This topic will consider both a general scheme for comparing decimal fractions and a detailed analysis of the principle of comparing finite and infinite fractions. Let us fix the theoretical part by solving typical problems. We will also analyze with examples the comparison of decimal fractions with natural or mixed numbers, and ordinary fractions.
Let's make a clarification: in the theory below, only positive decimal fractions will be compared.
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General principle for comparing decimal fractions
For each finite decimal and infinite recurring decimal fraction, there are certain common fractions corresponding to them. Therefore, the comparison of finite and infinite periodic fractions can be made as a comparison of their corresponding ordinary fractions. Actually, this statement is the general principle for comparing decimal periodic fractions.
Based on the general principle, the rules for comparing decimal fractions are formulated, adhering to which it is possible not to convert the compared decimal fractions into ordinary ones.
The same can be said about the cases when a periodic decimal fraction is compared with natural numbers or mixed numbers, ordinary fractions - the given numbers must be replaced with their corresponding ordinary fractions.
If we are talking about comparing infinite non-periodic fractions, then it is usually reduced to comparing finite decimal fractions. For consideration, such a number of signs of the compared infinite non-periodic decimal fractions is taken, which will make it possible to obtain the result of the comparison.
Equal and unequal decimals
Definition 1Equal Decimals- these are two final decimal fractions, which have the same ordinary fractions corresponding to them. Otherwise, decimals are unequal.
Based on this definition, it is easy to justify such a statement: if at the end of a given decimal fraction we sign or, conversely, discard several digits 0, then we get a decimal fraction equal to it. For example: 0 , 5 = 0 , 50 = 0 , 500 = ... . Or: 130 , 000 = 130 , 00 = 130 , 0 = 130 . In fact, adding or discarding zero at the end of the fraction on the right means multiplying or dividing by 10 the numerator and denominator of the corresponding ordinary fraction. Let's add to what has been said the main property of fractions (by multiplying or dividing the numerator and denominator of a fraction by the same natural number, we get a fraction equal to the original one) and we have a proof of the above statement.
For example, the decimal fraction 0, 7 corresponds to an ordinary fraction 7 10. Adding zero to the right, we get the decimal fraction 0, 70, which corresponds to the ordinary fraction 70 100, 7 70 100: 10 . I.e.: 0 , 7 = 0 , 70 . And vice versa: discarding zero in the decimal fraction 0, 70 on the right, we get the fraction 0, 7 - thus, from the decimal fraction 70 100 we go to the fraction 7 10, but 7 10 \u003d 70: 10 100: 10 Then: 0, 70 \u003d 0 , 7 .
Now consider the content of the concept of equal and unequal infinite periodic decimal fractions.
Definition 2
Equal infinite periodic fractions are infinite periodic fractions that have equal ordinary fractions corresponding to them. If the ordinary fractions corresponding to them are not equal, then the periodic fractions given for comparison are also unequal.
This definition allows us to draw the following conclusions:
If the records of the given periodic decimal fractions are the same, then such fractions are equal. For example, the periodic decimals 0, 21 (5423) and 0, 21 (5423) are equal;
If in the given decimal periodic fractions the periods start from the same position, the first fraction has a period of 0, and the second - 9; the value of the digit preceding period 0 is one greater than the value of the digit preceding period 9 , then such infinite periodic decimal fractions are equal. For example, periodic fractions 91 , 3 (0) and 91 , 2 (9) are equal, as well as fractions: 135 , (0) and 134 , (9) ;
Any two other periodic fractions are not equal. For example: 8 , 0 (3) and 6 , (32) ; 0 , (42) and 0 , (131) etc.
It remains to consider equal and unequal infinite non-periodic decimal fractions. Such fractions are irrational numbers and cannot be converted to ordinary fractions. Therefore, the comparison of infinite non-periodic decimal fractions is not reduced to the comparison of ordinary ones.
Definition 3
Equal infinite non-recurring decimals are non-periodic decimal fractions, the entries of which are exactly the same.
The question would be logical: how to compare records if it is impossible to see the “finished” record of such fractions? Comparing infinite non-periodic decimal fractions, it is necessary to consider only a certain finite number of signs of the fractions given for comparison so that this allows us to draw a conclusion. Those. in essence, comparing infinite non-recurring decimals is comparing finite decimals.
This approach makes it possible to assert the equality of infinite non-periodic fractions only up to the considered digit. For example, the fractions 6, 73451 ... and 6, 73451 ... are equal to within hundred thousandths, because the end decimals 6, 73451 and 6, 7345 are equal. Fractions 20, 47 ... and 20, 47 ... are equal to within hundredths, because the fractions 20, 47 and 20, 47 are equal, and so on.
