Examples for multiplication with decimal fractions. Fractions

To understand how to multiply decimals, 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.

Examples.

Find the product of decimals:

To multiply decimals, we multiply without paying attention to commas. That is, we do not multiply 6.8 and 3.4, but 68 and 34. As a result, we separate as many digits after the decimal point as there are after the commas in both factors together. In the first factor after the decimal point there is one digit, in the second there is also one. In total, we separate two digits after the decimal point. Thus, we got the final answer: 6.8∙3.4=23.12.

Multiplying 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 has two digits after the decimal point, the second has 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 decimals, we multiply the numbers without paying attention to the commas. That is, we multiply the natural numbers 2315 and 7. We get 16205. In this number, four digits must be separated after the decimal point - as many as there are in both factors together (two in each). Final answer: 23.15∙0.07=1.6205.

Multiplication decimal fraction on the natural number performed similarly. We multiply the numbers without paying attention to the comma, that is, we multiply 75 by 16. In the result obtained, after the comma there should be as many signs as there are in both factors together - one. Thus, 75∙1.6=120.0=120.

We begin the multiplication of decimal fractions by multiplying natural numbers, since we do not pay attention to commas. After that, we separate as many digits after the comma as there are in both factors together. The first number has two decimal places, and the second has two decimal places. In total, as a result, there should be four digits after the decimal point: 4.72∙5.04=23.7888.

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The purpose of the lesson:

AT fascinating form introduce students to the rule of multiplying a decimal fraction by a natural number, by a bit unit and the rule of expressing a decimal fraction as a percentage. Develop the ability to apply the acquired knowledge in solving examples and problems.
Develop and activate logical thinking students, the ability to identify patterns and generalize them, strengthen memory, cooperation, provide assistance, evaluate their work and the work of each other.
To cultivate interest in mathematics, activity, mobility, ability to communicate.

Equipment: interactive board, a poster with a cyphergram, posters with the 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 new material.
Homework assignment.
Mathematical physical education.
Generalization and systematization of the acquired knowledge in game form using a computer.
Grading.

2. Guys, today our lesson will be somewhat unusual, because I will not spend 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 he can talk. What's your name, friend? 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 an oral account for adding and subtracting 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

Komposha helps to decipher the received code. As a result of decoding, the word MULTIPLICATION is obtained. Multiplication is keyword topics of 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 the multiplication of natural numbers is performed. Today we are going to look at multiplication. decimal numbers to 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 \u003d 5.21 + 5, 21 + 5.21 \u003d 15.63 So 5.21 3 = 15.63. Representing 5.21 as an ordinary fraction of a natural number, we get

And in this case, we got the same result of 15.63. Now, ignoring the comma, let's take the number 521 instead of the number 5.21 and multiply by the given natural number. Here we must remember that in one of the factors the comma is moved two places to the right. When multiplying the numbers 5, 21 and 3, we get a product equal to 15.63. Now, in this example, we will move the comma to the left by two digits. Thus, by how many times one of the factors was increased, the product was reduced by so many times. Based on the similar points of these methods, we draw a conclusion.

To multiply a decimal 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 characters as there are in a decimal fraction.

The following examples are displayed on the monitor, which we analyze together with Komposha and the guys: 5.21 3 = 15.63 and 7.624 15 = 114.34. After I show multiplication by a round number 12.6 50 \u003d 630. Next, I turn to the multiplication of a decimal fraction by a bit unit. Showing the following examples: 7,423 100 \u003d 742.3 and 5.2 1000 \u003d 5200. So, I introduce the rule for multiplying a decimal fraction by a bit unit:

To multiply a decimal fraction by bit units 10, 100, 1000, etc., it is necessary 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 the expression of a decimal fraction as a percentage. I enter the rule:

To express a decimal as a percentage, multiply it by 100 and add the % sign.

I give an example on a computer 0.5 100 \u003d 50 or 0.5 \u003d 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 rest a little, to consolidate the topic, we do a mathematical physical education session together with Komposha. Everyone stands up, shows the class the solved examples and they must answer whether the example is correct or incorrect. If the example is solved correctly, then 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 your textbook to page 205, № 1029. in this task it is necessary to calculate the value of expressions:

Tasks appear on the computer. As they are solved, a picture appears with the image of a boat, which, when fully assembled, sails away.

