The decimal is used when you need to perform operations with non-integer numbers. This may seem irrational. But this type of numbers greatly simplifies the mathematical operations that need to be performed with them. This understanding comes over time, when writing them becomes familiar, and reading them does not cause difficulties, and the rules of decimal fractions have been mastered. Moreover, all actions repeat already known ones, which have been learned with natural numbers. You just need to remember some features.
Decimal definition
A decimal is a special representation of a non-integer number with a denominator that is divisible by 10, giving the answer as 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 digit 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 for using decimals
Mathematicians needed decimals for several reasons:
Simplifying recording. Such a fraction is located along one line without a dash between the denominator and numerator, while clarity does not suffer.
Simplicity in comparison. It is enough to simply correlate numbers that are in the same positions, while with ordinary fractions you would have to reduce them to a common denominator.
Simplify calculations.
Calculators are not designed to accept 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 exception is fractions without an integer value, then when reading you need to pronounce “zero integers.”
For example, 45/1000 should be pronounced as forty-five thousandths, at the same time 0.045 will sound like zero point forty five thousandths.
Mixed number with whole part equal to 7 and the fraction 17/100, which will be written as 7.17, in both cases it will be read as seven point seventeen.
The role of digits in writing fractions
Correctly marking the rank is what mathematics requires. Decimals and their meaning can change significantly if you write the digit in the wrong place. However, this was true before.
To read the digits of an integer part decimal you just need to use the rules known for natural numbers. And on the right side they are mirrored and read differently. If the whole part sounded “tens”, then after the decimal point it will be “tenths”.
This can be clearly seen in this table.
| Class | thousands | units | , | fraction | |||||||
| discharge | cell | dec. | units | cell | dec. | units | tenth | hundredth | thousandth | ten-thousandth | |
How to correctly write a mixed number as a decimal?
If the denominator contains a number equal to 10 or 100, and others, then the question of how to convert a fraction to a decimal is not difficult. To do this, it is enough to rewrite all its components differently. The following points will help with this:
write the numerator of the fraction a little to the side, at this moment the decimal point is located on the right, after the last digit;
move the comma to the left, the most important thing here is to count the numbers correctly - 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 the empty positions;
the zeros that were at the end of the numerator are now not needed and can be crossed out;
Before the comma, add 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 below about what to do in a situation where the denominator has a number not only consisting of ones and zeros, and how to convert a fraction to a decimal. This important information, which is definitely worth checking out.
How to convert a fraction to a decimal if the denominator is an arbitrary number?
There are two options here:
When the denominator can be represented as a number that is equal to ten to any power.
If such an operation cannot be performed.
How can I check this? You need to factor the denominator. If only 2 and 5 are present in the product, then everything is fine, and the fraction is easily converted to a final decimal. Otherwise, if 3, 7 and other prime numbers appear, the result will be infinite. It is customary to round such a decimal fraction for ease of use in mathematical operations. This will be discussed a little below.
Explores how decimals are made, 5th grade. Examples here will be very helpful.
Let the denominators contain the numbers: 40, 24 and 75. The decomposition into prime factors for them 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 represented as the final fraction.
Algorithm for converting a common fraction to a final decimal
Check the factorization of the denominator into prime factors and make sure that it will consist of 2 and 5.
Add as many 2s and 5s to these numbers so that there are an equal number of them. They will give the value of the additional multiplier.
Multiply the denominator and numerator by this number. The result will be an ordinary fraction, under the line of which there is 10 to some extent.
If in the problem these actions are performed with a mixed number, then it must first be represented as an improper fraction. And only then act according to the described scenario.
Representing a fraction as a rounded decimal
This method of converting a fraction to a decimal may seem even easier to some. Because it doesn't have large quantity actions. You just need to divide 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 is what you need to take advantage of.
First, write down the whole part and put a comma after it. If the fraction is correct, write zero.
Then you need to divide the numerator by the denominator. So that they have the same number of digits. That is, add the required number of zeros to the right of the numerator.
Perform long division until the required number of digits is reached. For example, if you need to round to hundredths, then the answer should be 3. In general, there should be one more number than you need to get in the end.
Write down the intermediate answer after the decimal point and round according to the rules. If last digit- from 0 to 4, then you just need to discard it. And when it is equal to 5-9, then the one in front of it needs to be increased by one, discarding the last one.
