In the last lesson, we learned how to add and subtract decimals (see lesson “Adding and subtracting decimals”). At the same time, we assessed how much calculations are simplified compared to ordinary “two-story” fractions.
Unfortunately, with multiplication and division decimals no such effect occurs. In some cases, decimal notation even complicates these operations.
First, let's introduce a new definition. We'll see him quite often, and not just in this lesson.
The significant part of a number is everything between the first and last non-zero digit, including the ends. It's about about numbers only, the decimal point is not taken into account.
The numbers included in significant part numbers are called significant figures. They can be repeated and even be equal to zero.
For example, consider several decimal fractions and write out the corresponding significant parts:
- 91.25 → 9125 (significant figures: 9; 1; 2; 5);
- 0.008241 → 8241 (significant figures: 8; 2; 4; 1);
- 15.0075 → 150075 (significant figures: 1; 5; 0; 0; 7; 5);
- 0.0304 → 304 (significant figures: 3; 0; 4);
- 3000 → 3 (there is only one significant figure: 3).
Please note: the zeros inside the significant part of the number do not go anywhere. We have already encountered something similar when we learned to convert decimal fractions to ordinary ones (see lesson “ Decimals”).
This point is so important, and mistakes are made here so often, that in the near future I will publish a test on this topic. Be sure to practice! And we, armed with the concept of the significant part, will proceed, in fact, to the topic of the lesson.
Multiplying Decimals
The multiplication operation consists of three successive steps:
- For each fraction, write down the significant part. You will get two ordinary integers - without any denominators and decimal points;
- Multiply these numbers in any convenient way. Directly, if the numbers are small, or in a column. We obtain the significant part of the desired fraction;
- Find out where and by how many digits the decimal point in the original fractions is shifted to obtain the corresponding significant part. Perform reverse shifts for the significant part obtained in the previous step.
Let me remind you once again that zeros on the sides of the significant part are never taken into account. Ignoring this rule leads to errors.
- 0.28 12.5;
- 6.3 · 1.08;
- 132.5 · 0.0034;
- 0.0108 1600.5;
- 5.25 · 10,000.
We work with the first expression: 0.28 · 12.5.
- Let's write out the significant parts for the numbers from this expression: 28 and 125;
- Their product: 28 · 125 = 3500;
- In the first factor the decimal point is shifted 2 digits to the right (0.28 → 28), and in the second it is shifted by 1 more digit. In total, you need a shift to the left by three digits: 3500 → 3,500 = 3.5.
Now let's look at the expression 6.3 · 1.08.
- Let's write down the significant parts: 63 and 108;
- Their product: 63 · 108 = 6804;
- Again, two shifts to the right: by 2 and 1 digit, respectively. Total - again 3 digits to the right, so the reverse shift will be 3 digits to the left: 6804 → 6.804. This time there are no trailing zeros.
We reached the third expression: 132.5 · 0.0034.
- Significant parts: 1325 and 34;
- Their product: 1325 · 34 = 45,050;
- In the first fraction, the decimal point moves to the right by 1 digit, and in the second - by as many as 4. Total: 5 to the right. We shift by 5 to the left: 45,050 → .45050 = 0.4505. The zero was removed at the end, and added at the front so as not to leave a “naked” decimal point.
The following expression is: 0.0108 · 1600.5.
- We write the significant parts: 108 and 16 005;
- We multiply them: 108 · 16,005 = 1,728,540;
- We count the numbers after the decimal point: in the first number there are 4, in the second there are 1. The total is again 5. We have: 1,728,540 → 17.28540 = 17.2854. At the end, the “extra” zero was removed.
Finally, the last expression: 5.25 10,000.
- Significant parts: 525 and 1;
- We multiply them: 525 · 1 = 525;
- The first fraction is shifted 2 digits to the right, and the second fraction is shifted 4 digits to the left (10,000 → 1.0000 = 1). Total 4 − 2 = 2 digits to the left. We perform a reverse shift by 2 digits to the right: 525, → 52,500 (we had to add zeros).
Note in the last example: since the decimal point moves in different directions, the total shift is found through the difference. This is very important point! Here's another example:
Consider the numbers 1.5 and 12,500. We have: 1.5 → 15 (shift by 1 to the right); 12,500 → 125 (shift 2 to the left). We “step” 1 digit to the right, and then 2 to the left. As a result, we stepped 2 − 1 = 1 digit to the left.
