Division of decimal fractions by natural numbers. Division by decimal
I. To divide a decimal fraction by a natural number, you need to divide the fraction by this number, as is divided integers and put in a private comma when the division of the whole part ends.
Examples.
Perform division: 1) 96,25: 5; 2) 4,78: 4; 3) 183,06: 45.
Solution.
Example 1) 96,25: 5.
Divide by "corner" as natural numbers are divided. After we demolish the figure 2 (the number of tenths is the first digit after the decimal point in the record of the dividend 96, 2 5), put a comma in the quotient and continue dividing.
Answer: 19,25.
Example 2) 4,78: 4.
Divide as natural numbers divide. In private, we put a comma as soon as we demolish 7
- the first digit after the decimal point in the dividend 4, 7
8. We continue the division further. Subtracting 38-36 gets 2, but the division is not complete. How do we act? We know that zeros can be added to the end of the 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, put a comma as soon as we demolish 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.
Output: when dividing a decimal fraction by a natural number in we put a comma in private right after we demolish the digit in the tenths of the dividend... Please note: all highlighted red numbers in these three examples refer to the category tenths of the dividend.
II... To divide a decimal fraction by 10, 100, 1000, etc., you need to move the comma 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.
Carrying a comma to the left depends on how many zeros after one are in the divisor. So, when dividing a decimal fraction by 10
we will transfer in the dividend comma left one digit; when divided by 100
- move the comma left two digits; when divided by 1000
carry over in the given decimal fraction a comma three digits to the left.
Division rule decimal fractions on natural numbers.
Four identical toys total 921 rubles 20 kopecks. How much does one toy cost (see Fig. 1)?
Rice. 1. Illustration for the problem
Solution
To find the cost of one toy, you need to divide this amount by four. Let's convert the amount into kopecks:
Answer: the cost of one toy is 23,030 kopecks, that is, 230 rubles 30 kopecks, or 230.3 rubles.
You can solve this problem without converting rubles into kopecks, that is, divide the decimal fraction by a natural number:.
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 in the quotient comma when the division of the integer part ends.
Divide in a column as natural numbers divide. After we demolish the number 2 (the number of tenths is the first digit after the decimal point in the record of the dividend 921.20), in the quotient we put a comma and continue the division:
Answer: 230.3 rubles.
Divide in a column as natural numbers divide. After we demolish the number 6 (the number of tenths is the number after the decimal point in the record of the dividend 437.6), in the quotient we put a comma and continue the division:
If the dividend is less than the divisor, then the quotient will start at zero.
1 is not divisible by 19, so we put zero in the quotient. The division of the whole part is over, we put a comma in the quotient. We demolish 7. 17 by 19 is not divisible, in the quotient we write zero. We demolish 6 and continue dividing:
Divide as natural numbers divide. In the quotient, we put a comma as soon as we demolish 8 - the first digit after the decimal point in the dividend 74.8. We continue the division further. Subtracting we get 8, but the division is not over. We know that zeros can be added to the end of the decimal fraction - this will not change the value of the fraction. We assign zero and divide 80 by 10. We get 8 - the division is over.
To divide a decimal fraction by 10, 100, 1000, etc., you need to move the comma in this fraction by as many digits to the left as there are zeros after one in the divisor.
In this lesson, we learned how to divide a decimal fraction by a natural number. We considered the option with the usual natural number, as well as the option in which division by bit unit occurs (10, 100, 1000, etc.).
Solve the equations:
To find the unknown divisor, you need to divide the dividend by the quotient. That is .
We divide in a column. After we demolish the number 4 (the number of tenths is the first digit after the decimal point in the record of the dividend 134.4), in the quotient we put a comma and continue the division:
1. Budaakai Nadezhda Duktugovna MBOU OOSh p. Ust-Khadyn Tandinsky kozhuun
2. Teacher of mathematics and physics
3. Mathematics
5. Division of decimal fractions by natural numbers. Lesson number 1
6. "Mathematics 5" N. Ya. Vilenkin, V. I. Zhokhov and others.
