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The purpose of the lesson:
- Introduce to students in a fun form the rule for multiplying a decimal fraction by a natural number, by a digit unit and the rule for expressing a decimal fraction as a percentage. Develop the ability to apply the knowledge gained when solving examples and problems.
- Develop and activate logical thinking students, the ability to identify patterns and generalize them, strengthen memory, the ability to cooperate, provide assistance, evaluate their work and the work of each other.
- To foster interest in mathematics, activity, mobility, the ability to communicate.
Equipment: interactive whiteboard, poster with cyphergram, posters with statements of mathematicians.
During the classes
- Organizing time.
- Oral counting is a generalization of previously studied material, preparation for the study of new material.
- Explanation of the new material.
- Home assignment.
- Mathematical physical education minute.
- Generalization and systematization of the knowledge gained in game form using a computer.
- Grading.
2. Guys, today our lesson will be somewhat unusual, because I will not teach it alone, but with my friend. And my friend is also unusual, now you will see him. (A cartoon computer appears on the screen). My friend has a name and can speak. What's your name, buddy? Komposha replies: "My name is Komposha." Are you ready to help me today? YES! Well then, let's start the lesson.
Today I received an encrypted cyphergram, guys, which we must solve and decipher together. (A poster is posted on the board with oral counting for addition and subtraction of decimal fractions, as a result of which the guys get the following code 523914687. )
| 5 | 2 | 3 | 9 | 1 | 4 | 6 | 8 | 7 |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
Composha helps to decipher the received code. As a result of decoding, the word MULTIPLICATION is obtained. Multiplication is the key word for today's lesson. The topic of the lesson is displayed on the monitor: "Multiplying a decimal fraction by a natural number"
Guys, we know how multiplication is done natural numbers... Today we will look at multiplication decimal numbers by a natural number. The multiplication of a decimal fraction by a natural number can be considered as the sum of terms, each of which is equal to this decimal fraction, and the number of terms is equal to this natural number. For example: 5.21 3 = 5.21 + 5.11 + 5.21 = 15.63 Hence, 5.21 3 = 15.63. Representing 5.21 as common fraction by a natural number, we get
And in this case we got the same result 15.63. Now, disregarding the comma, we will take the number 521 instead of the number 5.21 and multiply it by this natural number. Here we must remember that in one of the factors the comma has been moved two places to the right. When multiplying the numbers 5, 21 and 3, we get the product equal to 15.63. Now, in this example, we will move the comma to the left by two places. Thus, by how many times one of the factors was increased, the product was reduced by that many times. Based on the similarities of these methods, we draw a conclusion.
To multiply a decimal fraction by a natural number, you need:
1) ignoring the comma, perform the multiplication of natural numbers;
2) in the resulting product, separate with a comma on the right as many digits as there are in a decimal fraction.
The following examples are displayed on the monitor, which we analyze together with Kompoche and the guys: 5.21 · 3 = 15.63 and 7.624 · 15 = 114.34. Then I show the multiplication by the round number 12.6 50 = 630. Next, I turn to multiplying the decimal fraction by the digit unit. I show the following examples: 7,423 · 100 = 742.3 and 5.2 · 1000 = 5200. So, I introduce the rule for multiplying a decimal fraction by a digit unit:
To multiply a decimal fraction by 10, 100, 1000, etc., you need to move the comma to the right in this fraction by as many digits as there are zeros in the bit unit record.
I end the explanation with a decimal percentage. I introduce a rule:
To express a decimal fraction as a percentage, you need to multiply it by 100 and assign a% sign.
I give an example on a computer 0.5 · 100 = 50 or 0.5 = 50%.
4. At the end of the explanation, I give the guys homework, which is also displayed on the computer monitor: № 1030, № 1034, № 1032.
5. In order for the guys to have a little rest, to consolidate the topic, we do a mathematical physical education together with Komposha. Everyone stands up, I show the class solved examples and they must answer whether the example was solved correctly or not. If the example is correct, they raise their hands above their heads and clap their palms. If the example is not solved correctly, the guys stretch their arms to the sides and knead their fingers.
6. And now you have a little rest, you can solve the tasks. Open the tutorial to page 205, № 1029. in this task, you need to calculate the value of the expressions:
The tasks appear on the computer. As they are solved, a picture appears with the image of a boat, which, when fully assembled, floats away.
No. 1031 Calculate:
Solving this task on the computer, the rocket gradually develops, solving the last example, the rocket flies away. The teacher gives a little information to the students: “Every year from the Kazakh land from the Baikonur cosmodrome, spaceships take off to the stars. Kazakhstan is building its new Baiterek cosmodrome near Baikonur.
No. 1035. Problem.
What is the distance a passenger car will cover in 4 hours if the speed of a passenger car is 74.8 km / h.