The inequality of infinite non-periodic fractions is established quite concretely with obvious differences in the records. For example, fractions 6, 4135 ... and 6, 4176 ... or 4, 9824 ... and 7, 1132 ... and so on are unequal.
Rules for comparing decimal fractions. Solution of examples
If it is established that two decimal fractions are not equal, it is usually also necessary to determine which of them is greater and which is less. Consider the rules for comparing decimal fractions, which make it possible to solve the above problem.
Very often, it is enough just to compare the integer parts of the decimal fractions given for comparison.
Definition 4
That decimal fraction, which has a larger integer part, is larger. The smaller fraction is the one whose integer part is smaller.
This rule applies to both finite decimal fractions and infinite ones.
Example 1
It is necessary to compare decimal fractions: 7, 54 and 3, 97823 ....
Solution
It is quite obvious that the given decimal fractions are not equal. Their whole parts are equal respectively: 7 and 3 . Because 7 > 3, then 7, 54 > 3, 97823 … .
Answer: 7 , 54 > 3 , 97823 … .
In the case when the integer parts of the fractions given for comparison are equal, the solution of the problem is reduced to comparing the fractional parts. The fractional parts are compared bit by bit - from the tenth place to the lower ones.
Consider first the case when you need to compare trailing decimal fractions.
Example 2
You want to compare the end decimals 0.65 and 0.6411.
Solution
Obviously, the integer parts of the given fractions are (0 = 0) . Let's compare the fractional parts: in the tenth place, the values are (6 \u003d 6) , but in the hundredth place, the value of the fraction 0, 65 is greater than the value of the hundredth place in the fraction 0, 6411 (5 > 4) . So 0.65 > 0.6411 .
Answer: 0 , 65 > 0 , 6411 .
In some tasks for comparing final decimal fractions with a different number of decimal places, it is necessary to attribute the required number of zeros to the right to a fraction with fewer decimal places. It is convenient to equalize in this way the number of decimal places in given fractions even before the start of the comparison.
Example 3
It is necessary to compare the final decimals 67 , 0205 and 67 , 020542 .
Solution
These fractions are obviously not equal, because their records are different. Moreover, their integer parts are equal: 67 \u003d 67. Before proceeding to the bitwise comparison of the fractional parts of the given fractions, we equalize the number of decimal places by adding zeros to the right in fractions with fewer decimal places. Then we get fractions for comparison: 67, 020500 and 67, 020542. We carry out a bitwise comparison and see that in the hundred-thousandth place the value in the fraction 67 , 020542 is greater than the corresponding value in the fraction 67 , 020500 (4 > 0) . So 67.020500< 67 , 020542 , а значит 67 , 0205 < 67 , 020542 .
Answer: 67 , 0205 < 67 , 020542 .
If it is necessary to compare a finite decimal fraction with an infinite one, then the final fraction is replaced by an infinite one equal to it with a period of 0. Then a bitwise comparison is made.
Example 4
It is necessary to compare the final decimal fraction 6, 24 with an infinite non-periodic decimal fraction 6, 240012 ...
Solution
We see that the integer parts of the given fractions are (6 = 6) . In the tenth and hundredth places, the values of both fractions are also equal. To be able to draw a conclusion, we continue the comparison, replacing the final decimal fraction equal to it with an infinite one with a period of 0 and get: 6, 240000 ... . Having reached the fifth decimal place, we find the difference: 0< 1 , а значит: 6 , 240000 … < 6 , 240012 … . Тогда: 6 , 24 < 6 , 240012 … .
Answer: 6, 24< 6 , 240012 … .
When comparing infinite decimal fractions, a bitwise comparison is also used, which will end when the values in some digit of the given fractions turn out to be different.
Example 5
It is necessary to compare the infinite decimal fractions 7, 41 (15) and 7, 42172 ... .
Solution
In the given fractions, there are equal whole parts, the values of the tenths are also equal, but in the hundredth place we see the difference: 1< 2 . Тогда: 7 , 41 (15) < 7 , 42172 … .
Answer: 7 , 41 (15) < 7 , 42172 … .
Example 6
It is necessary to compare the infinite periodic fractions 4 , (13) and 4 , (131) .
Solution:
Equalities are clear and correct: 4 , (13) = 4 , 131313 … and 4 , (133) = 4 , 131131 … . We compare integer parts and bitwise fractional parts, and fix the discrepancy at the fourth decimal place: 3 > 1 . Then: 4 , 131313 … > 4 , 131131 … , and 4 , (13) > 4 , (131) .