7. The purpose of the lesson:
8. Planned results:
Personal : develop listening skills; clearly, accurately, competently express their thoughts in oral and written speech; develop creativity of thinking, initiative, resourcefulness, activity in solving mathematical problems; to form ideas about mathematics as a method of cognition;
Metasubject: develop the ability to see math problem in the context problem situation in other disciplines, in the surrounding life; to form the ability to work in groups;
Subject: develop the ability to work with mathematical text (analyze, extract the necessary information).
9. Lesson type: discovery of new knowledge
10. Forms of student work: group, individual
11. Necessary Technical equipment: multimedia projector, computer, handouts for group work.
12. Structure and course of the lesson
Download:
Preview:
Task for group work.
Perform the action:
A) 0.7: 25; e) 9.607: 10;
B) 543.4: 143; g) 0.0142: 100;
TEST
- Calculate: What is the quotient if the dividend is 199.5 and the divisor is 15
a) 133;
b) 13.3;
c) 1.33.
- Find the value of the expression 243.2: 8
a) 30.4;
b) 3.04;
c) 304.
- 0.76 * 0.7598. Between the numbers, instead of *, you must put a sign:
a) ">";
b) "
c) "=".
- Find the meaning of the expression 45: 60
a) 1.333;
b) 7 5;
c) 0.75.
Preview:
Topic: Division of decimal fractions by natural numbers.
- Budaakai Nadezhda Duktugovna MBOU OOSh p. Ust-Khadyn Tandinsky kozhuun
- Mathematics and Physics Teacher
- Maths
- Grade 5
- Division of decimal fractions by natural numbers. Lesson number 1
- "Mathematics 5" N. Ya. Vilenkin, V. I. Zhokhov and others.
- The purpose of the lesson:
- Planned results:
Personal : develop listening skills; clearly, accurately, competently express their thoughts in oral and written speech; develop creativity of thinking, initiative, resourcefulness, activity in solving mathematical problems; to form ideas about mathematics as a method of cognition;
Metasubject: develop the ability to see a mathematical problem in the context of a problem situation in other disciplines, in the surrounding life; to form the ability to work in groups;
Subject: develop the ability to work with mathematical text (analyze, extract the necessary information).
- Lesson type: discovering new knowledge
- Forms of student work: group, individual
- Required technical equipment: multimedia projector, computer, handouts for group work.
- Lesson structure and course
Technological lesson map
Lesson steps | Student activities | Teacher activity | Universal learning activities |
1. The stage of motivation (self-determination) to learning activities. | The mood for work. Student responses | Create conditions for the emergence of internal demand Emotional attitude to the lesson. Children, are you warm? (Yes!) Is it light in the classroom? (Yes!) Has the bell already rung? (Yes!) Is the lesson over yet? (No!) Lesson just started? (Yes!) Do you want to study? (Yes!) So everyone can sit down! Lesson motivation. Slide 1 And so that you guys are not bored in the lesson, everyone should take an active part. Each of you knows that the horse is the most beloved animal among the Tuvans. Do you love a horse? Let's remember what horses are like? Today we'll talk about the legendary horse, which won 5 times in a row. | Personal: self-determination; Regulatory: goal setting; Communicative:planning educational collaboration with teacher and peers |
2. Stage Updating basic knowledge | Checks and agrees. Exercise. Slide 1 | Communicative: Cognitive: choice of the most effective ways solving problems Brain teaser: - formulation of the problem. |
|
3. Stage actualization and trial training action. | Activated the corresponding mental operations (analysis, generalization, classification, etc.) and cognitive processes (attention, memory, etc.); Student response. Done using division Different answers. (The formula for finding the speed) Tried to do it on my own individual task and recorded the difficulty encountered in performing the trial action or justifying it. | It activates the knowledge of students and the preparation of the thinking of students and the organization of their awareness of their inner need to build a new way of action. How do we solve this problem?Presentation Slide 3 Do we know how to divide a decimal fraction by a natural number? The tutorial page 208 will help us | Communicative:planning educational collaboration with teacher and peers; Cognitive: independent selection and formulation of a cognitive goal. Brain teaser: - formulation of the problem. |
3. The stage of identifying the place and cause of the difficulty. | Analyzed, recorded what knowledge or skill is lacking to solve the original problem (cause of difficulty) | Presentation Slide 4 Analyzes the causes of difficulties and helps in choosing the knowledge that is lacking | Regulatory: goal setting, forecasting; Cognitive : choosing the most effective ways to solve problems |
4. The stage of setting the topic of the lesson and the educational goal. | V communicative form formulated a specific goal of their future educational actions, eliminating the cause of the difficulty (that is, they formulated what knowledge they need to build and what to learn); proposed and agreed on the topic of the lesson Division of decimal fractions by natural numbers. | Consults, checks, agrees, clarifies the topic of the lesson Questions?
