Rule for multiplying decimals by decimals. Multiplication of decimal fractions: rules, examples, solutions. Decimal multiplication

§ 1 Application of the multiplication rule decimal fractions

In this lesson, you will introduce and learn how to apply the rule for multiplying decimal fractions and the rule for multiplying a decimal fraction by a place unit such as 0.1, 0.01, etc. In addition, we will consider the properties of multiplication when finding the values of expressions containing decimal fractions.

Let's solve the problem:

The vehicle speed is 59.8 km/h.

How far will the car travel in 1.3 hours?

As you know, to find a path, you need to multiply the speed by the time, i.e. 59.8 times 1.3.

Let's write the numbers in a column and start multiplying them without noticing the commas: 8 times 3 will be 24, 4 we write 2 in our minds, 3 times 9 is 27, plus 2, we get 29, we write 9, 2 in our minds. Now we multiply 3 by 5, it will be 15 and add 2 more, we get 17.

Go to the second line: 1 times 8 is 8, 1 times 9 is 9, 1 times 5 is 5, add these two lines, we get 4, 9+8 is 17, 7 write 1 in your head, 7 +9 is 16 plus 1, it will be 17, 7 we write 1 in our mind, 1+5 plus 1 we get 7.

Now let's see how many decimal places are in both decimal fractions! The first fraction has one digit after the decimal point and the second fraction has one digit after the decimal point, two digits in total. So, on the right in the result you need to count two digits and put a comma, i.e. will be 77.74. So, when multiplying 59.8 by 1.3, we got 77.74. So the answer in the problem is 77.74 km.

Thus, to multiply two decimal fractions, you need:

First: do the multiplication, ignoring the commas

Second: in the resulting product, separate with a comma as many digits on the right as there are after the comma in both factors together.

If there are fewer digits in the resulting product than it is necessary to separate with a comma, then one or more zeros must be assigned in front.

For example: 0.145 times 0.03 we get 435 in the product, and we need to separate 5 digits on the right with a comma, so we add 2 more zeros before the number 4, put a comma and add one more zero. We get the answer 0.00435.

§ 2 Properties of multiplication of decimal fractions

When multiplying decimal fractions, all the same multiplication properties that apply to natural numbers are preserved. Let's do some tasks.

Task number 1:

Let's solve this example by applying the distributive property of multiplication with respect to addition.

5.7 (common factor) will be taken out of the brackets, 3.4 plus 0.6 will remain in brackets. The value of this sum is 4, and now 4 must be multiplied by 5.7, we get 22.8.

Task number 2:

Let's use the commutative property of multiplication.

We first multiply 2.5 by 4, we get 10 integers, and now we need to multiply 10 by 32.9 and we get 329.

In addition, when multiplying decimal fractions, you can notice the following:

When multiplying a number by an improper decimal fraction, i.e. greater than or equal to 1, it increases or does not change, for example:

When multiplying a number by a proper decimal fraction, i.e. less than 1, it decreases, for example:

Let's solve an example:

23.45 times 0.1.

We have to multiply 2,345 by 1 and separate three commas from the right, we get 2.345.

Now let's solve another example: 23.45 divided by 10, we have to move the comma to the left by one place, because 1 zero in a bit one, we get 2.345.

From these two examples, we can conclude that multiplying a decimal by 0.1, 0.01, 0.001, etc. means dividing the number by 10, 100, 1000, etc., i.e. in a decimal fraction, move the decimal point to the left by as many digits as there are zeros in front of 1 in the multiplier.

Using the resulting rule, find the values works:

13.45 times 0.01

there are 2 zeros in front of the number 1, so we move the comma to the left by 2 digits, we get 0.1345.

0.02 times 0.001

there are 3 zeros in front of the number 1, which means we move the comma three digits to the left, we get 0.00002.

Thus, in this lesson you have learned how to multiply decimal fractions. To do this, you just need to perform the multiplication, ignoring the commas, and in the resulting product, separate as many digits on the right with a comma as there are after the comma in both factors together. In addition, they got acquainted with the rule for multiplying a decimal fraction by 0.1, 0.01, etc., and also considered the properties of multiplying decimal fractions.

