Dividing a decimal fraction by a natural three-digit number. Dividing natural fractions

§ 1 The rule for dividing a decimal fraction by natural number

In this lesson we will look at the division rule decimals into natural numbers, and also learn how to easily and quickly divide by 10, 100, 1000, etc.

First, let's solve the problem:

The perimeter of an equilateral triangle is 16.2 in.

What is the length of the side of the triangle?

You know that an equilateral triangle is one in which all sides are equal. In order to solve this problem, you need to divide 16.2 by 3.

Let's convert 16.2 dm to centimeters, we get 162 cm.

Now divide 162 by 3, we get 54 cm.

Convert back to decimeters, i.e. 5.4 dm.

This means that when 16.2 is divided by 3, it becomes 5.4.

And in fact, if you multiply 5.4 by 3, you get 16.2.

Let's decide what it means to divide a decimal fraction by a natural number?

This means finding a fraction that, when multiplied by this natural number, gives the dividend.

The following rule applies to dividing a decimal fraction by a natural number:

1. Divide the decimal fraction by a natural number according to the rules of long division, not paying attention to the comma.

2. We put a comma in the quotient when the division of the whole part of the dividend ends.

Attention! If the integer part of the dividend is less than the divisor, then in the quotient we put 0 integers.

Let's get back to our task:

Consider dividing 16.2 by 3 with a column:

divide 16 by 3, take 5 each, we get 3 multiplied by 5, it will be 15, subtract 15 from 16, 1 will remain. Next, we finished dividing the whole part, so we must put a comma in the quotient. Now, without paying attention to the comma, we take away 2, we get 12, divide by 3, take 4 each, which means we multiply 3 by 4, it will be 12, and subtract 12 from 12, it will be zero. This means that the division is over and the answer when dividing 16.2 by 3 is 5.4.

Let's look at another example: 0.806 divided by 31.

Please note that the integer part of the decimal (for us it is 0) is less than the divisor (31).

Therefore, in the quotient we immediately put 0 in the integer part, separating it with a comma. Then we begin to divide according to the rules of division by a column, not paying attention to the comma.

So, the next number is 8, again less than the divisor, which means we write zero again after the decimal point. Then we take into consideration 80, after zero in the quotient we write 2, multiply 2 by 31, we get 62, from 80 we subtract 62, it becomes 18, we take away the six, we have 186, which means in the quotient after 2 we write six. 6 multiplied by 31 equals 186, so the answer is ready: 0.026.

§ 2 The rule for dividing a decimal fraction by 10,100,1000, etc.

Now let's divide 87.3 by 10.

If the resulting quotient is multiplied by 10, the result should be 87.3 again. But when multiplying a decimal fraction by 10, the decimal point is moved one digit to the right. This means that when dividing by 10, the decimal point must be moved one digit to the left: 87.3 divided by 10 will be 8.73. Check: 8.73. 10 equals 87.3.

How many places do you think the decimal point should be moved to the left when dividing by 100? Right! For 2 characters.

So, we got the following rule:

To divide a decimal by 10, 100, 1000, etc. you need to move the comma in this fraction as many digits to the left as there are zeros after the one in the divisor. However, sometimes you have to write before whole part zero or more zeros.

Let's look at two examples.

First: 213.84 must be divided by 10. The number of zeros after one is equal to one, which means the decimal point is moved to the left one place and the result is 21.384.

Second example: 8.765 needs to be divided by 100. There are two zeros after one, which means the decimal point needs to be moved to the left two places, for this you need to add the required number of zeros, i.e. Let's add two zeros in front of the eight 008.765 and divide by 100, move the comma to the left two places, we get 0.08765.

Thus, in this lesson we found out how to divide a decimal fraction by a natural number, and also got a rule that allows you to very easily and quickly divide a decimal fraction by 10, 100, 1000, etc.

List of used literature:

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

I. To divide a decimal fraction by a natural number, you need to divide the fraction by this number, as natural numbers are divided, and put a comma in the quotient when the division of the whole part is completed.

Examples.

Perform division: 1) 96,25: 5; 2) 4,78: 4; 3) 183,06: 45.

Solution.

Example 1) 96,25: 5.

We divide with a “corner” in the same way as natural numbers are divided. After we take down the number 2 (the number of tenths is the first digit after the decimal point in the dividend 96, 2 5), in the quotient we put a comma and continue the division.

Answer: 19,25.

Example 2) 4,78: 4.

We divide as natural numbers are divided. In the quotient we will put a comma as soon as we remove it 7 — the first digit after the decimal point in the dividend 4, 7 8. We continue the division further. When subtracting 38-36 we get 2, but the division is not completed. How do we proceed? We know that zeros can be added to the end of a decimal fraction - this will not change the value of the fraction. We assign zero and divide 20 by 4. We get 5 - the division is over.

