Ordinary fractions are divided into \textit (proper) and \textit (improper) fractions. This division is based on comparing the numerator and denominator.
Proper fractions
Proper fraction called common fraction$\frac(m)(n)$, whose numerator is less than the denominator, i.e. $m
Example 1
For example, the fractions $\frac(1)(3)$, $\frac(9)(123)$, $\frac(77)(78)$, $\frac(378567)(456298)$ are regular, so how in each of them the numerator is less than the denominator, which corresponds to the definition of a proper fraction.
There is a definition of a proper fraction, which is based on comparing a fraction with a unit.
correct, If she less than one:
Example 2
For example, the common fraction $\frac(6)(13)$ is proper because condition $\frac(6)(13)
Improper fractions
Improper fraction is an ordinary fraction $\frac(m)(n)$ whose numerator is greater than or equal to the denominator, i.e. $m\ge n$.
Example 3
For example, the fractions $\frac(5)(5)$, $\frac(24)(3)$, $\frac(567)(113)$, $\frac(100001)(100000)$ are improper, so how in each of them the numerator is greater than or equal to the denominator, which corresponds to the definition of an improper fraction.
Let's give the definition of an improper fraction, which is based on its comparison with the unit.
The ordinary fraction $\frac(m)(n)$ is wrong if it is equal to or greater than one:
\[\frac(m)(n)\ge 1\]
Example 4
For example, the common fraction $\frac(21)(4)$ is improper because the condition $\frac(21)(4) >1$ is satisfied;
the ordinary fraction $\frac(8)(8)$ is improper because the condition $\frac(8)(8)=1$ is satisfied.
Let us consider in more detail the concept of an improper fraction.
Let's take $\frac(7)(7)$ as an example. The value of this fraction is taken as seven parts of an object, which is divided into seven equal parts. Thus, from the seven shares that are available, you can make up the whole subject. Those. improper fraction$\frac(7)(7)$ describes the whole item and $\frac(7)(7)=1$. So don't proper fractions, whose numerator is equal to the denominator, describe one whole object and such a fraction can be replaced by a natural number $1$.
$\frac(5)(2)$ -- it's pretty obvious that these five second parts can make $2$ whole items (one whole item will make $2$ parts, and to make two whole items you need $2+2=4$ share) and one second share remains. That is, the improper fraction $\frac(5)(2)$ describes $2$ of an item and $\frac(1)(2)$ of that item.
$\frac(21)(7)$ -- twenty-one-sevenths can make $3$ whole items ($3$ items with $7$ shares each). Those. the fraction $\frac(21)(7)$ describes $3$ integers.
From the considered examples, the following conclusion can be drawn: an improper fraction can be replaced by a natural number if the numerator is completely divisible by the denominator (for example, $\frac(7)(7)=1$ and $\frac(21)(7)=3$) , or the sum of a natural number and a proper fraction if the numerator is not even divisible by the denominator (for example, $\ \frac(5)(2)=2+\frac(1)(2)$). Therefore, such fractions are called wrong.
Definition 1
The process of representing an improper fraction as the sum of a natural number and a proper fraction (for example, $\frac(5)(2)=2+\frac(1)(2)$) is called extracting the integer part from an improper fraction.
When working with improper fractions, there is a close relationship between them and mixed numbers.
An improper fraction is often written as a mixed number, a number that consists of a whole number and a fractional part.
To write an improper fraction as a mixed number, you must divide the numerator by the denominator with a remainder. The quotient will be the integer part of the mixed number, the remainder will be the numerator of the fractional part, and the divisor will be the denominator of the fractional part.
Example 5
Write the improper fraction $\frac(37)(12)$ as a mixed number.
Solution.
Divide the numerator by the denominator with a remainder:
\[\frac(37)(12)=37:12=3\ (remainder\ 1)\] \[\frac(37)(12)=3\frac(1)(12)\]
Answer.$\frac(37)(12)=3\frac(1)(12)$.
To write a mixed number as an improper fraction, you need to multiply the denominator by the integer part of the number, add the numerator of the fractional part to the product that turned out, and write the resulting amount into the numerator of the fraction. The denominator of the improper fraction will be equal to the denominator of the fractional part of the mixed number.
Example 6
Write the mixed number $5\frac(3)(7)$ as an improper fraction.
Solution.
Answer.$5\frac(3)(7)=\frac(38)(7)$.
