Decimal numbers are multiplied. Multiplication of decimal fractions: rules, examples, solutions. Multiplying a Decimal by a Common Fraction or a Mixed Number

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 in the first decimal fraction and the second one. 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.

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.

And a couple more examples for multiplying decimal fractions:

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Multiplication of decimal fractions, rules, examples, solutions.

We turn to the study of the next action with decimal fractions, now we will comprehensively consider multiplying decimals. First, let's discuss the general principles of multiplying decimal fractions. 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 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 the improper fraction, but more conveniently the resulting 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 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.

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 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.

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 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 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 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, the periodic fraction should be replaced with an ordinary 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, it 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 the infinite fraction to a certain digit, and then carry out the 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 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). A comma is placed in the quotient at the stage of division when the whole part of the fraction is divided.

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

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 4182.0 three digits to the left, we get the number 4.182, which is 1000 times less than the 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 multiplication properties of natural numbers hold 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.

Like regular numbers.

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

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

Rules for multiplying decimals.

1. Multiply without paying attention to the comma.

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

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

1. Multiply numbers, ignoring the comma;

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

Multiplication of decimal fractions by a column.

Let's look at an example:

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

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

2 + 2 = 4

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

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

Note:

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

for example:

70,1 . 10 = 701

0,023 . 100 = 2,3

5,6 . 1 000 = 5 600

Note:

To multiply a decimal by 0.1; 0.01; 0.001; and so on, you need to move the comma to the left in this fraction by as many characters as there are zeros in front of the unit.

We count zero integers!

For example:

12 . 0,1 = 1,2

0,05 . 0,1 = 0,005

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

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.

In the course of the average and high school The students went through the topic "Fractions". However, this concept is much broader than given in the learning process. Today, the concept of a fraction occurs quite often, and not everyone can calculate any expression, for example, multiplying fractions.

What is a fraction?

It so happened historically that fractional numbers appeared due to the need to measure. As practice shows, there are often examples for determining the length of a segment, the volume of a rectangular rectangle.

Initially, students are introduced to such a concept as a share. For example, if you divide a watermelon into 8 parts, then each will get one-eighth of a watermelon. This one part of eight is called a share.

A share equal to ½ of any value is called a half; ⅓ - third; ¼ - a quarter. Entries like 5/8, 4/5, 2/4 are called common fractions. An ordinary fraction is divided into a numerator and a denominator. Between them is a fractional line, or fractional line. A fractional bar can be drawn as either a horizontal or a slanted line. In this case, it stands for the division sign.

The denominator represents how many equal shares a value, an object is divided into; and the numerator is how many equal shares are taken. The numerator is written above the fractional bar, the denominator below it.

It is most convenient to show ordinary fractions on a coordinate ray. If you divide a single segment into 4 equal parts, designate each part with a Latin letter, then as a result you can get an excellent visual aid. So, point A shows a share equal to 1/4 of the entire unit segment, and point B marks 2/8 of this segment.

Varieties of fractions

Fractions are common, decimal, and mixed numbers. In addition, fractions can be divided into proper and improper. This classification is more suitable for ordinary fractions.

A proper fraction is a number whose numerator is less than the denominator. Accordingly, an improper fraction is a number whose numerator is greater than the denominator. The second kind is usually written as a mixed number. Such an expression consists of an integer part and a fractional part. For example, 1½. 1 - integer part, ½ - fractional. However, if you need to perform some manipulations with the expression (dividing or multiplying fractions, reducing or converting them), the mixed number is converted into an improper fraction.

correct fractional expression is always less than one, and wrong is always greater than or equal to 1.

As for this expression, they understand a record in which any number is represented, the denominator of the fractional expression of which can be expressed through one with several zeros. If the fraction is correct, then the integer part in the decimal notation will be zero.

To write a decimal, you must first write the integer part, separate it from the fractional with a comma, and then write the fractional expression. It must be remembered that after the comma the numerator must contain as many numeric characters as there are zeros in the denominator.

Example. Represent the fraction 7 21 / 1000 in decimal notation.

Algorithm for converting an improper fraction to a mixed number and vice versa

It is incorrect to write down an improper fraction in the answer of the problem, so it must be converted to a mixed number:

• divide the numerator by the existing denominator;
• in specific example incomplete quotient - whole;
• and the remainder is the numerator of the fractional part, with the denominator remaining unchanged.

Example. Convert improper fraction to mixed number: 47 / 5 .

Decision. 47: 5. The incomplete quotient is 9, the remainder = 2. Hence, 47 / 5 = 9 2 / 5.

Sometimes you need to represent a mixed number as an improper fraction. Then you need to use the following algorithm:

• the integer part is multiplied by the denominator of the fractional expression;
• the resulting product is added to the numerator;
• the result is written in the numerator, the denominator remains unchanged.