List of used literature:

Mathematics 5th grade. Vilenkin N.Ya., Zhokhov V.I. and others. 31st ed., ster. - M: 2013.
Didactic materials in mathematics grade 5. Author - Popov M.A. - year 2013
We calculate without errors. Work with self-examination in mathematics grades 5-6. Author - Minaeva S.S. - year 2014
Didactic materials in mathematics Grade 5. Authors: Dorofeev G.V., Kuznetsova L.V. - 2010
Control and independent work in mathematics grade 5. Authors - Popov M.A. - year 2012
Mathematics. Grade 5: textbook. for general education students. institutions / I. I. Zubareva, A. G. Mordkovich. - 9th ed., Sr. - M.: Mnemosyne, 2009

You already know that a * 10 = a + a + a + a + a + a + a + a + a + a. For example, 0.2 * 10 = 0.2 + 0.2 + 0.2 + 0.2 + 0.2 + 0.2 + 0.2 + 0.2 + 0.2 + 0.2 . It is easy to guess that this sum is equal to 2, i.e. 0.2 * 10 = 2.

Similarly, one can verify that:

5,2 * 10 = 52 ;

0,27 * 10 = 2,7 ;

1,253 * 10 = 12,53 ;

64,95 * 10 = 649,5 .

You probably guessed that when multiplying a decimal fraction by 10, you need to move the decimal point to the right by one digit in this fraction.

How do you multiply a decimal by 100?

We have: a * 100 = a * 10 * 10 . Then:

2,375 * 100 = 2,375 * 10 * 10 = 23,75 * 10 = 237,5 .

Arguing similarly, we get that:

3,2 * 100 = 320 ;

28,431 * 100 = 2843,1 ;

0,57964 * 100 = 57,964 .

Multiply the fraction 7.1212 by the number 1000.

We have: 7.1212 * 1000 = 7.1212 * 100 * 10 = 712.12 * 10 = 7121.2.

These examples illustrate the following rule.

To multiply a decimal fraction by 10, 100, 1,000, etc., you need to move the decimal point to the right in this fraction, respectively, by 1, 2, 3, etc. numbers.

So, if you move the comma to the right by 1, 2, 3, etc. numbers, then the fraction will increase by 10, 100, 1,000, etc., respectively. once.

Hence, if you move the comma to the left by 1, 2, 3, etc. numbers, then the fraction will decrease by 10, 100, 1,000, etc., respectively. once .

Let us show that the decimal form of notation of fractions makes it possible to multiply them, guided by the rule of multiplication of natural numbers.

Let's find, for example, the product 3.4 * 1.23. Let's increase the first multiplier by 10 times, and the second by 100 times. This means that we have increased the product by 1,000 times.

Therefore, the product of natural numbers 34 and 123 is 1,000 times greater than the desired product.

We have: 34 * 123 = 4182. Then, to get an answer, the number 4,182 must be reduced by 1,000 times. Let's write: 4 182 \u003d 4 182.0. Moving the comma in the number 4182.0 three digits to the left, we get the number 4.182, which is 1000 times less than number 4182 . So 3.4 * 1.23 = 4.182 .

The same result can be obtained using the following rule.

To multiply two decimals:

1) multiply them as natural numbers, ignoring commas;

2) in the resulting product, separate with a comma on the right as many digits as there are after the commas in both factors together.

In cases where the product contains fewer digits than is required to be separated by a comma, the required number of zeros is added to the left before this product, and then the comma is moved to the left by the required number of digits.

For example, 2 * 3 = 6, then 0.2 * 3 = 0.006; 25 * 33 = 825, then 0.025 * 0.33 = 0.00825.

In cases where one of the factors is equal to 0.1; 0.01; 0.001, etc., it is convenient to use the following rule.

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, respectively, by 1, 2, 3, etc. numbers.

For example, 1.58 * 0.1 = 0.158; 324.7 * 0.01 = 3.247.

The properties of multiplication of natural numbers are also valid for fractional numbers:

ab = ba − commutative property of multiplication,

(ab) c = a(b c) − the associative property of multiplication,

a(b + c) = ab + ac is the distributive property of multiplication with respect to addition.

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.

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.

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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, and 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.

Solution.

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) .

Solution.

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.

Solution.

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 as many digits on the right with a decimal point 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 on the left.

Consider examples of multiplying decimal fractions by a column.

Example.

Multiply the decimals 63.37 and 0.12.

Solution.

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 .

Solution.

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

Now in the product you need to separate 8 digits on the right with a comma, since the total number of decimal places of the 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 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 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 .

Solution.

Let's do the multiplication natural number to a decimal fraction in a column:

Answer:

15 2.27=34.05.

When multiplying a periodic decimal fraction by a natural number, the periodic fraction should be replaced with an ordinary fraction.

Example.

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

Solution.

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….

Solution.

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.