Answer: 4 , (13) > 4 , (131) .
To get the result of comparing a decimal fraction with a natural number, you need to compare the integer part of a given fraction with a given natural number. In this case, it should be taken into account that periodic fractions with periods of 0 or 9 must first be represented as final decimal fractions equal to them.
Definition 5
If the integer part of a given decimal fraction is less than a given natural number, then the whole fraction is smaller with respect to a given natural number. If the integer part of a given fraction is greater than or equal to a given natural number, then the fraction is greater than the given natural number.
Example 7
It is necessary to compare the natural number 8 and the decimal fraction 9, 3142 ... .
Solution:
The given natural number is less than the integer part of the given decimal fraction (8< 9) , а значит это число меньше заданной десятичной дроби.
Answer: 8 < 9 , 3142 … .
Example 8
It is necessary to compare the natural number 5 and the decimal fraction 5, 6.
Solution
The integer part of a given fraction is equal to a given natural number, then, according to the above rule, 5< 5 , 6 .
Answer: 5 < 5 , 6 .
Example 9
It is necessary to compare the natural number 4 and the periodic decimal fraction 3 , (9) .
Solution
The period of the given decimal fraction is 9, which means that before comparing, it is necessary to replace the given decimal fraction with a finite or natural number equal to it. In this case: 3 , (9) = 4 . Thus, the original data are equal.
Answer: 4 = 3 , (9) .
To compare a decimal fraction with an ordinary fraction or a mixed number, you must:
Write a common fraction or mixed number as a decimal and then compare the decimals or
- write the decimal fraction as a common fraction (except for infinite non-periodic), and then perform a comparison with a given common fraction or mixed number.
Example 10
It is necessary to compare the decimal fraction 0, 34 and the common fraction 1 3 .
Solution
Let's solve the problem in two ways.
- We write the given ordinary fraction 1 3 as a periodic decimal fraction equal to it: 0 , 33333 ... . Then it becomes necessary to compare the decimal fractions 0, 34 and 0, 33333…. We get: 0 , 34 > 0 , 33333 ... , which means 0 , 34 > 1 3 .
- Let's write the given decimal fraction 0, 34 in the form of an ordinary equal to it. I.e.: 0 , 34 = 34 100 = 17 50 . Let's compare ordinary fractions with different denominators and get: 17 50 > 1 3 . Thus, 0 , 34 > 1 3 .
Answer: 0 , 34 > 1 3 .
Example 11
You need to compare an infinite non-repeating decimal 4 , 5693 ... and a mixed number 4 3 8 .
Solution
An infinite non-periodic decimal fraction cannot be represented as a mixed number, but it is possible to convert the mixed number into an improper fraction, and this, in turn, can be written as a decimal fraction equal to it. Then: 4 3 8 = 35 8 and
Those.: 4 3 8 = 35 8 = 4, 375 . Let's compare decimal fractions: 4, 5693 ... and 4, 375 (4, 5693 ... > 4, 375) and get: 4, 5693 ... > 4 3 8 .
Answer: 4 , 5693 … > 4 3 8 .
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We will call a fraction one or more equal parts of one whole. A fraction is written using two natural numbers, which are separated by a line. For example, 1/2, 14/4, ¾, 5/9, etc.
The number above the bar is called the numerator of the fraction, and the number below the bar is called the denominator of the fraction.
For fractional numbers whose denominator is 10, 100, 1000, etc. agreed to write the number without a denominator. To do this, first write the integer part of the number, put a comma and write the fractional part of this number, that is, the numerator of the fractional part.
For example, instead of 6 * (7/10) they write 6.7.
Such a record is called decimal fraction.
How to compare two decimals
Let's figure out how to compare two decimal fractions. To do this, we first verify one auxiliary fact.
For example, the length of a certain segment is 7 centimeters or 70 mm. Also 7 cm = 7 / 10 dm or in decimal notation 0.7 dm.
On the other hand, 1 mm = 1/100 dm, then 70 mm = 70/100 dm, or in decimal notation 0.70 dm.
Thus, we get that 0.7 = 0.70.
From this we conclude that if zero is added or discarded at the end of the decimal fraction, then a fraction equal to the given one will be obtained. In other words, the value of the fraction will not change.
Fractions with the same denominators
Let's say we need to compare two decimals 4.345 and 4.36.