Adding a mixed number and a proper fraction
Adding a mixed number$a\frac(b)(c)$ and proper fraction$\frac(d)(e)$ performs by adding the fractional part of the given mixed number to the given fraction:
Example 7
Add the proper fraction $\frac(4)(15)$ and the mixed number $3\frac(2)(5)$.
Solution.
Let's use the formula for adding a mixed number and a proper fraction:
\[\frac(4)(15)+3\frac(2)(5)=3+\left(\frac(2)(5)+\frac(4)(15)\right)=3+\ left(\frac(2\cdot 3)(5\cdot 3)+\frac(4)(15)\right)=3+\frac(6+4)(15)=3+\frac(10)( fifteen)\]
By the criterion of division by the number \textit(5 ) one can determine that the fraction $\frac(10)(15)$ is reducible. Perform the reduction and find the result of the addition:
So, the result of adding the proper fraction $\frac(4)(15)$ and the mixed number $3\frac(2)(5)$ is $3\frac(2)(3)$.
Answer:$3\frac(2)(3)$
Adding a mixed number and an improper fraction
Adding an improper fraction and a mixed number reduce to the addition of two mixed numbers, for which it is enough to select the whole part from an improper fraction.
Example 8
Calculate the sum of the mixed number $6\frac(2)(15)$ and the improper fraction $\frac(13)(5)$.
Solution.
First, we extract the integer part from the improper fraction $\frac(13)(5)$:
Answer:$8\frac(11)(15)$.
At the word "fractions" many goosebumps run. Because I remember the school and the tasks that were solved in mathematics. This was a duty that had to be fulfilled. But what if we treat tasks containing proper and improper fractions as a puzzle? After all, many adults solve digital and Japanese crosswords. Understand the rules and that's it. Same here. One has only to delve into the theory - and everything will fall into place. And examples will turn into a way to train the brain.
What types of fractions are there?
Let's start with what it is. A fraction is a number that has some fraction of one. It can be written in two forms. The first is called ordinary. That is, one that has a horizontal or oblique stroke. It equates to the division sign.
In such a notation, the number above the dash is called the numerator, and below it is called the denominator.
Among ordinary fractions, right and wrong fractions are distinguished. For the former, the modulo numerator is always less than the denominator. The wrong ones are called that because they have the opposite. The value of a proper fraction is always less than one. While the wrong one is always greater than this number.
There are also mixed numbers, that is, those that have an integer and a fractional part.
The second type of record is decimal. About her separate conversation.
What is the difference between improper fractions and mixed numbers?
Basically, nothing. It's just a different notation of the same number. Improper fractions after simple operations easily become mixed numbers. And vice versa.
It all depends on specific situation. Sometimes in tasks it is more convenient to use an improper fraction. And sometimes it is necessary to translate it into a mixed number, and then the example will be solved very easily. Therefore, what to use: improper fractions, mixed numbers - depends on the observation of the solver of the problem.
The mixed number is also compared with the sum of the integer part and the fractional part. Moreover, the second is always less than unity.
How to represent a mixed number as an improper fraction?
If you want to perform some action with several numbers that are written in different types, then you need to make them the same. One method is to represent numbers as improper fractions.
For this purpose, you will need to follow the following algorithm:
- multiply the denominator by the integer part;
- add the value of the numerator to the result;
- write the answer above the line;
- leave the denominator the same.
Here are examples of how to write improper fractions from mixed numbers:
- 17 ¼ \u003d (17 x 4 + 1): 4 \u003d 69/4;
- 39 ½ \u003d (39 x 2 + 1): 2 \u003d 79/2.
How to write an improper fraction as a mixed number?
The next method is the opposite of the one discussed above. That is, when all mixed numbers are replaced with improper fractions. The algorithm of actions will be as follows:
- divide the numerator by the denominator to get the remainder;
- write the quotient in place of the integer part of the mixed;
- the remainder should be placed above the line;
- the divisor will be the denominator.
Examples of such a transformation:
76/14; 76:14 = 5 with a remainder of 6; the answer is 5 integers and 6/14; the fractional part in this example needs to be reduced by 2, you get 3/7; the final answer is 5 whole 3/7.
108/54; after division, the quotient 2 is obtained without a remainder; this means that not all improper fractions can be represented as a mixed number; the answer is an integer - 2.
How do you turn an integer into an improper fraction?
There are situations when such action is necessary. To get improper fractions with a predetermined denominator, you will need to perform the following algorithm:
- multiply an integer by the desired denominator;
- write this value above the line;
- place a denominator below it.
The simplest option is when the denominator is equal to one. Then there is no need to multiply. It is enough just to write an integer, which is given in the example, and place a unit under the line.