Answer: 1,195.

Example 3) 183,06: 45.

Divide as 18306 by 45. In the quotient we put a comma as soon as we remove the number 0 — the first digit after the decimal point in the dividend 183, 0 6. Just as in example 2), we had to assign zero to the number 36 - the difference between the numbers 306 and 270.

Answer: 4,068.

Conclusion: when dividing a decimal fraction by a natural number in private we put a comma immediately after we take down the figure in the tenths place of the dividend. Please note: all highlighted numbers in red in these three examples belong to the category tenths of the dividend.

II. To divide a decimal fraction by 10, 100, 1000, etc., you need to move the decimal point to the left by 1, 2, 3, etc. digits.

Examples.

Perform division: 1) 41,56: 10; 2) 123,45: 100; 3) 0,47: 100; 4) 8,5: 1000; 5) 631,2: 10000.

Solution.

Moving the decimal point to the left depends on how many zeros after the one are in the divisor. So, when dividing a decimal fraction by 10 we will carry over in the dividend comma to the left one digit; when divided by 100 - move the comma left two digits; when divided by 1000 convert to this decimal fraction comma three digits to the left.

The rule for dividing decimal fractions by natural numbers.

Four identical toys cost a total of 921 rubles 20 kopecks. How much does one toy cost (see Fig. 1)?

Rice. 1. Illustration for the problem

Solution

To find the cost of one toy, you need to divide this amount by four. Let's convert the amount into kopecks:

Answer: the cost of one toy is 23,030 kopecks, that is, 230 rubles 30 kopecks, or 230.3 rubles.

You can solve this problem without converting rubles to kopecks, that is, divide the decimal fraction by a natural number: .

To divide a decimal fraction by a natural number, you need to divide the fraction by this number, as natural numbers are divided, and put a comma in the quotient when the division of the whole part is completed.

We divide into a column in the same way as natural numbers are divided. After we remove the number 2 (the number of tenths is the first digit after the decimal point in the dividend 921.20), we put a comma in the quotient and continue the division:

Answer: 230.3 rubles.

We divide into a column in the same way as natural numbers are divided. After we remove the number 6 (the number of tenths is the number after the decimal point in the notation of the dividend 437.6), we put a comma in the quotient and continue the division:

If the dividend is less than the divisor, then the quotient will start from zero.

1 is not divisible by 19, so we put zero in the quotient. The division of the whole part is completed, we put a comma in the quotient. We take down 7. 17 is not divisible by 19, in the quotient we write zero. We take down 6 and continue division:

We divide as natural numbers are divided. In the quotient, we put a comma as soon as we remove 8 - the first digit after the decimal point in the dividend 74.8. We continue the division further. When subtracting, we get 8, but the division is not completed. We know that zeros can be added to the end of a decimal fraction - this will not change the value of the fraction. We assign zero and divide 80 by 10. We get 8 - the division is over.

To divide a decimal fraction by 10, 100, 1000, etc., you need to move the decimal point in this fraction as many digits to the left as there are zeros after the one in the divisor.

On this lesson We learned how to divide a decimal fraction by a natural number. We considered the option with an ordinary natural number, as well as the option in which division occurs by a digit unit (10, 100, 1000, etc.).

Solve the equations:

To find an unknown divisor, you need to divide the dividend by the quotient. That is .

We divide into a column. After we remove the number 4 (the number of tenths is the first digit after the decimal point in the dividend 134.4), we put a comma in the quotient and continue the division:

Let's write down the rule and consider its application using examples.

When dividing a decimal fraction by a natural number:

1) divide without paying attention to the comma;

2) when the division of the whole part ends, we put a comma in the quotient.

If the integer part is less than the divisor, then the integer part of the quotient is zero.

Examples of dividing decimal fractions by natural numbers.

We divide without paying attention to the comma, that is, we divide 348 by 6. When dividing 34 by 6, we take 5 each. 5∙6=30, 34-30=4, that is, the remainder is 4.

The difference between dividing a decimal fraction by a natural number and dividing integers is only that when the division of the integer part is completed, we put a comma in the quotient. That is, when passing through a comma, before taking it down to the remainder of the division of the integer part, 4, the number 8 from the fractional part, we write a comma in the quotient.

We take down 8. 48:6=8. In private we write 8.

So, 34.8:6=5.8.

Since 5 is not divisible by 12, we write zero in the quotient. The division of the whole part is completed, we put a comma in the quotient.

We take down 1. When dividing 51 by 12, we take 4. The remainder is 3.

We take down 6. 36:12=3.

Thus, 5.16:12=0.43.

3) 0,646:38=?

The integer part of the dividend contains zero. Since zero is not divisible by 38, we put 0 in the quotient. The division of the integer part is completed, in the quotient we write a comma.