First, you need to equalize the number of decimal places by adding or discarding zeros to the right. You get 4.345 and 4.360.
Now you need to write them as improper fractions:
- 4,345 = 4345 / 1000 ;
- 4,360 = 4360 / 1000 .
The resulting fractions have the same denominators. By the rule of comparing fractions, we know that in this case, the larger fraction is the one with the larger numerator. So the fraction 4.36 is greater than the fraction 4.345.
Thus, in order to compare two decimal fractions, you must first equalize their number of decimal places, assigning zeros to one of them on the right, and then discarding the comma to compare the resulting natural numbers.
Decimals can be represented as dots on a number line. And therefore, sometimes in the case when one number is greater than another, they say that this number is located to the right of the other, or if it is less, then to the left.
If two decimal fractions are equal, then they are depicted on the number line by the same point.
The segment AB is 6 cm, that is, 60 mm. Since 1 cm = dm, then 6 cm = dm. So AB is 0.6 dm. Since 1 mm = dm, then 60 mm = dm. Hence, AB = 0.60 dm.
Thus, AB \u003d 0.6 dm \u003d 0.60 dm. This means that decimal fractions 0.6 and 0.60 express the length of the same segment in decimeters. These fractions are equal to each other: 0.6 = 0.60.
If zero is added at the end of the decimal fraction or zero is discarded, then we get fraction, equal to the given one.
For example,
0,87 = 0,870 = 0,8700; 141 = 141,0 = 141,00 = 141,000;
26,000 = 26,00 = 26,0 = 26; 60,00 = 60,0 = 60;
0,900 = 0,90 = 0,9.
Let's compare two decimals 5.345 and 5.36. Let's equalize the number of decimal places by adding zero to the number 5.36 on the right. We get fractions 5.345 and 5.360.
We write them as improper fractions:
These fractions have the same denominators. This means that the one with the larger numerator is greater.
Since 5345< 5360, то 
which means 5.345< 5,360, то есть 5,345 < 5,36.
To compare two decimal fractions, you must first equalize their number of decimal places by assigning zeros to one of them on the right, and then, discarding the comma, compare the resulting integers.
Decimal fractions can be represented on the coordinate ray in the same way as ordinary fractions.
For example, in order to depict the decimal fraction 0.4 on the coordinate ray, we first represent it as an ordinary fraction: 0.4 = Then we set aside four tenths of a unit segment from the beginning of the ray. We get point A(0,4) (Fig. 141).
Equal decimal fractions are depicted on the coordinate ray by the same point.
For example, fractions 0.6 and 0.60 are represented by one point B (see Fig. 141).
The smallest decimal lies on coordinate beam to the left of the larger one, and the larger one to the right of the smaller one.
For example, 0.4< 0,6 < 0,8, поэтому точка A(0,4) лежит левее точки B(0,6), а точка С(0,8) лежит правее точки B(0,6) (см. рис. 141).
Will a decimal change if a zero is added to the end of it?
A6 zeros?
Formulate a comparison rule decimal fractions.
1172. Write a decimal fraction:
a) with four decimal places, equal to 0.87;
b) with five decimal places, equal to 0.541;
c) with three digits after busy, equal to 35;
d) with two decimal places, equal to 8.40000.
1173. Having assigned zeros to the right, equalize the number of decimal places in decimal fractions: 1.8; 13.54 and 0.789.
1174. Write shorter fractions: 2.5000; 3.02000; 20.010.
85.09 and 67.99; 55.7 and 55.7000; 0.5 and 0.724; 0.908 and 0.918; 7.6431 and 7.6429; 0.0025 and 0.00247.
1176. Arrange in ascending order the numbers:
3,456; 3,465; 8,149; 8,079; 0,453.
0,0082; 0,037; 0,0044; 0,08; 0,0091
arrange in descending order.
a) 1.41< х < 4,75; г) 2,99 < х < 3;
b) 0.1< х < 0,2; д) 7 < х < 7,01;
c) 2.7< х < 2,8; е) 0,12 < х < 0,13.
1184. Compare the values:
a) 98.52 m and 65.39 m; e) 0.605 t and 691.3 kg;
b) 149.63 kg and 150.08 kg; f) 4.572 km and 4671.3 m;
c) 3.55°C and 3.61°C; g) 3.835 ha and 383.7 a;
d) 6.781 h and 6.718 h; h) 7.521 l and 7538 cm3.
Is it possible to compare 3.5 kg and 8.12 m? Give some examples of quantities that cannot be compared.
