In the last lesson, we learned how to add and subtract decimal fractions (see the lesson " Adding and subtracting decimal fractions"). At the same time, they estimated how much the calculations are simplified compared to the usual “two-story” fractions.

Unfortunately, with multiplication and division of decimal fractions, this effect does not occur. In some cases, decimal notation even complicates these operations.

First, let's introduce a new definition. We will meet him quite often, and not only in this lesson.

The significant part of a number is everything between the first and last non-zero digit, including the trailers. We are only talking about numbers, the decimal point is not taken into account.

The digits included in the significant part of the number are called significant digits. They can be repeated and even be equal to zero.

For example, consider several decimal fractions and write out their corresponding significant parts:

91.25 → 9125 (significant figures: 9; 1; 2; 5);
0.008241 → 8241 (significant figures: 8; 2; 4; 1);
15.0075 → 150075 (significant figures: 1; 5; 0; 0; 7; 5);
0.0304 → 304 (significant figures: 3; 0; 4);
3000 → 3 (there is only one significant figure: 3).

Please note: zeros inside the significant part of the number do not go anywhere. We have already encountered something similar when we learned to convert decimal fractions to ordinary ones (see the lesson “ Decimal Fractions”).

This point is so important, and errors are made here so often that I will publish a test on this topic in the near future. Be sure to practice! And we, armed with the concept of a significant part, will proceed, in fact, to the topic of the lesson.

Decimal multiplication

The multiplication operation consists of three consecutive steps:

For each fraction, write down the significant part. You will get two ordinary integers - without any denominators and decimal points;
Multiply these numbers in any convenient way. Directly, if the numbers are small, or in a column. We get the significant part of the desired fraction;
Find out where and by how many digits the decimal point is shifted in the original fractions to obtain the corresponding significant part. Perform reverse shifts on the significant part obtained in the previous step.

Let me remind you once again that zeros on the sides of the significant part are never taken into account. Ignoring this rule leads to errors.

0.28 12.5;

6.3 1.08;

132.5 0.0034;

0.0108 1600.5;

5.25 10,000.

We work with the first expression: 0.28 12.5.

Let's write out the significant parts for the numbers from this expression: 28 and 125;
Their product: 28 125 = 3500;
In the first multiplier, the decimal point is shifted 2 digits to the right (0.28 → 28), and in the second - by another 1 digit. In total, a shift to the left by three digits is needed: 3500 → 3.500 = 3.5.

Now let's deal with the expression 6.3 1.08.

Let's write out the significant parts: 63 and 108;
Their product: 63 108 = 6804;
Again, two shifts to the right: by 2 and 1 digits, respectively. In total - again 3 digits to the right, so the reverse shift will be 3 digits to the left: 6804 → 6.804. This time there are no zeros at the end.

We got to the third expression: 132.5 0.0034.

Significant parts: 1325 and 34;
Their product: 1325 34 = 45,050;
In the first fraction, the decimal point goes to the right by 1 digit, and in the second - by as many as 4. Total: 5 to the right. We perform a shift by 5 to the left: 45050 → .45050 = 0.4505. Zero was removed at the end, and added to the front so as not to leave a “bare” decimal point.

The following expression: 0.0108 1600.5.

We write significant parts: 108 and 16 005;
We multiply them: 108 16 005 = 1 728 540;
We count the numbers after the decimal point: in the first number there are 4, in the second - 1. In total - again 5. We have: 1,728,540 → 17.28540 = 17.2854. At the end, the “extra” zero was removed.

Finally, the last expression: 5.25 10,000.

Significant parts: 525 and 1;
We multiply them: 525 1 = 525;
The first fraction is shifted 2 digits to the right, and the second fraction is shifted 4 digits to the left (10,000 → 1.0000 = 1). Total 4 − 2 = 2 digits to the left. We perform a reverse shift by 2 digits to the right: 525, → 52 500 (we had to add zeros).