This task is accompanied by sound design and a brief condition of the task being displayed on the monitor. If the problem is solved correctly, then the car begins to move forward to the finish flag.
№ 1033. Write it down decimals in percentages.
0,2 = 20%; 0,5 = 50%; 0,75 = 75%; 0,92 = 92%; 1,24 =1 24%; 3,5 = 350%; 5,61= 561%.
Solving each example, when the answer appears, a letter appears, resulting in the word Well done.
The teacher asks Composha, what would this word appear for? Komposha replies: "Well done, guys!" and says goodbye to everyone.
The teacher summarizes the lesson and gives marks.
1 lesson
1. Organizational moment
Check the readiness of students for the lesson.
(Availability of educational supplies for the lesson)
I .Knowledge update
Oral work.
Target: To systematize the previous knowledge necessary when studying new material.
Students orally complete tasks on multiplying a decimal fraction by a natural number and multiplying ordinary fractions.
Calculate:
The teacher then asks the question, “Formulate how to multiply a decimal by a natural number?” Students recall the definition. The lesson topic and lesson objectives are communicated.
II .Simultaneous division into groups and into pairs.
Students select one card at a time from the teacher's desk. Some of them contain examples of actions with ordinary fractions, and others have corresponding answers. They will have to find matches, and will split into pairs, but if they work in groups, they will split in this way:
Group 1 are students who have come across examples, Group 2 is those students who will have the appropriate answers. (See Appendix # 1)
III .Study of new material
Target: Introduce new material to students.
Explanation of the teacher:
3.1 Group work.
Target: Having independently solved the problem in two ways, formulate the rule for multiplying a decimal fraction by a decimal fraction.
Students are offered the following task:
The length of the rectangle is 6.3 cm, the width is 2.8 cm. Find its area.
Each group performs this task according to the proposed method indicated to it.
Method 1: Write down numerical values measurements of a rectangle as natural numbers, expressed in millimeters. Calculate the area and express the answer in square centimeters.
Method 2: Present the measurements of a rectangle as fractions, find the area by multiplying fractions and convert to a decimal.
Then a representative from each group explains the decision. this example students of the other group at the blackboard. Students exchange opinions and from the results of solving the problem conclude:
How many decimal places are in multipliers, the same number of decimal places in their product.
Then the teacher comments on the work of the groups, summarizes and concludes.
Students write in notebooks for notes.
Conclusion: To multiply decimal fractions you need:
1) perform multiplication, ignoring the commas;
2) separate in the resulting product with a comma as many digits to the right as they are after the comma in both factors together.
3.2 Analysis of various examples.
Target: Further development of skills to perform multiplication of decimal fractions.
We multiply these numbers without paying attention to the commas, we get the number 20 496 in the product. There are three decimal places in the two multipliers after the decimal point. Therefore, in the work, you need to separate the three numbers on the right, so the work is 20.496.
VI .Solution of tasks
Target: Practice skills to apply the rule of multiplication of decimal fractions when solving problems.
Students work in pairs.
Perform tasks: # 812, # 814
Vii . Summing up the lesson. Reflection
Target: Determine if the students have met the lesson objectives to consider when planning the next lesson.
Student actions : Summarizing your knowledge answer questions.
Summing up questions .(Orally).
1. What have we learned in the lesson today?
2. What purpose did we learn in today's lesson?
3. Let's repeat the rule for multiplying decimal fractions.
At the end of the lesson, students give reflection:
Lesson liked / disliked
The purpose of the lesson understood / did not understand
What I learned, what I learned ______________________________
What I didn't fully understand _______________________________
What needs to be worked on _______________________________
Assessment: The teacher encourages students to answer and work.
Home assignment:№813 № 815
Like ordinary numbers.
2. We count the number of decimal places in the 1st decimal fraction and in the 2nd. We add up their number.
3. In the final result, count from right to left as many digits as you get in the paragraph above, and put a comma.
Decimal multiplication rules.
1. Multiply, ignoring the comma.
2. In the product, separate as many digits after the comma as there are after the commas in both factors together.
Multiplying a decimal fraction by a natural number, you need:
1. Multiply numbers, ignoring the comma;
2. As a result, we put the comma so that to the right of it there are as many digits as there are in decimal fraction.
Multiplication of decimal fractions by a column.
Let's take an example:
We write decimal fractions in a column and multiply them as natural numbers, ignoring the commas. Those. We regard 3.11 as 311, and 0.01 as 1.
The result is 311. Next, we count the number of decimal places for both fractions. In the 1st decimal fraction there are 2 digits and in the 2nd - 2. Total number digits after commas:
2 + 2 = 4
We count from right to left four characters in the result. In the final result, there are fewer numbers than you need to separate with a comma. In this case, it is necessary to add the missing number of zeros to the left.