Slide 5 What tasks do we face today? Summarize the subtotal. | Communicative: planning educational collaboration with teacher and peers Personal : planning educational activities |
5.The stage of discovering new knowledge | Apply new way actions to solve the problem that caused the difficulty; fix in a generalized form a new way of acting in speech and recording fractions; fix the overcoming of the previously encountered difficulty. | Let's compose an algorithm for dividing decimal fractions by a natural number Slide 6 Slide 7.8 Slide 9, 10 Learn how to divide a decimal fraction by 10, 100, ... and so on. Physical minute. Slide 11 | Communicative: developing the ability to work in a group Cognitive: building logical circuits, analysis, ability to structure knowledge |
6. The stage of primary reinforcement with pronunciation in external speech. | We solved (frontally) several typical tasks for a new method of action; while speaking aloud the steps taken and their rationale Group work. | Organizes the solution of typical tasks (frontally) There was a custom: a winning horse is awarded a nickname if he takes first place three times in a row. At the republican races in honor of Naadim, the main annual holiday of livestock breeders, the black horse of Soyan Sandanmaa became the winner three times in a row: in 1934, 1935 and 1936. Slide 12,13,14,15 | Regulatory: highlighting and awareness of what has been learned, what is still to be learned Subject: developing skills in building mathematical models and solving practical problems |
7. Stage of group work. | Group work. Present the finished result of the work to the class (analyze, systematize) | Slide 16 A) 0.7: 25; e) 9.607: 10; B) 543.4: 143; g) 0.0142: 100; Problem Slide 17 The mass of a foal is 0.86 centners, and the mass of 2 horses is more than the mass of 4 foals by 1.36 centners. What is the mass of one horse. | Communicative:managing the behavior of a partner, resolving conflicts, the ability to fully and accurately express your thoughts Cognitive: analysis, synthesis, generalization, analogy, comparison, classification and construction of a logical chain of reasoning Regulatory: be able to plan and carry out activities aimed at solving research problems Subject: development of ideas about the number |
8 stage independent work self-test | Independently perform typical tasks for a new method of action Perform self-test Identify the causes of errors and their correction | Organizes independent execution learners of typical assignments to a new way of acting; organizes students' self-examination of their decisions; creates (if possible) a situation of success for each child; for students who have made mistakes, provides an opportunity to identify the causes of mistakes and correct them Individually (test) | Communicative:planning educational collaboration with teacher and peers Regulatory: control, assessment, highlighting and awareness of what has been learned, what is still to be learned Subject: development of ideas about number and number systems from natural to rational, the ability to apply the studied material |
9. Reflection of educational activities, summing up the results of the lesson | Carries out a self-assessment of his own educational activities, correlates the goal and results Choose a statement that matches the mood in the lesson Outline the prospect of further work Homework recording | Organizes reflection and self-assessment by students of their own educational activities in the classroom; Slide 19 goals for further activities are outlined and tasks for self-preparation are determined ( homework with elements of creative activity) Slide 20 |
Let's look at examples of dividing decimal fractions in this light.
Example.
Divide the decimal 1.2 by the decimal 0.48.
Solution.
Answer:
1,2:0,48=2,5 .
Example.
Divide the periodic decimal 0, (504) by the decimal 0.56.
Solution.
Let's convert a periodic decimal fraction to a common one:. We also translate the final decimal fraction 0.56 into an ordinary one, we have 0.56 = 56/100. Now we can go from dividing the original decimal fractions to dividing ordinary fractions and finish the calculations:.