Return from decimal to common 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 there is no need to write anything;
draw a fraction line;
above it, write down the numbers from the right side, if the zeros come first, then they need to be crossed out;
Under the line, write a unit with as many zeros as there are digits after the decimal point in the original fraction.
That's all you need to do to convert a decimal to a fraction.
What can you do with decimals?
In mathematics, these will be certain operations with decimals that were previously performed 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 whole part. If they turn out to be equal, then they move on to the fractional and also compare them by digits. The number with the largest digit in the most significant digit will be the answer.
Adding and subtracting decimals
These are perhaps the simplest steps. Because they are carried out according to the rules for natural numbers.
So, in order to add decimal fractions, they need to be written one below the other, placing 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, moving the comma down. You need to start adding from the smallest digit of the fractional part of the number. If there are not enough numbers in the right half, then zeros are added.
The same applies to subtraction. And here there is a rule that describes the possibility of taking a unit from the highest rank. If in the fraction being reduced there is a decimal point fewer numbers than that of the subtrahend, then zeros are simply assigned to it.
The situation is a little more complicated with tasks where you need to multiply and divide decimal fractions.
How to multiply a decimal fraction in different examples?
The rule for multiplying decimal fractions by natural number, like this:
write them down in a column, ignoring the comma;
multiply as if they were naturals;
Separate with a comma as many digits as there were in the fractional part of the original number.
A special case is the example in which a natural number is equal to 10 to any power. Then to get the answer you just need to move the decimal point to the right by as many positions as there are zeros in the other factor. In other words, when multiplied by 10, the decimal point moves by one digit, by 100 - there will already be two of them, and so on. If there are not enough numbers in the fractional part, then you need to write zeros in the empty positions.
The rule that is used when a task requires multiplying decimal fractions by another same number:
write them down one after another, not paying attention to commas;
multiply as if they were natural;
Separate with a comma as many digits as there were in the fractional parts of both original fractions together.
A special case are examples in which one of the multipliers is equal to 0.1 or 0.01 and so on. In them you need to move the decimal point to the left by the number of digits in the presented factors. That is, if it is multiplied by 0.1, then the decimal point is shifted by one position.
How to divide a decimal fraction in different tasks?
Dividing decimal fractions by a natural number is performed according to the following rule:
write them down for division in a column as if they were natural ones;
divide according to the usual rule until the whole part is over;
put a comma in the answer;
continue dividing the fractional component until the remainder is zero;
if necessary, you can add 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, hundred, and so on. In such problems, you need to move the decimal point to the left by the number of zeros in the divisor. It happens that there are not enough numbers in a whole part, then zeros are used instead. You can see that this operation is similar to multiplying by 0.1 and similar numbers.
To divide decimals, you need to use this rule:
turn the divisor into a natural number, and to do this, move the comma in it to the right to the end;
move the decimal point in the dividend by the same number of digits;
act according to the previous scenario.
The division by 0.1 is highlighted; 0.01 and other similar numbers. In such examples, the decimal point is shifted to the right by the number of digits in the fractional part. If they run out, then you need to add 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 in learning comes easy or without effort. Reliably mastering new material takes time and practice. Mathematics is no exception.
To ensure 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 adding natural numbers was a dead end. And now everything is fine.
Therefore, to paraphrase famous phrase: decide, decide and decide again. Then tasks with such numbers will be completed 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. It’s the same in mathematical examples: having walked along the same path several times, then you will no longer think about where to turn.
Back forward
Attention! Slide previews are for informational purposes only and may not represent all the features of the presentation. If you are interested this work, please download the full version.
The purpose of the lesson:
- In a fun way, introduce to students the rule for multiplying a decimal fraction by a natural number, by a place value unit, and the rule for expressing a decimal fraction as a percentage. Develop the ability to apply acquired knowledge 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 own work and the work of each other.
- Cultivate interest in mathematics, activity, mobility, and communication skills.
Equipment: interactive board, a poster with a cyphergram, posters with statements by mathematicians.
During the classes
- Organizing time.
- Oral arithmetic – generalization of previously studied material, preparation for studying new material.
- Explanation of new material.
- Homework assignment.
- Mathematical physical education.
- Generalization and systematization of acquired knowledge in game form using a computer.