In our case, the 1st digit is missing, so we add 1 zero to the left.
Note:
Multiplying any decimal fraction by 10, 100, 1000, and so on, the decimal point is moved to the right by as many digits as there are zeros after one.
for instance:
70,1 . 10 = 701
0,023 . 100 = 2,3
5,6 . 1 000 = 5 600
Note:
To multiply a decimal by 0.1; 0.01; 0.001; and so on, you need to move the comma to the left in this fraction by as many digits as there are zeros in front of the unit.
We count zero integers!
For instance:
12 . 0,1 = 1,2
0,05 . 0,1 = 0,005
1,256 . 0,01 = 0,012 56
Decimal multiplication occurs in three stages.
Decimal fractions are written in a column and multiplied like ordinary numbers.
We count the number of decimal places in the first decimal fraction and in the second. We add up their number.
In the resulting result, we count from right to left as many digits as we got them in the paragraph above and put a comma.
How to multiply decimal fractions
We write decimal fractions in a column and multiply them as natural numbers, ignoring the commas. That is, we consider 3.11 as 311, and 0.01 as 1.
Received 311. Now we count the number of digits (digits) after the decimal point for both fractions. The first decimal has two digits and the second has two. Total number of digits after commas:
We count from right to left 4 characters (numbers) from the resulting number. In the resulting result, there are fewer digits than you need to separate with a comma. In this case, you need left assign the missing number of zeros.
We are missing one digit, so we assign one zero to the left.
When multiplying any decimal on 10; one hundred; 1000, etc. the decimal point is moved to the right by as many digits as there are zeros after one.
To multiply a decimal by 0.1; 0.01; 0.001, etc., it is necessary to move the comma to the left in this fraction by as many digits as there are zeros in front of the unit.
We count zero integers!
- 12 0.1 = 1.2
- 0.05 0.1 = 0.005
- 1.256 0.01 = 0.012 56
- ignoring the commas, perform multiplication according to all the rules of multiplication with a column of natural numbers;
- in the resulting number, separate the decimal point as many digits on the right as there are decimal places in both factors together, and if there are not enough digits in the product, then on the left you need to add the right amount zeros.
To understand how to multiply decimal fractions, let's look at specific examples.
Decimal multiplication rule
1) We multiply, ignoring the comma.
2) As a result, we separate as many digits after the comma as there are after the commas in both factors together.
Find the product of decimal fractions:
To multiply decimal fractions, we multiply, ignoring the commas. That is, we are not multiplying 6.8 and 3.4, but 68 and 34. As a result, we separate as many digits after the comma as there are after the commas in both factors together. The first multiplier after the decimal point has one digit, the second - also one. In total, we separate two digits after the decimal point. Thus, we got the final answer: 6.8 ∙ 3.4 = 23.12.
Multiply decimals without taking into account the comma. That is, in fact, instead of multiplying 36.85 by 1.14, we multiply 3685 by 14. We get 51590. Now, in this result, we need to separate as many digits with a comma as there are in both factors together. The first number after the decimal point has two digits, the second - one. In total, we separate three digits with a comma. Since there is a zero at the end of the entry after the decimal point, we do not write it in response: 36.85 ∙ 1.4 = 51.59.
To multiply these decimal fractions, we multiply the numbers, ignoring the commas. That is, we multiply the natural numbers 2315 and 7. We get 16205. In this number, you need to separate four digits after the decimal point - as many as there are in both factors together (two in each). The final answer: 23.15 ∙ 0.07 = 1.6205.
Multiplication of a decimal fraction by a natural number is performed in the same way. We multiply the numbers, not paying attention to the comma, that is, we multiply 75 by 16. In the result, after the comma, there should be as many digits as there are in both factors together - one. Thus, 75 ∙ 1.6 = 120.0 = 120.
We start multiplying decimal fractions by multiplying natural numbers, since we do not pay attention to commas. After that, we separate as many digits after the decimal point as there are in both factors together. In the first number after the decimal point, there are two digits, in the second - also two. In total, as a result, there should be four digits after the decimal point: 4.72 ∙ 5.04 = 23.7888.
And a couple more examples for multiplying decimal fractions:
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Decimal multiplication, rules, examples, solutions.
Moving on to the study of the next action with decimal fractions, now we will comprehensively consider decimal multiplication... Let's talk first general principles multiplication of decimal fractions. After that, we will move on to multiplying a decimal fraction by a decimal fraction, show how the multiplication of decimal fractions by a column is performed, consider the solutions of examples. Next, we will analyze the multiplication of decimal fractions by natural numbers, in particular by 10, 100, etc. In conclusion, let's talk about multiplying decimal fractions by fractions and mixed numbers.
