The rule for dividing decimal fractions by natural numbers.
Four identical toys cost 921 rubles 20 kopecks in total. How much does one toy cost (see Fig. 1)?
Rice. 1. Illustration for the problem
Decision
To find the cost of one toy, you need to divide this amount by four. Let's convert the amount to kopecks:
Answer: the cost of one toy is 23,030 kopecks, that is, 230 rubles 30 kopecks, or 230.3 rubles.
You can solve this problem without converting rubles into kopecks, that is, dividing decimal on the natural number: .
To divide a decimal fraction by a natural number, you need to divide the fraction by this number, as natural numbers are divided, and put in a private comma when the division of the whole part is over.
We divide in a column as we divide natural numbers. After we demolish the number 2 (the number of tenths is the first digit after the decimal point in the record of the dividend 921.20), put a comma in the quotient and continue the division:
Answer: 230.3 rubles.
We divide in a column as we divide natural numbers. After we take down the number 6 (the number of tenths is the number after the decimal point in the record of the dividend 437.6), put a comma in the quotient and continue the division:
If the dividend is less than the divisor, then the quotient will start from zero.
1 is not divisible by 19, so we put zero in the quotient. The division of the integer part is over, in the private we put a comma. We demolish 7. 17 is not divisible by 19, in private we write zero. We demolish 6 and continue the division:
We divide as we divide natural numbers. In the quotient, we put a comma as soon as we take down 8 - the first digit after the decimal point in the dividend 74.8. Let's continue the division. When subtracting, we get 8, but the division is not over. We know that zeros can be added at the end of a decimal fraction - this will not change the value of the fraction. We assign zero and divide 80 by 10. We get 8 - the division is over.
To divide a decimal fraction by 10, 100, 1000, etc., you need to move the comma in this fraction as many digits to the left as there are zeros after one in the divisor.
In this lesson, we learned how to divide a decimal fraction by a natural number. We considered a variant with an ordinary natural number, as well as a variant in which division by a bit unit occurs (10, 100, 1000, etc.).
Solve the equations:
To find unknown divisor, it is necessary to divide the dividend by the quotient. I.e .
We divide into a column. After we demolish the number 4 (the number of tenths is the first digit after the decimal point in the record of the dividend 134.4), put a comma in the quotient and continue the division:
each part.
Decision. To solve the problem, let's express the length of the tape in decimeters: 19.2 m = 192 dm. But 192: 8 = 24. Hence, the length of each part is 24 dm,
that is, 2.4 m. If we multiply 2.4 by 8, we get 19.2. So 2.4 is the quotient of 19.2 divided by 8.
They write: 19.2: 8 = 2.4.
The same answer can be obtained without converting meters to decimeters. To do this, you need to divide 19.2 by 8, ignoring the comma, and put a comma in the quotient when the division of the whole part ends:
To divide a decimal fraction by a natural number means to find a fraction that, when multiplied by this natural number, gives the dividend.
To divide a decimal by a natural number, you need:
1) divide the fraction by this number, ignoring the comma;
2) put a comma in the private when the division of the whole part ends;
If the integer part is less than the divisor, then the quotient starts from zero integers:
Divide 96.1 by 10. If you multiply the quotient by 10, you should get 96.1 again.
In other words, with the help of division, an ordinary fraction is converted into a decimal.
Example. Let's convert the fraction to a decimal.
Decision. The fraction is the quotient of 3 divided by 4. Dividing 3 by 4, we get the decimal fraction 0.75. Hence, = 0.75.
What does it mean to divide a decimal by a natural number?
How do you divide a decimal by a natural number?
How to divide a decimal by 10, 100, 1000?
How to convert a common fraction to a decimal?
1340. Perform division:
a) 20.7: 9;
b) 243.2: 8;
c) 88.298: 7;
d) 772.8: 12;
e) 93.15: 23;
e) 0.644: 92;
g) 1: 80;
h) 0.909: 45;
i) 3:32;
j) 0.01242: 69;
k) 1.016: 8;
m) 7.368: 24.