Example: Make 5 an improper fraction with a denominator of 3. After multiplying 5 by 3, you get 15. This number will be the denominator. The answer to the task is a fraction: 15/3.
Two approaches to solving tasks with different numbers
In the example, it is required to calculate the sum and difference, as well as the product and quotient of two numbers: 2 integers 3/5 and 14/11.
In the first approach the mixed number will be represented as an improper fraction.
After performing the steps described above, you get the following value: 13/5.
In order to find out the sum, you need to reduce the fractions to the same denominator. 13/5 multiplied by 11 becomes 143/55. And 14/11 after multiplying by 5 will take the form: 70/55. To calculate the sum, you only need to add the numerators: 143 and 70, and then write down the answer with one denominator. 213/55 - this improper fraction is the answer to the problem.
When finding the difference, these same numbers are subtracted: 143 - 70 = 73. The answer is a fraction: 73/55.
When multiplying 13/5 and 14/11, you do not need to reduce to a common denominator. Just multiply the numerators and denominators in pairs. The answer will be: 182/55.
Likewise with division. For right decision you need to replace division with multiplication and flip the divisor: 13/5: 14/11 \u003d 13/5 x 11/14 \u003d 143/70.
In the second approach An improper fraction becomes a mixed number.
After performing the actions of the algorithm, 14/11 will turn into a mixed number with an integer part of 1 and a fractional part of 3/11.
When calculating the sum, you need to add the integer and fractional parts separately. 2 + 1 = 3, 3/5 + 3/11 = 33/55 + 15/55 = 48/55. The final answer is 3 whole 48/55. In the first approach there was a fraction 213/55. You can check the correctness by converting it to a mixed number. After dividing 213 by 55, the quotient is 3 and the remainder is 48. It is easy to see that the answer is correct.
When subtracting, the "+" sign is replaced by "-". 2 - 1 = 1, 33/55 - 15/55 = 18/55. To check the answer from the previous approach, you need to convert it to a mixed number: 73 is divided by 55 and you get a quotient of 1 and a remainder of 18.
To find the product and the quotient, it is inconvenient to use mixed numbers. Here it is always recommended to switch to improper fractions.
This article is about common fractions. Here we will get acquainted with the concept of a fraction of a whole, which will lead us to the definition of an ordinary fraction. Next, we will dwell on the accepted notation for ordinary fractions and give examples of fractions, say about the numerator and denominator of a fraction. After that, we will give definitions of correct and improper, positive and negative fractions, and also consider the position of fractional numbers on coordinate beam. In conclusion, we list the main actions with fractions.
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Shares of the whole
First we introduce share concept.
Let's assume that we have some object made up of several absolutely identical (that is, equal) parts. For clarity, you can imagine, for example, an apple cut into several equal parts, or an orange, consisting of several equal slices. Each of these equal parts that make up the whole object is called share of the whole or simply shares.
Note that the shares are different. Let's explain this. Let's say we have two apples. Let's cut the first apple into two equal parts, and the second one into 6 equal parts. It is clear that the share of the first apple will be different from the share of the second apple.
Depending on the number of shares that make up the whole object, these shares have their own names. Let's analyze share names. If the object consists of two parts, any of them is called one second part of the whole object; if the object consists of three parts, then any of them is called one third part, and so on.
One second beat has a special name - half. One third is called third, and one quadruple - quarter.
For the sake of brevity, the following share designations. One second share is designated as or 1/2, one third share - as or 1/3; one fourth share - like or 1/4, and so on. Note that the notation with a horizontal bar is used more often. To consolidate the material, let's give one more example: the entry denotes one hundred and sixty-seventh of the whole.
The concept of a share naturally extends from objects to magnitudes. For example, one of the measures of length is the meter. To measure lengths less than a meter, fractions of a meter can be used. So you can use, for example, half a meter or a tenth or thousandth of a meter. Shares of other quantities are applied similarly.
Common fractions, definition and examples of fractions
To describe the number of shares are used common fractions. Let's give an example that will allow us to approach the definition of ordinary fractions.
Let an orange consist of 12 parts. Each share in this case represents one twelfth of a whole orange, that is, . Let's denote two beats as , three beats as , and so on, 12 beats as . Each of these entries is called an ordinary fraction.
Now let's give a general definition of common fractions.
The voiced definition of ordinary fractions allows us to bring examples of common fractions: 5/10 , , 21/1 , 9/4 , . And here are the records 
do not fit the voiced definition of ordinary fractions, that is, they are not ordinary fractions.
