The rule for multiplying an integer by a decimal. Multiplying a decimal by a natural number

In this lesson, we'll look at converting fractions to common denominator and solve problems on this topic. Let us define the concept of a common denominator and an additional factor, recall the mutual prime numbers. Let's define the concept of the least common denominator (LCD) and solve a number of problems to find it.

Topic: Adding and subtracting fractions with different denominators

Lesson: Reducing fractions to a common denominator

Repetition. Basic property of a fraction.

If the numerator and denominator of a fraction are multiplied or divided by the same natural number, then you get a fraction equal to it.

For example, the numerator and denominator of a fraction can be divided by 2. We get a fraction. This operation is called fraction reduction. Can be done and inverse transformation, multiplying the numerator and denominator of the fraction by 2. In this case, we say that we have reduced the fraction to a new denominator. The number 2 is called an additional factor.

Conclusion. A fraction can be reduced to any denominator that is a multiple of the denominator of the given fraction. In order to bring a fraction to a new denominator, its numerator and denominator are multiplied by an additional factor.

1. Bring the fraction to the denominator 35.

The number 35 is a multiple of 7, that is, 35 is divisible by 7 without a remainder. So this transformation is possible. Let's find an additional factor. To do this, we divide 35 by 7. We get 5. We multiply the numerator and denominator of the original fraction by 5.

2. Bring the fraction to the denominator 18.

Let's find an additional factor. To do this, we divide the new denominator by the original one. We get 3. We multiply the numerator and denominator of this fraction by 3.

3. Bring the fraction to the denominator 60.

By dividing 60 by 15, we get an additional multiplier. It is equal to 4. Let's multiply the numerator and denominator by 4.

4. Bring the fraction to the denominator 24

In simple cases, reduction to a new denominator is performed in the mind. It is customary to only indicate an additional factor behind the bracket a little to the right and above the original fraction.

A fraction can be reduced to a denominator of 15 and a fraction can be reduced to a denominator of 15. Fractions have a common denominator of 15.

The common denominator of fractions can be any common multiple of their denominators. For simplicity, fractions are reduced to the lowest common denominator. It is equal to the least common multiple of the denominators of the given fractions.

Example. Reduce to the least common denominator of the fraction and .

First, find the least common multiple of the denominators of these fractions. This number is 12. Let's find an additional factor for the first and second fractions. To do this, we divide 12 by 4 and by 6. Three is an additional factor for the first fraction, and two for the second. We bring the fractions to the denominator 12.

We reduced the fractions to a common denominator, that is, we found fractions that are equal to them and have the same denominator.

Rule. To bring fractions to the lowest common denominator,

First, find the least common multiple of the denominators of these fractions, which will be their least common denominator;

Secondly, divide the least common denominator by the denominators of these fractions, that is, find an additional factor for each fraction.

Thirdly, multiply the numerator and denominator of each fraction by its additional factor.

a) Reduce the fractions and to a common denominator.

The lowest common denominator is 12. The additional factor for the first fraction is 4, for the second - 3. We bring the fractions to the denominator 24.

b) Reduce the fractions and to a common denominator.

The lowest common denominator is 45. Dividing 45 by 9 by 15, we get 5 and 3, respectively. We bring the fractions to the denominator 45.

c) Reduce the fractions and to a common denominator.

The common denominator is 24. The additional factors are 2 and 3, respectively.

Sometimes it is difficult to verbally find the least common multiple for the denominators of given fractions. Then the common denominator and additional factors are found by expanding into prime factors.

Reduce to a common denominator of the fraction and .

Let's decompose the numbers 60 and 168 into prime factors. Let's write out the expansion of the number 60 and add the missing factors 2 and 7 from the second expansion. Multiply 60 by 14 and get a common denominator of 840. The additional factor for the first fraction is 14. The additional factor for the second fraction is 5. Let's reduce the fractions to a common denominator of 840.

Bibliography

1. Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S. and others. Mathematics 6. - M.: Mnemozina, 2012.

2. Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics 6th grade. - Gymnasium, 2006.

3. Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook. - Enlightenment, 1989.

4. Rurukin A.N., Chaikovsky I.V. Tasks for the course of mathematics grade 5-6. - ZSH MEPhI, 2011.

5. Rurukin A.N., Sochilov S.V., Chaikovsky K.G. Mathematics 5-6. A manual for students of the 6th grade of the MEPhI correspondence school. - ZSH MEPhI, 2011.

6. Shevrin L.N., Gein A.G., Koryakov I.O. etc. Mathematics: Interlocutor textbook for grades 5-6 high school. Library of the teacher of mathematics. - Enlightenment, 1989.