1341. 3 tractors, weighing 1.2 tons each, and 7 snowmobiles were loaded into the plane for the polar expedition. The mass of all snowmobiles is 2 tons more than the mass of tractors. What is the mass of one aerosleigh?
a) 4x - x = 8.7; c) a + a + 8.154 = 32;
b) Zu + bу = 9.6; d) 7k - 4k - 55.2 = 63.12.
1349. Two baskets contain 16.8 kg of tomatoes. There are twice as many tomatoes in one basket as in the other. How many kilograms of tomatoes are in each basket?
1350. The area of the first field is 5 times more area second. What is the area of each field if square second on 23.2 ha less area first?
1351. For the preparation of compote, a mixture was made of 8 parts (by weight) of dry apples, 4 parts of apricots and 3 parts of raisins. How many kilograms of each of the dried fruits were needed for 2.7 kg of such a mixture?
1352. In two bags 1.28 centners of flour. In the first bag there is 0.12 centners more flour than in the second. How many quintals of flour are in each sack?
1353. There are 18.6 kg of apples in two baskets. There are 2.4 kg fewer apples in the first basket than in the second. How many kilograms of apples are in each basket?
1354. Express as a decimal fraction:
1355. To collect 100 g of honey, a bee delivers 16,000 loads of nectar to the hive. What is one load of nectar?
1356. There is 30 g of medicine in a vial. Find the mass of one drop of medicine if there are 1500 drops in the vial.
1357. Convert a common fraction to a decimal and do the following:
1358. Solve the equation:
a) (x - 5.46) -2 = 9;
b) (y + 0.5): 2 = 1.57.
1359. Find the value of the expression:
a) 91.8: (10.56 - 1.56) + 0.704; e) 15.3 -4:9 + 3.2;
b) (61.5 - 5.16): 30 + 5.05; f) (4.3 + 2.4: 8) 3;
c) 66.24 - 16.24: (3.7 + 4.3); g) 280.8: 12 - 0.3 24;
d) 28.6 + 11.4: (6.595 + 3.405); h) (17.6 13 - 41.6): 12.
1360. Calculate orally:
a) 2.5 - 1.6; b) 1.8 + 2.5; c) 3.4 - 0.2; d) 5 + 0.35;
3,2 - 1,4; 2,7 + 1,6; 2,6 - 0,05; 3,7 + 0,24;
0,47 - 0,27; 0,63 + 0,17; 4,52 - 1,2; 0,46 + 1,8;
0,64-0,15; 0,38 + 0,29; 4-0,8; 0,57 + 3;
0,71 - 0,28; 0,55 + 0,45; 1 - 0,45; 1,64 + 0,36.
a) 0.3 2; d) 2.3 3; g) 3.7 10; i) 0.185;
b) 0.8 3; e) 0.214; h) 0.096; j) 0.87 0.
c) 1.2 2; e) 1.6 5;
1362. Guess what are the roots of the equation:
a) 2.9x = 2.9; c) 3.7x = 37; e) a 3 \u003d a;
b) 5.25x = 0; d) x 2 \u003d x e) m 2 \u003d m 3.
1363. How will the value of the expression 2.5a change if a: is increased by 1? increase by 2? double up?
1364. Tell us how to mark the number on the coordinate ray: 0.25; 0 5; 0.75. Think about which of the given numbers are equal. What fraction with denominator 4 is equal to 0.5? Add up: 
1365. Think about the rule by which a series of numbers is composed, and write down two more numbers of this series:
a) 1.2; 1.8; 2.4; 3; ... c) 0.9; 1.8; 3.6; 7.2; ...
b) 9.6; 8.9; 8.2; 7.5; ... d) 1.2; 0.7; 2.2; 1.4; 3.2; 2.1; ...
1366. Follow these steps:
a) (37.8 - 19.1) 4; c) (64.37 + 33.21 - 21.56) 14;
b) (14.23 + 13.97) 31; d) (33.56 - 18.29) (13.2 + 24.9 - 38.1).
a) 3.705; 62.8; 0.5 to 10 times;
b) 2.3578; 0.0068; 0.3 100 times.
