In this article we will look at such an important operation with decimals as division. First, we will formulate general principles, then we will analyze how to correctly divide decimal fractions in a column both by other fractions and by natural numbers. Next, we will analyze the division of ordinary fractions into decimals and vice versa, and at the end we will look at how to correctly divide fractions ending in 0, 1, 0, 01, 100, 10, etc.

Here we will take only cases with positive fractions. If there is a minus in front of the fraction, then to operate with it you need to study material about dividing rational and real numbers.

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All decimal fractions, both finite and periodic, are just a special form of writing ordinary fractions. Therefore, they are subject to the same principles as their corresponding ordinary fractions. Thus, we reduce the entire process of dividing decimal fractions to replacing them with ordinary ones, followed by calculation using methods already known to us. Let's take a specific example.

Example 1

Divide 1.2 by 0.48.

Solution

Let's write decimal fractions as ordinary fractions. We will get:

1 , 2 = 12 10 = 6 5

0 , 48 = 48 100 = 12 25 .

Thus, we need to divide 6 5 by 12 25. We count:

1, 2: 0, 48 = 6 2: 12 25 = 6 5 25 12 = 6 25 5 12 = 5 2

From the resulting improper fraction, you can select the whole part and get the mixed number 2 1 2, or you can present it as a decimal fraction so that it corresponds to the original numbers: 5 2 = 2, 5. We have already written about how to do this earlier.

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

Example 2

Calculate how much 0 , (504) 0 , 56 will be.

Solution

First, we need to convert a periodic decimal fraction into a common fraction.

0 , (504) = 0 , 504 1 - 0 , 001 = 0 , 504 0 , 999 = 504 999 = 56 111

After this, we will also convert the final decimal fraction into another form: 0, 56 = 56,100. Now we have two numbers with which it will be easy for us to carry out the necessary calculations:

0 , (504) : 1 , 11 = 56 111: 56 100 = 56 111 100 56 = 100 111

We have a result that we can also convert to decimal form. To do this, divide the numerator by the denominator using the column method:

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

If in the division example we encountered non-periodic decimal fractions, then we will act a little differently. We cannot reduce them to the usual ordinary fractions, so when dividing we have to first round them to a certain digit. This action must be performed with both the dividend and the divisor: we will also round the existing finite or periodic fraction in the interests of accuracy.

Example 3

Find how much 0.779... / 1.5602 is.

Solution

First, we round both fractions to the nearest hundredth. This is how we move from infinite non-periodic fractions to finite decimal ones:

0 , 779 … ≈ 0 , 78

1 , 5602 ≈ 1 , 56

We can continue the calculations and get an approximate result: 0, 779 ...: 1, 5602 ≈ 0, 78: 1, 56 = 78,100: 156,100 = 78,100 100,156 = 78,156 = 1 2 = 0, 5.

The accuracy of the result will depend on the degree of rounding.

Answer: 0 , 779 … : 1 , 5602 ≈ 0 , 5 .

How to divide a natural number by a decimal and vice versa

The approach to division in this case is almost the same: we replace finite and periodic fractions with ordinary ones, and round off infinite non-periodic ones. Let's start with the example of division with a natural number and a decimal fraction.

Example 4

Divide 2.5 by 45.

Solution

Let's reduce 2, 5 to the form of an ordinary fraction: 255 10 = 51 2. Next we just need to divide it by a natural number. We already know how to do this:

25, 5: 45 = 51 2: 45 = 51 2 1 45 = 17 30

If we convert the result to decimal notation, we get 0.5 (6).

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

The long division method is good not only for natural numbers. By analogy, we can use it for fractions. Below we indicate the sequence of actions that need to be carried out for this.

Definition 1

To divide a column of decimal fractions by natural numbers you need:

1. Add a few zeros to the decimal fraction on the right (for division we can add any number of them that we need).

2. Divide a decimal fraction by a natural number using an algorithm. When the division of the whole part of the fraction comes to an end, we put a comma in the resulting quotient and count further.

The result of such division can be either a finite or an infinite periodic decimal fraction. It depends on the remainder: if it is zero, then the result will be finite, and if the remainders begin to repeat, then the answer will be a periodic fraction.

Let's take several problems as an example and try to perform these steps with specific numbers.

Example 5

Calculate how much 65, 14 4 will be.

Solution

We use the column method. To do this, add two zeros to the fraction and get the decimal fraction 65, 1400, which will be equal to the original one. Now we write a column for dividing by 4:

The resulting number will be the result we need from dividing the integer part. We put a comma, separating it, and continue:

We have reached zero remainder, therefore the division process is complete.

Answer: 65 , 14: 4 = 16 , 285 .

Example 6

Divide 164.5 by 27.

Solution

We first divide the fractional part and get:

Separate the resulting number with a comma and continue dividing:

We see that the remainders began to repeat periodically, and in the quotient the numbers nine, two and five began to alternate. We will stop here and write the answer in the form of a periodic fraction 6, 0 (925).

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

This division can be reduced to the process of finding the quotient of a decimal fraction and a natural number, already described above. To do this, we need to multiply the dividend and divisor by 10, 100, etc. so that the divisor turns into a natural number. Next we carry out the sequence of actions described above. This approach is possible due to the properties of division and multiplication. We wrote them down like this:

a: b = (a · 10) : (b · 10) , a: b = (a · 100) : (b · 100) and so on.

Let's formulate a rule:

Definition 2

To divide one final decimal fraction by another:

1. Move the comma in the dividend and divisor to the right by the number of digits necessary to turn the divisor into a natural number. If there are not enough signs in the dividend, we add zeros to it on the right side.

2. After this, divide the fraction by a column by the resulting natural number.

Let's look at a specific problem.