Pay attention to the last example: since the decimal point moves in different directions, the total shift is through the difference. This is very important point! Here's another example:

Consider the numbers 1.5 and 12,500. We have: 1.5 → 15 (shift by 1 to the right); 12 500 → 125 (shift 2 to the left). We “step” 1 digit to the right, and then 2 digits to the left. As a result, we stepped 2 − 1 = 1 digit to the left.

Decimal division

Division is perhaps the most difficult operation. Of course, here you can act by analogy with multiplication: divide the significant parts, and then “move” the decimal point. But in this case, there are many subtleties that negate the potential savings.

So let's look at a generic algorithm that is a little longer, but much more reliable:

Convert all decimals to common fractions. With a little practice, this step will take you a matter of seconds;
Divide the resulting fractions in the classical way. In other words, multiply the first fraction by the "inverted" second (see the lesson " Multiplication and division of numerical fractions");
If possible, return the result as a decimal. This step is also fast, because often the denominator already has a power of ten.

Task. Find the value of the expression:

3,51: 3,9;

1,47: 2,1;

6,4: 25,6:

0,0425: 2,5;

0,25: 0,002.

We consider the first expression. First, let's convert obi fractions to decimals:

We do the same with the second expression. The numerator of the first fraction is again decomposed into factors:

There is an important point in the third and fourth examples: after getting rid of the decimal notation, cancellable fractions appear. However, we will not perform this reduction.

The last example is interesting because the numerator of the second fraction is a prime number. There is simply nothing to factorize here, so we consider it “blank through”:

Sometimes division results in an integer (I'm talking about the last example). In this case, the third step is not performed at all.

In addition, when dividing, “ugly” fractions often appear that cannot be converted to decimals. This is where division differs from multiplication, where the results are always expressed in decimal form. Of course, in this case, the last step is again not performed.

Pay also attention to the 3rd and 4th examples. In them, we deliberately do not reduce common fractions derived from decimals. Otherwise, it will complicate the inverse problem - representing the final answer again in decimal form.

Remember: the basic property of a fraction (like any other rule in mathematics) in itself does not mean that it must be applied everywhere and always, at every opportunity.

Division by decimal comes down to dividing by natural number.

Rule for dividing a number by a decimal fraction

To divide a number by a decimal fraction, it is necessary both in the dividend and in the divisor to move the comma as many digits to the right as there are in the divisor after the decimal point. After that, divide by a natural number.

Examples.

Perform division by decimal:

To divide by a decimal fraction, you need to move the comma as many digits to the right in both the dividend and the divisor as there are after the decimal point in the divisor, that is, by one sign. We get: 35.1: 1.8 \u003d 351: 18. Now we perform division by a corner. As a result, we get: 35.1: 1.8 = 19.5.

2) 14,76: 3,6

To perform the division of decimal fractions, both in the dividend and in the divisor, move the comma to the right by one sign: 14.76: 3.6 \u003d 147.6: 36. Now we perform on a natural number. Result: 14.76: 3.6 = 4.1.

To perform division by a decimal fraction of a natural number, it is necessary both in the dividend and in the divisor to move as many characters to the right as there are in the divisor after the decimal point. Since the comma is not written in the divisor in this case, we fill in the missing number of characters with zeros: 70: 1.75 \u003d 7000: 175. We divide the resulting natural numbers with a corner: 70: 1.75 \u003d 7000: 175 \u003d 40.

4) 0,1218: 0,058

To divide one decimal fraction into another, we move the comma to the right both in the dividend and in the divisor by as many digits as there are in the divisor after the decimal point, that is, by three digits. Thus, 0.1218: 0.058 \u003d 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

37. Decimal division

Task. The area of the rectangle is 2.88 dm 2 and its width is 0.8 dm. What is the length of the rectangle?

Solution. Since 2.88 dm 2 \u003d 288 cm 2, and 0.8 dm \u003d 8 cm, the length of the rectangle is 288: 8, that is, 36 cm \u003d 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.

The answer 3.6 can be obtained without converting decimeters to centimeters. To do this, multiply the divisor 0.8 and the dividend 2.88 by 10 (that is, move the comma one digit to the right in them) 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 that perform division by a natural number.