1368. Round the number 82,719.364:
a) up to units; c) up to tenths; e) up to thousands.
b) up to hundreds; d) up to hundredths;
1369. Take action:
1370. Compare:
1371. Kolya, Petya, Zhenya and Senya weighed on the scales. The results were: 37.7 kg; 42.5 kg; 39.2 kg; 40.8 kg. Find the mass of each boy if it is known that Kolya is heavier than Senya and lighter than Petya, and Zhenya is lighter than Senya.
1372. Simplify the expression and find its value:
a) 23.9 - 18.55 - mt if m = 1.64;
b) 16.4 + k + 3.8 if k = 2.7.
1373. Solve the equation:
a) 16.1 - (x - 3.8) = 11.3;
b) 25.34 - (2.7 + y) = 15.34.
1374. Find the value of the expression:
1) (1070 - 104 040: 2312) 74 + 6489;
2) (38 529 + 205 87) : 427 - 119.
1375. Perform division:
a) 53.5: 5; e) 0.7: 25; i) 9.607: 10;
b) 1.75: 7; e) 7.9: 316; j) 14.706: 1000;
c) 0.48: 6; g) 543.4: 143; k) 0.0142: 100;
d) 13.2: 24; h) 40.005: 127; m) 0.75: 10,000.
1376. The car walked along the highway for 3 hours at a speed of 65.8 km / h, and then for 5 hours it walked along a dirt road. At what speed did she walk along the dirt road if her entire path is 324.9 km?
1377. There were 180.4 tons of coal in the warehouse. This coal was supplied for heating schools. How many tons of coal are left in the warehouse?
1378. Plowed fields. Find the area of this field if 32.5 hectares were plowed.
1379. Solve the equation:
a) 15x = 0.15; e) 8p - 2p - 14.21 = 75.19;
b) 3.08: y = 4; g) 295.1: (n - 3) = 13;
c) Za + 8a = 1.87; h) 34 (m + 1.2) = 61.2;
d) 7z - 3z = 5.12; i) 15 (k - 0.2) = 21.
e) 2t + 5t + 3.18 = 25.3;
1380. Find the value of the expression:
a) 0.24: 4 + 15.3: 5 + 12.4: 8 + 0.15: 30;
b) (1.24 + 3.56): 16;
c) 2.28 + 3.72: 12;
d) 3.6 4-2.4: (11.7 - 3.7).
1381. 19.7 tons of hay were collected from three meadows. Hay was harvested equally from the first and second meadows, and hay was harvested from the third by 1.1 tons more than from each of the first two. How much hay was harvested from each meadow?
1382. The store sold 1240.8 kg of sugar in 3 days. On the first day, 543 kg were sold, on the second - 2 times more than on the third. How many kilograms of sugar were sold on the third day?
1383. The car passed the first section of the path in 3 hours, and the second section - in 2 hours. The length of both sections together is 267 km. What was the speed of the car in each section if the speed in the second section was 8.5 km / h more than in the first?
1384. Convert to decimal fractions;
1385. Build a figure equal to the figure shown in Figure 151.
1386. A cyclist left the city at a speed of 13.4 km / h. After 2 hours, another cyclist followed him, whose speed was 17.4 km / h. Through
how many hours after his departure will the second cyclist catch up with the first?
1387. The boat, moving against the current, traveled 177.6 km in 6 hours. Find the own speed of the boat if the speed of the current is 2.8 km/h.
1388. A tap that delivers 30 liters of water per minute fills a bath in 5 minutes. Then the tap was closed and a drain hole was opened, through which all the water poured out in b minutes. How many liters of water were poured out in 1 minute?
1389. Solve the equation:
a) 26 (x + 427) = 15 756; c) 22 374: (k - 125) = 1243;
b) 101 (351 + y) = 65 549; d) 38 007: (4223 - t) = 9.
N.Ya. VILENKIN, V. I. ZHOKHOV, A. S. CHESNOKOV, S. I. SHVARTSBURD, Mathematics Grade 5, Textbook for educational institutions
Mathematics video download, homework help teachers and students
You know that dividing a natural number a by a natural number b means finding a natural number c that, when multiplied by b, gives the number a. This statement remains true if at least one of the numbers a, b, c is a decimal fraction.