We translate the resulting ordinary fraction into a decimal fraction by dividing the numerator by the denominator in a column:
Answer:
0,(504):0,56=0,(900) .
The principle of division of infinite non-periodic decimal fractions differs from the principle of division of final and periodic decimal fractions, since non-periodic decimal fractions cannot be converted to ordinary fractions. Division of infinite non-periodic decimal fractions is reduced to the division of finite decimal fractions, for which it is carried out rounding numbers to a certain level. Moreover, if one of the numbers with which the division is carried out is a final 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 final decimal 1.5602.
Solution.
First, you need to round the decimal fractions in order to go from dividing an infinite non-periodic decimal fraction to dividing the final decimal fractions. 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 .
Division of 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.
For illustration, consider an example of dividing a decimal fraction by a natural number.
Example.
Divide the decimal fraction 25.5 by the natural number 45.
Solution.
Replacing the decimal fraction 25.5 with an ordinary fraction 255/10 = 51/2, division is reduced to dividing an ordinary fraction by a natural number:. The resulting fraction in decimal notation has the form 0.5 (6).
Answer:
25,5:45=0,5(6) .
Column division of a decimal by a natural number
It is convenient to divide finite decimal fractions by natural numbers in a column, by analogy with division by a column of natural numbers. Here is the division rule.
To divide a decimal by a natural number in a column, necessary:
- add several digits 0 to the right in the divisible decimal fraction (in the process of division, 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 for division by a column of natural numbers, but when the division of the integer part of the decimal fraction ends, 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 the final decimal fraction by a natural number, either a final decimal fraction or an infinite periodic decimal fraction can be obtained. Indeed, after the division of all non-0 decimal places ends divisible fraction, you can get either the remainder of 0, and we get the final decimal fraction, or the remainders will start repeating periodically, and we get the periodic decimal fraction.
Let's figure out all the intricacies of dividing decimal fractions by natural numbers in a column when solving examples.
Example.
Divide the decimal 65.14 by 4.
Solution.
Let's divide the decimal fraction by a natural number in a column. Let's add a couple of zeros to the right in the fraction 65.14, and we get an equal decimal fraction 65.1400 (see equal and unequal decimal fractions). Now you can start dividing the whole 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 come to a remainder of 0, at this stage long division 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 in a column. After dividing the whole part, we get the following picture: 
Now we put a comma in the private and continue division with a column: 
Now you can clearly see that the remainders 25, 7 and 16 have begun to repeat, while in the quotient the numbers 9, 2 and 5 are repeated. So dividing the decimal 164.5 by 27 brings us to the periodic decimal 6.0 (925).
Answer:
164,5:27=6,0(925) .
Long division of decimal fractions
To dividing a decimal fraction by a natural number with a column, you can reduce the division of a decimal fraction by a decimal fraction. To do this, the dividend and the divisor must be multiplied by such a number 10, or 100, or 1,000, etc., so that the divisor becomes a natural number, and then divide by a natural number in a column. We can do this by virtue of 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 the final decimal by the final decimal, necessary:
- in the dividend and the divisor, move the comma to the right by as many digits as there are after the comma in the divisor, if at the same time the dividend does not have enough signs to carry the comma, then you need to add required amount zeros to the right;
- after that, divide with a column of a decimal fraction by a natural number.
Consider, when solving an example, the application of this rule of division by a decimal fraction.
Example.
Perform long division 7.287 by 2.1.
Solution.
Move the comma in these decimal fractions one digit to the right, this will allow us from dividing the decimal fraction 7.287 by the decimal fraction 2.1 to go to dividing the decimal fraction 72.87 by the natural number 21. Let's do long division: 
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 by 3 characters. Obviously, the divisor does not have enough digits to carry the comma, so we add the required number of zeros to the right. Now let's perform column division of the fraction 16300.0 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 voiced rule allows you to divide a natural number by a final decimal fraction by a column.
Example.
Divide the natural number 3 by the decimal 5.4.
Solution.
After moving the comma 1 digit to the right, we come to the division of the number 30.0 by 54. Let's do long division: 
.