You can download the books specified in clause 1.2. this lesson.

Homework

Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S. and others. Mathematics 6. - M .: Mnemozina, 2012. (see link 1.2)

Homework: No. 297, No. 298, No. 300.

Other tasks: #270, #290

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.

Multiplying a decimal fraction by a natural number is done in the same way. 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.

Decimal multiplication takes place in three stages.

Decimals are written in a column and multiplied as ordinary numbers.

We count the number of decimal places for the first decimal and the second. We add their number.

In the result obtained, we count from right to left as many digits as they turned out in the paragraph above and put a comma.

How to multiply decimals

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 signs (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) of the resulting number. There are fewer digits in the result than you need to separate with a comma. In that case, you need left assign the missing number of zeros.

We are missing one digit, so we attribute one zero to the left.

When multiplying any decimal fraction on 10; 100; 1000 etc. the decimal point moves to the right as many digits as there are zeros after the one.

70.1 10 = 701

0.023 100 = 2.3

5.6 1000 = 5600

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

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.

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 multiplier there is one digit after the decimal point, 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.

And a couple more examples for multiplying decimal fractions:

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Multiplication of decimal fractions, rules, examples, solutions.

Let's move on to the next step with decimals, we will now take a closer look at 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 discussed in the articles multiplication 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 trailing decimals and infinite periodic fractions are the decimal form of common fractions, multiplying such decimals is essentially multiplying common 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 ones.

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

Perform the multiplication of decimals 1.5 and 0.75.

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 improper fraction, but more conveniently obtained common fraction 1 125/1 000 write as a decimal fraction 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 the next paragraph.

Consider an example of multiplying periodic decimal fractions.

Compute the product of the periodic decimals 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 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.

Multiply the decimals 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. 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.

Multiplication of decimal fractions by a column

Multiplication of finite decimal fractions can be performed by a column, similar to multiplication by a column 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 as many digits on the right with a decimal point 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.

Multiply the decimals 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 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.

Calculate the product of decimals 3.2601 and 0.0254 .

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.

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 multiplying 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 add 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.

Calculate the product 15 2.27 .

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

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

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

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

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

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

Do the multiplication 4 2.145….

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.

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 the 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.

Multiply the decimal 0.0783 by 100.

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.

Multiply the decimal fraction 0.02 by 10,000.

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, which is the result of multiplying the decimal fraction 0.02 by 10,000.

The stated rule is also valid 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 multiplication.

Multiply the periodic decimal 5.32(672) by 1000 .

Before multiplication, we write 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, a periodic decimal fraction is obtained 5 326, (726) .

5.32(672) 1000=5326,(726) .

When multiplying infinite non-periodic fractions by 10, 100, ... you must first round infinite fraction to a certain digit, after which to carry out multiplication.

Multiplying a Decimal by a Common Fraction or a Mixed Number

To multiply a finite decimal or an infinite periodic decimal by a fraction or mixed number, you need to represent the decimal fraction as an ordinary fraction, and then carry out the multiplication.

Multiply the decimal fraction 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 a common fraction or a mixed number, the common fraction or mixed number should be replaced by a decimal fraction, then round the multiplied fractions and finish the calculation.

Since 2/3 \u003d 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 the multiplication in a column:

The result obtained should be rounded to thousandths, since the multiplied fractions were taken with an accuracy of thousandths, we have 2.379856≈2.380.

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29. Multiplication of decimal fractions. rules

Find the area of a rectangle with equal sides
1.4 dm and 0.3 dm. Convert decimeters to centimeters:

1.4 dm = 14 cm; 0.3 dm = 3 cm.

Now let's calculate the area in centimeters.

S \u003d 14 3 \u003d 42 cm 2.

Convert square centimeters to square
decimeters:

d m 2 \u003d 0.42 d m 2.

Hence, S \u003d 1.4 dm 0.3 dm \u003d 0.42 dm 2.

Multiplying two decimals is done like this:
1) numbers are multiplied without taking into account commas.
2) the comma in the product is placed so as to separate on the right
as many signs as separated in both factors
taken together. 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 the numbers, ignoring the comma;

2) in the resulting product, put a comma so that on the right
from it there were as many digits as in a decimal fraction.

Let's find the product 3.12 10 . According to the above rule
first multiply 312 by 10 . We get: 312 10 \u003d 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 fraction by 10, 100, 1000, etc., you need to
in this fraction, move the comma to the right as many characters as there are zeros
is in the multiplier. For example:

0,065 1000 = 0065, = 65 ;

2,9 1000 = 2,900 1000 = 2900, = 2900 .