No. 1031 Calculate:

Solving this task on a computer, the rocket gradually develops, solving the last example, the rocket flies away. The teacher gives a little information to the students: “Every year, spaceships take off to the stars from the Kazakhstani land from the Baikonur Cosmodrome. Near Baikonur, Kazakhstan is building its new Baiterek cosmodrome.

No. 1035. Task.

How far will a car travel in 4 hours if the speed of the car is 74.8 km/h.

This task is accompanied by sound design and displaying a brief condition of the task on the monitor. If the problem is solved, right, then the car starts to move forward to the finish flag.

№ 1033. Write decimals as 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 Komposha, why would this word appear? Komposha replies: “Well done, guys!” and say goodbye to everyone.

The teacher sums up the lesson and assigns grades.

1 lesson

1. Organizational moment

Check student readiness for the lesson.

(Availability of study supplies for the lesson)

I .Knowledge update

oral work.

Target: To systematize the previous knowledge necessary for the study of new material.

Students verbally perform tasks on multiplying a decimal fraction by a natural number and multiplying ordinary fractions.

Calculate:

Then the teacher asks the question: Formulate how to multiply a decimal fraction by a natural number? Students remember the definition. The topic of the lesson and the objectives of the lesson are reported.

II .Simultaneous division into groups and pairs.

Students choose one card from the teacher's table. Some of them contain examples of actions with ordinary fractions, while others have the corresponding answers. They will have to find matches, and will be divided into pairs. If they work in groups, they will be divided in this way:

Group 1 - these are the students who came across examples, group 2 - these are the students who will have the appropriate answers. (See Appendix No. 1)

III .Studying new material

Target: Introduce students to new material.

Teacher's explanation:

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 given 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: burn numerical values measurements of a rectangle in the form of natural numbers, expressed in millimeters. Calculate the area and express the answer in square centimeters.

Method 2: Express the dimensions of the rectangle as common fractions, find the area by multiplying the common fractions and convert to a decimal.

Then a representative from each group explains the solution. this example students of another group at the blackboard. Students exchange opinions and from the results of solving the problem they conclude:

How many decimal places in factors, the same number of decimal places in their product.

Then the teacher comments on the work of the groups, summarizes and draws a conclusion.

Students write in notebooks for notes.

Conclusion: To multiply decimal fractions you need:

1) perform multiplication, ignoring commas;

2) to separate in the resulting product with a comma as many digits on the right as there 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 commas, we get the number 20 496 in the product. There are three decimal places in two factors after the decimal point. Therefore, in the product, three digits must be separated on the right. So, the product is 20.496.

VI .Problem solving

Target: Development of skills to apply the rule of multiplication of decimal fractions in solving problems.

Students work in pairs.

Perform tasks: No. 812, No. 814

VII . Summing up the lesson. Reflection

Target: Find out if the students achieved the objectives of the lesson to take into account when planning the next lesson.

Student actions : Summarizing your knowledge , answer questions.

Questions for debriefing .(Orally).

1. What have we learned in the lesson today?

2. What goal did we study today in the lesson?

3. Let's repeat the rule for multiplying decimal fractions.

At the end of the lesson, students give a reflection:

Lesson liked / disliked

The purpose of the lesson understood / did not understand

What did I learn, what did I learn?

What I don't fully understand

What needs to be worked on?

Evaluation: The teacher encourages student responses and work.

Homework:№813 № 815

Like regular numbers.

2. We count the number of decimal places for the 1st decimal fraction and for the 2nd. We add up their number.

3. In the final result, we count from right to left such a number of digits as they turned out in the paragraph above, and put a comma.

Rules for multiplying decimals.

1. Multiply without paying attention to the comma.

2. In the product, we separate as many digits after the decimal point as there are after the commas in both factors together.

Multiplying a decimal fraction by a natural number, you must:

1. Multiply numbers, ignoring the comma;

2. As a result, we put a comma so that there are as many digits to the right of it as in a decimal fraction.

Multiplication of decimal fractions by a column.