We take down 6. Since 6 is not divisible by 38, we write one more zero in the quotient.

We take down 4. When dividing 64 by 38, we take 1. The remainder is 26.

We take down 6. 266:38=7.

So, 0.646:38=0.017.

4) 14917,5:325=?

When dividing 1491 by 325, we take 4 each. The remainder is 191. We take away 7. When dividing 1917 by 325, we take 5 each. The remainder is 292.

Since the division of the whole part is completed, we write a comma in the quotient.

Let's look at examples of dividing decimals in this light.

Example.

Divide the decimal fraction 1.2 by the decimal fraction 0.48.

Solution.

Answer:

1,2:0,48=2,5 .

Example.

Divide the periodic decimal fraction 0.(504) by the decimal fraction 0.56.

Solution.

Let's convert the periodic decimal fraction into an ordinary fraction: . We also convert the final decimal fraction 0.56 into an ordinary fraction, we have 0.56 = 56/100. Now we can move from dividing the original decimal fractions to dividing ordinary fractions and finish the calculations: .

We will translate the received common fraction to a decimal fraction by dividing the numerator by the denominator with a column:

Answer:

0,(504):0,56=0,(900) .

The principle of dividing infinite non-periodic decimal fractions differs from the principle of dividing finite and periodic decimal fractions, since non-periodic decimal fractions cannot be converted to ordinary fractions. The division of infinite non-periodic decimal fractions is reduced to the division of finite decimal fractions, for which we carry out rounding numbers up to a certain level. Moreover, if one of the numbers with which the division is carried out is a finite or periodic decimal fraction, then it is also rounded to the same digit as the non-periodic decimal fraction.

Example.

Divide the infinite non-periodic decimal 0.779... by the finite decimal 1.5602.

Solution.

First you need to round decimals so that you can move from dividing infinite non-periodic decimals to dividing finite decimals. We can round to the nearest hundredth: 0.779…≈0.78 and 1.5602≈1.56. Thus, 0.779…:1.5602≈0.78:1.56= 78/100:156/100=78/100·100/156= 78/156=1/2=0,5 .

Answer:

0,779…:1,5602≈0,5 .

Dividing a natural number by a decimal fraction and vice versa

The essence of the approach to dividing a natural number by a decimal fraction and to dividing a decimal fraction by a natural number is no different from the essence of dividing decimal fractions. That is, finite and periodic fractions are replaced by ordinary fractions, and infinite non-periodic fractions are rounded.

To illustrate, consider the example of dividing a decimal fraction by a natural number.

Example.

Divide the decimal fraction 25.5 by the natural number 45.

Solution.

By replacing the decimal fraction 25.5 with the common fraction 255/10=51/2, division is reduced to dividing the common fraction by a natural number:. The resulting fraction in decimal notation has the form 0.5(6) .

Answer:

25,5:45=0,5(6) .

Dividing a decimal fraction by a natural number with a column

It is convenient to divide finite decimal fractions into natural numbers by a column, by analogy with the division by a column of natural numbers. Let us present the division rule.

To divide a decimal fraction by a natural number using a column, necessary:

• add several digits 0 to the right of the decimal fraction being divided (during the division process, if necessary, you can add any number of zeros, but these zeros may not be needed);
• perform division by a column of a decimal fraction by a natural number according to all the rules of division by a column of natural numbers, but when the division of the whole part of the decimal fraction is completed, then in the quotient you need to put a comma and continue the division.

Let's say right away that as a result of dividing a finite decimal fraction by a natural number, you can get either a finite decimal fraction or an infinite periodic decimal fraction. Indeed, after the division of all non-0 decimal places is completed divisible fraction, either the remainder may be 0, and we will get a final decimal fraction, or the remainders will begin to repeat periodically, and we will get a periodic decimal fraction.

Let's understand all the intricacies of dividing decimal fractions by natural numbers in a column when solving examples.

Example.

Divide the decimal fraction 65.14 by 4.

Solution.

Let's divide a decimal fraction by a natural number using a column. Let's add a couple of zeros to the right in the notation of the fraction 65.14, and we will get an equal decimal fraction 65.1400 (see equal and unequal decimal fractions). Now you can begin to divide with a column the integer part of the decimal fraction 65.1400 by the natural number 4:

This completes the division of the integer part of the decimal fraction. Here in the quotient you need to put a decimal point and continue the division:

We have reached a remainder of 0, at this stage the division by the column ends. As a result, we have 65.14:4=16.285.

Answer:

65,14:4=16,285 .

Example.

Divide 164.5 by 27.

Solution.

Let's divide the decimal fraction by a natural number using a column. After dividing the whole part we get the following picture:

Now we put a comma in the quotient and continue dividing with a column:

Now it is clearly visible that the residues 25, 7 and 16 have begun to repeat, while in the quotient the numbers 9, 2 and 5 are repeated. Thus, dividing the decimal 164.5 by 27 gives us the periodic decimal 6.0(925) .