Numerator and denominator
For convenience, in ordinary fractions we distinguish numerator and denominator.
Definition.
Numerator ordinary fraction (m / n) is a natural number m.
Definition.
Denominator ordinary fraction (m / n) is a natural number n.
So, the numerator is located above the fraction bar (to the left of the slash), and the denominator is below the fraction bar (to the right of the slash). For example, let's take an ordinary fraction 17/29, the numerator of this fraction is the number 17, and the denominator is the number 29.
It remains to discuss the meaning contained in the numerator and denominator of an ordinary fraction. The denominator of the fraction shows how many shares one item consists of, the numerator, in turn, indicates the number of such shares. For example, the denominator 5 of the fraction 12/5 means that one item consists of five parts, and the numerator 12 means that 12 such parts are taken.
Natural number as a fraction with denominator 1
The denominator of an ordinary fraction can be equal to one. In this case, we can assume that the object is indivisible, in other words, it is something whole. The numerator of such a fraction indicates how many whole items are taken. Thus, an ordinary fraction of the form m/1 has the meaning of a natural number m. This is how we substantiated the equality m/1=m .
Let's rewrite the last equality like this: m=m/1 . This equality allows us to represent any natural number m as an ordinary fraction. For example, the number 4 is the fraction 4/1, and the number 103498 is the fraction 103498/1.
So, any natural number m can be represented as an ordinary fraction with denominator 1 as m/1 , and any ordinary fraction of the form m/1 can be replaced by a natural number m.
Fraction bar as division sign
The representation of the original object in the form of n shares is nothing more than a division into n equal parts. After the item is divided into n shares, we can divide it equally among n people - each will receive one share.
If we initially have m identical objects, each of which is divided into n shares, then we can equally divide these m objects among n people, giving each person one share from each of the m objects. In this case, each person will have m shares 1/n, and m shares 1/n gives an ordinary fraction m/n. Thus, the common fraction m/n can be used to represent the division of m items among n people.
So we got an explicit connection between ordinary fractions and division (see the general idea of the division of natural numbers). This relationship is expressed as follows: The bar of a fraction can be understood as a division sign, that is, m/n=m:n.
With the help of an ordinary fraction, you can write the result of dividing two natural numbers, for which integer division is not performed. For example, the result of dividing 5 apples by 8 people can be written as 5/8, that is, each will get five eighths of an apple: 5:8=5/8.
Equal and unequal ordinary fractions, comparison of fractions
A fairly natural action is comparison of common fractions, because it is clear that 1/12 of an orange is different from 5/12, and 1/6 of an apple is the same as the other 1/6 of this apple.
As a result of comparing two ordinary fractions, one of the results is obtained: the fractions are either equal or not equal. In the first case we have equal common fractions, and in the second unequal common fractions. Let's give a definition of equal and unequal ordinary fractions.
Definition.
equal, if the equality a d=b c is true.
Definition.
Two common fractions a/b and c/d not equal, if the equality a d=b c is not satisfied.
Here are some examples of equal fractions. For example, the common fraction 1/2 is equal to the fraction 2/4, since 1 4=2 2 (if necessary, see the rules and examples of multiplication of natural numbers). For clarity, you can imagine two identical apples, the first is cut in half, and the second - into 4 shares. It is obvious that two-fourths of an apple is 1/2 a share. Other examples of equal common fractions are the fractions 4/7 and 36/63, and the pair of fractions 81/50 and 1620/1000.
And ordinary fractions 4/13 and 5/14 are not equal, since 4 14=56, and 13 5=65, that is, 4 14≠13 5. Another example of unequal common fractions are the fractions 17/7 and 6/4.
If, when comparing two ordinary fractions, it turns out that they are not equal, then you may need to find out which of these ordinary fractions less another, and which more. To find out, the rule for comparing ordinary fractions is used, the essence of which is to bring the compared fractions to a common denominator and then compare the numerators. Detailed information on this topic is collected in the article comparison of fractions: rules, examples, solutions.
Fractional numbers
Each fraction is a record fractional number. That is, a fraction is just a “shell” of a fractional number, its appearance, and the entire semantic load is contained precisely in a fractional number. However, for brevity and convenience, the concept of a fraction and a fractional number are combined and simply called a fraction. Here it is appropriate to paraphrase a well-known saying: we say a fraction - we mean a fractional number, we say a fractional number - we mean a fraction.
Fractions on the coordinate beam
All fractional numbers corresponding to ordinary fractions have their own unique place on , that is, there is a one-to-one correspondence between fractions and points of the coordinate ray.