Let's say right away that in this article we will only talk about multiplying positive decimal fractions (see positive and negative numbers). The rest of the cases are discussed in the articles multiplication rational numbers and multiplication of real numbers.
Page navigation.
General principles of multiplying decimal fractions
Let's discuss the general principles that should be followed when carrying out multiplication with decimal fractions.
Since finite decimal fractions and infinite periodic fractions are the decimal form of writing common fractions, the multiplication of such decimal fractions is essentially the multiplication of common fractions. In other words, end decimal multiplication, multiplication of final and periodic decimal fractions, as well as multiplication of periodic decimal fractions is reduced to the multiplication of ordinary fractions after converting decimal fractions to ordinary ones.
Let's consider examples of using the sounded principle of multiplying decimal fractions.
Multiply the decimal fractions 1.5 and 0.75.
Replace the decimal fractions to be multiplied with the corresponding common fractions. Since 1.5 = 15/10 and 0.75 = 75/100, then. You can reduce the fraction, then select the whole part from the improper fraction, and it is more convenient to write the resulting ordinary fraction 1 125/1000 in the form of a decimal fraction 1.125.
It should be noted that it is convenient to multiply final decimal fractions in a column, we will talk about this method of multiplying decimal fractions in the next paragraph.
Let's look at an example of multiplying periodic decimal fractions.
Calculate the product of the periodic decimal fractions 0, (3) and 2, (36).
Let's translate periodic decimal fractions into ordinary fractions:
Then. You can convert the resulting ordinary fraction into a decimal fraction: 
If among the multiplied decimal fractions there are infinite non-periodic fractions, then all multiplied fractions, including finite and periodic ones, should be rounded to a certain digit (see rounding numbers), and then multiply the final decimal fractions obtained after rounding.
Perform the decimal multiplication 5.382 ... and 0.2.
First, we round off an infinite non-periodic decimal fraction, rounding can be done to hundredths, we have 5.382 ... ≈5.38. There is no need to round the final decimal 0.2 to hundredths. Thus, 5.382 ... · 0.2≈5.38 · 0.2. It remains to calculate the product of final decimal fractions: 5.38 · 0.2 = 538/100 · 2/10 = 1,076/1000 = 1.076.
Column Decimal Multiplication
Multiplication of final decimal fractions can be performed in a column, similar to multiplication in a column of natural numbers.
Let's formulate column decimal multiplication rule... To multiply decimal fractions with a column, you need:
Let's consider examples of multiplying decimal fractions with a column.
Multiply the decimal fractions 63.37 and 0.12.
Let's carry out the multiplication of decimal fractions by a column. First, we multiply the numbers, ignoring the commas: 
It remains to put a comma in the resulting product. She needs to separate 4 digits from the right, since the factors add up to four decimal places (two in the fraction 3.37 and two in the fraction 0.12). There are enough numbers, so there is no need to add zeros to the left. Let's finish recording: 
As a result, we have 3.37 0.12 = 7.6044.
Calculate the product of the decimal fractions 3.2601 and 0.0254.
After multiplying with a column without taking into account commas, we get the following picture: 
Now in the product you need to separate the 8 digits on the right with a comma, since total the decimal places of the multiplied fractions is eight. But there are only 7 digits in the product, therefore, you need to assign so many zeros to the left so that you can separate 8 digits with a comma. In our case, you need to assign two zeros: 
This completes the multiplication of decimal fractions by a column.
Multiplying decimal fractions by 0.1, 0.01, etc.
Quite often, you have to multiply decimal fractions by 0.1, 0.01, and so on. Therefore, it is advisable to formulate a rule for multiplying a decimal fraction by these numbers, which follows from the principles of multiplying decimal fractions discussed above.
So, multiplying the given decimal fraction by 0.1, 0.01, 0.001, and so on gives a fraction, which is obtained from the original, if in its entry the comma is moved to the left by 1, 2, 3 and so on digits, respectively, while if there are not enough digits to carry the comma, then you need to add it to the left required amount zeros.
For example, to multiply the decimal fraction 54.34 by 0.1, you need to move the comma to the left by 1 digit in the fraction 54.34, and you get the fraction 5.434, that is, 54.34 · 0.1 = 5.434. Let's give one more example. Multiply the decimal 9.3 by 0.0001. To do this, we need to move the comma 4 digits to the left in the decimal fraction 9.3 to be multiplied, but the fraction 9.3 does not contain that many digits. Therefore, we need to assign so many zeros in the fraction 9.3 on the left so that we can easily carry out the transfer of the comma by 4 digits, we have 9.3 · 0.0001 = 0.00093.
Note that the voiced rule for multiplying a decimal fraction by 0.1, 0.01, ... is also valid for infinite decimal fractions. For example, 0, (18) · 0.01 = 0.00 (18) or 93.938 ... · 0.1 = 9.3938….