- Grading.
2. Guys, today our lesson will be somewhat unusual, because I will not be teaching it alone, but with my friend. And my friend is also unusual, you will see him now. (A cartoon computer appears on the screen.) My friend has a name and he can talk. 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 hung on the board with an oral calculation for adding and subtracting decimal fractions, as a result of which the children receive the following code 523914687. )
| 5 | 2 | 3 | 9 | 1 | 4 | 6 | 8 | 7 |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
Komposha helps decipher the received code. The result of decoding is the word MULTIPLICATION. 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 to multiply natural numbers. Today we will look at multiplication decimal numbers to a natural number. Multiplying a decimal fraction by a natural number can be considered as a 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 + 5.21 + 5.21 = 15.63 This means 5.21·3 = 15.63. Presenting 5.21 as a common fraction to a natural number, we get
And in this case we got the same result: 15.63. Now, ignoring the comma, instead of the number 5.21, take the number 521 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 a product equal to 15.63. Now in this example we move the comma to the left two places. Thus, by how many times one of the factors was increased, by how many times the product was decreased. Based on the similarities of these methods, we will draw a conclusion.
To multiply a decimal fraction by a natural number, you need to:
1) without paying attention to the comma, multiply natural numbers;
2) in the resulting product, separate as many digits from the right with a comma as there are in the 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. Afterwards I show multiplication by round number 12.6·50 = 630. Next, I move on to multiplying a decimal fraction by a place value 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 digit units 10, 100, 1000, etc., you need to move the decimal point in this fraction to the right by as many places as there are zeros in the digit unit.
I finish my explanation by expressing the decimal fraction as a percentage. I introduce the rule:
To express a decimal fraction as a percentage, you must multiply it by 100 and add the % sign.
I’ll 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 rest a little, we are doing a mathematical physical education session together with Komposha to consolidate the topic. Everyone stands up, shows the solved examples to the class, and they must answer whether the example was solved correctly or incorrectly. If the example is solved correctly, then they raise their arms above their heads and clap their palms. If the example is not solved correctly, the guys stretch their arms to the sides and stretch their fingers.
6. And now you have rested a little, you can solve the tasks. Open your textbook 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 that floats away when fully assembled.
No. 1031 Calculate:
By solving this task on a computer, the rocket gradually folds up; after solving the last example, the rocket flies away. The teacher gives a little information to the students: “Every year, spaceships take off from the Baikonur Cosmodrome from Kazakhstan’s soil to the stars. Kazakhstan is building its new Baiterek cosmodrome near Baikonur.
No. 1035. Problem.
How far will a passenger car travel in 4 hours if the speed of the passenger car is 74.8 km/h.
This task is accompanied by sound design and a brief condition of the task displayed on the monitor. If the problem is solved, correctly, then the car begins to move forward until the finish flag.
№ 1033. Write the 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%.
By solving each example, when the answer appears, a letter appears, resulting in a word Well done.
The teacher asks Komposha why this word would appear? Komposha replies: “Well done, guys!” and says goodbye to everyone.
The teacher sums up the lesson and gives grades.
Multiplying Decimals occurs in three stages.
Decimal fractions are written in a column and multiplied like ordinary numbers.
We count the number of decimal places for the first decimal fraction and the second. We add up their number.
In the resulting result, we count from right to left the same number of numbers as we got in the paragraph above and put a comma.
How to Multiply Decimals
We write the 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.
We received 311. Now we count the number of signs (digits) after the decimal point for both fractions. The first decimal has two digits and the second has two. Total number of decimal places:
We count from right to left 4 signs (digits) of the resulting number. The resulting result contains fewer numbers than need to be separated by a comma. In this case you need left add the missing number of zeros.
We are missing one digit, so we add one zero to the left.
When multiplying any decimal fraction on 10; 100; 1000, etc. The decimal point moves to the right by as many places as there are zeros after the one.
To multiply a decimal by 0.1; 0.01; 0.001, etc., you need to move the decimal point in this fraction to the left by as many places as there are zeros before the one.
We count zero integers!
- 12 0.1 = 1.2
- 0.05 · 0.1 = 0.005
- 1.256 · 0.01 = 0.012 56
- without paying attention to commas, perform multiplication according to all the rules of multiplication with a column of natural numbers;
- in the resulting number, separate with a 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 the required number of zeros must be added to the left.