In order to get to the point corresponding to the fraction m / n on the coordinate ray, it is necessary to postpone m segments from the origin in the positive direction, the length of which is 1 / n of the unit segment. Such segments can be obtained by dividing a single segment into n equal parts, which can always be done using a compass and ruler.
For example, let's show the point M on the coordinate ray, corresponding to the fraction 14/10. The length of the segment with ends at the point O and the point closest to it, marked with a small dash, is 1/10 of the unit segment. The point with coordinate 14/10 is removed from the origin by 14 such segments.
Equal fractions correspond to the same fractional number, that is, equal fractions are the coordinates of the same point on the coordinate ray. For example, one point corresponds to the coordinates 1/2, 2/4, 16/32, 55/110 on the coordinate ray, since all written fractions are equal (it is located at a distance of half the unit segment, laid down from the origin in the positive direction).
On a horizontal and right-directed coordinate ray, the point whose coordinate is a large fraction is located to the right of the point whose coordinate is a smaller fraction. Similarly, the point with the smaller coordinate lies to the left of the point with the larger coordinate.
Proper and improper fractions, definitions, examples
Among ordinary fractions, there are proper and improper fractions. This division basically has a comparison of the numerator and denominator.
Let's give a definition of proper and improper ordinary fractions.
Definition.
Proper fraction is an ordinary fraction, the numerator of which is less than the denominator, that is, if m Definition.
Improper fraction is an ordinary fraction in which the numerator is greater than or equal to the denominator, that is, if m≥n, then the ordinary fraction is improper. Here are some examples of proper fractions: 1/4 , , 32 765/909 003 . Indeed, in each of the written ordinary fractions, the numerator is less than the denominator (if necessary, see the article comparison of natural numbers), so they are correct by definition. And here are examples of improper fractions: 9/9, 23/4,. Indeed, the numerator of the first of the written ordinary fractions is equal to the denominator, and in the remaining fractions the numerator is greater than the denominator. There are also definitions of proper and improper fractions based on comparing fractions with one. Definition. correct if it is less than one. Definition. The common fraction is called wrong, if it is either equal to one or greater than 1 . So the ordinary fraction 7/11 is correct, since 7/11<1
, а обыкновенные дроби 14/3
и 27/27
– неправильные, так как 14/3>1 , and 27/27=1 . Let's think about how ordinary fractions with a numerator greater than or equal to the denominator deserve such a name - "wrong". Let's take the improper fraction 9/9 as an example. This fraction means that nine parts of an object are taken, which consists of nine parts. That is, from the available nine shares, we can make up a whole subject. That is, the improper fraction 9/9 essentially gives a whole object, that is, 9/9=1. In general, improper fractions with a numerator equal to the denominator denote one whole object, and such a fraction can be replaced by a natural number 1. Now consider the improper fractions 7/3 and 12/4. It is quite obvious that from these seven thirds we can make two whole objects (one whole object is 3 shares, then to compose two whole objects we need 3 + 3 = 6 shares) and there will still be one third share. That is, the improper fraction 7/3 essentially means 2 items and even 1/3 of the share of such an item. And from twelve quarters we can make three whole objects (three objects with four parts each). That is, the fraction 12/4 essentially means 3 whole objects. The considered examples lead us to the following conclusion: improper fractions can be replaced either by natural numbers, when the numerator is divided entirely by the denominator (for example, 9/9=1 and 12/4=3), or the sum of a natural number and a proper fraction, when the numerator is not evenly divisible by the denominator (for example, 7/3=2+1/3 ). Perhaps this is precisely what improper fractions deserve such a name - “wrong”. Of particular interest is the representation of an improper fraction as the sum of a natural number and a proper fraction (7/3=2+1/3). This process is called the extraction of an integer part from an improper fraction, and deserves a separate and more careful consideration. It is also worth noting that there is a very close relationship between improper fractions and mixed numbers. Each ordinary fraction corresponds to a positive fractional number (see the article positive and negative numbers). That is, ordinary fractions are positive fractions. For example, ordinary fractions 1/5, 56/18, 35/144 are positive fractions. When it is necessary to emphasize the positiveness of a fraction, then a plus sign is placed in front of it, for example, +3/4, +72/34. If you put a minus sign in front of an ordinary