Consider several examples in which the divisor is a natural number.
1.2: 4 \u003d 0.3, since 0.3 * 4 \u003d 1.2;
2.5: 5 \u003d 0.5, since 0.5 * 5 \u003d 2.5;
1 : 2 = 0.5 since 0.5 * 2 = 1.
But what about in cases where the division cannot be performed orally?
For example, how do you divide 43.52 by 17?
Increasing the dividend 43.52 by 100 times, we get the number 4352. Then the value of the expression 4352:17 is 100 times greater than the value of the expression 43.52:17. After dividing with a corner, you can easily establish that 4352: 17 = 256. Here the dividend is magnified 100 times. So, 43.52: 17 = 2.56. Note that 2.56 * 17 = 43.52 , which confirms that the division is correct.
The quotient 2.56 can be obtained differently. We will divide 4352 by 17 corners, ignoring the comma. In this case, the comma in the private should be placed immediately before the first digit after the decimal point in the dividend is used:
If the dividend is less than the divisor, then the integer part of the quotient is zero. For example:
Let's consider one more example. Let's find the quotient 3.1:5. We have:
We stopped the division process because the digits of the dividend ran out, and we didn’t get zero in the remainder. You know that a decimal does not change if you add any number of zeros to the right of it. Then it becomes clear that the numbers of the dividend cannot end. We have:
Now we can find the quotient of two natural numbers when the dividend is not evenly divisible by the divisor. For example, let's find the quotient 31:5. Obviously, the number 31 is not divisible by 5:
We stopped the division process because the numbers of the dividend are over. However, if you represent the dividend as a decimal fraction, then the division can be continued.
We have: 31: 5 \u003d 31.0: 5. Next, let's do the division by a corner:
Therefore, 31: 5 = 6.2.
In the previous paragraph, we found out that if the comma is moved to the right by 1, 2, 3, etc. numbers, then the fraction will increase, respectively, by 10, 100, 1,000, etc. times, and if the comma is moved to the left by 1, 2, 3, etc. figures, then the fraction will decrease, respectively, by 10, 100, 1,000 and etc. times.
Therefore, in cases where the divisor is 10, 100, 1,000, etc., the following rule is used.
To divide a decimal by 10, 100, 1,000, etc., you need to move the decimal point to the left in this fraction by 1, 2, 3, etc. digits.
For example: 4.23: 10 = 0.423; 2: 100 = 0.02; 58.63 : 1000 = 0.05863 .
So, we have learned how to divide a decimal fraction by a natural number.
Let us show how division by a decimal fraction can be reduced to division by a natural number.
$\frac(2)(5) km = 400 m$
,$\frac(20)(50) km = 400 m$
,$\frac(200)(500) km = 400 m$
.We get that
$\frac(2)(5) = \frac(20)(50) = \frac(200)(500)$
Those. 2:5 = 20:50 = 200:500.
This example illustrates the following: if the dividend and divisor are increased simultaneously by 10, 100, 1,000, etc. times, then the quotient will not change .
Let's find the quotient 43.52: 1.7.
Let's increase both the dividend and the divisor by 10 times. We have:
43,52 : 1,7 = 435,2 : 17 .
Let's increase both the dividend and the divisor by 10 times. We have: 43.52: 1.7 = 25.6.
To divide a decimal by a decimal:
1) move the commas in the dividend and in the divisor to the right by as many digits as they are contained after the decimal point in the divisor;
2) perform division by a natural number.
Example 1 . Vanya collected 140 kg of apples and pears, of which 0.24 were pears. How many kilograms of pears did Vanya collect?
Decision. We have:
$0.24=\frac(24)(100)$
.1) 140 : 100 = 1.4 (kg) - is
Apples and pears.
2) 1.4 * 24 = 33.6 (kg) - pears were harvested.
Answer: 33.6 kg.
Example 2 . For breakfast, Winnie the Pooh ate 0.7 barrels of honey. How many kilograms of honey were in the barrel if Winnie the Pooh ate 4.2 kg?