Decimal multiplication by a natural number
At its core decimal multiplication by natural numbers is no different from multiplying a decimal by a decimal.
It is most convenient to multiply the final decimal fraction by a natural number in a column, while you should adhere to the rules for multiplying with a column of decimal fractions discussed in one of the previous paragraphs.
Calculate the product 15 · 2.27.
Let's multiply a natural number by a decimal fraction in a column: 
When multiplying a periodic decimal fraction by a natural number, replace the periodic fraction with an ordinary fraction.
Multiply the decimal 0, (42) by the natural number 22.
First, we convert the periodic decimal fraction to an ordinary fraction:
Now let's do the multiplication:. This result in decimal form is 9, (3).
And when multiplying an infinite non-periodic decimal fraction by a natural number, you must first round.
Perform multiplication 4 · 2.145….
Having rounded up the original infinite decimal fraction to hundredths, we arrive at the multiplication of a natural number and a final decimal fraction. We have 4 · 2.145 ... ≈4 · 2.15 = 8.60.
Decimal multiplication by 10, 100, ...
Quite often you have to multiply decimal fractions by 10, 100, ... Therefore, it is advisable to dwell on these cases in detail.
We will sound the rule for multiplying a decimal fraction by 10, 100, 1,000, etc. When multiplying a decimal fraction by 10, 100, ... in its record, you need to move the comma to the right by 1, 2, 3, ... numbers, respectively, and discard the extra zeros on the left; if there are not enough digits in the record of the multiplied fraction to carry the comma, then you need to add the required number of zeros to the right.
Multiply the decimal 0.0783 by 100.
Move the fraction 0.0783 two digits to the right in the record, and we get 007.83. Dropping two zeros from the left, we get the decimal fraction 7.38. Thus, 0.0783 100 = 7.83.
Multiply the decimal 0.02 by 10,000.
To multiply 0.02 by 10,000, we need to move the comma 4 digits to the right. Obviously, the fraction 0.02 does not have enough digits to transfer a comma to 4 digits, so we will add a few zeros to the right so that we can carry a comma transfer. In our example, it is enough to add three zeros, we have 0.02000. After transferring the comma, we get the entry 00200.0. Discarding the zeros on the left, we have the number 200.0, which is equal to the natural number 200, which is the result of multiplying the decimal fraction 0.02 by 10,000.
The stated rule is also true for multiplying infinite decimal fractions by 10, 100, ... When multiplying periodic decimal fractions, you need to be careful with the period of the fraction, which is the result of multiplication.
Multiply the periodic decimal 5.32 (672) by 1,000.
Before multiplying, let's write down the periodic decimal fraction as 5.32672672672 ..., this will allow us to avoid mistakes. Now let's move the comma to the right by 3 digits, we have 5 326.726726…. Thus, after multiplication, the periodic decimal fraction 5 326, (726) is obtained.
5.32 (672) 1000 = 5 326, (726).
When multiplying infinite non-periodic fractions by 10, 100, ..., you must first round off the infinite fraction to a certain digit, and then multiply.
Decimal multiplication by a fraction or mixed number
To multiply a finite decimal fraction or an infinite periodic decimal fraction by an ordinary fraction or a mixed number, you need to represent the decimal fraction as an ordinary fraction, and then multiply.
Multiply the decimal 0.4 by the mixed number.
Since 0.4 = 4/10 = 2/5 and, then. The resulting number can be written as a periodic decimal fraction 1.5 (3).
When multiplying an infinite non-periodic decimal fraction by an ordinary fraction or mixed number, the ordinary fraction or mixed number should be replaced with a decimal fraction, then round the multiplied fractions and finish the calculations.
Since 2/3 = 0.6666 ..., then. After rounding the multiplied fractions to thousandths, we come to the product of two final decimal fractions 3.568 and 0.667. Let's do long multiplication: 
The result should be rounded up to thousandths, since the fractions to be multiplied were taken to the nearest thousandths, we have 2.379856≈2.380.
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29. Multiplication of decimal fractions. Rules
Find the area of a rectangle with sides equal
1.4 dm and 0.3 dm. Let's convert decimeters to centimeters:
1.4 dm = 14 cm; 0.3 dm = 3 cm.
Now let's calculate the area in centimeters.
S = 14 3 = 42 cm 2.
Convert square centimeters to square centimeters
decimeters:
dm 2 = 0.42 dm 2.
Hence, S = 1.4 dm 0.3 dm = 0.42 dm 2.
Multiplication of two decimal fractions is done like this:
1) numbers are multiplied without regard to commas.
2) the comma in the work is put so as to separate from the right
as many signs as separated in both factors
put together. For instance:
1,1 0,2 = 0,22 ; 1,1 1,1 = 1,21 ; 2,2 0,1 = 0,22 .