Decimal division
Division is perhaps the most difficult operation. Of course, here you can act by analogy with multiplication: divide the significant parts, and then “move” the decimal point. But in this case there are many subtleties that negate potential savings.
Therefore, let's look at a universal algorithm, which is a little longer, but much more reliable:
- Convert all decimal fractions to ordinary fractions. With a little practice, this step will take you a matter of seconds;
- Divide the resulting fractions in the classical way. In other words, multiply the first fraction by the “inverted” second (see lesson “Multiplying and dividing numerical fractions");
- If possible, present the result again as a decimal fraction. This step is also quick, since the denominator is often already a power of ten.
Task. Find the meaning of the expression:
- 3,51: 3,9;
- 1,47: 2,1;
- 6,4: 25,6:
- 0,0425: 2,5;
- 0,25: 0,002.
Let's consider the first expression. First, let's convert fractions to decimals:
Let's do the same with the second expression. The numerator of the first fraction will again be factorized:
There is an important point in the third and fourth examples: after getting rid of the decimal notation, reducible fractions appear. However, we will not perform this reduction.
The last example is interesting because the numerator of the second fraction contains a prime number. There is simply nothing to factorize here, so we consider it straight ahead:
Sometimes division results in an integer (I'm talking about the last example). In this case, the third step is not performed at all.
In addition, when dividing, “ugly” fractions often arise that cannot be converted to decimals. This distinguishes division from multiplication, where the results are always represented in decimal form. Of course, in this case the last step is again not performed.
Pay also attention to the 3rd and 4th examples. In them, we deliberately do not reduce ordinary fractions obtained from decimals. Otherwise, this will complicate the inverse task - representing the final answer again in decimal form.
Remember: the basic property of a fraction (like any other rule in mathematics) in itself does not mean that it must be applied everywhere and always, at every opportunity.
I. To divide a decimal fraction by natural number, you need to divide the fraction by this number, as you divide natural numbers, 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.
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 convert the result to decimal notation, 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 to perform the division, in this case it is enough to simply move the decimal point to required quantity numbers 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 common fraction or a mixed number with its corresponding decimal fraction.
If you notice an error in the text, please highlight it and press Ctrl+Enter
Let's look at examples of dividing decimals in this light.
Example.
Divide the decimal fraction 1.2 by the decimal fraction 0.48.
Solution.
Answer:
1,2:0,48=2,5 .
Example.
Divide the periodic decimal fraction 0.(504) by the decimal fraction 0.56.
Solution.
Let's convert the periodic decimal fraction into an ordinary fraction: . We also convert the final decimal fraction 0.56 into an ordinary fraction, we have 0.56 = 56/100. Now we can move from dividing the original decimal fractions to dividing ordinary fractions and finish the calculations: .
Let's convert the resulting ordinary fraction into a decimal fraction by dividing the numerator by the denominator with a column:
Answer:
0,(504):0,56=0,(900) .
The principle of dividing infinite non-periodic decimal fractions differs from the principle of dividing finite and periodic decimal fractions, since non-periodic decimal fractions cannot be converted to ordinary fractions. The division of infinite non-periodic decimal fractions is reduced to the division of finite decimal fractions, for which we carry out rounding numbers up to a certain level. Moreover, if one of the numbers with which the division is carried out is a finite or periodic decimal fraction, then it is also rounded to the same digit as the non-periodic decimal fraction.
Example.
Divide the infinite non-periodic decimal 0.779... by the finite decimal 1.5602.
Solution.
First you need to round decimals so that you can move from dividing infinite non-periodic decimals to dividing finite decimals. We can round to the nearest hundredth: 0.779…≈0.78 and 1.5602≈1.56. Thus, 0.779…:1.5602≈0.78:1.56= 78/100:156/100=78/100·100/156= 78/156=1/2=0,5 .
Answer:
0,779…:1,5602≈0,5 .
Dividing a natural number by a decimal fraction and vice versa
The essence of the approach to dividing a natural number by a decimal fraction and to dividing a decimal fraction by a natural number is no different from the essence of dividing decimal fractions. That is, finite and periodic fractions are replaced by ordinary fractions, and infinite non-periodic fractions are rounded.
To illustrate, consider the example of dividing a decimal fraction by a natural number.
Example.
Divide the decimal fraction 25.5 by the natural number 45.
Solution.