1185. Calculate orally:
1186. Restore the chain of calculations
1187. Is it possible to say how many digits after the decimal point are in a decimal fraction if its name ends with the word:
a) hundredths; b) ten thousandths; c) tenths; d) millions?
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In the section you will learn:
what is a decimal fraction and what is its structure;
how to compare decimals;
what are the rules for adding and subtracting decimal fractions;
how to find the product and the quotient of two decimal fractions;
what is rounding a number and how to round numbers;
how to apply the learned material in practice
§ 29. WHAT IS A DECIMAL FRACTION. COMPARISON OF DECIMAL FRACTIONS
Look at Figure 220. You can see that the length of the segment AB is 7 mm, and the length of the segment DC is 18 mm. To give the lengths of these segments in centimeters, you need to use fractions: 
You know many other examples where fractions with denominators 10,100, 1000, and the like are used. So, 
Such fractions are called decimals. To record them, they use a more convenient form, which is suggested by the ruler from your accessories. Let's look at the example in question.
You know that the length of the segment DC (Fig. 220) can be expressed as a mixed number
If we put a comma after the integer part of this number, and after it the numerator of the fractional part, then we get a more compact notation: 1.8 cm. For the segment AB, then we get: 0.7 cm. Indeed, the fraction is correct, it is less than one, therefore its integer part is 0. The numbers 1.8 and 0.7 are examples of decimals.
The decimal fraction 1.8 is read like this: "one point eight", and the fraction 0.7 - "zero point seven".
How to write fractions 
in decimal form? To do this, you need to know the structure of the decimal notation.
In decimal notation, there is always an integer and a fractional part. they are separated by a comma. In the integer part, classes and digits are the same as for natural numbers. You know that these are classes of units, thousands, millions, etc., and each of them has 3 digits - units, tens and hundreds. In the fractional part of a decimal fraction, classes are not distinguished, and there can be any number of digits, their names correspond to the names of the denominators of fractions - tenths, hundredths, thousandths, ten thousandths, hundred thousandths, millionths, ten millionths, etc. The tenth place is the oldest in the fractional part of a decimal.
In table 40 you see the names of the decimal places and the number "one hundred and twenty-three integers and four thousand five hundred and six hundred thousandths" or 
The name of the fractional part of "hundred-thousandths" in an ordinary fraction determines its denominator, and in decimal - the last digit of its fractional part. You see that in the numerator of the fractional part of the number 
one fewer digits than zeros in the denominator. If this is not taken into account, then we will get an error in writing the fractional part - instead of 4506 hundred-thousandths we will write 4506 ten-thousandths, but 
Therefore, in writing this number as a decimal fraction, you must put 0 after the decimal point (in the tenth place): 123.04506.
Note:
in a decimal fraction, there should be as many digits after the decimal point as there are zeros in the denominator of the corresponding ordinary fraction.
We can now write fractions
in the form of decimals.
Decimals can be compared in the same way as natural numbers. If there are many digits in decimal fractions, then special rules are used. Consider examples.
A task. Compare fractions: 1) 96.234 and 830.123; 2) 3.574 and 3.547.
Solutions. 1, The integer part of the first fraction is the two-digit number 96, and the integer part of the fraction of the second is the three-digit number 830, so:
96,234 < 830,123.
2. In the entries of fractions 3.574 and 3.547 and the whole parts are equal. Therefore, we compare their fractional parts bit by bit. To do this, we write these fractions one below the other:
Each fraction has 5 tenths. But in the first fraction there are 7 hundredths, and in the second - only 4 hundredths. Therefore, the first fraction is greater than the second: 3.574 > 3.547.
Rules for comparing decimal fractions.
1. Of two decimal fractions, the one with the larger integer part is greater.
2. If the integer parts of decimal fractions are equal, then their fractional parts are compared bit by bit, starting from the most significant digit.
Like common fractions, decimal fractions can be placed on the coordinate line. In Figure 221, you see that points A, B and C have coordinates: A (0.2), B (0.9), C (1.6).
Find out more
Decimals are related to the decimal positional number system. However, their appearance has a longer history and is associated with the name of the outstanding mathematician and astronomer al-Kashi (full name - Jamshid ibn-Masudal-Kashi). In his work "The Key to Arithmetic" (XV centuries), he first formulated the rules for actions with decimal fractions, gave examples of performing actions with them. Knowing nothing about the discovery of al-Kashi, the Flemish mathematician and engineer Simon Stevin “discovered” decimal fractions for the second time approximately 150 years later. In the work "Decimal" (1585 p.), S. Stevin outlined the theory of decimal fractions. He promoted them in every possible way, emphasizing the convenience of decimal fractions for practical calculations.