Example. Express the number in mixed form as an improper fraction: 9 8 / 10 .

Decision. 9 x 10 + 8 = 90 + 8 = 98 is the numerator.

Answer: 98 / 10.

Multiplication of ordinary fractions

You can perform various algebraic operations on ordinary fractions. To multiply two numbers, you need to multiply the numerator with the numerator, and the denominator with the denominator. Moreover, the multiplication of fractions with different denominators does not differ from the product of fractional numbers with the same denominators.

It happens that after finding the result, you need to reduce the fraction. It is imperative to simplify the resulting expression as much as possible. Of course, it cannot be said that an improper fraction in the answer is a mistake, but it is also difficult to call it the correct answer.

Example. Find the product of two ordinary fractions: ½ and 20 / 18.

As can be seen from the example, after finding the product, a reducible fractional notation is obtained. Both the numerator and the denominator in this case are divisible by 4, and the result is the answer 5 / 9.

Multiplying decimal fractions

The product of decimal fractions is quite different from the product of ordinary fractions in its principle. So, multiplying fractions is as follows:

• two decimal fractions must be written under each other so that the rightmost digits are one under the other;
• you need to multiply the written numbers, despite the commas, that is, as natural ones;
• count the number of digits after the comma in each of the numbers;
• in the result obtained after multiplication, you need to count as many digital characters on the right as are contained in the sum in both factors after the decimal point, and put a separating sign;
• if there are fewer digits in the product, then so many zeros must be written in front of them to cover this number, put a comma and assign an integer part equal to zero.

Example. Calculate the product of two decimals: 2.25 and 3.6.

Decision.

Multiplication of mixed fractions

To calculate the product of two mixed fractions, you need to use the rule for multiplying fractions:

• convert mixed numbers to improper fractions;
• find the product of numerators;
• find the product of the denominators;
• write down the result;
• simplify the expression as much as possible.

Example. Find the product of 4½ and 6 2 / 5.

Multiplying a number by a fraction (fractions by a number)

In addition to finding the product of two fractions, mixed numbers, there are tasks where you need to multiply by a fraction.

So, to find the product of a decimal fraction and a natural number, you need:

• write the number under the fraction so that the rightmost digits are one above the other;
• find the work, despite the comma;
• in the result obtained, separate the integer part from the fractional part using a comma, counting to the right the number of characters that is after the decimal point in the fraction.

To multiply an ordinary fraction by a number, you must find the product of the numerator and the natural factor. If the answer is a reducible fraction, it should be converted.

Example. Calculate the product of 5 / 8 and 12.

Decision. 5 / 8 * 12 = (5*12) / 8 = 60 / 8 = 30 / 4 = 15 / 2 = 7 1 / 2.

Answer: 7 1 / 2.

As you can see from the previous example, it was necessary to reduce the resulting result and convert the incorrect fractional expression into a mixed number.

Also, the multiplication of fractions also applies to finding the product of a number in mixed form and a natural factor. To multiply these two numbers, you should multiply the integer part of the mixed factor by the number, multiply the numerator by the same value, and leave the denominator unchanged. If necessary, you need to simplify the result as much as possible.

Example. Find the product of 9 5 / 6 and 9.

Decision. 9 5 / 6 x 9 \u003d 9 x 9 + (5 x 9) / 6 \u003d 81 + 45 / 6 \u003d 81 + 7 3 / 6 \u003d 88 1 / 2.

Answer: 88 1 / 2.

Multiplication by factors 10, 100, 1000 or 0.1; 0.01; 0.001

The following rule follows from the previous paragraph. To multiply a decimal fraction by 10, 100, 1000, 10000, etc., you need to move the comma to the right by as many digit characters as there are zeros in the multiplier after one.

Example 1. Find the product of 0.065 and 1000.

Decision. 0.065 x 1000 = 0065 = 65.

Answer: 65.

Example 2. Find the product of 3.9 and 1000.

Decision. 3.9 x 1000 = 3.900 x 1000 = 3900.

Answer: 3900.

If you need to multiply a natural number and 0.1; 0.01; 0.001; 0.0001, etc., you should move the comma to the left in the resulting product by as many digit characters as there are zeros before one. If necessary, before natural number zeros are written in sufficient quantity.

Example 1. Find the product of 56 and 0.01.

Decision. 56 x 0.01 = 0056 = 0.56.

Answer: 0,56.

Example 2. Find the product of 4 and 0.001.

Decision. 4 x 0.001 = 0004 = 0.004.

Answer: 0,004.

So, finding the product of various fractions should not cause difficulties, except perhaps the calculation of the result; In this case, you simply cannot do without a calculator.