When dividing by 0.1 (etc.) or multiplying by 10 (etc.), the decimal point 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 added to the left (in the whole part) or to the right (after the decimal point).
Division of periodic fractions
In this case, it will not be possible to obtain an accurate answer when dividing into a column. How to solve an example if you encounter a fraction with a period? Here we need to move on to ordinary fractions. And then divide them according to the previously learned rules.
For example, you need to divide 0.(3) by 0.6. The first fraction is periodic. It converts to the fraction 3/9, which when reduced gives 1/3. The second fraction is the final decimal. It’s even easier to write it down as usual: 6/10, which is equal to 3/5. The rule for dividing ordinary fractions prescribes replacing division with multiplication and divisor - reciprocal number. That is, the example comes down to multiplying 1/3 by 5/3. The answer will be 5/9.
If the example contains different fractions...
Then several solutions are possible. Firstly, common fraction You can try to convert it to decimal. Then divide two decimals using the above algorithm.
Secondly, every final decimal fraction can be written as a common fraction. But 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 more preferable.
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, the result of division will be 0. Write 0 under the divisor - this is the first digit of the quotient.
Multiply the result by the divisor. Write the result of the multiplication under the first digit of the dividend, since this is the digit you just divided by the divisor.
- In our example, 0 × 12 = 0, so write 0 under 3.
Subtract the result of the multiplication from the first digit of the dividend. Write your answer on a new line.
- In our example: 3 - 0 = 3. Write 3 directly below 0.
Move down the second digit of the dividend. To do this, write down the next digit of the dividend next to the result of the subtraction.
- In our example, the dividend is 30. The second digit of the dividend is 0. Move it down by writing a 0 next to the 3 (the result of the subtraction). You will receive the number 30.
Divide the result by the divisor. You will find the second digit of the quotient. To do this, divide the number located on the bottom line by the divisor.
- In our example, divide 30 by 12. 30 ÷ 12 = 2 plus some remainder (since 12 x 2 = 24). Write 2 after 0 under the divisor - this is the second digit of the quotient.
- If you can't find a suitable digit, go through the digits until the result of multiplying a digit by a divisor is smaller and closest to the number located last in the column. In our example, consider the number 3. Multiply it by the divisor: 12 x 3 = 36. Since 36 is greater than 30, the number 3 is not suitable. Now consider the number 2. 12 x 2 = 24. 24 is less than 30, so the number 2 is the correct solution.
Repeat the steps above to find the next number. The described algorithm is used in any long division problem.
- Multiply the second digit of the quotient by the divisor: 2 x 12 = 24.
- Write the result of multiplication (24) below last number in column (30).
- Subtract the smaller number from the larger one. In our example: 30 - 24 = 6. Write the result (6) on a new line.
If there are digits left in the dividend that can be moved down, continue the calculation process. Otherwise, continue to the next step.
- In our example, you moved down the last digit of the dividend (0). So move on to the next step.
If necessary, use a decimal point to expand the dividend. If the dividend is divisible by the divisor, then on the last line you will get the number 0. This means that the problem has been solved, and the answer (in the form of an integer) is written under the divisor. But if at the very bottom of the column there is any figure other than 0, it is necessary to expand the dividend by adding a decimal point and adding 0. Let us remember that this does not change the value of the dividend.
- In our example, the last line contains the number 6. Therefore, to the right of 30 (the dividend), write a decimal point, and then write 0. Also, place a decimal point after the found digits of the quotient, which you write under the divisor (don’t write anything after this comma yet!) .
Repeat the steps described above to find the next number. The main thing is not to forget to put a decimal point both after the dividend and after the found digits of the quotient. The rest of the process is similar to the process described above.
- In our example, move down the 0 (which you wrote after the decimal point). You will get the number 60. Now divide this number by the divisor: 60 ÷ 12 = 5. Write 5 after the 2 (and after the decimal point) under the divisor. This is the third digit of the quotient. So the final answer is 2.5 (the zero before the 2 can be ignored).
In this article we will look at such an important operation with decimals as division. First let's formulate general principles, then we’ll look at how to correctly divide decimal fractions by columns both by other fractions and by natural numbers. Next, we will analyze the division of ordinary fractions into decimals and vice versa, and at the end we will look at how to correctly divide fractions ending in 0, 1, 0, 01, 100, 10, etc.
Here we will take only cases with positive fractions. If there is a minus in front of the fraction, then to operate with it you need to study material about dividing rational and real numbers.