Example 7

Divide 7.287 by 2.1.

Solution: To make the divisor a natural number, we need to move the decimal place one place to the right. So we moved on to dividing the decimal fraction 72, 87 by 21. Let's write the resulting numbers in a column and calculate

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

Example 8

Calculate 16.30.021.

Solution

We will have to move the comma three places. There are not enough digits in the divisor for this, which means you need to use additional zeros. We think the result will be:

We see periodic repetition of residues 4, 19, 1, 10, 16, 13. In the quotient, 1, 9, 0, 4, 7 and 5 are repeated. Then our result is the periodic decimal fraction 776, (190476).

Answer: 16 , 3: 0 , 021 = 776 , (190476) ​​​​​​

The method we described allows you to do the opposite, that is, divide a natural number by the final decimal fraction. Let's see how it's done.

Example 9

Calculate how much 3 5, 4 is.

Solution

Obviously, we will have to move the comma to the right one place. After this we can proceed to divide 30, 0 by 54. Let's write the data in a column and calculate the result:

Repeating the remainder gives us the final number 0, (5), which is a periodic decimal fraction.

Answer: 3: 5 , 4 = 0 , (5) .

How to divide decimals by 1000, 100, 10, etc.

According to the already studied rules for dividing ordinary fractions, dividing a fraction by tens, hundreds, thousands is similar to multiplying it by 1/1000, 1/100, 1/10, etc. It turns out that to perform the division, in this case it is enough to simply move the decimal point to the required amount numbers If there are not enough values ​​in the number to transfer, you need to add the required number of zeros.

Example 10

So, 56, 21: 10 = 5, 621, and 0, 32: 100,000 = 0, 0000032.

In the case of infinite decimal fractions, we do the same.

Example 11

For example, 3, (56): 1,000 = 0, 003 (56) and 593, 374...: 100 = 5, 93374....

How to divide decimals by 0.001, 0.01, 0.1, etc.

Using the same rule, we can also divide fractions into the indicated values. This action will be similar to multiplying by 1000, 100, 10, respectively. To do this, we move the comma to one, two or three digits, depending on the conditions of the problem, and add zeros if there are not enough digits in the number.

Example 12

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

This rule also applies to infinite decimal fractions. We only advise you to be careful with the period of the fraction that appears in the answer.

So, 7, 5 (716) : 0, 01 = 757, (167) because after we moved the comma in the decimal fraction 7, 5716716716... two places to the right, we got 757, 167167....

If we have non-periodic fractions in the example, then everything is simpler: 394, 38283...: 0, 001 = 394382, 83....

How to divide a mixed number or fraction by a decimal and vice versa

We also reduce this action to operations with ordinary fractions. To do this, you need to replace the decimal numbers with the corresponding ordinary fractions, and write the mixed number as an improper fraction.

If we divide a non-periodic fraction by an ordinary or mixed number, we need to do the opposite, replacing the ordinary fraction or mixed number with the corresponding decimal fraction.

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37. Division by decimal fraction

Task. The area of ​​the rectangle is 2.88 dm2, and its width is 0.8 dm. What is the length of the rectangle?

Solution. Since 2.88 dm 2 = 288 cm 2, and 0.8 dm = 8 cm, then the length of the rectangle is 288: 8, that is, 36 cm = 3.6 dm. We found the number 3.6 such that 3.6 0.8 = 2.88. It is the quotient of 2.88 divided by 0.8.

The answer 3.6 can be obtained without converting decimeters to centimeters. To do this, you need to multiply the divisor 0.8 and the dividend 2.88 by 10 (that is, move the comma in them one digit to the right) and divide 28.8 by 8. Again we get: .

To divide a number by a decimal, necessary:
1) in the dividend and divisor, move the comma to the right by as many digits as there are after the decimal point in the divisor;
2) after this, divide by a natural number.

Example 1. Divide 12.096 by 2.24. Move the comma in the dividend and divisor 2 digits to the right. We get the numbers 1209.6 and 224.

Since , then and .

Example 2. Divide 4.5 by 0.125. Here you need to move the comma in the dividend and divisor 3 digits to the right. Since the dividend has only one digit after the decimal point, we will add two zeros to the right of it. After moving the comma, we get the numbers 4500 and 125.

Since , then and .

From examples 1 and 2 it is clear that when dividing a number by an improper fraction, this number decreases or does not change, but when dividing by a proper decimal fraction it increases: , a .

Divide 2.467 by 0.01. After moving the comma in the dividend and divisor by 2 digits to the right, we find that the quotient is equal to 246.7: 1, that is, 246.7. This means 2.467: 0.01 = 246.7. From here we get the rule:

To divide a decimal by 0.1; 0.01; 0.001, you need to move the comma in it to the right by as many digits as there are zeros before one in the divisor (that is, multiply it by 10, 100, 1000).

If there are not enough numbers, you must first add a few zeros to the end of the fraction.

For example, .

1443. Find the quotient and check by multiplication:

a) 0.8: 0.5; b) 3.51: 2.7; c) 14.335: 0.61.

1444. Find the quotient and check by division:

a) 0.096: 0.12; 6)0.126:0.9; c) 42.105: 3.5.

1445. Perform division:

1446. Write down the expressions:

a) the quotient of dividing the sum of a and 2.6 by the difference of b and 8.5;
b) the sum of the quotient x and 3.7 and the quotient 3.1 and y.

1447. Read the expression:

a) m: 12.8 - n: 4.9; b) (x + 0.7) : (y + 3.4); c) (a: b) (8: c).