fraction, then this entry will correspond to a negative fractional number. In this case, one can speak of negative fractions. Here are some examples of negative fractions: −6/10 , −65/13 , −1/18 . The positive and negative fractions m/n and −m/n are opposite numbers. For example, the fractions 5/7 and −5/7 are opposite fractions. Positive fractions, like positive numbers in general, denote an increase, income, a change in some value upwards, etc. Negative fractions correspond to an expense, a debt, a change in any value in the direction of decrease. For example, a negative fraction -3/4 can be interpreted as a debt, the value of which is 3/4. On the horizontal and right-directed negative fractions are located to the left of the reference point. The points of the coordinate line whose coordinates are the positive fraction m/n and the negative fraction −m/n are located at the same distance from the origin, but on opposite sides of the point O . Here it is worth mentioning fractions of the form 0/n. These fractions are equal to the number zero, that is, 0/n=0 . Positive fractions, negative fractions, and 0/n fractions combine to form rational numbers. One action with ordinary fractions - comparing fractions - we have already considered above. Four more arithmetic are defined operations with fractions- addition, subtraction, multiplication and division of fractions. Let's dwell on each of them. The general essence of actions with fractions is similar to the essence of the corresponding actions with natural numbers. Let's draw an analogy. Multiplication of fractions can be considered as an action in which a fraction is found from a fraction. To clarify, let's take an example. Suppose we have 1/6 of an apple and we need to take 2/3 of it. The part we need is the result of multiplying the fractions 1/6 and 2/3. The result of multiplying two ordinary fractions is an ordinary fraction (which in a particular case is equal to a natural number). Further we recommend to study the information of the article multiplication of fractions - rules, examples and solutions. Bibliography. We encounter fractions in life much earlier than they begin to study at school. If you cut a whole apple in half, then we get a piece of fruit - ½. Cut it again - it will be ¼. This is what fractions are. And everything, it would seem, is simple. For an adult. For a child (and they begin to study this topic at the end of elementary school), abstract mathematical concepts are still frighteningly incomprehensible, and the teacher must explain in an accessible way what a proper fraction and improper, ordinary and decimal are, what operations can be performed with them and, most importantly, why all this is needed. Acquaintance with a new topic at school begins with ordinary fractions. They are easy to recognize by the horizontal line separating the two numbers - above and below. The top is called the numerator, the bottom is called the denominator. There is also a lower case spelling of improper and proper ordinary fractions - through a slash, for example: ½, 4/9, 384/183. This option is used when the line height is limited and it is not possible to apply the "two-story" form of the entry. Why? Yes, because it is more convenient. A little later we will verify this. In addition to ordinary, there are also decimal fractions. It is very easy to distinguish between them: if in one case a horizontal or slash is used, then in the other - a comma separating sequences of numbers. Let's see an example: 2.9; 163.34; 1.953. We deliberately used the semicolon as a delimiter to delimit the numbers. The first of them will be read like this: "two whole, nine tenths." Let's go back to ordinary fractions. They are of two kinds. The definition of a proper fraction is as follows: it is such a fraction, the numerator of which is less than the denominator. Why is it important? Now we'll see! You have several apples cut into halves. In total - 5 parts. How do you say: you have "two and a half" or "five second" apples? Of course, the first option sounds more natural, and when talking with friends, we will use it. But if you need to calculate how much fruit each will get, if there are five people in the company, we will write down the number 5/2 and divide it by 5 - from the point of view of mathematics, this will be clearer. So, for naming regular and improper fractions, the rule is as follows: if an integer part (14/5, 2/1, 173/16, 3/3) can be distinguished in a fraction, then it is incorrect. If this cannot be done, as in the case of ½, 13/16, 9/10, it will be correct. If the numerator and denominator of a fraction are simultaneously multiplied or divided by the same number, its value will not change. Imagine: the cake was cut into 4 equal parts and they gave you one. The same cake was cut into eight pieces and given you two. Isn't it all the same? After all, ¼ and 2/8 are the same thing! Authors of problems and examples in math textbooks often try to confuse students by offering fractions that are cumbersome to write and can actually be reduced. Here is an example of a proper fraction: 