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All decimal fractions, both finite and periodic, are just a special form of writing ordinary fractions. Consequently, they are subject to the same principles as their corresponding ordinary fractions. Thus, we reduce the entire process of dividing decimal fractions to replacing them with ordinary ones, followed by calculation using methods already known to us. Let's take a specific example.
Example 1
Divide 1.2 by 0.48.
Solution
Let's write decimal fractions as ordinary fractions. We will get:
1 , 2 = 12 10 = 6 5
0 , 48 = 48 100 = 12 25 .
Thus, we need to divide 6 5 by 12 25. We count:
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 corresponds to the original numbers: 5 2 = 2, 5. We have already written about how to do this earlier.
Answer: 1 , 2: 0 , 48 = 2 , 5 .
Example 2
Calculate how much 0 , (504) 0 , 56 will be.
Solution
First, we need to convert a periodic decimal fraction into a common fraction.
0 , (504) = 0 , 504 1 - 0 , 001 = 0 , 504 0 , 999 = 504 999 = 56 111
After this, we will also convert 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 form. 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 encountered non-periodic decimal fractions, then we will act a little differently. We cannot reduce them to the usual ordinary fractions, so when dividing we have to first round them to a certain digit. This action must be performed with both the dividend and the divisor: we will also round the existing finite or periodic fraction in the interests of accuracy.
Example 3
Find how much 0.779... / 1.5602 is.
Solution
First, we round both fractions to the nearest hundredth. This is how we move from infinite non-periodic fractions to finite decimal ones:
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 reduce 2, 5 to the form of an ordinary fraction: 255 10 = 51 2. Next we just need to divide it by a 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 long division method is good not only for natural numbers. By analogy, we can use it for fractions. Below we 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 need:
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 whole part of the fraction comes to an end, we put a comma in the resulting quotient and count further.
The result of such 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 several problems as an example and try to perform these steps with specific numbers.
Example 5
Calculate how much 65, 14 4 will be.
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 one. Now we write a column for dividing by 4:
The resulting number will be the result we need from dividing the integer part. We put a comma, separating it, and continue:
We have reached zero remainder, therefore the division process is complete.
Answer: 65 , 14: 4 = 16 , 285 .
Example 6
Divide 164.5 by 27.
Solution
We first divide the fractional part and get:
Separate the resulting number with a comma and continue dividing:
We see that the remainders began to repeat periodically, and in the quotient the numbers nine, two and five began to alternate. We will stop here and write the answer in the form of a periodic fraction 6, 0 (925).
Answer: 164 , 5: 27 = 6 , 0 (925) .
This division can be reduced to the process of finding the quotient of a decimal fraction and a natural number, already described above. To do this, we need to multiply the dividend and divisor by 10, 100, etc. so that the divisor turns into a natural number. Next we carry out the sequence of actions described above. This approach is possible due to the properties of division and multiplication. We wrote them down like this:
a: b = (a · 10) : (b · 10) , a: b = (a · 100) : (b · 100) and so on.
Let's formulate a rule:
Definition 2
To divide one final decimal fraction by another:
1. Move the comma in the dividend and divisor to the right by the number of digits 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 this, divide the fraction by a column by the resulting natural number.
Let's 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 decimal place one place to the right. So we moved on to dividing the decimal fraction 72, 87 by 21. Let's write the resulting numbers in a column and calculate
Answer: 7 , 287: 2 , 1 = 3 , 47
Example 8
Calculate 16.30.021.
Solution
We will have to move the comma three places. There are not enough digits in the divisor for this, which means you need to use additional zeros. We think the result will be:
We see periodic repetition of residues 4, 19, 1, 10, 16, 13. In the quotient, 1, 9, 0, 4, 7 and 5 are repeated. Then our result is the periodic decimal fraction 776, (190476).
Answer: 16 , 3: 0 , 021 = 776 , (190476)
The method we described allows you to do the opposite, that is, divide a natural number by the final decimal fraction. Let's see how it's done.
Example 9
Calculate how much 3 5, 4 is.
Solution
Obviously, we will have to move the comma to the right one place. After this we can proceed to divide 30, 0 by 54. Let's write the data in a column and calculate the result:
Repeating the remainder gives us the final number 0, (5), which is a periodic decimal fraction.
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 by 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 Simply move the comma to the required number of 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 decimal fractions, we do the same.
Example 11
For example, 3, (56): 1,000 = 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 into the indicated 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 decimal fractions. We only advise you to be careful with the period of the fraction that appears in the answer.
So, 7, 5 (716) : 0, 01 = 757, (167) because after we moved the comma in the decimal fraction 7, 5716716716... two places 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 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 the ordinary fraction or mixed number with the corresponding decimal fraction.
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