Separating the integer part from the fractional decimal fraction was proposed in different ways. So, al-Kashi wrote the integer and fractional parts in different ink or put a vertical line between them. S. Stevin put a zero in a circle to separate the integer part from the fractional part. The comma accepted in our time was proposed by the famous German astronomer Johannes Kepler (1571 - 1630).
SOLVE THE CHALLENGES
1173. Write down in centimeters the length of the segment AB if:
1)AB = 5mm; 2)AB = 8mm; 3)AB = 9mm; 4)AB = 2mm.
1174. Read fractions:
1)12,5; 3)3,54; 5)19,345; 7)1,1254;
2)5,6; 4)12,03; 6)15,103; 8)12,1065.
Name: a) the whole part of the fraction; b) the fractional part of the fraction; c) digits of a fraction.
1175. Give an example of a decimal fraction in which the decimal point is:
1) one digit; 2) two digits; 3) three digits.
1176. How many decimal places does a decimal fraction have if the denominator of the corresponding ordinary fraction is equal to:
1)10; 2)100; 3)1000; 4) 10000?
1177. Which of the fractions has the greater integer part:
1) 12.5 or 115.2; 4) 789.154 or 78.4569;
2) 5.25 or 35.26; 5) 1258.00265 or 125.0333;
3) 185.25 or 56.325; 6) 1269.569 or 16.12?
1178. In the number 1256897, separate the last digit with a comma and read the number you got. Then sequentially rearrange the comma one digit to the left and name the fractions that you received.
1179. Read the fractions and write them as a decimal fraction:
1180 Read the fractions and write them as a decimal:
1181. Write in ordinary fraction:
1) 2,5; 4)0,5; 7)315,89; 10)45,089;
2)125,5; 5)12,12; 8)0,15; 11)258,063;
3)0,9; 6)25,36; 9) 458;,025; 12)0,026.
1182. Write in ordinary fraction:
1)4,6; 2)34,45; 3)0,05; 4)185,342.
1183. Write down in decimal fraction:
1) 8 whole 3 tenths; 5) 145 point 14;
2) 12 whole 5 tenths; 6) 125 point 19;
3) 0 whole 5 tenths; 7) 0 whole 12 hundredths;
4) 12 whole 34 hundredths; 8) 0 whole 3 hundredths.
1184. Write in decimal fraction:
1) zero as many as eight thousandths;
2) twenty point four hundredths;
3) thirteen point five hundredths;
4) one hundred and forty-five point two hundredths.
1185. Write the share as a fraction, and then as a decimal:
1)33:100; 3)567:1000; 5)8:1000;
2)5:10; 4)56:1000; 6)5:100.
1186. Write as a mixed number and then as a decimal:
1)188:100; 3)1567:1000; 5)12548:1000;
2)25:10; 4)1326:1000; 6)15485:100.
1187. Write as a mixed number and then as a decimal:
1)1165:100; 3)2546:1000; 5)26548:1000;
2) 69: 10; 4) 1269: 1000; 6) 3569: 100.
1188. Express in hryvnias:
1) 35 k.; 2) 6 k.; 3) 12 UAH 35 kopecks; 4) 123k.
1189. Express in hryvnias:
1) 58 k.; 2) 2 to.; 3) 56 UAH 55 kopecks; 4) 175k.
1190. Write down in hryvnias and kopecks:
1) 10.34 UAH; 2) UAH 12.03; 3) 0.52 UAH; 4) UAH 126.05
1191. Express in meters and write down the answer as a decimal fraction: 1) 5 m 7 dm; 2) 15m 58cm; 3) 5 m 2 mm; 4) 12 m 4 dm 3 cm 2 mm.
1192. Express in kilometers and write down the answer in decimal fraction: 1) 3 km 175 m; 2) 45 km 47 m; 3) 15 km 2 m.
1193. Write down in meters and centimeters:
1) 12.55 m; 2) 2.06 m; 3) 0.25 m; 4) 0.08 m.
1194. The greatest depth of the Black Sea is 2.211 km. Express the depth of the sea in meters.
1195. Compare fractions:
1) 15.5 and 16.5; 5) 4.2 and 4.3; 9) 1.4 and 1.52;
2) 12.4 and 12.5; 6) 14.5 and 15.5; 10) 4.568 and 4.569;
3) 45.8 and 45.59; 7) 43.04 and 43.1; 11)78.45178.458;
4) 0.4 and 0.6; 8) 1.23 and 1.364; 12) 2.25 and 2.243.