Let's look at an example:

We write decimal fractions in a column and multiply them as natural numbers, ignoring the commas. Those. We consider 3.11 as 311, and 0.01 as 1.

The result is 311. Next, we count the number of decimal places (digits) for both fractions. The 1st decimal has 2 digits and the 2nd decimal has 2. Total number digits after commas:

2 + 2 = 4

We count from right to left four characters of the result. In the final result, there are fewer digits than you need to separate with a comma. In this case, it is necessary to add the missing number of zeros on the left.

In our case, the 1st digit is missing, so we add 1 zero on the left.

Note:

Multiplying any decimal fraction by 10, 100, 1000, and so on, the comma in the decimal fraction is moved to the right by as many places as there are zeros after the one.

for example:

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 characters as there are zeros in front of the unit.

We count zero integers!

For example:

12 . 0,1 = 1,2

0,05 . 0,1 = 0,005

1,256 . 0,01 = 0,012 56

Let's move on to studying the next action with decimal fractions, now we will comprehensively consider multiplying decimals. Let's discuss first general principles multiplying decimals. After that, let's 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 ordinary 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 remaining cases are analyzed in the articles multiplication of rational numbers and multiplication of real numbers.

Page navigation.

General principles for multiplying decimals

Let's discuss the general principles that should be followed when performing multiplication with decimal fractions.

Since finite decimals and infinite periodic fractions are the decimal form of ordinary fractions, the multiplication of such decimal fractions is essentially the multiplication of ordinary fractions. In other words, multiplication of final decimals, multiplication of final and periodic decimal fractions, as well as multiplying periodic decimals comes down to multiplying ordinary fractions after converting decimal fractions to ordinary.

Consider examples of the application of the voiced principle of multiplying decimal fractions.

Example.

Perform the multiplication of decimals 1.5 and 0.75.

Decision.

Let us replace the multiplied decimal fractions with the corresponding ordinary fractions. Since 1.5=15/10 and 0.75=75/100, then . You can reduce the fraction, and then select the whole part from the wrong fraction, but more conveniently the resulting common fraction 1 125/1 000 write as a decimal fraction 1.125.

Answer:

1.5 0.75=1.125.

It should be noted that it is convenient to multiply the final decimal fractions in a column; we will talk about this method of multiplying decimal fractions in.

Consider an example of multiplying periodic decimal fractions.

Example.

Compute the product of the periodic decimals 0,(3) and 2,(36) .

Decision.

Let's convert periodic decimal fractions to ordinary fractions:

Then . You can convert the resulting ordinary fraction to a decimal fraction:

Answer:

0,(3) 2,(36)=0,(78) .

If there are infinite non-periodic fractions among the multiplied decimal fractions, then all multiplied fractions, including finite and periodic ones, should be rounded up to a certain digit (see rounding numbers), and then perform the multiplication of the final decimal fractions obtained after rounding.

Example.

Multiply the decimals 5.382… and 0.2.

Decision.

First, we round off an infinite non-periodic decimal fraction, rounding can be done to hundredths, we have 5.382 ... ≈5.38. The final decimal fraction 0.2 does not need to be rounded 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 \u003d 538 / 100 2 / 10 \u003d 1,076/1,000 \u003d 1.076.

Answer:

5.382… 0.2≈1.076.

Multiplication of decimal fractions by a column

Multiplication of trailing decimals can be done by a column, similar to column multiplication of natural numbers.

Let's formulate multiplication rule for decimal fractions. To multiply decimal fractions by a column, you need:

ignoring commas, perform multiplication according to all the rules of multiplication by a column of natural numbers;
in the resulting number, separate with a decimal point as many digits on the right as there are decimal digits in both factors together, and if there are not enough digits in the product, then you need to add on the left right amount zeros.

Consider examples of multiplying decimal fractions by a column.

Example.

Multiply the decimals 63.37 and 0.12.

Decision.

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 on the right, since there are four decimal places in the factors (two in the fraction 3.37 and two in the fraction 0.12). There are enough numbers there, so you don’t have to add zeros on the left. Let's finish the record:

As a result, we have 3.37 0.12 = 7.6044.