To understand how to multiply decimals, let's look at specific examples.
Rule for multiplying decimals
1) Multiply without paying attention to the comma.
2) As a result, we separate as many digits after the decimal point as there are after the decimal points in both factors together.
Find the product of decimal fractions:
To multiply decimal fractions, we multiply without paying attention to commas. That is, we multiply not 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 decimal points in both factors together. In the first factor there is one digit after the decimal point, in the second there is also one. In total, we separate two numbers after the decimal point. Thus, we got the final answer: 6.8∙3.4=23.12.
We multiply decimals without taking into account the decimal point. 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 after the decimal point at the end of the entry, we do not write it in the answer: 36.85∙1.4=51.59.
To multiply these decimals, let's 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, you need to separate four digits after the decimal point - as many as there are in both factors together (two in each). Final answer: 23.15∙0.07=1.6205.
Multiplying a decimal fraction by a natural number is done in the same way. We multiply the numbers without paying attention to the decimal point, that is, we multiply 75 by 16. The resulting result should contain the same number of signs after the decimal point as there are in both factors together - one. Thus, 75∙1.6=120.0=120.
We begin multiplying decimal fractions by multiplying natural numbers, since we do not pay attention to commas. After this, we separate as many digits after the decimal point as there are in both factors together. The first number has two decimal places, the second also has two. In total, the result should be four digits after the decimal point: 4.72∙5.04=23.7888.
And a couple more examples on multiplying decimal fractions:
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Multiplying decimals, rules, examples, solutions.
Let's move on to studying next action with decimal fractions, we will now take a comprehensive look multiplying decimals. Let's talk first general principles multiplying decimal fractions. After this, we will move on to multiplying a decimal fraction by a decimal fraction, we will show how to multiply decimal fractions by a column, and we will consider solutions to examples. Next, we will look at multiplying decimal fractions by natural numbers, in particular by 10, 100, etc. Finally, let's talk about multiplying decimals 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 remaining cases are discussed in the articles multiplication of rational numbers and multiplying real numbers.
Page navigation.
General principles of multiplying decimals
Let's discuss the general principles that should be followed when multiplying with decimals.
Since finite decimals and infinite periodic fractions are the decimal form of common fractions, multiplying such decimals is essentially multiplying common fractions. In other words, multiplying finite decimals, multiplying finite and periodic decimal fractions, and multiplying periodic decimals comes down to multiplying ordinary fractions after converting decimal fractions to ordinary ones.
Let's look at examples of applying the stated principle of multiplying decimal fractions.
Multiply the decimals 1.5 and 0.75.
Let us replace the decimal fractions being multiplied with the corresponding ordinary fractions. Since 1.5=15/10 and 0.75=75/100, then. You can reduce a fraction and then select the whole part from improper fraction, and it is more convenient to write the resulting ordinary fraction 1 125/1 000 as 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 convert periodic decimal fractions to ordinary fractions:
Then. You can convert the resulting ordinary fraction to a decimal fraction: 
If among the multiplied decimal fractions there are infinite non-periodic ones, 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.
Multiply the decimals 5.382... and 0.2.
First, let's 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 the nearest hundredth. 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= 1,076/1,000=1.076.
Multiplying decimal fractions by column
Multiplying finite decimal fractions can be done in a column, similar to multiplying natural numbers in a column.
Let's formulate rule for multiplying decimal fractions by column. To multiply decimal fractions by column, you need to:
Let's look at examples of multiplying decimal fractions by columns.
Multiply the decimals 63.37 and 0.12.
Let's multiply decimal fractions in a column. First, we multiply the numbers, ignoring commas: 
All that remains is to add a comma to the resulting product. She needs to separate 4 digits to the right because the factors have a total of four decimal places (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 to the left. Let's finish recording: 
As a result, we have 3.37·0.12=7.6044.
Calculate the product of the decimals 3.2601 and 0.0254.
Having performed multiplication in 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 fractions being multiplied are equal to eight. But there are only 7 digits in the product, therefore, you need to add as many zeros to the left so that you can separate 8 digits with a comma. In our case, we need to assign two zeros: 
This completes the multiplication of decimal fractions by column.