167/334, which, it would seem, looks very "scary". But in fact, we can write it as ½. The number 334 is divisible by 167 without a remainder - having done this operation, we get 2. An improper fraction can be represented as a mixed number. This is when the whole part is brought forward and written at the level of the horizontal line. In fact, the expression takes the form of a sum: 11/2 = 5 + ½; 13/6 = 2 + 1/6 and so on. To take out the whole part, you need to divide the numerator by the denominator. Write the remainder of the division above, above the line, and the whole part before the expression. Thus, we get two structural parts: whole units + proper fraction. You can also carry out the reverse operation - for this you need to multiply the integer part by the denominator and add the resulting value to the numerator. Nothing complicated. Oddly enough, multiplying fractions is easier than adding them. All that is required is to extend the horizontal line: (2/3) * (3/5) = 2*3 / 3*5 = 2/5. With division, everything is also simple: you need to multiply the fractions crosswise: (7/8) / (14/15) \u003d 7 * 15 / 8 * 14 \u003d 15/16. What if you need to perform addition or if they have different numbers in the denominator? It will not work in the same way as with multiplication - here one should understand the definition of a proper fraction and its essence. It is necessary to bring the terms to a common denominator, that is, the same numbers should appear at the bottom of both fractions. To do this, you should use the basic property of a fraction: multiply both parts by the same number. For example, 2/5 + 1/10 = (2*2)/(5*2) + 1/10 = 5/10 = ½. How to choose which denominator to bring the terms to? This must be the smallest multiple of both denominators: for 1/3 and 1/9 it will be 9; for ½ and 1/7 - 14, because there is no smaller value divisible by 2 and 7 without a remainder. What are improper fractions for? After all, it is much more convenient to immediately select the whole part, get a mixed number - and that's it! It turns out that if you need to multiply or divide two fractions, it is more profitable to use the wrong ones. Let's take the following example: (2 + 3/17) / (37 / 68). It would seem that there is nothing to cut at all. But what if we write the result of the addition in the first brackets as an improper fraction? Look: (37/17) / (37/68) Now everything falls into place! Let's write the example in such a way that everything becomes obvious: (37 * 68) / (17 * 37). Let's reduce the 37 in the numerator and denominator, and finally divide the top and bottom parts by 17. Do you remember the basic rule for proper and improper fractions? We can multiply and divide them by any number, as long as we do it for the numerator and denominator at the same time. So, we get the answer: 4. The example looked complicated, and the answer contains only one digit. This often happens in mathematics. The main thing is not to be afraid and follow simple rules. When exercising, the student can easily make one of the popular mistakes. Usually they occur due to inattention, and sometimes due to the fact that the studied material has not yet been properly deposited in the head. Often the sum of the numbers in the numerator causes a desire to reduce its individual components. Suppose, in the example: (13 + 2) / 13, written without brackets (with a horizontal line), many students, due to inexperience, cross out 13 from above and below. But this should not be done in any case, because this is a gross mistake! If instead of addition there was a multiplication sign, we would get the number 2 in the answer. But when performing addition, no operations with one of the terms are allowed, only with the entire sum. Children often make mistakes when dividing fractions. Let's take two regular irreducible fractions and divide by each other: (5/6) / (25/33). The student can confuse and write the resulting expression as (5*25) / (6*33). But this would have happened with multiplication, and in our case everything will be a little different: (5 * 33) / (6 * 25). We reduce what is possible, and in the answer we will see 11/10. We write the resulting improper fraction as a decimal - 1.1. Remember that in any mathematical expression, the order of operations is determined by the precedence of operation signs and the presence of brackets. Other things being equal, the sequence of actions is counted from left to right. This is also true for fractions - the expression in the numerator or denominator is calculated strictly according to this rule. It is the result of dividing one number by another. If they do not divide completely, it turns out a fraction - that's all. Since standard tools do not always allow you to create a fraction consisting of two "tiers", students sometimes go for various tricks. For example, they copy the numerators and denominators into the Paint editor and glue them together, drawing a horizontal line between them. Of course, there is a simpler option, which, by the way, also provides a lot of additional features that will be useful to you in the future. Open