Solution.

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.

Solution.

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.

In this article, we will consider such an action as multiplying decimal fractions. Let's start with the formulation of general principles, then we will show how to multiply one decimal fraction by another and consider the method of multiplication by a column. All definitions will be illustrated with examples. Then we will analyze how to correctly multiply decimal fractions by ordinary, as well as by mixed and natural numbers (including 100, 10, etc.)

As part of this material, we will only touch on the rules for multiplying positive fractions. Cases with negative numbers are discussed separately in the articles on the multiplication of rational and real numbers.

Let us formulate the general principles that must be followed when solving problems on the multiplication of decimal fractions.

To begin with, let us recall that decimal fractions are nothing more than a special form of writing ordinary fractions, therefore, the process of their multiplication can be reduced to the same for ordinary fractions. This rule works for both finite and infinite fractions: after converting them to ordinary fractions, it is easy to perform multiplication with them according to the rules we have already studied.

Let's see how such tasks are solved.

Example 1

Compute the product of 1.5 and 0.75.

Solution: First, replace the decimal fractions with ordinary ones. We know that 0.75 is 75/100 and 1.5 is 1510. We can reduce the fraction and extract the whole part. We will write the result 125 1000 as 1 , 125 .

Answer: 1 , 125 .

We can use the column counting method as we do for natural numbers.

Example 2

Multiply one periodic fraction 0 , (3) by another 2 , (36) .

First, let's reduce the original fractions to ordinary ones. We will be able to:

0 , (3) = 0 , 3 + 0 , 03 + 0 , 003 + 0 , 003 + . . . = 0 , 3 1 - 0 , 1 = 0 , 3 9 = 3 9 = 1 3 2 , (36) = 2 + 0 , 36 + 0 , 0036 + . . . = 2 + 0 , 36 1 - 0 , 01 = 2 + 36 99 = 2 + 4 11 = 2 4 11 = 26 11

Therefore, 0 , (3) 2 , (36) = 1 3 26 11 = 26 33 .

The resulting ordinary fraction can be reduced to decimal form by dividing the numerator by the denominator in a column:

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

If we have infinite non-periodic fractions in the condition of the problem, then we need to perform their preliminary rounding (see the article on rounding numbers if you forgot how this is done). After that, you can perform the multiplication operation with already rounded decimal fractions. Let's take an example.

Example 3

Compute the product of 5 , 382 ... and 0 , 2 .

Solution

We have an infinite fraction in the problem, which must first be rounded to hundredths. It turns out that 5, 382 ... ≈ 5, 38. Rounding the second factor to hundredths does not make sense. Now you can calculate the desired product and write down the answer: 5, 38 0, 2 = 538 100 2 10 = 1 076 1000 = 1, 076.

Answer: 5.382… 0.2 ≈ 1.076.

The column counting method can be applied not only to natural numbers. If we have decimals, we can multiply them in exactly the same way. Let's derive the rule:

Definition 1

Multiplication of decimal fractions by a column is performed in 2 steps:

1. We perform multiplication by a column, not paying attention to commas.

2. We put a decimal point in the final number, separating it as many digits on the right side as both factors contain decimal places together. If as a result there are not enough numbers for this, we add zeros on the left.

We will analyze examples of such calculations in practice.

Example 4

Multiply the decimals 63, 37 and 0, 12 by a column.

Solution

First of all, let's do the multiplication of numbers, ignoring the decimal points.

Now we need to put a comma on Right place. It will separate the four digits on the right side since the sum of the decimal places in both factors is 4 . You don't have to add zeros, because signs are enough.

Answer: 3.37 0.12 = 7.6044.

Example 5

Calculate how much is 3.2601 times 0.0254.

Solution

We count without commas. We get the following number:

We will put a comma separating 8 digits on the right side, because the original fractions together have 8 decimal places. But our result has only seven digits, and we can't do without extra zeros:

Answer: 3.2601 0.0254 = 0.08280654.

How to multiply a decimal by 0.001, 0.01, 01, etc

You often have to multiply decimals by such numbers, so it is important to be able to do this quickly and accurately. We write down a special rule that we will use in such multiplication:

Definition 2

If we multiply a decimal by 0, 1, 0, 01, etc., we end up with a number that looks like the original fraction, with the decimal point moved to the left by the required number of places. If there are not enough digits to transfer, you need to add zeros on the left.

So, to multiply 45, 34 by 0, 1, the comma must be moved in the original decimal fraction by one sign. We end up with 4,534.