Tasks on the topic "Multiplication of decimal fractions"

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Addition, subtraction, multiplication and division of decimals

Adding and subtracting decimals is similar to adding and subtracting natural numbers, but with certain conditions.

Rule. is made by the digits of the integer and fractional parts as natural numbers.

When written adding and subtracting decimals the comma separating the integer part from the fractional part must be in the terms and the sum or in the minuend, subtrahend and difference in one column (a comma under a comma from 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 decimal fractions requires an upper extra line to write numbers when the sum of the digit goes through a ten. Subtracting decimals requires the top extra line to mark the digit in which the 1 is being borrowed.

If there are not enough digits of the fractional part to the right of the term or reduced, then as many zeros can be added to the right in the fractional part (increase the bit depth of the fractional part) as there are digits in another term or reduced.

Decimal multiplication is performed in the same way as the multiplication of 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 factors is the number of digits after the decimal point for the factors taken together).

At multiplying decimals 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 multiplying decimals in a column:

Recording decimal division in a column:

The underlined characters are comma wrapping characters because the divisor must be an integer.

Rule. At division of fractions the divisor of a decimal fraction increases by as many digits as there are digits in its fractional part. So that the fraction does not change, the dividend increases by the same number of digits (in the dividend and divisor, the comma is transferred to the same number of characters). The comma is placed in the quotient at the stage of division when whole part fractions are divided.

For decimal fractions, as well as for natural numbers, the rule is preserved: You can't divide a decimal by zero!

In the last lesson, we learned how to add and subtract decimal fractions (see the lesson " Adding and subtracting decimal fractions"). At the same time, they estimated how much the calculations are simplified compared to the usual “two-story” fractions.

Unfortunately, with multiplication and division of decimal fractions, this effect does not occur. In some cases, decimal notation even complicates these operations.

First, let's introduce a new definition. We will meet him quite often, and not only in this lesson.

The significant part of a number is everything between the first and last non-zero digit, including the trailers. It's about only about numbers, the decimal point is not taken into account.

The digits included in the significant part of the number are called significant digits. They can be repeated and even be equal to zero.

For example, consider several decimal fractions and write out their 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: 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 the lesson “ Decimal Fractions”).

This point is so important, and errors are made here so often that I will publish a test on this topic in the near future. Be sure to practice! And we, armed with the concept of a significant part, will proceed, in fact, to the topic of the lesson.

Decimal multiplication

The multiplication operation consists of three consecutive steps:

For each fraction, write down the significant part. You will get two ordinary integers - without any denominators and decimal points;
Multiply these numbers in any convenient way. Directly, if the numbers are small, or in a column. We get the significant part of the desired fraction;
Find out where and by how many digits the decimal point is shifted in the original fractions to obtain the corresponding significant part. Perform reverse shifts on 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 multiplier, the decimal point is shifted 2 digits to the right (0.28 → 28), and in the second - by another 1 digit. In total, a shift to the left by three digits is needed: 3500 → 3.500 = 3.5.

Now let's deal with 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 digits, respectively. In 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 zeros at the end.

We got to 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 goes to the right by 1 digit, and in the second - by as many as 4. Total: 5 to the right. We perform a shift by 5 to the left: 45050 → .45050 = 0.4505. Zero was removed at the end, and added to the front so as not to leave a “bare” decimal point.

The following expression: 0.0108 1600.5.

We write 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 - 1. In total - 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).

pay attention to last example: since the decimal point moves in different directions, the total shift is 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 digits to the left. As a result, we stepped 2 − 1 = 1 digit to the left.

Decimal division

Division is perhaps the most complicated 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 the potential savings.

So let's look at a generic algorithm that is a little longer, but much more reliable:

Convert all decimals to common 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 the lesson " Multiplication and division of numerical fractions");
If possible, return the result as a decimal. This step is also fast, because often the denominator already has a power of ten.

Task. Find the value of the expression:

3,51: 3,9;

1,47: 2,1;

6,4: 25,6:

0,0425: 2,5;

0,25: 0,002.

We consider the first expression. First, let's convert obi fractions to decimals:

We do the same with the second expression. The numerator of the first fraction is again decomposed into factors:

There is an important point in the third and fourth examples: after getting rid of the decimal notation, cancellable fractions appear. However, we will not perform this reduction.

The last example is interesting because the numerator of the second fraction is a prime number. There is simply nothing to factorize here, so we consider it “blank through”:

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 appear that cannot be converted to decimals. This is where division differs from multiplication, where the results are always expressed in decimal form. Of course, in this case, the last step is again not performed.

Pay also attention to the 3rd and 4th examples. In them, we deliberately do not reduce common fractions derived from decimals. Otherwise it will make it harder inverse problem- representation of 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.