1448. A person’s step is 0.8 m. How many steps does he need to take to cover a distance of 100 m?

1449. Alyosha traveled 162.5 km by train in 2.6 hours. How fast was the train going?

1450. Find the mass of 1 cm 3 of ice if the mass of 3.5 cm 3 of ice is 3.08 g.

1451. The rope was cut into two parts. The length of one part is 3.25 m, and the length of the other part is 1.3 times less than the first. What was the length of the rope?

1452. The first package contained 6.72 kg of flour, which is 2.4 times more than the second package. How many kilograms of flour are in both bags?

1453. Borya spent 3.5 times less time preparing his lessons than taking a walk. How long did it take Bori to walk and prepare his homework if the walk took 2.8 hours?

§ 107. Addition of decimal fractions.

Adding decimals is the same as adding whole numbers. Let's see this with examples.

1) 0.132 + 2.354. Let's label the terms one below the other.

Here, adding 2 thousandths to 4 thousandths resulted in 6 thousandths;
from adding 3 hundredths with 5 hundredths you get 8 hundredths;
from adding 1 tenth with 3 tenths -4 tenths and
from adding 0 integers with 2 integers - 2 integers.

2) 5,065 + 7,83.

There are no thousandths in the second term, so it is important not to make mistakes when labeling terms one after another.

3) 1,2357 + 0,469 + 2,08 + 3,90701.

Here, when adding thousandths, the result is 21 thousandths; we wrote 1 under the thousandths, and added 2 to the hundredths, so in the hundredths place we got the following terms: 2 + 3 + 6 + 8 + 0; in total they give 19 hundredths, we signed 9 under hundredths, and 1 counted as tenths, etc.

Thus, when adding decimal fractions, the following order must be observed: sign the fractions one below the other so that in all terms the same digits are located under each other and all commas are in the same vertical column; To the right of the decimal places of some terms, such a number of zeros are added, at least mentally, so that all terms after the decimal point have the same number of digits. Then they perform addition by digits, starting from the right side, and in the resulting sum they put a comma in the same vertical column in which it is located in these terms.

§ 108. Subtraction of decimal fractions.

Subtracting decimals works the same way as subtracting whole numbers. Let's show this with examples.

1) 9.87 - 7.32. Let's sign the subtrahend under the minuend so that units of the same digit are under each other:

2) 16.29 - 4.75. Let's sign the subtrahend under the minuend, as in the first example:

To subtract tenths, you had to take one whole unit from 6 and split it into tenths.

3) 14.0213- 5.350712. Let's sign the subtrahend under the minuend:

The subtraction was performed as follows: since we cannot subtract 2 millionths from 0, we should turn to the nearest digit on the left, i.e., hundred thousandths, but in place of hundred thousandths there is also zero, so we take 1 ten thousandth from 3 ten thousandths and We split it into hundred thousandths, we get 10 hundred thousandths, of which we leave 9 hundred thousandths in the hundred thousandths category, and we break 1 hundred thousandth into millionths, we get 10 millionths. Thus, in the last three digits we have: millionths 10, hundred thousandths 9, ten thousandths 2. For greater clarity and convenience (so as not to forget), these numbers are written above the corresponding fractional digits of the minuend. Now you can start subtracting. From 10 millionths we subtract 2 millionths, we get 8 millionths; from 9 hundred thousandths we subtract 1 hundred thousandth, we get 8 hundred thousandths, etc.

Thus, when subtracting decimal fractions, the following order is observed: sign the subtrahend under the minuend so that the same digits are located under each other and all commas are in the same vertical column; on the right they add, at least mentally, so many zeros in the minuend or subtrahend so that they have the same number of digits, then they subtract by digits, starting from the right side, and in the resulting difference they put a comma in the same vertical column in which it is located in minuend and subtract.

§ 109. Multiplication of decimal fractions.

Let's look at some examples of multiplying decimal fractions.

To find the product of these numbers, we can reason as follows: if the factor is increased by 10 times, then both factors will be integers and we can then multiply them according to the rules for multiplying integers. But we know that when one of the factors increases several times, the product increases by the same amount. This means that the number that is obtained from multiplying the integer factors, i.e. 28 by 23, is 10 times greater than the true product, and in order to obtain the true product, the found product must be reduced by 10 times. Therefore, here you will have to multiply by 10 once and divide by 10 once, but multiplying and dividing by 10 is done by moving the decimal point to the right and left by one place. Therefore, you need to do this: in the factor, move the comma to the right one place, this will make it equal to 23, then you need to multiply the resulting integers:

This product is 10 times larger than the true one. Therefore, it must be reduced by 10 times, for which we move the comma one place to the left. Thus, we get

28 2,3 = 64,4.

For verification purposes, you can write a decimal fraction with a denominator and perform the action according to the rule for multiplying ordinary fractions, i.e.

2) 12,27 0,021.

The difference between this example and the previous one is that here both factors are represented as decimal fractions. But here, in the process of multiplication, we will not pay attention to commas, i.e. we will temporarily increase the multiplicand by 100 times, and the multiplier by 1,000 times, which will increase the product by 100,000 times. Thus, multiplying 1,227 by 21, we get:

1 227 21 = 25 767.

Considering that the resulting product is 100,000 times larger than the true product, we must now reduce it by 100,000 times by properly placing a comma in it, then we get:

32,27 0,021 = 0,25767.

Let's check:

Thus, in order to multiply two decimal fractions, it is enough, without paying attention to commas, to multiply them as whole numbers and in the product to separate as many decimal places with a comma on the right side as there were in the multiplicand and in the multiplier together.

The last example resulted in a product with five decimal places. If such great precision is not required, then the decimal fraction is rounded. When rounding, you should use the same rule as was indicated for integers.

§ 110. Multiplication using tables.

Multiplying decimals can sometimes be done using tables. For this purpose, you can, for example, use those multiplication tables for two-digit numbers, the description of which was given earlier.

1) Multiply 53 by 1.5.

We will multiply 53 by 15. In the table, this product is equal to 795. We found the product 53 by 15, but our second factor was 10 times smaller, which means the product must be reduced by 10 times, i.e.

53 1,5 = 79,5.

2) Multiply 5.3 by 4.7.

First, we find in the table the product of 53 by 47, it will be 2,491. But since we increased the multiplicand and the multiplier by a total of 100 times, the resulting product is 100 times larger than it should be; so we must reduce this product by 100 times:

5,3 4,7 = 24,91.

3) Multiply 0.53 by 7.4.

First, we find in the table the product 53 by 74; it will be 3,922. But since we increased the multiplicand by 100 times, and the multiplier by 10 times, the product increased by 1,000 times; so we now have to reduce it by 1,000 times:

0,53 7,4 = 3,922.

§ 111. Division of decimal fractions.

We will look at dividing decimal fractions in this order:

1. Dividing a decimal fraction by a whole number,

1. Divide a decimal fraction by a whole number.

1) Divide 2.46 by 2.

We divided by 2 first whole, then tenths and finally hundredths.

2) Divide 32.46 by 3.

32,46: 3 = 10,82.

We divided 3 tens by 3, then began to divide 2 units by 3; since the number of units of the dividend (2) is less than the divisor (3), we had to put 0 in the quotient; further, to the remainder we took 4 tenths and divided 24 tenths by 3; received 8 tenths in the quotient and finally divided 6 hundredths.

3) Divide 1.2345 by 5.

1,2345: 5 = 0,2469.

Here in the quotient the first place is zero integers, since one integer is not divisible by 5.

4) Divide 13.58 by 4.

The peculiarity of this example is that when we received 9 hundredths in the quotient, we discovered a remainder equal to 2 hundredths, we split this remainder into thousandths, got 20 thousandths and completed the division.

Rule. Dividing a decimal fraction by an integer is performed in the same way as dividing integers, and the resulting remainders are converted into decimal fractions, smaller and smaller; Division continues until the remainder is zero.

2. Divide a decimal by a decimal.

1) Divide 2.46 by 0.2.

We already know how to divide a decimal fraction by a whole number. Let's think, is it possible to reduce this new case of division to the previous one? At one time, we considered the remarkable property of a quotient, which consists in the fact that it remains unchanged when the dividend and divisor simultaneously increase or decrease by the same number of times. We could easily divide the numbers given to us if the divisor were an integer. To do this, it is enough to increase it by 10 times, and to obtain the correct quotient, it is necessary to increase the dividend by the same amount, i.e., 10 times. Then the division of these numbers will be replaced by the division of the following numbers:

Moreover, there will no longer be any need to make any amendments to the particulars.

Let's do this division:

So 2.46: 0.2 = 12.3.

2) Divide 1.25 by 1.6.

We increase the divisor (1.6) by 10 times; so that the quotient does not change, we increase the dividend by 10 times; 12 integers are not divisible by 16, so we write 0 in the quotient and divide 125 tenths by 16, we get 7 tenths in the quotient and the remainder 13. We split 13 tenths into hundredths by assigning zero and divide 130 hundredths by 16, etc. Please note to the following:

a) when there are no integers in a particular, then zero integers are written in their place;

b) when, after adding the digit of the dividend to the remainder, a number is obtained that is not divisible by the divisor, then zero is written in the quotient;

c) when, after removing the last digit of the dividend, the division does not end, then, adding zeros to the remainder, the division continues;

d) if the dividend is an integer, then when dividing it by a decimal fraction, it is increased by adding zeros to it.

Thus, to divide a number by a decimal fraction, you need to discard the comma in the divisor, and then increase the dividend by as many times as the divisor increased when discarding the comma in it, and then perform the division according to the rule for dividing a decimal fraction by a whole number.

§ 112. Approximate quotients.

In the previous paragraph, we looked at the division of decimal fractions, and in all the examples we solved the division was completed, i.e., an exact quotient was obtained. However, in most cases, an exact quotient cannot be obtained, no matter how far we continue the division. Here is one such case: divide 53 by 101.

We have already received five digits in the quotient, but the division has not yet ended and there is no hope that it will ever end, since in the remainder we begin to have numbers that have already been encountered before. In the quotient, numbers will also be repeated: it is obvious that after the number 7 the number 5 will appear, then 2, etc. endlessly. In such cases, the division is interrupted and limited to the first few digits of the quotient. This quotient is called close ones. We will show with examples how to perform division.

Let the requirement be 25 divided by 3. Obviously, an exact quotient, expressed as an integer or a decimal fraction, cannot be obtained from such a division. Therefore, we will look for an approximate quotient:

25: 3 = 8 and remainder 1

The approximate quotient is 8; it is, of course, less than the exact quotient, because there is a remainder 1. To obtain the exact quotient, you need to add the fraction that is obtained by dividing the remainder equal to 1 by 3 to the found approximate quotient, i.e., to 8; this will be a fraction 1/3. This means that the exact quotient will be expressed as a mixed number 8 1/3. Since 1/3 is a proper fraction, i.e. a fraction, less than one, then, discarding it, we will allow error, which less than one. The quotient 8 will be approximate quotient up to unity with a disadvantage. If instead of 8 we take 9 in the quotient, then we will also allow an error that is less than one, since we will not add the whole unit, but 2/3. Such a private will approximate quotient to within one with excess.

Let's now take another example. Let’s say we need to divide 27 by 8. Since here we won’t get an exact quotient expressed as an integer, we will look for an approximate quotient:

27: 8 = 3 and remainder 3.

Here the error is equal to 3/8, it is less than one, which means that the approximate quotient (3) was found accurate to one with a disadvantage. Let's continue the division: split the remainder 3 into tenths, we get 30 tenths; divide them by 8.

We got 3 in the quotient in place of tenths and 6 tenths in the remainder. If we limit ourselves to the number 3.3 and discard the remainder 6, then we will allow an error of less than one tenth. Why? Because the exact quotient would be obtained when we added to 3.3 the result of dividing 6 tenths by 8; this division would yield 6/80, which is less than one tenth. (Check!) Thus, if in the quotient we limit ourselves to tenths, then we can say that we have found the quotient accurate to one tenth(with a disadvantage).

Let's continue division to find another decimal place. To do this, we split 6 tenths into hundredths and get 60 hundredths; divide them by 8.

In the quotient in third place it turned out to be 7 and the remainder 4 hundredths; if we discard them, we will allow an error of less than one hundredth, because 4 hundredths divided by 8 is less than one hundredth. In such cases they say that the quotient has been found accurate to one hundredth(with a disadvantage).

In the example we are now looking at, we can get the exact quotient expressed as a decimal fraction. To do this, it is enough to split the last remainder, 4 hundredths, into thousandths and divide by 8.

However, in the vast majority of cases it is impossible to obtain an exact quotient and one has to limit oneself to its approximate values. We will now look at this example:

40: 7 = 5,71428571...

The dots placed at the end of the number indicate that the division is not completed, i.e. the equality is approximate. Usually the approximate equality is written as follows:

40: 7 = 5,71428571.

We took the quotient with eight decimal places. But if such great accuracy is not required, you can limit yourself to only the whole part of the quotient, i.e., the number 5 (more precisely 6); for greater accuracy, one could take into account tenths and take the quotient equal to 5.7; if for some reason this accuracy is insufficient, then you can stop at hundredths and take 5.71, etc. Let’s write out the individual quotients and name them.

The first approximate quotient accurate to one 6.

Second » » » to one tenth 5.7.

Third » » » to one hundredth 5.71.

Fourth » » » to one thousandth 5.714.

Thus, in order to find an approximate quotient accurate to some, for example, 3rd decimal place (i.e., up to one thousandth), stop division as soon as this sign is found. In this case, you need to remember the rule set out in § 40.

§ 113. The simplest problems involving percentages.

After learning about decimals, we'll do a few more percent problems.

These problems are similar to those we solved in the fractions department; but now we will write hundredths in the form of decimal fractions, that is, without an explicitly designated denominator.

First of all, you need to be able to easily move from an ordinary fraction to a decimal with a denominator of 100. To do this, you need to divide the numerator by the denominator:

The table below shows how a number with a % (percentage) symbol is replaced by a decimal fraction with a denominator of 100:

Let us now consider several problems.

1. Finding the percentage of a given number.

Task 1. Only 1,600 people live in one village. The number of school-age children makes up 25% of the total population. How many school-age children are there in this village?

In this problem you need to find 25%, or 0.25, of 1,600. The problem is solved by multiplying:

1,600 0.25 = 400 (children).

Therefore, 25% of 1,600 is 400.

To clearly understand this task, it is useful to recall that for every hundred of the population there are 25 school-age children. Therefore, to find the number of all school-age children, you can first find out how many hundreds there are in the number 1,600 (16), and then multiply 25 by the number of hundreds (25 x 16 = 400). This way you can check the validity of the solution.

Task 2. Savings banks provide depositors with a 2% return annually. How much income will a depositor receive in a year if he puts in the cash register: a) 200 rubles? b) 500 rubles? c) 750 rubles? d) 1000 rubles?

In all four cases, to solve the problem you will need to calculate 0.02 of the indicated amounts, i.e. each of these numbers will have to be multiplied by 0.02. Let's do it:

a) 200 0.02 = 4 (rub.),

b) 500 0.02 = 10 (rub.),

c) 750 0.02 = 15 (rub.),

d) 1,000 0.02 = 20 (rub.).

Each of these cases can be verified by the following considerations. Savings banks give investors 2% income, i.e. 0.02 of the amount deposited in savings. If the amount was 100 rubles, then 0.02 of it would be 2 rubles. This means that every hundred brings the investor 2 rubles. income. Therefore, in each of the cases considered, it is enough to figure out how many hundreds there are in a given number, and multiply 2 rubles by this number of hundreds. In example a) there are 2 hundreds, which means

2 2 = 4 (rub.).

In example d) there are 10 hundreds, which means

2 10 = 20 (rub.).

2. Finding a number by its percentage.

Task 1. The school graduated 54 students in the spring, representing 6% of its total enrollment. How many students were there in the school last school year?

Let us first clarify the meaning of this task. The school graduated 54 students, which is 6% of the total number of students, or, in other words, 6 hundredths (0.06) of all students at the school. This means that we know the part of the students expressed by the number (54) and the fraction (0.06), and from this fraction we must find the entire number. Thus, we have before us an ordinary task of finding a number from its fraction (§90, paragraph 6). Problems of this type are solved by division:

This means that there were only 900 students in the school.

It is useful to check such problems by solving the inverse problem, i.e. after solving the problem, you should, at least in your head, solve a problem of the first type (finding the percentage of a given number): take the found number (900) as given and find the percentage of it indicated in the solved problem , namely:

900 0,06 = 54.

Task 2. The family spends 780 rubles on food during the month, which is 65% of the father’s monthly earnings. Determine his monthly income.

This task has the same meaning as the previous one. It gives part of the monthly earnings, expressed in rubles (780 rubles), and indicates that this part is 65%, or 0.65, of the total earnings. And what you are looking for is all the earnings:

780: 0,65 = 1 200.

Therefore, the required income is 1200 rubles.

3. Finding the percentage of numbers.

Task 1. There are only 6,000 books in the school library. Among them are 1,200 books on mathematics. What percentage of math books make up the total number of books in the library?

We have already considered (§97) problems of this kind and came to the conclusion that to calculate the percentage of two numbers, you need to find the ratio of these numbers and multiply it by 100.

In our problem we need to find the percentage ratio of the numbers 1,200 and 6,000.

Let's first find their ratio, and then multiply it by 100:

Thus, the percentage of the numbers 1,200 and 6,000 is 20. In other words, math books make up 20% of the total number of all books.

To check, let’s solve the inverse problem: find 20% of 6,000:

6 000 0,2 = 1 200.

Task 2. The plant should receive 200 tons of coal. 80 tons have already been delivered. What percentage of coal has been delivered to the plant?

This problem asks what percentage one number (80) is of another (200). The ratio of these numbers will be 80/200. Let's multiply it by 100:

This means that 40% of the coal has been delivered.

Rectangle?

Solution. Since 2.88 dm2 = 288 cm2, and 0.8 dm = 8 cm, then the length of the rectangle is 288: 8, that is, 36 cm = 3.6 dm. We found a number 3.6 such that 3.6 0.8 = 2.88. It is the quotient of 2.88 divided by 0.8.

They write: 2.88: 0.8 = 3.6.

The answer 3.6 can be obtained without converting decimeters to centimeters. To do this, you need to multiply the divisor 0.8 and the dividend 2.88 by 10 (that is, move the comma one digit to the right) and divide 28.8 by 8. Again we get: 28.8: 8 = 3.6.

To divide a number by a decimal fraction, you need to:

1) in the dividend and divisor, move the comma to the right by as many digits as there are after the decimal point in the divisor;
2) after this, divide by a natural number.

Example 1. Divide 12.096 by 2.24. Move the comma in the dividend and divisor 2 digits to the right. We get the numbers 1209.6 and 224. Since 1209.6: 224 = 5.4, then 12.096: 2.24 = 5.4.

Example 2. Divide 4.5 by 0.125. Here you need to move the comma in the dividend and divisor 3 digits to the right. Since the dividend has only one digit after the decimal point, we will add two zeros to the right of it. After moving the comma we get numbers 4500 and 125. Since 4500: 125 = 36, then 4.5: 0.125 = 36.

From examples 1 and 2 it is clear that when dividing a number by an improper fraction, this number decreases or does not change, and when dividing by a proper decimal fraction it increases: 12.096 > 5.4, and 4.5< 36.

Divide 2.467 by 0.01. After moving the comma in the dividend and divisor by 2 digits to the right, we find that the quotient is equal to 246.7: 1, that is, 246.7.

This means 2.467: 0.01 = 246.7. From here we get the rule:

To divide a decimal by 0.1; 0.01; 0.001, you need to move the comma in it to the right by as many digits as there are zeros before one in the divisor (that is, multiply it by 10, 100, 1000).

If there are not enough numbers, you must first add them at the end fractions a few zeros.

For example, 56.87: 0.0001 = 56.8700: 0.0001 = 568,700.

Formulate the rule for dividing a decimal fraction: by a decimal fraction; by 0.1; 0.01; 0.001.
By multiplying by what number can you replace division by 0.01?

1443. Find the quotient and check by multiplication:

a) 0.8: 0.5; b) 3.51: 2.7; c) 14.335: 0.61.

1444. Find the quotient and check by division:

a) 0.096: 0.12; b) 0.126: 0.9; c) 42.105: 3.5.

a) 7.56: 0.6; g) 6.944: 3.2; m) 14.976: 0.72;
b) 0.161: 0.7; h) 0.0456: 3.8; o) 168.392: 5.6;
c) 0.468: 0.09; i) 0.182: 1.3; n) 24.576: 4.8;
d) 0.00261: 0.03; j) 131.67: 5.7; p) 16.51: 1.27;
e) 0.824: 0.8; k) 189.54: 0.78; c) 46.08: 0.384;
e) 10.5: 3.5; m) 636: 0.12; t) 22.256: 20.8.

1446. Write down the expressions:

a) 10 - 2.4x = 3.16; e) 4.2р - р = 5.12;
b) (y + 26.1) 2.3 = 70.84; e) 8.2t - 4.4t = 38.38;
c) (z - 1.2): 0.6 = 21.1; g) (10.49 - s): 4.02 = 0.805;
d) 3.5m + t = 9.9; h) 9k - 8.67k = 0.6699.

1460. There were 119.88 tons of gasoline in two tanks. The first tank contained 1.7 times more gasoline than the second. How much gasoline was in each tank?

1461. 87.36 tons of cabbage were collected from three plots. At the same time, 1.4 times more was collected from the first plot, and 1.8 times more from the second plot than from the third plot. How many tons of cabbage were collected from each plot?

1462. A kangaroo is 2.4 times shorter than a giraffe, and a giraffe is 2.52 m taller than a kangaroo. What is the height of a giraffe and what is the height of a kangaroo?

1463. Two pedestrians were at a distance of 4.6 km from each other. They went towards each other and met after 0.8 hours. Find the speed of each pedestrian if the speed of one of them is 1.3 times the speed of the other.

1464. Follow these steps:

a) (130.2 - 30.8) : 2.8 - 21.84:
b) 8.16: (1.32 + 3.48) - 0.345;
c) 3.712: (7 - 3.8) + 1.3 (2.74 + 0.66);
d) (3.4: 1.7 + 0.57: 1.9) 4.9 + 0.0825: 2.75;
e) (4.44: 3.7 - 0.56: 2.8) : 0.25 - 0.8;
e) 10.79: 8.3 0.7 - 0.46 3.15: 6.9.

1465. Represent a fraction as a decimal and find the value expressions:

1466. Calculate orally:

a) 25.5: 5; b) 9 0.2; c) 0.3: 2; d) 6.7 - 2.3;
1,5: 3; 1 0,1; 2:5; 6- 0,02;
4,7: 10; 16 0,01; 17,17: 17; 3,08 + 0,2;
0,48: 4; 24 0,3; 25,5: 25; 2,54 + 0,06;
0,9:100; 0,5 26; 0,8:16; 8,2-2,2.

1467. Find the work:

a) 0.1 0.1; d) 0.4 0.4; g) 0.7 0.001;
b) 1.3 1.4; e) 0.06 0.8; h) 100 0.09;
c) 0.3 0.4; e) 0.01 100; i) 0.3 0.3 0.3.

1468. Find: 0.4 of the number 30; 0.5 of the number 18; 0.1 numbers 6.5; 2.5 numbers 40; 0.12 number 100; 0.01 of the number 1000.

1469. What is the value of the expression 5683.25a when a = 10; 0.1; 0.01; 100; 0.001; 1000; 0.00001?

1470. Think about which of the numbers can be exact and which can be approximate:

a) there are 32 students in the class;
b) the distance from Moscow to Kyiv is 900 km;
c) the parallelepiped has 12 edges;
d) table length 1.3 m;
e) the population of Moscow is 8 million people;
e) in a bag 0.5 kg of flour;
g) the area of ​​the island of Cuba is 105,000 km2;
h) the school library has 10,000 books;
i) one span is equal to 4 vershok, and a vershok is equal to 4.45 cm (vershok
length of the phalanx of the index finger).

1471. Find three solutions to the inequality:

a) 1.2< х < 1,6; в) 0,001 < х < 0,002;
b) 2.1< х< 2,3; г) 0,01 <х< 0,011.

1472. Compare, without calculating, the values ​​of the expressions:

a) 24 0.15 and (24 - 15) : 100;

b) 0.084 0.5 and (84 5) : 10,000.
Explain your answer.

1473. Round the numbers:

1474. Perform division:

a) 22.7: 10; 23.3:10; 3.14:10; 9.6:10;
b) 304: 100; 42.5: 100; 2.5: 100; 0.9: 100; 0.03: 100;
c) 143.4: 12; 1.488: 124 ; 0.3417: 34; 159.9: 235; 65.32: 568.

1475. A cyclist left the village at a speed of 12 km/h. After 2 hours, another cyclist rode out from the same village in the opposite direction,
and the speed of the second is 1.25 times greater than the speed of the first. What will be the distance between them 3.3 hours after the second cyclist leaves?

1476. The boat's own speed is 8.5 km/h, and the speed of the current is 1.3 km/h. How far will the boat travel downstream in 3.5 hours? How far will the boat travel against the current in 5.6 hours?

1477. The plant produced 3.75 thousand parts and sold them at a price of 950 rubles. a piece. The plant's expenses for the production of one part amounted to 637.5 rubles. Find the profit received by the factory from the sale of these parts.

1478. The width of a rectangular parallelepiped is 7.2 cm, which is Find the volume of this parallelepiped and round the answer to whole numbers.

1479. Papa Carlo promised to give Piero 4 soldi every day, and Buratino 1 soldi on the first day, and 1 soldi more on each subsequent day if he behaves well. Pinocchio was offended: he decided that, no matter how hard he tried, he would never be able to get as much soldi as Pierrot. Think about whether Pinocchio is right.

1480. For 3 cabinets and 9 bookshelves, 231 m of boards were used, and 4 times more material is used for the cabinet than for the shelf. How many meters of boards go on a cabinet and how many on a shelf?

1481. Solve the problem:
1) The first number is 6.3 and makes up the second number. The third number makes up the second. Find the second and third numbers.

2) The first number is 8.1. The second number is from the first number and from the third number. Find the second and third numbers.

1482. Find the meaning of the expression:

1) (7 - 5,38) 2,5;

2) (8 - 6,46) 1,5.

1483. Find the value of the quotient:

a) 17.01: 6.3; d) 1.4245: 3.5; g) 0.02976: 0.024;
b) 1.598: 4.7; e) 193.2: 8.4; h) 11.59: 3.05;
c) 39.156: 7.8; e) 0.045: 0.18; i) 74.256: 18.2.

1484. The distance from home to school is 1.1 km. The girl covers this path in 0.25 hours. How fast is the girl walking?

1485. In a two-room apartment, the area of ​​one room is 20.64 m2, and the area of ​​the other room is 2.4 times less. Find the area of ​​these two rooms together.

1486. ​​The engine consumes 111 liters of fuel in 7.5 hours. How many liters of fuel will the engine consume in 1.8 hours?
1487. A metal part with a volume of 3.5 dm3 has a mass of 27.3 kg. Another part made of the same metal has a mass of 10.92 kg. What is the volume of the second part?

1488. 2.28 tons of gasoline were poured into a tank through two pipes. Through the first pipe, 3.6 tons of gasoline flowed per hour, and it was open for 0.4 hours. Through the second pipe, 0.8 tons of gasoline flowed per hour less than through the first. How long was the second pipe open?

1489. Solve the equation:

a) 2.136: (1.9 - x) = 7.12; c) 0.2t + 1.7t - 0.54 = 0.22;
b) 4.2 (0.8 + y) = 8.82; d) 5.6g - 2z - 0.7z + 2.65 = 7.

1490. Goods weighing 13.3 tons were distributed among three vehicles. The first car was loaded 1.3 times more, and the second car was loaded 1.5 times more than the third car. How many tons of goods were loaded onto each vehicle?

1491. Two pedestrians left the same place at the same time in opposite directions. After 0.8 hours, the distance between them became 6.8 km. The speed of one pedestrian was 1.5 times the speed of the other. Find the speed of each pedestrian.