Multiplying decimals 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 a given decimal by 0.1, 0.01, 0.001, and so on gives a fraction that is obtained from the original one if in its notation 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 the required number of zeros to the left.
For example, to multiply the decimal fraction 54.34 by 0.1, you need to move the decimal point in the fraction 54.34 to the left by 1 digit, which will give you the fraction 5.434, that is, 54.34·0.1=5.434. Let's give another example. Multiply the decimal fraction 9.3 by 0.0001. To do this, we need to move the decimal point 4 digits to the left in the multiplied decimal fraction 9.3, but the notation of the fraction 9.3 does not contain that many digits. Therefore, we need to assign so many zeros to the left of the fraction 9.3 so that we can easily move the decimal point to 4 digits, we have 9.3·0.0001=0.00093.
Note that the stated 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 no different from multiplying a decimal by a decimal.
It is most convenient to multiply a final decimal fraction by a natural number in a column; in this case, you should adhere to the rules for multiplying decimal fractions in a column, 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, the periodic fraction should be replaced by an ordinary fraction.
Multiply the decimal fraction 0.(42) by the natural number 22.
First, let's convert the periodic decimal fraction into an ordinary fraction:
Now let's do the multiplication: . This result as a decimal is 9,(3) .
And when multiplying an infinite non-periodic decimal fraction by a natural number, you must first perform rounding.
Multiply 4·2.145….
Having rounded 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.
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 it rule for multiplying a decimal fraction by 10, 100, 1,000, etc. When multiplying a decimal fraction by 10, 100, ... in its notation, you need to move the decimal point to the right to 1, 2, 3, ... digits, respectively, and discard the extra zeros on the left; if the notation of the fraction being multiplied does not have enough digits to move the decimal point, then you need to add the required number of zeros to the right.
Multiply the decimal fraction 0.0783 by 100.
Let's move the fraction 0.0783 two digits to the right, and we get 007.83. Dropping the two zeros on the left gives the decimal fraction 7.38. Thus, 0.0783·100=7.83.
Multiply the decimal fraction 0.02 by 10,000.
To multiply 0.02 by 10,000, we need to move the decimal point 4 digits to the right. Obviously, in the fraction 0.02 there are not enough digits to move the decimal point by 4 digits, so we will add a few zeros to the right so that the decimal point can be moved. In our example, it is enough to add three zeros, we have 0.02000. After moving 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 that is the result of the multiplication.
Multiply the periodic decimal fraction 5.32(672) by 1,000.
Before multiplying, let's write the periodic decimal fraction as 5.32672672672..., this will allow us to avoid mistakes. Now move the comma to the right by 3 places, we have 5 326.726726…. Thus, after multiplication, the periodic decimal fraction 5 326,(726) is obtained.
5.32(672)·1,000=5,326,(726) .
When multiplying infinite non-periodic fractions by 10, 100, ..., you must first round infinite fraction up to a certain digit, after which multiplication is carried out.
Multiplying a decimal by a fraction or mixed number
To multiply a finite decimal fraction or an infinite periodic decimal fraction by a common fraction or mixed number, you need to represent the decimal fraction as a common fraction, and then perform the multiplication.
Multiply the decimal fraction 0.4 by a 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 a fraction or mixed number, replace the fraction or mixed number with a decimal fraction, then round the multiplied fractions and finish the calculation.
Since 2/3=0.6666..., then. After rounding the multiplied fractions to thousandths, we arrive at the product of two final decimal fractions 3.568 and 0.667. Let's do columnar multiplication: 
The result obtained should be rounded to the nearest thousandth, since the multiplied fractions were taken accurate to the thousandth, we have 2.379856≈2.380.
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29. Multiplying decimals. Rules
Find the area of a rectangle with equal sides
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:
d m 2 = 0.42 d m 2.
This means S = 1.4 dm 0.3 dm = 0.42 dm 2.
Multiplying two decimal fractions is done like this:
1) numbers are multiplied without taking commas into account.
2) the comma in the product is placed so as to separate it on the right
the same number of signs as are separated in both factors
combined. For example:
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; 100 ; or 1000 respectively.
For example:
22 0,1 = 2,2 ; 22: 10 = 2,2 .
When multiplying a decimal fraction by a natural number, we must:
1) multiply numbers without paying attention to the comma;
2) in the resulting product, place a comma so that on the right
it had the same number of digits as a decimal fraction.