Decision. We have:
$0.7=\frac(7)(10)$
.1) 4.2: 7 = 0.6 (kg) - is
Whole honey.
2) 0.6 * 10 = 6 (kg) - there was honey in the barrel.
Answer: 6 kg.
§ 107. Addition of decimal fractions.Adding decimals is done in the same way as adding whole numbers. Let's see this with examples.
1) 0.132 + 2.354. Let's sign the terms one under the other.
Here, from the addition of 2 thousandths with 4 thousandths, 6 thousandths were obtained;
from the addition of 3 hundredths with 5 hundredths, it turned out 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 signing the terms under each other.
3) 1,2357 + 0,469 + 2,08 + 3,90701.
Here, when adding thousandths, we get 21 thousandths; we wrote 1 under the thousandths, and 2 added to the hundredths, so in the hundredth place we got the following terms: 2 + 3 + 6 + 8 + 0; in sum, they give 19 hundredths, we signed 9 under hundredths, and 1 was counted as tenths, etc.
Thus, when adding decimal fractions, the following order must be observed: fractions are signed one under the other so that in all terms the same digits are under each other and all commas are in the same vertical column; to the right of the decimal places of some terms, they attribute, at least mentally, such a number of zeros so that all terms after the decimal point have the same number digits. Then perform the addition by digits, starting with right side, and in the resulting amount put a comma in the same vertical column in which it is in these terms.
§ 108. Subtraction of decimal fractions.
Subtracting decimals is done in 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 the 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, one 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 refer to the nearest digit to the left, i.e., to hundred-thousandths, but there is also zero in place of hundred-thousandths, so we take 1 ten-thousandth from 3 ten-thousandths and we split it into hundred-thousandths, we get 10 hundred-thousandths, of which 9 hundred-thousandths are left in the category of hundred-thousandths, and 1 hundred-thousandth is crushed 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 (not to forget), these numbers are written on top of the corresponding fractional digits of the reduced. Now we can start subtracting. We subtract 2 millionths from 10 millionths, we get 8 millionths; subtract 1 hundred-thousandth from 9 hundred-thousandths, we get 8 hundred-thousandths, etc.
Thus, when subtracting decimal fractions, the following order is observed: the subtrahend is signed under the reduced so that the same digits are one under the other and all the commas are in the same vertical column; on the right, they attribute, at least mentally, in the reduced or subtracted so many zeros so that they have the same number of digits, then subtract by digits, starting from the right side, and in the resulting difference put a comma in the same vertical column in which it is located in reduced and subtracted.
§ 109. Multiplication of decimal fractions.
Consider a few 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 is increased several times, the product increases by the same amount. This means that the number that results from multiplying integer factors, i.e. 28 by 23, is 10 times greater than the true product, and in order to get the true product, you need to reduce the found product by 10 times. Therefore, here you have to perform a multiplication by 10 once and a division by 10 once, but multiplication and division by 10 is performed by moving the comma to the right and left by one sign. Therefore, you need to do this: in the multiplier, move the comma to the right by one sign, from this it will be 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 character 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 an 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 by decimal fractions. But here, in the process of multiplication, we will not pay attention to commas, that is, we will temporarily increase the multiplier by 100 times, and the multiplier by 1,000 times, which will increase the product by 100,000 times. Thus, multiplying 1227 by 21, we get:
1 227 21 = 25 767.
Considering that the resulting product is 100,000 times the true product, we must now reduce it by a factor of 100,000 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 integers and in the product to separate with a comma on the right side as many decimal places as there were in the multiplicand and in the factor together.
AT last example product with five decimal places. If such greater accuracy is not required, then rounding of the decimal fraction is done. When rounding, you should use the same rule that 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 of 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 of 53 by 15, but our second factor was 10 times less, which means that 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 2491. But since we increased the multiplicand and the multiplier by a total of 100 times, then the resulting product is 100 times larger than it should be; so we have to reduce this product by a factor of 100:
5,3 4,7 = 24,91.
3) Multiply 0.53 by 7.4.