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

Since , then and .

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

Since , then and .

From examples 1 and 2 it can be seen that when dividing a number by an improper fraction, this number decreases or does not change, and when divided 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 get that the quotient is 246.7: 1, that is, 246.7. Hence, and 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 in front of the unit 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 test by multiplication:

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

1444. Find the quotient and test 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 man's step is 0.8 m. How many steps does he need to take to walk a distance of 100 m?

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

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 included 6.72 kg of flour, which is 2.4 times more than the second package. How many kilograms of flour were included in both bags?

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

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

Examples.

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

Solution.

Example 1) 96,25: 5.

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

Answer: 19,25.

Example 2) 4,78: 4.

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

Answer: 1,195.

Example 3) 183,06: 45.

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

Answer: 4,068.

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

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

Examples.

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

Solution.

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

Rectangle?

Solution. Since 2.88 dm2 \u003d 288 cm2, and 0.8 dm \u003d 8 cm, the length of the rectangle is 288: 8, that is, 36 cm \u003d 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.

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

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 that perform division by a natural number.

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

From examples 1 and 2, it can be seen that when a number is divided by an improper fraction, this number decreases or does not change, and when divided by a proper decimal fraction, it increases: 12.096\u003e 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 get that the quotient is 246.7: 1, that is, 246.7.

Hence, and 2.467: 0.01 = 246.7. From here we get the rule:

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

If there are not enough numbers, you must first attribute 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.
What number can be multiplied to replace division by 0.01?

1443. Find the quotient and test by multiplication:

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

1444. Find the quotient and test 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) 4.2p - p = 5.12;
b) (y + 26.1) 2.3 = 70.84; f) 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 + m = 9.9; h) 9k - 8.67k = 0.6699.

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

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

1462. A kangaroo is 2.4 times lower than a giraffe, and a giraffe is 2.52 m higher 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 in 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. Do the following:

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;
f) 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; f) 0.01 100; i) 0.3 0.3 0.3.

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

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

1470. Think about which of the numbers can be exact, which are 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;
f) 0.5 kg of flour in a bag;
g) the area of the island of Cuba is 105,000 km2;
h) there are 10,000 books in the school library;
i) one span is equal to 4 vershoks, and an vershok is equal to 4.45 cm (vershok
the 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 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 left the same village in the opposite direction,
and the speed of the second is 1.25 times the speed of the first. What is the distance between them 3.3 hours after the second cyclist leaves?

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

1477. The plant manufactured 3.75 thousand parts and sold them at a price of 950 rubles. a piece. The cost of the plant for the manufacture of one part amounted to 637.5 rubles. Find the profit made 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 box and round your answer to the nearest integer.

1479. Pope Carlo promised to give Piero 4 soldi every day, and Pinocchio 1 soldi on the first day, and 1 soldi more every next 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 solido in total as Pierrot. Think about whether Pinocchio is right.

1480. 231 m of boards went to 3 cabinets and 9 bookshelves, and 4 times more material goes to the cabinet than to the shelf. How many meters of boards go to the cabinet and how many - to the shelf?

1481. Solve the problem:
1) The first number is 6.3 and is the second number. The third number is 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 value of the expression:

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

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

1483. Find the value of the private:

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 path 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 m 2, 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 use in 1.8 hours?
1487. A metal part with a volume of 3.5 dm3 has a mass of 27.3 kg. Another item 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 the tank through two pipes. 3.6 tons of gasoline per hour came through the first pipe, and it was open for 0.4 hours. 0.8 tons of gasoline less than through the first pipe came in per hour through the second pipe. 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 - 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 equal to 6.8 km. The speed of one pedestrian was 1.5 times the speed of the other. Find the speed of each pedestrian.

1492. Do the following:

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 children can he give injections if each injection requires 0.002 kg of serum?

1494. 2.8 tons of gingerbread were brought 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 off from a piece of cloth. How many meters of cloth 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 educational institutions

Decimal multiplication

Decimal division

37. Decimal division

You may be interested