Let's find the product 3.12 10. According to the above rule
First we multiply 312 by 10. We get: 312 10 = 3120.
Now we separate the two digits on the right with a comma and get:
3,12 10 = 31,20 = 31,2 .
This means that when multiplying 3.12 by 10, we moved the decimal point 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 fraction by 10, 100, 1000, etc., you must
in this fraction move the decimal point to the right by as many places as there are zeros
is worth the multiplier. For example:
0,065 1000 = 0065, = 65 ;
2,9 1000 = 2,900 1000 = 2900, = 2900 .
Problems on the topic “Multiplying decimals”
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Adding, subtracting, multiplying and dividing decimals
Adding and subtracting decimals is similar to adding and subtracting natural numbers, but with certain conditions.
Rule. is performed according to the digits of the integer and fractional parts as natural numbers.
In writing adding and subtracting decimals the comma separating the integer part from the fractional part should be located at the addends and the sum or at the minuend, subtrahend and difference in one column (a comma under the comma from writing the condition to the end of the calculation).
Adding and subtracting decimals 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 decimals in a column:
Adding decimals requires an additional top line to record numbers when the sum of the place value goes beyond ten. Subtracting decimals requires an extra top line to mark the place where the 1 is borrowed.
If there are not enough digits of the fractional part to the right of the addend or minuend, then to the right in the fractional part you can add as many zeros (increase the digit of the fractional part) as there are digits in the other addend or minuend.
Multiplying Decimals is performed in the same way as multiplying natural numbers, according to the same rules, but in the product a comma is placed 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 multipliers is the number of digits after the decimal point of the factors taken together).
At multiplying decimals in the column first from the right significant figure signed under the first significant digit on the right, as in natural numbers:
Record multiplying decimals in a column:
Record division of decimals in a column:
The underlined characters are the characters that are followed by a comma because the divisor must be an integer.
Rule. At dividing fractions The decimal divisor is increased by as many digits as there are digits in the fractional part. To ensure that the fraction does not change, the dividend is increased by the same number of digits (in the dividend and divisor, the decimal point is moved to the same number of digits). A comma is placed in the quotient at that stage of division when the whole part of the fraction is divided.
For decimal fractions, as for natural numbers, the rule remains: You cannot divide a decimal fraction by zero!
In the last lesson, we learned how to add and subtract decimals (see lesson “Adding and subtracting decimals”). At the same time, we assessed how much calculations are simplified compared to ordinary “two-story” fractions.
Unfortunately, this effect does not occur with multiplying and dividing decimals. In some cases, decimal notation even complicates these operations.
First, let's introduce a new definition. We'll see him quite often, and not just in this lesson.
The significant part of a number is everything between the first and last non-zero digit, including the ends. We are talking about numbers only, the decimal point is not taken into account.
The digits included in the significant part of a number are called significant digits. They can be repeated and even be equal to zero.
For example, consider several decimal fractions and write out the corresponding significant parts:
- 91.25 → 9125 (significant figures: 9; 1; 2; 5);
- 0.008241 → 8241 (significant figures: 8; 2; 4; 1);
- 15.0075 → 150075 (significant figures: 1; 5; 0; 0; 7; 5);
- 0.0304 → 304 (significant figures: 3; 0; 4);
- 3000 → 3 (there is only one significant figure: 3).
Please note: the zeros inside the significant part of the number do not go anywhere. We have already encountered something similar when we learned to convert decimal fractions to ordinary ones (see lesson “ Decimals”).
This point is so important, and mistakes are made here so often, that in the near future I will publish a test on this topic. Be sure to practice! And we, armed with the concept of the significant part, will proceed, in fact, to the topic of the lesson.
Multiplying Decimals
The multiplication operation consists of three successive steps:
- For each fraction, write down the significant part. You will get two ordinary integers - without any denominators and decimal points;
- Multiply these numbers by any in a convenient way. Directly, if the numbers are small, or in a column. We obtain the significant part of the desired fraction;
- Find out where and by how many digits the decimal point in the original fractions is shifted to obtain the corresponding significant part. Perform reverse shifts for the significant part obtained in the previous step.
Let me remind you once again that zeros on the sides of the significant part are never taken into account. Ignoring this rule leads to errors.
