How to divide hundredths. Drawing up a system of equations

Division by a decimal fraction is reduced to division by a natural number.

The rule for dividing a number by a decimal fraction

To divide a number by a decimal fraction, you need to move the decimal point in both the dividend and the divisor by as many digits to the right as there are in the divisor after the decimal point. After this, divide by a natural number.

Examples.

Divide by decimal fraction:

To divide by a decimal, you need to move the decimal point in both the dividend and the divisor by as many digits to the right as there are after the decimal point in the divisor, that is, by one digit. We get: 35.1: 1.8 = 351: 18. Now we perform the division with a corner. As a result, we get: 35.1: 1.8 = 19.5.

2) 14,76: 3,6

To divide decimal fractions, in both the dividend and the divisor we move the decimal point to the right one place: 14.76: 3.6 = 147.6: 36. Now we perform a natural number. Result: 14.76: 3.6 = 4.1.

To divide a natural number by a decimal fraction, you need to move both the dividend and the divisor to the right as many places as there are in the divisor after the decimal point. Since a comma is not written in the divisor in this case, we fill in the missing number of characters with zeros: 70: 1.75 = 7000: 175. Divide the resulting natural numbers with a corner: 70: 1.75 = 7000: 175 = 40.

4) 0,1218: 0,058

To divide one decimal fraction by another, we move the decimal point to the right in both the dividend and the divisor by as many digits as there are in the divisor after the decimal point, that is, by three decimal places. Thus, 0.1218: 0.058 = 121.8: 58. Division by a decimal fraction was replaced by division by a natural number. We share a corner. We have: 0.1218: 0.058 = 121.8: 58 = 2.1.

5) 0,0456: 3,8

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; 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; 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

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 .

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?

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 more, 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 is 1/7 of a 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 it? 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.

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 if 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.