Microsoft Word. One of the panels at the top of the screen is called "Insert" - click it. On the right, on the side where the icons for closing and minimizing the window are located, there is a Formula button. This is exactly what we need! If you use this function, a rectangular area will appear on the screen in which you can use any mathematical symbols that are not available on the keyboard, as well as write fractions in the classic form. That is, separating the numerator and denominator with a horizontal bar. You may even be surprised that such a proper fraction is so easy to write down. If you are in grades 5-6, then soon knowledge of mathematics (including the ability to work with fractions!) Will be required in many school subjects. In almost any problem in physics, when measuring the mass of substances in chemistry, in geometry and trigonometry, fractions cannot be dispensed with. Soon you will learn to calculate everything in your mind, without even writing expressions on paper, but more and more complex examples will appear. Therefore, learn what a proper fraction is and how to work with it, keep up with the curriculum, do your homework on time, and then you will succeed. 326. Fill in the gaps. 1) If the numerator of a fraction is equal to the denominator, then the fraction is equal to 1. 327. Write out from fractions 1/20, 16/9, 7/2, 14/28.10/10, 5/32.11/2: 1) proper fractions; 2) improper fractions. 1) 1/20, 14/23, 5/32 2) 19/9, 7/2, 10/10, 11/2 328. Think up and write down: 1) 5 correct fractions; 2) improper fractions. 1) ½, 1/3, ¼, 1/5, 1/6 2) 3/2, 4/2, 5/2Yu 6/2, 7/2 329. Write down all the correct fractions with a denominator of 9. 1/9, 2/9, 3/9, 4/9, 5/9, 6/9, 7/9, 8/9. 330. Write down all improper fractions with numerator 9. 9/1,9/2, 9/3, 9/4, 9/5, 9/6, 9/7, 9/8, 9/9. 331. Two identical strips were divided into 7 equal parts. Paint over 4/7 of one strip and 6/7 of the other. Compare the resulting fractions: 4/7< 6/7. Formulate a rule for comparing fractions with the same denominators: of two fractions with the same denominators, the one with the larger numerator is larger. 332. Two identical strips were divided into parts. One strip was divided into 7 equal parts, and the other into 5 equal parts. Paint over 3/7 of the first strip and 3/5 of the second. Compare the resulting fractions: 3/7< /5. Formulate a rule for comparing fractions with the same numerators: of two fractions with the same numerators, the one with the smaller denominator is greater. 333. Fill in the gaps. 1) All proper fractions are less than 1, and improper ones are greater than 1 or equal to 1. 2) Each improper fraction is greater than any proper fraction, and each proper fraction less than any wrong. 3) On a coordinate beam of two fractions, the larger fraction is located to the right of the smaller one. 334. Circle the correct statements. 335. Compare numbers. 2)17/25>14/25 4)24/51>24/53 336. Which of the fractions 10/11, 16/4, 18/17, 24/24, 2005/207, 310/303, 39/40 is greater than 1? Answer: 16/4, 18/17, 310/303 337. Arrange fractions 5/29, 7/29, 4/29, 25/29, 17/29, 13/29. Answer: 29/29, 17/29, 13/29, 7/29, 5/29, 4/29. 338. Mark on the coordinate beam all the numbers that are fractions with a denominator of 5, located between the numbers 0 and 3. Which of the marked numbers are correct and which are incorrect? 0 1/5 2/5 3/5 4/5 5/5 6/5 7/5 8/5 9/5 10/5 11/5 12/5 13/5 14/5 Answer: 1) proper fractions: 1/5, 2/5, 3/5, 4/5. 2) improper fractions: 5/5, 6/5, 7/5, 8/5, 9/5, 10/5, 11/5, 12/5, 13/5, 14/5. 339. Find all natural values of x for which the fraction x/8 is correct. Answer: 1,2,3,4,5,6,7 340. Find natural expressions x for which the fraction 11/x will be improper. Answer: 1,2,3,4,5,6,7,8,9,10,11 341. 1) Write the numbers in the empty cells so that a correct fraction is formed. 2) Enter the numbers in the empty cells so that an improper fraction is formed. 342. Construct and designate a segment, the length of which is: 1) 9/8 of the length of segment AB; 2) 10/8 of the length of segment AB; 3) 7/4 of the length of segment AB; 4) the length of the segment AB. Sasha read 42:6*7= 49 pages Answer: 49 pages 344. Find all natural values of x for which the inequality is true: 1) x/15<7/15; 2)10/x>10/9. Answer: 1) 1,2,3,4,5,6; 2) 1,2,3,4,5,6,7,8. 345. Using the numbers 1,4,5,7 and the line of a fraction, write down all possible proper fractions. Answer: ¼, 1/5.1/7.4/5.4/7.5/7. 346. Find all natural values of m for which 4m+5/17 is correct. 4m+5<17; 4m<12; m<3. Answer: m =1; 2. 347. Find all natural values of a for which the fraction 10/a is improper and the fraction 7/a is correct. a≤10 and a >7, i.e. 7 Answer: a = 8,9,10 348. Natural numbers a, b, c and d such that a Positive and negative fractions
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2) A fraction a/b (a and b are natural numbers) is called correct if a< b
3) The fraction a/b (a and b are natural numbers) is called improper if a >b or a =b.
4) 9/14 is a proper fraction because 9< 14.
5) 7/5 is an improper fraction because 7 > 5.
6) 16/16 is an improper fraction because 16=16.