Answer:

3.37 0.12=7.6044.

Example.

Calculate the product of decimals 3.2601 and 0.0254 .

Decision.

Having performed multiplication by a column without taking into account commas, we get the following picture:

Now in the work you need to separate the 8 digits on the right with a comma, since total decimal places of multiplied fractions is eight. But there are only 7 digits in the product, therefore, you need to assign as many zeros on the left so that 8 digits can be separated by a comma. In our case, we need to assign two zeros:

This completes the multiplication of decimal fractions by a column.

Answer:

3.2601 0.0254=0.08280654 .

Multiplying decimals by 0.1, 0.01, etc.

Quite often you have to multiply decimals 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 multiplication of decimal fractions discussed above.

So, multiplying a given decimal by 0.1, 0.01, 0.001, and so on gives a fraction, which is obtained from the original one, if in its entry the comma is moved to the left by 1, 2, 3 and so on digits, respectively, and if there are not enough digits to move the comma, then you need to add required amount zeros.

For example, to multiply the decimal fraction 54.34 by 0.1, you need to move the decimal point to the left by 1 digit in the fraction 54.34, and you get the fraction 5.434, that is, 54.34 0.1 \u003d 5.434. Let's take another example. Multiply the decimal fraction 9.3 by 0.0001. To do this, we need to move the comma 4 digits to the left in the multiplied decimal fraction 9.3, but the record of the fraction 9.3 does not contain such a number of characters. Therefore, we need to assign as many zeros in the record of the fraction 9.3 on the left so that we can easily transfer the comma to 4 digits, we have 9.3 0.0001 \u003d 0.00093.

Note that the announced 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… .

Multiplying a decimal by a natural number

At its core multiplying decimals by natural numbers is no different from multiplying a decimal by a decimal.

It is most convenient to multiply a finite decimal fraction by a natural number by a column, while you should follow the rules for multiplying by a column of decimal fractions discussed in one of the previous paragraphs.

Example.

Calculate the product 15 2.27 .

Decision.

Let's carry out the multiplication of a natural number by a decimal fraction in a column:

Answer:

15 2.27=34.05.

When multiplying a periodic decimal fraction by a natural number, periodic fraction should be replaced with a common fraction.

Example.

Multiply the decimal fraction 0,(42) by the natural number 22.

Decision.

First, let's convert the periodic decimal to a common fraction:

Now let's do the multiplication: . This decimal result is 9,(3) .

Answer:

0,(42) 22=9,(3) .

And when multiplying an infinite non-periodic decimal fraction by a natural number, you must first round off.

Example.

Do the multiplication 4 2.145….

Decision.

Rounding up to hundredths the original infinite decimal fraction, we will come to the multiplication of a natural number and a final decimal fraction. We have 4 2.145…≈4 2.15=8.60.

Answer:

4 2.145…≈8.60.

Multiplying a decimal 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.

Let's voice rule for multiplying a decimal by 10, 100, 1,000, etc. When multiplying a decimal fraction by 10, 100, ... in its entry, you need to move the comma to the right by 1, 2, 3, ... digits, respectively, and discard extra zeros on the left; if there are not enough digits in the record of the multiplied fraction to transfer the comma, then you need to add the required number of zeros to the right.

Example.

Multiply the decimal 0.0783 by 100.

Decision.

Let's transfer the fraction 0.0783 two digits to the right into the record, and we get 007.83. Dropping two zeros on the left, we get the decimal fraction 7.38. Thus, 0.0783 100=7.83.

Answer:

0.0783 100=7.83.

Example.

Multiply the decimal fraction 0.02 by 10,000.

Decision.

To multiply 0.02 by 10,000 we need to move the comma 4 digits to the right. Obviously, in the record of the fraction 0.02 there are not enough digits to transfer the comma to 4 digits, so we will add a few zeros to the right so that the comma can be transferred. In our example, it is enough to add three zeros, we have 0.02000. After moving the comma, we get the entry 00200.0 . Dropping the zeros on the left, we have the number 200.0, which is equal to the natural number 200, it is the result of multiplying the decimal fraction 0.02 by 10,000.