1196. Compare fractions:
1) 78.5 and 79.5; 3) 78.3 and 78.89; 5) 25.03 and 25.3;
2) 22.3 and 22.7; 4) 0.3 and 0.8; 6) 23.569 and 23.568.
1197. Write down the decimal fractions in ascending order:
1) 15,3; 6,9; 18,1; 9,3; 12,45; 36,85; 56,45; 36,2;
2) 21,35; 21,46; 21,22; 21,56; 21,59; 21,78; 21,23; 21,55.
1198. Write down the decimal fractions in descending order:
15,6; 15,9; 15,5; 15,4; 15,45; 15,95; 15,2; 15,35.
1199. Express in square meters and write as a decimal fraction:
1) 5 dm2; 2) 15 cm2; 3)5dm212cm2.
1200 . The room is in the shape of a rectangle. Its length is 90 dm, and its width is 40 dm. Find the area of the room. Write your answer in square meters.
1201 . Compare fractions:
1) 0.04 and 0.06; 5) 1.003 and 1.03; 9) 120.058 and 120.051;
2) 402.0022 and 40.003; 6) 1.05 and 1.005; 10) 78.05 and 78.58;
3) 104.05 and 105.05; 7) 4.0502 and 4.0503; 11) 2.205 and 2.253;
4) 40.04 and 40.01; 8) 60.4007і60.04007; 12) 20.12 and 25.012.
1202. Compare fractions:
1) 0.03 and 0.3; 4) 6.4012 and 6.404;
2) 5.03 and 5.003; 5) 450.025 and 450.2054;
1203. Write down five decimal fractions that are between the fractions on the coordinate beam:
1) 6.2 and 6.3; 2) 9.2 and 9.3; 3) 5.8 and 5.9; 4) 0.4 and 0.5.
1204. Write down five decimal fractions that are between the fractions on the coordinate beam: 1) 3.1 and 3.2; 2) 7.4 and 7.5.
1205. Between which two adjacent natural numbers is a decimal fraction placed:
1)3,5; 2)12,45; 3)125,254; 4)125,012?
1206. Write down five decimal fractions for which the inequality is true:
1)3,41 <х< 5,25; 3) 1,59 < х < 9,43;
2) 15,25 < х < 20,35; 4) 2,18 < х < 2,19.
1207. Write down five decimal fractions for which the inequality is true:
1) 3 < х < 4; 2) 3,2 < х < 3,3; 3)5,22 <х< 5,23.
1208. Write down the largest decimal fraction:
1) with two digits after the decimal point, less than 2;
2) with one digit after the decimal point less than 3;
3) with three digits after the decimal point, less than 4;
4) with four digits after the decimal point, less than 1.
1209. Write down the smallest decimal fraction:
1) with two digits after the decimal point, which is greater than 2;
2) with three digits after the decimal point, which is greater than 4.
1210. Write down all the numbers that can be put instead of an asterisk to get the correct inequality:
1) 0, *3 >0,13; 3) 3,75 > 3, *7; 5) 2,15 < 2,1 *;
2) 8,5* < 8,57; 4) 9,3* < 9,34; 6)9,*4>9,24.
1211. What number can be put instead of an asterisk to get the correct inequality:
1)0,*3 >0,1*; 2) 8,5* <8,*7; 3)3,7*>3,*7?
1212. Write down all decimal fractions, the whole part of which is 6, and the fractional part contains three decimal places, written as 7 and 8. Write these fractions in descending order.
1213. Write down six decimal fractions, the whole part of which is 45, and the fractional part consists of four different numbers: 1, 2, 3, 4. Write these fractions in ascending order.
1214. How many decimal fractions can be formed, the whole part of which is equal to 86, and the fractional part consists of three different digits: 1,2,3?
1215. How many decimal fractions can be formed, the whole part of which is equal to 5, and the fractional part is three-digit, written as 6 and 7? Write these fractions in descending order.
1216. Cross out three zeros in the number 50.004007 so that it forms:
1) the largest number; 2) the smallest number.
APPLY IN PRACTICE
1217. Measure the length and width of your notebook in millimeters and write down your answer in decimeters.
1218. Write down your height in meters using a decimal fraction.
1219. Measure the dimensions of your room and calculate its perimeter and area. Write your answer in meters and square meters.
REPETITION TASKS
1220. For what values of x is a fraction improper?
1221. Solve the equation:
1222. The store had to sell 714 kg of apples. For the first day, all apples were sold, and for the second - from what was sold on the first day. How many apples were sold in 2 days?