First we find in the table the product of 53 by 74; it will be 3,922. But since we have increased the multiplier by 100 times, and the multiplier by 10 times, the product has increased by 1,000 times; so we now have to reduce it by a factor of 1,000:
0,53 7,4 = 3,922.
§ 111. Division of decimals.
We will look at decimal division in this order:
1. Decimal division by integer,
1. Division of a decimal fraction by an integer.
1) Divide 2.46 by 2.
We divided by 2 first integers, then tenths and finally hundredths.
2) Divide 32.46 by 3.
32,46: 3 = 10,82.
We divided 3 tens by 3, then we 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 demolished 4 tenths and divided 24 tenths by 3; received in private 8 tenths and finally divided 6 hundredths.
3) Divide 1.2345 by 5.
1,2345: 5 = 0,2469.
Here, in the quotient in the first place, zero integers turned out, 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 private, then a remainder equal to 2 hundredths was found, we split this remainder into thousandths, got 20 thousandths and brought the division to the end.
Rule. The division of a decimal fraction by an integer is carried out in the same way as the division of integers, and the resulting remainders are converted into decimal fractions, more and more small; division continues until the remainder is zero.
2. Division of a decimal fraction by a decimal fraction.
1) Divide 2.46 by 0.2.
We already know how to divide a decimal fraction by an integer. Let's think about whether this new case of division can also be reduced to the previous one? At one time, we considered the remarkable property of the quotient, which consists in the fact that it remains unchanged while increasing or decreasing the dividend and divisor by the same number of times. We would easily perform the division of the numbers offered to us if the divisor were an integer. To do this, it is enough to increase it 10 times, and to obtain the correct quotient, it is necessary to increase the dividend by the same number of times, that is, 10 times. Then the division of these numbers will be replaced by the division of such numbers:
and there is no need to make any amendments in private.
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 quotient 0 and divide 125 tenths by 16, we get 7 tenths in quotient and the remainder is 13. We split 13 tenths into hundredths by assigning zero and divide 130 hundredths by 16, etc. Pay attention to the following:
a) when integers are not obtained in the quotient, then zero integers are written in their place;
b) when, after taking 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 the removal of the last digit of the dividend, the division does not end, then, by assigning zeros to the remainders, the division continues;
d) if the dividend is an integer, then when dividing it by a decimal fraction, its increase is carried out by assigning zeros to it.
Thus, in order to divide a number by a decimal fraction, you need to drop a comma in the divisor, and then increase the dividend as many times as the divisor increased when the comma was dropped in it, and then perform the division according to the rule of dividing the decimal fraction by an integer.
§ 112. Approximate quotient.
In the previous paragraph, we considered the division of decimal fractions, and in all the examples we solved, the division was brought to the end, i.e., an exact quotient was obtained. However, in most cases the exact quotient cannot be obtained, no matter how far we extend 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 the numbers that we have met before begin to appear in the remainder. Numbers will also be repeated in the quotient: obviously, after the number 7, the number 5 will appear, then 2, and so on without end. In such cases, division is interrupted and limited to the first few digits of the quotient. This private is called approximate. How to perform division in this case, we will show with examples.
Let it be required to divide 25 by 3. It is obvious that the exact quotient, expressed as an integer or 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 of 1. To get the exact quotient, you need to add to the found approximate quotient, that is, to 8, the fraction that results from dividing the remainder, equal to 1, by 3; it will be a fraction 1/3. So the exact quotient is expressed mixed number 8 1 / 3 . Since 1/3 is proper fraction, i.e. fraction, less than one, then, discarding it, we assume error, which less than one. Private 8 will approximate quotient up to one with a disadvantage. If we take 9 instead of 8, then we also allow an error that is less than one, since we will add not a whole unit, but 2 / 3. Such a private will approximate quotient up to one with an excess.
Let's take another example now. Let it be required to divide 27 by 8. Since here we will not 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 3 / 8 , it is less than one, which means that the approximate quotient (3) is found up to one with a drawback. We continue the division: we split the remainder of 3 into tenths, we get 30 tenths; Let's divide them by 8.