Answer:

164,5:27=6,0(925) .

Column division of decimal fractions

The division of a decimal fraction by a decimal fraction can be reduced to dividing a decimal fraction by a natural number with a column. To do this, the dividend and the divisor must be multiplied by such a number as 10, or 100, or 1,000, etc., so that the divisor becomes a natural number, and then divide by a natural number with a column. We can do this due to the properties of division and multiplication, since a:b=(a·10):(b·10) , a:b=(a·100):(b·100) and so on.

In other words, to divide a trailing decimal by a trailing decimal, need to:

• in the dividend and divisor, move the comma to the right by as many places as there are after the decimal point in the divisor; if in the dividend there are not enough signs to move the comma, then you need to add the required number of zeros to the right;
• After this, divide with a decimal column by a natural number.

When solving an example, consider the application of this rule of division by a decimal fraction.

Example.

Divide with a column 7.287 by 2.1.

Solution.

Let's move the comma in these decimal fractions one digit to the right, this will allow us to move from dividing the decimal fraction 7.287 by the decimal fraction 2.1 to dividing the decimal fraction 72.87 by the natural number 21. Let's do the division by column:

Answer:

7,287:2,1=3,47 .

Example.

Divide the decimal 16.3 by the decimal 0.021.

Solution.

Move the comma in the dividend and divisor to the right three places. Obviously, the divisor does not have enough digits to move the decimal point, so we will add the required number of zeros to the right. Now let’s divide the fraction 16300.0 with a column by the natural number 21:

From this moment, the remainders 4, 19, 1, 10, 16 and 13 begin to repeat, which means that the numbers 1, 9, 0, 4, 7 and 6 in the quotient will also be repeated. As a result, we get the periodic decimal fraction 776,(190476) .

Answer:

16,3:0,021=776,(190476) .

Note that the announced rule allows you to divide a natural number by a column into a final decimal fraction.

Example.

Divide the natural number 3 by the decimal fraction 5.4.

Solution.

After moving the decimal point one digit to the right, we arrive at dividing the number 30.0 by 54. Let's do the division by column:
.

This rule can also be applied when dividing infinite decimal fractions by 10, 100, .... For example, 3,(56):1,000=0.003(56) and 593.374…:100=5.93374… .

Dividing decimals by 0.1, 0.01, 0.001, etc.

Since 0.1 = 1/10, 0.01 = 1/100, etc., then from the rule of dividing by a common fraction it follows that divide the decimal fraction by 0.1, 0.01, 0.001, etc. . it's the same as multiplying a given decimal by 10, 100, 1,000, etc. respectively.

In other words, to divide a decimal fraction by 0.1, 0.01, ... you need to move the decimal point to the right by 1, 2, 3, ... digits, and if the digits in the decimal fraction are not enough to move the decimal point, then you need to add the required number to the right zeros.

For example, 5.739:0.1=57.39 and 0.21:0.00001=21,000.

The same rule can be applied when dividing infinite decimal fractions by 0.1, 0.01, 0.001, etc. In this case, you should be very careful when dividing periodic fractions so as not to make a mistake with the period of the fraction that is obtained as a result of division. For example, 7.5(716):0.01=757,(167), since after moving the decimal point in the decimal fraction 7.5716716716... two places to the right, we have the entry 757.167167.... With infinite non-periodic decimal fractions everything is simpler: 394,38283…:0,001=394382,83… .

Dividing a fraction or mixed number by a decimal and vice versa

Dividing a fraction or mixed number by a finite or periodic decimal, and dividing a finite or periodic decimal by a fraction or mixed number comes down to dividing ordinary fractions. To do this, decimal fractions are replaced by the corresponding ordinary fractions, and the mixed number is represented as an improper fraction.

When dividing an infinite non-periodic decimal fraction by a common fraction or mixed number and vice versa, you should proceed to dividing decimal fractions, replacing the common fraction or mixed number with the corresponding decimal fraction.

Bibliography.

• Mathematics: textbook for 5th grade. general education institutions / N. Ya. Vilenkin, V. I. Zhokhov, A. S. Chesnokov, S. I. Shvartsburd. - 21st ed., erased. - M.: Mnemosyne, 2007. - 280 pp.: ill. ISBN 5-346-00699-0.
• Mathematics. 6th grade: educational. for general education institutions / [N. Ya. Vilenkin and others]. - 22nd ed., rev. - M.: Mnemosyne, 2008. - 288 p.: ill. ISBN 978-5-346-00897-2.
• Algebra: textbook for 8th grade. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 16th ed. - M.: Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
• Gusev V. A., Mordkovich A. G. Mathematics (a manual for those entering technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.