By replacing the decimal fraction 25.5 with the common fraction 255/10=51/2, division is reduced to dividing the common fraction by a natural number:. The resulting fraction in decimal notation has the form 0.5(6) .
Answer:
25,5:45=0,5(6) .
Dividing a decimal fraction by a natural number with a column
It is convenient to divide finite decimal fractions into natural numbers by a column, by analogy with the division by a column of natural numbers. Let us present the division rule.
To divide a decimal fraction by a natural number using a column, necessary:
- add several digits 0 to the right of the decimal fraction being divided (during the division process, if necessary, you can add any number of zeros, but these zeros may not be needed);
- perform division by a column of a decimal fraction by a natural number according to all the rules of division by a column of natural numbers, but when the division of the whole part of the decimal fraction is completed, then in the quotient you need to put a comma and continue the division.
Let's say right away that as a result of dividing a finite decimal fraction by a natural number, you can get either a finite decimal fraction or an infinite periodic decimal fraction. Indeed, after the division of all non-0 decimal places is completed divisible fraction, either the remainder may be 0, and we will get a final decimal fraction, or the remainders will begin to repeat periodically, and we will get a periodic decimal fraction.
Let's understand all the intricacies of dividing decimal fractions by natural numbers in a column when solving examples.
Example.
Divide the decimal fraction 65.14 by 4.
Solution.
Let's divide a decimal fraction by a natural number using a column. Let's add a couple of zeros to the right in the notation of the fraction 65.14, and we will get an equal decimal fraction 65.1400 (see equal and unequal decimal fractions). Now you can begin to divide with a column the integer part of the decimal fraction 65.1400 by the natural number 4: 
This completes the division of the integer part of the decimal fraction. Here in the quotient you need to put a decimal point and continue the division: 
We have reached a remainder of 0, at this stage the division by the column ends. As a result, we have 65.14:4=16.285.
Answer:
65,14:4=16,285 .
Example.
Divide 164.5 by 27.
Solution.
Let's divide the decimal fraction by a natural number using a column. After dividing the whole part we get the following picture: 
Now we put a comma in the quotient and continue dividing with a column: 
Now it is clearly visible that the residues 25, 7 and 16 have begun to repeat, while in the quotient the numbers 9, 2 and 5 are repeated. Thus, dividing the decimal 164.5 by 27 gives us the periodic decimal 6.0(925) .
Answer:
164,5:27=6,0(925) .
Column division of decimal fractions
The division of a decimal fraction by a decimal fraction can be reduced to dividing a decimal fraction by a natural number with a column. To do this, the dividend and the divisor must be multiplied by such a number as 10, or 100, or 1,000, etc., so that the divisor becomes a natural number, and then divide by a natural number with a column. We can do this due to the properties of division and multiplication, since a:b=(a·10):(b·10) , a:b=(a·100):(b·100) and so on.
In other words, to divide a trailing decimal by a trailing decimal, need to:
- in the dividend and divisor, move the comma to the right by as many places as there are after the decimal point in the divisor; if in the dividend there are not enough signs to move the comma, then you need to add the required number of zeros to the right;
- After this, divide with a decimal column by a natural number.
When solving an example, consider the application of this rule of division by a decimal fraction.
Example.
Divide with a column 7.287 by 2.1.
Solution.
Let's move the comma in these decimal fractions one digit to the right, this will allow us to move from dividing the decimal fraction 7.287 by the decimal fraction 2.1 to dividing the decimal fraction 72.87 by the natural number 21. Let's do the division by column: 
Answer:
7,287:2,1=3,47 .
Example.
Divide the decimal 16.3 by the decimal 0.021.
Solution.
Move the comma in the dividend and divisor to the right three places. Obviously, the divisor does not have enough digits to move the decimal point, so we will add the required number of zeros to the right. Now let’s divide the fraction 16300.0 with a column by the natural number 21: 
From this moment, the remainders 4, 19, 1, 10, 16 and 13 begin to repeat, which means that the numbers 1, 9, 0, 4, 7 and 6 in the quotient will also be repeated. As a result, we get the periodic decimal fraction 776,(190476) .
Answer:
16,3:0,021=776,(190476) .
Note that the announced rule allows you to divide a natural number by a column into a final decimal fraction.
Example.
Divide the natural number 3 by the decimal fraction 5.4.
Solution.
After moving the decimal point one digit to the right, we arrive at dividing the number 30.0 by 54. Let's do the division by column: 
.