Examples of multiplying decimal fractions in a column:
Instead of multiplying any number by 0.1; 0.01; 0.001,
you can divide this number by 10; one hundred ; or 1000 respectively.
For instance:
22 0,1 = 2,2 ; 22: 10 = 2,2 .
When multiplying a decimal fraction by a natural number, we must:
1) multiply the numbers, ignoring the comma;
2) in the resulting work, put a comma so that on the right
there were as many digits from it as there were in a decimal fraction.
Find the product 3.12 10. According to the above rule
first we multiply 312 by 10. We get: 312 10 = 3120.
And now we separate the two digits on the right with a comma and get:
3,12 10 = 31,20 = 31,2 .
So, when multiplying 3.12 by 10, we moved the comma by one
number to the right. If we multiply 3.12 by 100, we get 312, that is
the comma was moved two digits to the right.
3,12 100 = 312,00 = 312 .
When multiplying a decimal by 10, 100, 1000, etc., you must
in this fraction, move the comma to the right by as many digits as there are zeros
stands in the multiplier. For instance:
0,065 1000 = 0065, = 65 ;
2,9 1000 = 2,900 1000 = 2900, = 2900 .
Decimal Multiplication Tasks
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Addition, subtraction, multiplication, and division of decimal fractions
Adding and subtracting decimal fractions is similar to adding and subtracting natural numbers, but with certain conditions.
Rule. is produced by the digits of the integer and fractional parts as natural numbers.
In writing addition and subtraction of decimal fractions the comma separating the integer part from the fractional part must be in the terms and the sum or in the reduced, subtracted and difference in one column (a comma under the comma from the condition record to the end of the calculation).
Adding and subtracting decimal fractions to the line:
243,625 + 24,026 = 200 + 40 + 3 + 0,6 + 0,02 + 0,005 + 20 + 4 + 0,02 + 0,006 = 200 + (40 + 20) + (3 + 4)+ 0,6 + (0,02 + 0,02) + (0,005 + 0,006) = 200 + 60 + 7 + 0,6 + 0,04 + 0,011 = 200 + 60 + 7 + 0,6 + (0,04 + 0,01) + 0,001 = 200 + 60 + 7 + 0,6 + 0,05 + 0,001 = 267,651
843,217 - 700,628 = (800 - 700) + 40 + 3 + (0,2 - 0,6) + (0,01 - 0,02) + (0,007 - 0,008) = 100 + 40 + 2 + (1,2 - 0,6) + (0,01 - 0,02) + (0,007 - 0,008) = 100 + 40 + 2 + 0,5 + (0,11 - 0,02) + (0,007 - 0,008) = 100 + 40 + 2 + 0,5 + 0,09 + (0,007 - 0,008) = 100 + 40 + 2 + 0,5 + 0,08 + (0,017 - 0,008) = 100 + 40 + 2 + 0,5 + 0,08 + 0,009 = 142,589
Adding and subtracting decimal fractions in a column:
Adding decimal fractions requires an additional upper line to write numbers when the sum of the digit goes over ten. Subtraction of decimals requires an additional top line to mark the digit in which 1 is borrowed.
If there are not enough digits of the fractional part to the right of the addend or the reduced one, then on the right in the fractional part, you can add as many zeros (increase the digit capacity of the fractional part) as there are digits in the other addend or the reduced one.
Decimal multiplication It is performed in the same way as the multiplication of natural numbers, according to the same rules, but a comma is put in the product according to the sum of the digits of the factors in the fractional part, counting from right to left (the sum of the digits of the factors is the number of digits after the decimal point of the factors combined).
At decimal multiplication in a column, the first significant digit on the right is signed under the first significant digit on the right, as in natural numbers:
Recording decimal multiplication in a column:
Recording division of decimal fractions in a column:
Underlined characters are characters that carry a comma because the divisor must be an integer.
Rule. At dividing fractions the divisor of the decimal fraction increases by as many digits as there are digits in its fractional part. So that the fraction does not change, the dividend is also increased by the same number of digits (in the dividend and the divisor, the comma is transferred by the same number of digits). The comma is placed in the quotient at the stage of division when the integer part of the fraction is divided.
For decimal fractions, as well as for natural numbers, the rule remains: You cannot divide a decimal fraction by zero!
The decimal fraction is used when you need to perform actions with non-integers. This may seem irrational. But this kind of numbers greatly facilitates the mathematical operations that must be performed with them. This understanding comes with time, when their writing becomes familiar, and reading is not difficult, and the rules of decimal fractions are mastered. Moreover, all actions are repeated already known, which are mastered with natural numbers. You just need to remember some features.