- 0.28 12.5;
- 6.3 · 1.08;
- 132.5 · 0.0034;
- 0.0108 1600.5;
- 5.25 · 10,000.
We work with the first expression: 0.28 · 12.5.
- Let's write out the significant parts for the numbers from this expression: 28 and 125;
- Their product: 28 · 125 = 3500;
- In the first factor the decimal point is shifted 2 digits to the right (0.28 → 28), and in the second it is shifted by 1 more digit. In total, you need a shift to the left by three digits: 3500 → 3,500 = 3.5.
Now let's look at the expression 6.3 · 1.08.
- Let's write out the significant parts: 63 and 108;
- Their product: 63 · 108 = 6804;
- Again, two shifts to the right: by 2 and 1 digit, respectively. Total - again 3 digits to the right, so the reverse shift will be 3 digits to the left: 6804 → 6.804. This time there are no trailing zeros.
We reached the third expression: 132.5 · 0.0034.
- Significant parts: 1325 and 34;
- Their product: 1325 · 34 = 45,050;
- In the first fraction, the decimal point moves to the right by 1 digit, and in the second - by as many as 4. Total: 5 to the right. We shift by 5 to the left: 45,050 → .45050 = 0.4505. The zero was removed at the end, and added at the front so as not to leave a “bare” decimal point.
The following expression is: 0.0108 · 1600.5.
- We write the significant parts: 108 and 16 005;
- We multiply them: 108 · 16,005 = 1,728,540;
- We count the numbers after the decimal point: in the first number there are 4, in the second there are 1. The total is again 5. We have: 1,728,540 → 17.28540 = 17.2854. At the end, the “extra” zero was removed.
Finally, the last expression: 5.25 10,000.
- Significant parts: 525 and 1;
- We multiply them: 525 · 1 = 525;
- The first fraction is shifted 2 digits to the right, and the second fraction is shifted 4 digits to the left (10,000 → 1.0000 = 1). Total 4 − 2 = 2 digits to the left. We perform a reverse shift by 2 digits to the right: 525, → 52,500 (we had to add zeros).
Note in the last example: since the decimal point moves in different directions, the total shift is found through the difference. This is very important point! Here's another example:
Consider the numbers 1.5 and 12,500. We have: 1.5 → 15 (shift by 1 to the right); 12,500 → 125 (shift 2 to the left). We “step” 1 digit to the right, and then 2 to the left. As a result, we stepped 2 − 1 = 1 digit to the left.
Decimal division
Division is perhaps the most difficult operation. Of course, here you can act by analogy with multiplication: divide the significant parts, and then “move” the decimal point. But in this case there are many subtleties that negate potential savings.
Therefore, let's look at a universal algorithm, which is a little longer, but much more reliable:
- Convert all decimal fractions to ordinary fractions. With a little practice, this step will take you a matter of seconds;
- Divide the resulting fractions in the classical way. In other words, multiply the first fraction by the “inverted” second (see lesson “Multiplying and dividing numerical fractions");
- If possible, present the result again as a decimal fraction. This step is also quick, since the denominator is often already a power of ten.
Task. Find the meaning of the expression:
- 3,51: 3,9;
- 1,47: 2,1;
- 6,4: 25,6:
- 0,0425: 2,5;
- 0,25: 0,002.
Let's consider the first expression. First, let's convert fractions to decimals:
Let's do the same with the second expression. The numerator of the first fraction will again be factorized:
There is an important point in the third and fourth examples: after getting rid of the decimal notation, reducible fractions appear. However, we will not perform this reduction.
The last example is interesting because the numerator of the second fraction contains a prime number. There is simply nothing to factorize here, so we consider it straight ahead:
Sometimes division results in an integer (I'm talking about the last example). In this case, the third step is not performed at all.
In addition, when dividing, “ugly” fractions often arise that cannot be converted to decimals. This distinguishes division from multiplication, where the results are always represented in decimal form. Of course, in this case the last step is again not performed.
Pay also attention to the 3rd and 4th examples. In them we do not intentionally shorten ordinary fractions, derived from decimals. Otherwise, this will complicate the inverse task - representing the final answer again in decimal form.
Remember: the basic property of a fraction (like any other rule in mathematics) in itself does not mean that it must be applied everywhere and always, at every opportunity.
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