This rule can also be applied when dividing infinite decimal fractions by 10, 100,…. For example, 3, (56): 1000 = 0.003 (56) and 593.374…: 100 = 5.93374….
Division of decimal fractions by 0.1, 0.01, 0.001, etc.
Since 0.1 = 1/10, 0.01 = 1/100, etc., from the rule of division by an ordinary fraction it follows that to divide the decimal fraction by 0.1, 0.01, 0.001, etc. ... it's like multiplying the given decimal by 10, 100, 1,000, etc. respectively.
In other words, to divide the decimal fraction by 0.1, 0.01, ... you need to move the comma to the right by 1, 2, 3, ... digits, while if the digits in the decimal notation are not enough to carry the comma, then you need to add the required amount 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, one should be very careful with the division of periodic fractions, so as not to be mistaken with the period of the fraction, which is obtained as a result of division. For example, 7.5 (716): 0.01 = 757, (167), since after moving the comma in the decimal fraction 7.5716716716 ... two digits to the right, we have the record 757.167167 .... With infinite non-periodic decimal fractions, everything is simpler: 394,38283…:0,001=394382,83… .
Division of a fraction or mixed number by a decimal and vice versa
Dividing an ordinary fraction or a mixed number by a finite or periodic decimal fraction, as well as dividing a finite or periodic decimal fraction by an ordinary fraction or mixed number, is reduced to dividing ordinary fractions. To do this, decimal fractions are replaced with the corresponding ordinary fractions, and the mixed number is represented as an improper fraction.
When dividing an infinite non-periodic decimal fraction by an ordinary fraction or a mixed number and vice versa, you should go to the division of decimal fractions, replacing the ordinary fraction or mixed number with the corresponding decimal fraction.
Bibliography.
- Maths: textbook. for 5 cl. general education. institutions / N. Ya. Vilenkin, V. I. Zhokhov, A. S. Chesnokov, S. I. Shvartsburd. - 21st ed., Erased. - M .: Mnemosina, 2007 .-- 280 p .: ill. ISBN 5-346-00699-0.
- Maths. Grade 6: textbook. for general education. institutions / [N. Ya. Vilenkin and others]. - 22nd ed., Rev. - M .: Mnemosina, 2008 .-- 288 p.: Ill. ISBN 978-5-346-00897-2.
- Algebra: study. for 8 cl. general education. institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M.: Education, 2008 .-- 271 p. : ill. - ISBN 978-5-09-019243-9.
- Gusev V.A., Mordkovich A.G. Mathematics (manual for applicants to technical schools): Textbook. manual. - M .; Higher. shk., 1984.-351 p., ill.
Let's write down the rule and consider its application with 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, put a comma in the quotient.
If whole part is less than the divisor, then the integer part of the quotient is zero.
Examples of dividing decimal fractions by natural numbers.
Divide, not paying attention to the comma, that is, 348 is divided by 6. When dividing 34 by 6, we take 5. 5 ∙ 6 = 30, 34-30 = 4, that is, the remainder is 4.
The only difference between dividing a decimal fraction by a natural number and dividing integers is that when the division of the integer part is over, we put a comma in the quotient. That is, when passing through a comma, before removing to the remainder of the division of the integer part, 4, the number 8 from the fractional part, in the private we write a comma.
We demolish 8.48: 6 = 8. In the 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 over, we put a comma in the quotient.
We demolish 1. When dividing 51 by 12 we take 4. In the remainder - 3.
We demolish 6.36: 12 = 3.
So 5.16: 12 = 0.43.
3) 0,646:38=?
The integer part of the dividend is zero. Since zero is not divisible by 38, in the quotient we put 0. The division of the integer part is over, in the quotient we write a comma.
We demolish 6. Since 6 is not divisible by 38, we write one more zero in the quotient.
We demolish 4. When dividing 64 by 38 we take 1. The remainder is 26.
We demolish 6.266: 38 = 7.
So 0.646: 38 = 0.017.
4) 14917,5:325=?
When dividing 1491 by 325 we take 4. The remainder is 191. We demolish 7. When dividing 1917 by 325 we take 5. The remainder is 292.
Since the division of the whole part is over, we write a comma in the quotient.