This rule can also be applied when dividing infinite decimal fractions by 10, 100, .... For example, 3,(56):1,000=0.003(56) and 593.374…:100=5.93374… .
Dividing decimals by 0.1, 0.01, 0.001, etc.
Since 0.1 = 1/10, 0.01 = 1/100, etc., then from the rule of dividing by a common fraction it follows that divide the decimal fraction by 0.1, 0.01, 0.001, etc. . it's the same as multiplying a given decimal by 10, 100, 1,000, etc. respectively.
In other words, to divide a decimal fraction by 0.1, 0.01, ... you need to move the decimal point to the right by 1, 2, 3, ... digits, and if the digits in the decimal fraction are not enough to move the decimal point, then you need to add the required number to the right zeros.
For example, 5.739:0.1=57.39 and 0.21:0.00001=21,000.
The same rule can be applied when dividing infinite decimal fractions by 0.1, 0.01, 0.001, etc. In this case, you should be very careful when dividing periodic fractions so as not to make a mistake with the period of the fraction that is obtained as a result of division. For example, 7.5(716):0.01=757,(167), since after moving the decimal point in the decimal fraction 7.5716716716... two places to the right, we have the entry 757.167167.... With infinite non-periodic decimal fractions everything is simpler: 394,38283…:0,001=394382,83… .
Dividing a fraction or mixed number by a decimal and vice versa
Dividing a common fraction or mixed number by a finite or periodic decimal fraction, as well as dividing a finite or periodic decimal fraction by a common fraction or mixed number, comes down to dividing common fractions. To do this, decimal fractions are replaced by the corresponding ordinary fractions, and the mixed number is represented as an improper fraction.
When dividing an infinite non-periodic decimal fraction by a common fraction or mixed number and vice versa, you should proceed to dividing decimal fractions, replacing the common fraction or mixed number with the corresponding decimal fraction.
Bibliography.
- Mathematics: textbook for 5th grade. general education institutions / N. Ya. Vilenkin, V. I. Zhokhov, A. S. Chesnokov, S. I. Shvartsburd. - 21st ed., erased. - M.: Mnemosyne, 2007. - 280 pp.: ill. ISBN 5-346-00699-0.
- Mathematics. 6th grade: educational. for general education institutions / [N. Ya. Vilenkin and others]. - 22nd ed., rev. - M.: Mnemosyne, 2008. - 288 p.: ill. ISBN 978-5-346-00897-2.
- Algebra: textbook for 8th grade. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 16th ed. - M.: Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
- Gusev V. A., Mordkovich A. G. Mathematics (a manual for those entering technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.
Let's write down the rule and consider its application using examples.
When dividing a decimal fraction by a natural number:
1) divide without paying attention to the comma;
2) when the division of the whole part ends, we put a comma in the quotient.
If the whole part less than divisor, then the integer part of the quotient is equal to zero.
Examples of dividing decimal fractions by natural numbers.
We divide without paying attention to the comma, that is, we divide 348 by 6. When dividing 34 by 6, we take 5 each. 5∙6=30, 34-30=4, that is, the remainder is 4.
The difference between dividing a decimal fraction by a natural number and dividing integers is only that when the division of the integer part is completed, we put a comma in the quotient. That is, when passing through a comma, before taking it down to the remainder of the division of the integer part, 4, the number 8 from the fractional part, we write a comma in the quotient.
We take down 8. 48:6=8. In private we write 8.
So, 34.8:6=5.8.
Since 5 is not divisible by 12, we write zero in the quotient. The division of the whole part is completed, we put a comma in the quotient.
We take down 1. When dividing 51 by 12, we take 4. The remainder is 3.
We take down 6. 36:12=3.
Thus, 5.16:12=0.43.
3) 0,646:38=?
The integer part of the dividend contains zero. Since zero is not divisible by 38, we put 0 in the quotient. The division of the integer part is completed, in the quotient we write a comma.
We take down 6. Since 6 is not divisible by 38, we write one more zero in the quotient.
We take down 4. When dividing 64 by 38, we take 1. The remainder is 26.
We take down 6. 266:38=7.
So, 0.646:38=0.017.
4) 14917,5:325=?
When dividing 1491 by 325, we take 4 each. The remainder is 191. We take away 7. When dividing 1917 by 325, we take 5 each. The remainder is 292.
Since the division of the whole part is completed, we write a comma in the quotient.