We got in private on the spot tenths 3 and in the remainder b tenths. If we confine ourselves to the number 3.3 in particular, and discard the remainder 6, then we will allow an error 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; from this division would be 6/80, which is less than one tenth. (Check!) Thus, if we limit ourselves to tenths in the quotient, then we can say that we have found the quotient accurate to one tenth(with disadvantage).
Let's continue the division to find one more decimal place. To do this, we split 6 tenths into hundredths and get 60 hundredths; Let's divide them by 8.
In private in third place it turned out 7 and in the remainder 4 hundredths; if we discard them, then we allow an error of less than one hundredth, because 4 hundredths divided by 8 is less than one hundredth. In such cases, the quotient is said to be found. accurate to one hundredth(with disadvantage).
In the example that we are now considering, you 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 consider such an example:
40: 7 = 5,71428571...
The dots at the end of the number indicate that the division is not completed, that is, the equality is approximate. Usually approximate equality is written like this:
40: 7 = 5,71428571.
We took the quotient with eight decimal places. But if such great precision is not required, one can confine oneself to the whole part of the quotient, i.e., the number 5 (more precisely, 6); for greater accuracy, tenths could be taken into account and the quotient taken equal to 5.7; if for some reason this accuracy is insufficient, then we can stop at hundredths and take 5.71, etc. Let's write out the individual quotients and name them.
The first approximate quotient up to one 6.
The second » » » to one tenth 5.7.
Third » » » up to one hundredth 5.71.
Fourth » » » up to one thousandth of 5.714.
Thus, in order to find an approximate quotient with an accuracy of some, for example, the 3rd decimal place (i.e., up to one thousandth), division is stopped as soon as this sign is found. In this case, one must remember the rule set forth in § 40.
§ 113. The simplest problems for interest.
After studying decimal fractions, we will solve a few more percentage problems.
These problems are similar to those we solved in the department of ordinary fractions; 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 switch from common fraction to a decimal with a denominator of 100. To do this, divide the numerator by the denominator:
The table below shows how a number with a % (percentage) symbol is replaced by a decimal with a denominator of 100:
Let's now consider a few problems.
1. Finding percentages given number.
Task 1. Only 1,600 people live in one village. Number of children school age accounts for 25% of the total population. How many school-age children are 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.
For a clear understanding of 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 give depositors 2% of income annually. How much income per year will be received by a depositor who has deposited: a) 200 rubles? b) 500 rubles? c) 750 rubles? d) 1000 rubles?
In all four cases, to solve the problem, it will be necessary 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 (rubles),
b) 500 0.02 = 10 (rubles),
c) 750 0.02 = 15 (rubles),
d) 1,000 0.02 = 20 (rubles).
Each of these cases can be verified by the following considerations. Savings banks give depositors 2% of income, that is, 0.02 of the amount put into savings. If the amount were 100 rubles, then 0.02 of it would be 2 rubles. This means that every hundred brings the depositor 2 rubles. income. Therefore, in each of the cases considered, it is enough to figure out how many hundreds are in a given number, and multiply 2 rubles by this number of hundreds. In example a) hundreds of 2, so
2 2 \u003d 4 (rubles).
In example d) hundreds are 10, which means
2 10 \u003d 20 (rubles).
2. Finding a number by its percentage.
Task 1. In the spring, the school graduated 54 students, which is 6% of the total number of students. How many students were in the school during the last academic year?
Let us first clarify the meaning of this problem. 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 in 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 whole number. Thus, before us is an ordinary problem of finding a number by its fraction (§ 90 p. 6). Problems of this type are solved by division:
This means that there were 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 mind, solve the problem of the first type (finding the percentage of a given number): take the found number (900) as given and find the percentage indicated in the solved problem from it , 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 income. 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 the desired is the entire earnings:
780: 0,65 = 1 200.
Therefore, the desired earnings is 1200 rubles.
3. Finding the percentage of numbers.
Task 1. The school library has a total of 6,000 books. 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 task, we need to find the percentage of the numbers 1,200 and 6,000.
We 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, we 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.
Division by a decimal is the same as division by a 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, we 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