Example 6

Multiply 9.4 by 0.0001.

Solution

We will have to move the comma to four digits according to the number of zeros in the second factor, but the numbers in the first are not enough for this. We assign the necessary zeros and get that 9, 4 0, 0001 = 0, 00094.

Answer: 0 , 00094 .

For infinite decimals, we use the same rule. So, for example, 0 , (18) 0 , 01 = 0 , 00 (18) or 94 , 938 … 0 , 1 = 9 , 4938 … . and etc.

The process of such a multiplication is no different from the action of multiplying two decimal fractions. It is convenient to use the multiplication method in a column if the condition of the problem contains a final decimal fraction. In this case, it is necessary to take into account all the rules that we talked about in the previous paragraph.

Example 7

Calculate how much will be 15 2, 27.

Solution

Multiply the original numbers by a column and separate the two commas.

Answer: 15 2.27 = 34.05.

If we perform the multiplication of a periodic decimal fraction by a natural number, we must first change the decimal fraction to an ordinary one.

Example 8

Compute the product of 0 , (42) and 22 .

We bring the periodic fraction to the form of an ordinary fraction.

0 , (42) = 0 , 42 + 0 , 0042 + 0 , 000042 + . . . = 0 , 42 1 - 0 , 01 = 0 , 42 0 , 99 = 42 99 = 14 33

0, 42 22 = 14 33 22 = 14 22 3 = 28 3 = 9 1 3

The final result can be written as a periodic decimal fraction as 9 , (3) .

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

Infinite fractions must be rounded before counting.

Example 9

Calculate how much will be 4 2 , 145 ... .

Solution

Let's round up to hundredths the original infinite decimal fraction. After that, we will come to the multiplication of a natural number and a final decimal fraction:

4 2, 145 ... ≈ 4 2, 15 = 8, 60.

Answer: 4 2.145 ... ≈ 8.60.

How to multiply a decimal by 1000, 100, 10, etc.

Multiplying a decimal fraction by 10, 100, etc. is often found in problems, so we will analyze this case separately. The basic multiplication rule is:

Definition 3

To multiply a decimal by 1000, 100, 10, etc., you need to move its comma by 3, 2, 1 digits depending on the multiplier and discard extra zeros on the left. If there are not enough digits to move the comma, we add as many zeros to the right as we need.

Let's show an example how to do it.

Example 10

Do the multiplication of 100 and 0.0783.

Solution

To do this, we need to move the decimal point by 2 digits to the right. We end up with 007 , 83 The zeros on the left can be discarded and the result can be written as 7 , 38 .

Answer: 0.0783 100 = 7.83.

Example 11

Multiply 0.02 by 10 thousand.

Solution: we will move the comma four digits to the right. In the original decimal fraction, we do not have enough signs for this, so we have to add zeros. In this case, three 0's will suffice. As a result, it turned out 0, 02000, move the comma and get 00200, 0. Ignoring the zeros on the left, we can write the answer as 200 .

Answer: 0.02 10,000 = 200.

The rule we have given will also work in the case of infinite decimal fractions, but here you should be very careful about the period of the final fraction, since it is easy to make a mistake in it.

Example 12

Compute the product of 5.32 (672) times 1000 .

Solution: first of all, we will write the periodic fraction as 5, 32672672672 ..., so the probability of making a mistake will be less. After that, we can move the comma to the desired number of characters (three). As a result, we get 5326 , 726726 ... Let's enclose the period in brackets and write the answer as 5 326 , (726) .

Answer: 5 . 32 (672) 1 000 = 5 326 . (726) .

If in the conditions of the problem there are infinite non-periodic fractions that must be multiplied by ten, one hundred, one thousand, etc., do not forget to round them before multiplying.

To perform this type of multiplication, you need to represent the decimal fraction as an ordinary fraction and then follow the already familiar rules.

Example 13

Multiply 0 , 4 by 3 5 6

Solution

Let's first convert the decimal to a common fraction. We have: 0 , 4 = 4 10 = 2 5 .

We received a response in the form mixed number. You can write it as a periodic fraction 1, 5 (3) .

Answer: 1 , 5 (3) .

If an infinite non-periodic fraction is involved in the calculation, you need to round it up to a certain number and only then multiply it.

Example 14

Calculate the product of 3.5678. . . 2 3

Solution

We can represent the second factor as 2 3 = 0, 6666 …. Next, we round both factors to the thousandth place. After that, we will need to calculate the product of two final decimal fractions 3.568 and 0.667. Let's count the column and get the answer: