How to correctly divide numbers with commas. Multiplying and dividing decimals

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 improper fraction this number decreases or does not change, and when divided by the correct decimal 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; n) 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; l) 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. Imagine common 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
phalanx length 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. per 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 Pinocchio 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 he needs 0.002 kg of serum for each injection?

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

§ 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 got: 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 diminished and subtracted.

§ 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, 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 ones 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 are simultaneously increased or decreased 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 in the quotient 0 and divide 125 tenths by 16, we get 7 tenths in the quotient and the remainder is 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, in order to divide a number by a decimal fraction, you need to drop the comma in the divisor, and then increase the dividend by as many times as the divisor increased when dropping 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 unity, 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 looking at now, 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 some 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 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 this:

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 depositors 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 salary.

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.

At school these actions are studied from simple to complex. Therefore, it is imperative to thoroughly understand the algorithm for performing these operations using simple examples. So that later there will be no difficulties with dividing decimal fractions into a column. After all, this is the most difficult version of such tasks.

This subject requires consistent study. Gaps in knowledge are unacceptable here. Every student should learn this principle already in the first grade. Therefore, if you miss several lessons in a row, you will have to master the material yourself. Otherwise, later problems will arise not only with mathematics, but also with other subjects related to it.

The second prerequisite for successfully studying mathematics is to move on to examples of long division only after addition, subtraction and multiplication have been mastered.

It will be difficult for a child to divide if he has not learned the multiplication table. By the way, it is better to teach it using the Pythagorean table. There is nothing superfluous, and multiplication is easier to learn in this case.

How are natural numbers multiplied in a column?

If there is difficulty in solving examples in a column for division and multiplication, then you should begin to solve the problem with multiplication. Since division is the inverse operation of multiplication:

1. Before multiplying two numbers, you need to look at them carefully. Choose the one with more digits (longer) and write it down first. Place the second one under it. Moreover, the numbers of the corresponding category must be under the same category. That is, the rightmost digit of the first number should be above the rightmost digit of the second.
2. Multiply the rightmost digit of the bottom number by each digit of the top number, starting from the right. Write the answer below the line so that its last digit is under the one you multiplied by.
3. Repeat the same with another digit of the lower number. But the result of multiplication must be shifted one digit to the left. In this case, its last digit will be under the one by which it was multiplied.

Continue this multiplication in a column until the numbers in the second factor run out. Now they need to be folded. This will be the answer you are looking for.

Algorithm for multiplying decimals

First, you need to imagine that the given fractions are not decimals, but natural ones. That is, remove the commas from them and then proceed as described in the previous case.

The difference begins when the answer is written down. At this moment, it is necessary to count all the numbers that appear after the decimal points in both fractions. That's exactly how many of them you need to count from the end of the answer and put a comma there.

It is convenient to illustrate this algorithm using an example: 0.25 x 0.33:

Where to start learning division?

Before solving long division examples, you need to remember the names of the numbers that appear in the long division example. The first of them (the one that is divided) is divisible. The second (divided by) is the divisor. The answer is private.

After this, using a simple everyday example, we will explain the essence of this mathematical operation. For example, if you take 10 sweets, then it’s easy to divide them equally between mom and dad. But what if you need to give them to your parents and brother?

After this, you can become familiar with the division rules and master them using specific examples. First simple ones, and then move on to more and more complex ones.

Algorithm for dividing numbers into a column

First, let us present the procedure for natural numbers divisible by a single-digit number. They will also be the basis for multi-digit divisors or decimal fractions. Only then should you make small changes, but more on that later:

• Before you do long division, you need to figure out where the dividend and divisor are.
• Write down the dividend. To the right of it is the divider.
• Draw a corner on the left and bottom near the last corner.
• Determine the incomplete dividend, that is, the number that will be minimal for division. Usually it consists of one digit, maximum of two.
• Choose the number that will be written first in the answer. It should be the number of times the divisor fits into the dividend.
• Write down the result of multiplying this number by the divisor.
• Write it under the incomplete dividend. Perform subtraction.
• Add to the remainder the first digit after the part that has already been divided.
• Choose the number for the answer again.
• Repeat multiplication and subtraction. If the remainder is zero and the dividend is over, then the example is done. Otherwise, repeat the steps: remove the number, pick up the number, multiply, subtract.

How to solve long division if the divisor has more than one digit?

The algorithm itself completely coincides with what was described above. The difference will be the number of digits in the incomplete dividend. Now there should be at least two of them, but if they turn out to be less than the divisor, then you have to work with the first three digits.

There is one more nuance in this division. The fact is that the remainder and the number added to it are sometimes not divisible by the divisor. Then you have to add another number in order. But the answer must be zero. If you are dividing three-digit numbers into a column, you may need to remove more than two digits. Then a rule is introduced: there should be one less zero in the answer than the number of digits removed.

You can consider this division using the example - 12082: 863.

• The incomplete dividend in it turns out to be the number 1208. The number 863 is placed in it only once. Therefore, the answer is supposed to be 1, and under 1208 write 863.
• After subtraction, the remainder is 345.
• You need to add the number 2 to it.
• The number 3452 contains 863 four times.
• Four must be written down as an answer. Moreover, when multiplied by 4, this is exactly the number obtained.
• The remainder after subtraction is zero. That is, the division is completed.

The answer in the example would be the number 14.

What if the dividend ends in zero?

Or a few zeros? In this case, the remainder is zero, but the dividend still contains zeros. There is no need to despair, everything is simpler than it might seem. It is enough to simply add to the answer all the zeros that remain undivided.

For example, you need to divide 400 by 5. The incomplete dividend is 40. Five fits into it 8 times. This means that the answer should be written as 8. When subtracting, there is no remainder left. That is, the division is completed, but a zero remains in the dividend. It will have to be added to the answer. Thus, dividing 400 by 5 equals 80.

What to do if you need to divide a decimal fraction?

Again, this number looks like a natural number, if not for the comma separating the whole part from the fractional part. This suggests that the division of decimal fractions into a column is similar to that described above.

The only difference will be the semicolon. It is supposed to be put in the answer as soon as the first digit from the fractional part is removed. Another way to say this is this: if you have finished dividing the whole part, put a comma and continue the solution further.

When solving examples of long division with decimal fractions, you need to remember that any number of zeros can be added to the part after the decimal point. Sometimes this is necessary in order to complete the numbers.

Dividing two decimals

It may seem complicated. But only at the beginning. After all, how to divide a column of fractions by a natural number is already clear. This means that we need to reduce this example to an already familiar form.

It's easy to do. You need to multiply both fractions by 10, 100, 1,000 or 10,000, and maybe by a million if the problem requires it. The multiplier is supposed to be chosen based on how many zeros are in the decimal part of the divisor. That is, the result will be that you will have to divide the fraction by a natural number.

And this will be in the worst case scenario. After all, it may happen that the dividend from this operation becomes an integer. Then the solution to the example with column division of fractions will be reduced to the simplest option: operations with natural numbers.

As an example: divide 28.4 by 3.2:

• They must first be multiplied by 10, since the second number has only one digit after the decimal point. Multiplying will give 284 and 32.
• They are supposed to be separated. Moreover, the whole number is 284 by 32.
• The first number chosen for the answer is 8. Multiplying it gives 256. The remainder is 28.
• The division of the whole part has ended, and a comma is required in the answer.
• Remove to remainder 0.
• Take 8 again.
• Remainder: 24. Add another 0 to it.
• Now you need to take 7.
• The result of multiplication is 224, the remainder is 16.
• Take down another 0. Take 5 each and you get exactly 160. The remainder is 0.

The division is complete. The result of example 28.4:3.2 is 8.875.

What if the divisor is 10, 100, 0.1, or 0.01?

Just like with multiplication, long division is not needed here. It is enough to simply move the comma in the desired direction for a certain number of digits. Moreover, using this principle, you can solve examples with both integers and decimal fractions.

So, if you need to divide by 10, 100 or 1,000, then the decimal point is moved to the left by the same number of digits as there are zeros in the divisor. That is, when a number is divisible by 100, the decimal point must move to the left by two digits. If the dividend is a natural number, then it is assumed that the comma is at the end.

This action gives the same result as if the number were to be multiplied by 0.1, 0.01 or 0.001. In these examples, the comma is also moved to the left by a number of digits equal to the length of the fractional part.

When dividing by 0.1 (etc.) or multiplying by 10 (etc.), the decimal point should move to the right by one digit (or two, three, depending on the number of zeros or the length of the fractional part).

It is worth noting that the number of digits given in the dividend may not be sufficient. Then the missing zeros can be added to the left (in the whole part) or to the right (after the decimal point).

Division of periodic fractions

In this case, it will not be possible to obtain an accurate answer when dividing into a column. How to solve an example if you encounter a fraction with a period? Here we need to move on to ordinary fractions. And then divide them according to the previously learned rules.

For example, you need to divide 0.(3) by 0.6. The first fraction is periodic. It converts to the fraction 3/9, which when reduced gives 1/3. The second fraction is the final decimal. It’s even easier to write it down as usual: 6/10, which is equal to 3/5. The rule for dividing ordinary fractions requires replacing division with multiplication and divisor with the reciprocal. That is, the example comes down to multiplying 1/3 by 5/3. The answer will be 5/9.

If the example contains different fractions...

Then several solutions are possible. Firstly, you can try to convert a common fraction to a decimal. Then divide two decimals using the above algorithm.

Secondly, every final decimal fraction can be written as a common fraction. But this is not always convenient. Most often, such fractions turn out to be huge. And the answers are cumbersome. Therefore, the first approach is considered more preferable.

If your child can't seem to figure out how to divide decimals, that's no reason to think he's incapable of math.

Most likely, they simply did not clearly explain to him how this was done. We need to help the child and tell him about fractions and operations with them in the simplest, almost playful way possible. And for this we need to remember something ourselves.

Fractional expressions are used when talking about non-integer numbers. If a fraction is less than one, then it describes a part of something; if it is larger, it describes several whole parts and another piece. Fractions are described by 2 values: a denominator, which explains how many equal parts the number is divided into, and a numerator, which tells us how many such parts we mean.

Let's say you cut the pie into 4 equal parts and gave 1 of them to your neighbors. The denominator will be equal to 4. And the numerator depends on what we want to describe. If we talk about how much was given to neighbors, then the numerator is 1, and if we are talking about how much was left, then 3.

In the pie example, the denominator is 4, and in the expression “1 day - 1/7 week” it is 7. A fraction expression with any denominator is a common fraction.

Mathematicians, like everyone else, try to make their lives easier. And that's why decimal fractions were invented. In them, the denominator is equal to 10 or numbers that are multiples of 10 (100, 1000, 10,000, etc.), and they are written as follows: the integer component of the number is separated from the fractional component by a comma. For example, 5.1 is 5 whole and 1 tenth, and 7.86 is 7 whole and 86 hundredth.

A small retreat is not for your children, but for yourself. It is customary in our country to separate the fractional part with a comma. Abroad, according to an established tradition, it is customary to separate it with a dot. Therefore, if you come across similar markup in a foreign text, do not be surprised.

Division of fractions

Each arithmetic operation with similar numbers has its own characteristics, but now we will try to learn how to divide decimal fractions. It is possible to divide a fraction by a natural number or by another fraction.

To make it easier to master this arithmetic operation, it is important to remember one simple thing.

Once you learn how to use commas, you can use the same division rules as for whole numbers.

Consider dividing a fraction by a natural number. The technology of dividing into a column should already be known to you from previously covered material. The procedure is similar. The dividend is divided sign by sign by the divisor. As soon as the turn reaches the last sign before the comma, a comma is placed in the quotient, and then the division proceeds in the usual manner.

That is, apart from the removal of the comma, this is the most common division, and the comma is not very difficult.

Dividing a fraction by a fraction

Examples where you need to divide one fractional value by another seem very complex. But in fact, they are no more difficult to deal with. Dividing one decimal fraction by another will be much easier if you get rid of the comma in the divisor.

How to do this? If you need to put 90 pencils into 10 boxes, how many pencils will be in each box? 9. Let's multiply both numbers by 10 - 900 pencils and 100 boxes. How many in each? 9. The same principle applies when you need to divide a decimal fraction.

The divisor gets rid of the comma altogether, and the dividend's comma is moved to the right by as many places as there were previously in the divisor. And then the usual division into a column is carried out, which we discussed above. For example:

25,6/6,4 = 256/64 = 4;

10,24/1,6 = 102,4/16 =6,4;

100,725/1,25 =10072,5/125 =80,58.

The dividend must be multiplied and multiplied by 10 until the divisor becomes a whole number. Therefore, it may have extra zeros on the right.

40,6/0,58 =4060/58=70.

There's nothing wrong with that. Remember the example with pencils - the answer will not change if you increase both numbers by the same amount. Common fractions are more difficult to divide, especially when there are no common factors in the numerator and denominator.

Dividing a decimal is much more convenient in this regard. The most difficult trick here is the comma wrapping trick, but as we have seen, it is easy to handle. By being able to convey this to your child, you will be teaching him how to divide decimals.

Having mastered this simple rule, your son or daughter will feel much more confident in mathematics lessons and, who knows, maybe he will become interested in this subject. A mathematical mindset rarely manifests itself from early childhood; sometimes a push and interest are needed.

By helping your child with homework, you will not only improve his academic performance, but also expand his range of interests, for which over time he will be grateful to you.