1223. The edge of a cube was reduced by 10 cm and a cube was obtained, the volume of which is 8 dm3. Find the volume of the first cube.
The purpose of the lesson:
- create conditions for the derivation of the rule for comparing decimal fractions and the ability to apply it;
- repeat writing ordinary fractions as decimals, rounding decimals;
- develop logical thinking, the ability to generalize, research skills, speech.
During the classes
Guys, let's remember what we did with you in previous lessons?
Answer: studied decimal fractions, wrote ordinary fractions as decimals and vice versa, rounded decimal fractions.
What would you like to do today?
(Students answer.)
But still, what we will do in the lesson, you will find out in a few minutes. Open your notebooks, write down the date. A student will go to the board and work from the back of the board. I will offer you tasks that you complete orally. Write down the answers in a notebook in a line separated by a semicolon. The student at the blackboard writes in a column.
I read tasks that are pre-written on the board:
Let's check. Who has other answers? Remember the rules.
Got: 1,075; 2,175; 3,275; 4,375; 5,475; 6,575; 7,675.
Set the pattern and continue the resulting series for another 2 numbers. Let's check.
Take the transcript and under each number (the person answering at the board puts a letter next to the number) put the corresponding letter. Read the word.
Decryption:
So what are we going to do in class?
Answer: comparison.
By comparison! Well, for example, I will now start comparing my hands, 2 textbooks, 3 rulers. What do you want to compare?
Answer: decimal fractions.
What is the topic of the lesson?
I write the topic of the lesson on the board, and the students in the notebook: "Comparison of decimal fractions."
Exercise: compare the numbers (written on the board)
| 18.625 and 5.784 | 15.200 and 15.200 | |
| 3.0251 and 21.02 | 7.65 and 7.8 | |
| 23.0521 and 0.0521 | 0.089 and 0.0081 |
First, open the left side. Whole parts are different. We draw a conclusion about comparing decimal fractions with different integer parts. Open the right side. Whole parts are equal numbers. How to compare?
Sentence: write decimal fractions as common fractions and compare.
Write a comparison of ordinary fractions. If each decimal is converted to a common fraction and the 2 fractions are compared, it will take a long time. Can we derive a comparison rule? (Students suggest.) I wrote out the rule for comparing decimal fractions, which the author suggests. Let's compare.
There are 2 rules printed on a piece of paper:
- If the integer parts of decimal fractions are different, then that fraction is greater, which has a larger integer part.
- If the integer parts of the decimal fractions are the same, then the larger fraction is the one that has the larger first of the mismatched digits after the decimal point.
We have made a discovery. And this discovery is the rule for comparing decimal fractions. It coincided with the rule proposed by the author of the textbook.
I've noticed that the rules say which of the 2 fractions is greater. Can you tell me which of the 2 decimals is smaller.
Complete in notebook No. 785 (1, 2) on page 172. The task is written on the board. The students comment, and the teacher puts signs.
Exercise: compare
3.4208 and 3.4028
So what have we learned to do today? Let's check ourselves. Work on sheets of paper with carbon paper.
Students compare decimals by using > signs.<, =. Когда ученики выполнят задание, то листок сверху оставляют себе, а листок снизу сдают учителю.
Independent work.
(Check-answers on the back of the board.)
Compare
148.05 and 14.805
6.44806 and 6.44863
35.601 and 35.6010
The first one to do it gets the task (performs from the back of the board) No. 786 (1, 2):
Find a pattern and write down the next number in the sequence. In what sequences are the numbers arranged in ascending order, in which in descending order?
Answer:
- 0.1; 0.02; 0.003; 0.0004; 0.00005; (0.000006) - decreasing
- 0.1; 0.11; 0.111; 0.1111; 0.11111; (0.111111) - increases.
After the last student submits the work - check.
Students compare their answers.
Those who did everything right will mark themselves as “5”, those who made 1-2 mistakes - “4”, 3 mistakes - “3”. Find out in which comparisons errors were made, for which rule.
Write down your homework: No. 813, No. 814 (item 4, p. 171). Comment. If there is time, execute No. 786(1, 3), No. 793(a).
Summary of the lesson.
- What did you guys learn to do in class?
- Did you like or dislike?
- What were the difficulties?
Take the leaflets and fill them out, indicating the degree of your assimilation of the material:
- fully mastered, I can perform;
- learned completely, but find it difficult to apply;
- acquired partially;
- not acquired.
Thank you for the lesson.