1492. Follow these steps:

a) (21.2544: 0.9 + 1.02 3.2) : 5.6;
b) 4.36: (3.15 + 2.3) + (0.792 - 0.78) 350;
c) (3.91: 2.3 5.4 - 4.03) 2.4;
d) 6.93: (0.028 + 0.36 4.2) - 3.5.

1493. A doctor came to school and brought 0.25 kg of serum for vaccination. How many guys can he give injections to if each injection requires 0.002 kg of serum?

1494. 2.8 tons of gingerbread were delivered to the store. Before lunch these gingerbread cookies were sold. How many tons of gingerbread are left to sell?

1495. 5.6 m were cut from a piece of fabric. How many meters of fabric were in the piece if this piece was cut off?

N.Ya. VILENKIN, V. I. ZHOKHOV, A. S. CHESNOKOV, S. I. SHVARTSBURD, Mathematics grade 5, Textbook for general education institutions

Many schoolchildren forget how to do long division by the time they reach high school. Computers, calculators, mobile phones and other devices have become so integral to our lives that elementary mathematical operations sometimes leave us stunned. And how did people manage without all these benefits a few decades ago? First, you need to remember the main mathematical concepts that are needed for division. So, the dividend is the number that will be divided. Divisor – the number to be divided by. What results as a result is called a quotient. To divide into a line, use a symbol similar to a colon - “:”, and when dividing into a column, use the “∟” icon; it is also called a corner.

It is also worth recalling that any division can be checked by multiplication. To check the result of division, just multiply it by the divisor; the result should be a number that corresponds to the dividend (a: b=c; therefore, c*b=a). Now about what a decimal fraction is. A decimal fraction is obtained by dividing the unit by 0.0, 1000, and so on. The recording of these numbers and mathematical operations with them are exactly the same as with integers. When dividing decimal fractions, there is no need to remember where the denominator is located. Everything becomes clear when writing down the number. First, the whole number is written, and after the decimal point its tenths, hundredths, thousandths are written. The first digit after the decimal point corresponds to tens, the second to hundreds, the third to thousands, etc.

Every student should know how to divide decimals by decimals. If both the dividend and the divisor are multiplied by the same number, then the answer, i.e., the quotient, will not change. If a decimal fraction is multiplied by 0.0, 1000, etc., then the comma after the whole number will change its position - it will move to the right by the same number of digits as there are zeros in the number that was multiplied by. For example, when multiplying a decimal by 10, the decimal point will move one number to the right. 2.9: 6.7 – we multiply both the divisor and the dividend by 100, we get 6.9: 3687. It is best to multiply so that when multiplied by it, at least one number (divisor or dividend) has no digits left after the decimal point, i.e. make at least one number an integer. A few more examples of moving commas after an integer: 9.2: 1.5 = 2492: 2.5; 5.4:4.8 = 5344:74598.

Attention, the decimal fraction will not change its value if zeros are added to its right side, for example 3.8 = 3.0. Also, the value of the fraction will not change if the zeros at the very end of the number are removed from the right: 3.0 = 3.3. However, you cannot remove zeros in the middle of the number - 3.3. How to divide a decimal fraction by a natural number in a column? To divide a decimal fraction by a natural number in a column, you need to make the appropriate notation with a corner, divide. In the quotient, a comma must be placed when the division of the integer ends. For example, 5.4|2 14 7.2 18 18 0 4 4 0If the first digit of the number in the dividend is less than the divisor, then subsequent digits are used until it is possible to perform the first action.

In this case, the first digit of the dividend is 1, it cannot be divided by 2, so two digits 1 and 5 are used for division at once: 15 by 2 is divided with the remainder, it turns out to be the quotient of 7, and the remainder remains 1. Then we use the next digit of the dividend - 8. We lower it down to 1 and divide 18 by 2. In the quotient we write down the number 9. There is nothing left in the remainder, so we write 0. We lower the remaining number 4 of the dividend down and divide by the divisor, i.e. by 2. In the quotient We write down 2, and the remainder is again 0. The result of this division is the number 7.2. It's called private. It's quite easy to solve the question of how to divide a decimal by a decimal if you know a few tricks. Dividing decimals mentally is sometimes quite difficult, so long division is used to make the process easier.

With this division, all the same rules apply as when dividing a decimal fraction by an integer or when dividing into a string. On the left side of the line they write the dividend, then put the “corner” symbol and then write the divisor and begin division. To facilitate division and move a comma after a whole number to a convenient place, you can multiply by tens, hundreds or thousands. For example, 9.2: 1.5 = 24920: 125. Attention, both fractions are multiplied by 0.0, 1000. If the dividend was multiplied by 10, then the divisor is also multiplied by 10. In this example, both the dividend and the divisor were multiplied by 100. Next, the calculation is performed in the same way as shown in the example of dividing a decimal fraction by a natural number. In order to divide by 0.1; 0.1; 0.1, etc. it is necessary to multiply both the divisor and the dividend by 0.0, 1000.

Quite often, when dividing in a quotient, i.e., in the answer, infinite fractions are obtained. In this case, it is necessary to round the number to tenths, hundredths or thousandths. In this case, the rule applies: if after the number to which the answer needs to be rounded is less than or equal to 5, then the answer is rounded down, but if it is more than 5, it is rounded up. For example, you want to round the result of 5.5 to thousandths. This means that the answer after the decimal point should end with the number 6. After 6 there is 9, which means we round the answer up and get 5.7. But if the answer 5.5 needed to be rounded not to thousandths, but to tenths, then the answer would look like this - 5.2. In this case, 2 was not rounded up because 3 comes after it, and it is less than 5.