Decimal Definition
A decimal fraction is a special representation of a non-integer number with a denominator that is divisible by 10, and the answer is obtained in the form of one and possibly zeros. In other words, if the denominator is 10, 100, 1000, and so on, then it is more convenient to rewrite the number using a comma. Then the whole part will be located before it, and then the fractional part. Moreover, the recording of the second half of the number will depend on the denominator. The number of digits that are in the fractional part must be equal to the place of the denominator.
The above can be illustrated with these numbers:
9/10=0,9; 178/10000=0,0178; 3,05; 56 003,7006.
Reasons why you need to use decimal fractions
Decimals were needed by mathematicians for several reasons:
Simplification of the recording. Such a fraction is located along one line without a dash between the denominator and the numerator, while the clarity does not suffer.
Simplicity in comparison. It is enough just to correlate the numbers that are in the same positions, while with ordinary fractions it would be necessary to bring them to a common denominator.
Simplification of calculations.
Calculators are not designed for the introduction of common fractions, they use decimal notation for all operations.
How to read such numbers correctly?
The answer is simple: just like an ordinary mixed number with a denominator that is a multiple of 10. The only exceptions are fractions without an integer value, then when reading it is necessary to pronounce “zero integers”.
For example, 45/1000 should be pronounced as forty five thousandths, at the same time 0.045 would sound like zero point forty-five thousandths.
A mixed number with the integer part equal to 7 and the fraction 17/100, which will be written as 7.17, in both cases will be read as seven point seventeen hundredths.
The role of digits in writing fractions
To correctly mark the rank is what the mathematician demands. Decimal fractions and their meaning can change significantly if you write the number in the wrong place. However, this was true before.
To read the digits of the integer part of a decimal fraction, you just need to use the rules known for natural numbers. And on the right side, they are mirrored and read differently. If "tens" sounded in the whole part, then after the decimal point it will be "tenths".
This can be clearly seen in this table.
| Class | thousand | units | , | fraction | |||||||
| discharge | honeycomb | dess. | units | honeycomb | dess. | units | tenth | hundredth | thousandth | ten thousandth | |
What is the correct way to write a mixed number as a decimal fraction?
If the denominator contains a number equal to 10 or 100, and others, then the question of how to convert a fraction to decimal is not difficult. To do this, it is enough to rewrite all its constituent parts in a different way. The following points will help with this:
a little aside write the numerator of the fraction, at this moment decimal point located on the right, after the last digit;
move the comma to the left, the most important thing here is to correctly count the numbers - you need to move it by as many positions as there are zeros in the denominator;
if there are not enough of them, then there should be zeros in empty positions;
zeros that were at the end of the numerator are no longer needed and can be crossed out;
in front of the comma, assign the whole part, if it was not there, then there will also be zero here.
Attention. You cannot cross out zeros that are surrounded by other numbers.
You can read about how to be in a situation when the denominator contains not only ones and zeros, how to convert a fraction into decimal, you can read below. This is important information that you should definitely read.
How can a fraction be converted to a decimal if the denominator is an arbitrary number?
Two options are possible here:
When the denominator can be represented as a number that is equal to ten to any power.
If such an operation cannot be done.
How can I check this? You need to factor the denominator. If the product contains only 2 and 5, then everything is fine, and the fraction is easily converted to the final decimal. Otherwise, if 3, 7 and other prime numbers appear, then the result will be infinite. For ease of use in mathematical operations, it is customary to round off such a decimal fraction. This will be discussed a little below.
Studying how such decimal fractions are obtained, grade 5. Examples will come in handy here.
Let the denominators contain numbers: 40, 24 and 75. Their prime factorization will be as follows:
- 40 = 2 2 2 5;
- 24 = 2 2 2 3;
- 75 = 5 5 3.
In these examples, only the first fraction can be finalized.
Algorithm for converting an ordinary fraction to a final decimal
Check the prime factorization of the denominator and make sure that it will consist of 2 and 5.
Add to these numbers as many 2 and 5 so that they become equal. They will give the value of the extra multiplier.
Multiply the denominator and numerator by this number. As a result, you get an ordinary fraction, under the line which has 10 to some extent.
If in a problem these actions are performed with a mixed number, then it must first be represented as an improper fraction. And only then proceed according to the described scenario.
Rounded Decimal Representation of a Fraction
This way of converting a fraction to a decimal will seem even easier to some. Because there is no a large number action. You just need to divide the value of the numerator by the denominator.
Any number with a decimal part to the right of the decimal point can be assigned an infinite number of zeros. This property should be used.
First write down the whole part followed by a comma. If the fraction is correct, then write zero.
Then it is supposed to perform division of the numerator by the denominator. So that they have the same number of digits. That is, assign the required number of zeros to the right of the numerator.
Perform long division until the required number of digits is entered. For example, if you need to round up to hundredths, then the answer should be 3. In general, there should be one more numbers than you need to get in the end.
Write down the intermediate answer after the comma and round according to the rules. If the last digit is from 0 to 4, then you just need to discard it. And when it is 5-9, then the one standing in front of it needs to be increased by one, discarding the last one.
Back from decimal to fraction
In mathematics, there are problems when it is more convenient to represent decimal fractions in the form of ordinary fractions, in which there is a numerator with a denominator. You can breathe a sigh of relief: this operation is always possible.
For this procedure, you need to do the following:
write down the whole part, if it is equal to zero, then you do not need to write anything;
draw a fractional line;
write down the numbers from the right side above it, if zeros come first, then they need to be crossed out;
under the line, write a unit with as many zeros as the number of digits after the decimal point in the initial fraction.
That's all you need to do to convert a decimal to a fraction.
What can you do with decimal fractions?
In mathematics it will be certain actions with decimal fractions that were previously done for other numbers.
They are:
comparison;
addition and subtraction;
multiplication and division.
The first action, comparison, is similar to how it was done for natural numbers. To determine which is greater, you need to compare the digits of the integer part. If they turn out to be equal, then they go to fractional ones and compare them in the same way. The number where it will be big figure in the high order, and will be the answer.
Adding and subtracting decimal fractions
This is perhaps the most simple actions... Because they are performed according to the rules for natural numbers.
So, to perform the addition of decimal fractions, they need to be written under each other, placing the commas in a column. With this notation, whole parts appear to the left of the commas, and fractional parts to the right. And now you need to add the numbers bit by bit, as is done with natural numbers, dropping a comma down. You need to start addition with the smallest digit of the fractional part of the number. If there are not enough digits in the right half, then zeros are added.
The same applies for subtraction. And here there is a rule that describes the possibility of borrowing one from the most significant bit. If there are fewer digits in the reduced fraction after the decimal point than in the subtracted fraction, then zeros are simply assigned in it.
The situation is a little more complicated with tasks where you need to perform multiplication and division of decimal fractions.
How to multiply decimal in different examples?
The rule by which decimal fractions are multiplied by a natural number is as follows:
write them down in a column, ignoring the comma;
multiply as if they were natural;
separate as many digits with a comma as there were in the fractional part of the original number.
A special case is an example in which a natural number is equal to 10 to any power. Then, to get an answer, you just need to move the comma to the right by as many positions as there are zeros in another factor. In other words, when multiplied by 10, the comma is shifted by one digit, by 100 - there will already be two of them, and so on. If there are not enough digits in the fractional part, then you need to write zeros in empty positions.
The rule that is used when in a task you need to multiply decimal fractions by another of the same number:
write them under each other, ignoring the commas;
multiply as if they were natural;
separate as many digits with a comma as there were in the fractional parts of both original fractions together.
Examples are highlighted as a special case in which one of the factors is 0.1 or 0.01 and so on. In them, you need to move the comma to the left by the number of digits in the presented multipliers. That is, if it is multiplied by 0.1, then the comma is shifted by one position.
How do I split a decimal in different tasks?
Division of decimal fractions by a natural number is performed according to the following rule:
write them down for long division, as if they were natural;
divide according to the usual rule until the whole part ends;
put a comma in response;
continue dividing the fractional component until the remainder is zero;
if necessary, you can assign the required number of zeros.
If the integer part is equal to zero, then it will not be in the answer either.
Separately, there is division into numbers equal to ten, one hundred, and so on. In such problems, you need to move the comma to the left by the number of zeros in the divisor. It happens that there are not enough digits in the whole part, then zeros are used instead. You may notice that this operation is similar to multiplying by 0.1 and similar numbers.
To perform decimal division, you need to use this rule:
turn the divisor into a natural number, and for this move the comma in it to the right to the end;
move a comma and in a divisible by the same number of digits;
proceed according to the previous scenario.
Division by 0.1 is highlighted; 0.01 and others similar numbers... In such examples, the comma is shifted to the right by the number of digits in the fractional part. If they are over, then you need to assign the missing number of zeros. It is worth noting that this action repeats division by 10 and similar numbers.
Conclusion: it's all about practice
Nothing about learning comes easily or effortlessly. It takes time and practice to master new material reliably. Mathematics is no exception.
So that the topic about decimal fractions does not cause difficulties, you need to solve as many examples with them as possible. After all, there was a time when the addition of natural numbers was perplexing. And now everything is fine.
Therefore, to paraphrase a well-known phrase: decide, decide and decide again. Then tasks with such numbers will be performed easily and naturally, like another puzzle.
By the way, puzzles are difficult to solve at first, and then you need to do the usual movements. The same is in mathematical examples: after walking along the same path several times, then you will no longer think about where to turn.